Presented at 8th International Conference of the Society for Computational Economics(Aix en Provence, France, June 2002)

Digital Security Tokens in Network Commerce:
Modeling and Derivative Application
Kanta MatsuuraInstitute of Industrial Science, University of Tokyo (Japan)

Abstract

Digital objects in network commerce cause new credit risks (e.g. by an unpredictable compromise of a secret key and the resultant revocation of the corresponding public-key certicate). We recognize that we need to study a new nancial risk
management in this respect. The study rstly needs models. This paper describes
a full framework of modeling uncertain digital objects distributed in a network society. We have made an abstraction of the objects and dened \security token",which is abbreviated into a word coinage setok. Each setok has its price, values,
and timestamp on it as well as the main contents. Not only the price but also thevalues can be uncertain and may cause risks. A number of properties of the setok
are dened. They include value response to compromise, price response to compromise, refundability, tradability, online divisibility, and o ine divisibility. Thena derivative written not on the price but on the value is introduced. With thehelp of popular stochastic theory, we derive several option-pricing formulae. These
formulae do not require any divisibility of the underlying setok.
In search of applications, an inverse estimation of compromise probability is
studied. We derive a partial di erential equation (PDE) to price a call option
given a set of parameters including the compromise probability, the PDE can tellus the option price. By making an inverse use of this, we can estimate the risk ofcompromise.
In addition to related works and future directions, a comprehensive discussion
on the relation between information-security technologies and economics is given.
The discussion also shows how nancial tools can help information security in thereal world, and thus enriches the implications of this framework.

Key words: network commerce, information security, public-key certicate, trust,setok, option pricing, credit risk.

Correspondence to: Dr. K. Matsuura, Institute of Industrial Science, University of Tokyo, Komaba
4-6-1, Meguro-ku, Tokyo 153-8505, JAPAN. Tel. +81-3-5452-6284, Fax.: +81-3-5452-6285, E-mail:kanta@iis.u-tokyo.ac.jp1

Contents

1 Introduction
2 Security Token

3
4

3 Call Option on a Simple Setok

14

4 Compromise and Revocation

34

5 Related Works

41

6 Concluding Remarks

48

2.1
2.2
2.3
2.4
2.5

Network Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Setok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Price and Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Resale of Setoks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Simple Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Pricing in Discrete-Time Models . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Single-Period Binomial Model . . . . . . . . . . . . . . . . . . . . 16
3.2.2 Hedging Probability and Martingale Measure . . . . . . . . . . . . 20
3.2.3 Multiple-Period Binomial Model . . . . . . . . . . . . . . . . . . . 22
3.3 Pricing in a Continuous-Time Model . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.3 Analysis of the Option-Pricing Formula . . . . . . . . . . . . . . . 31
4.1
4.2
4.3
4.4
4.5

5.1
5.2
5.3
5.4

Observation and Implication . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Extended Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Inverse Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Eect of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Foreign Derivatives and International Issues . . . . . . . . . . . . . . . . . 41
Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Information/Network Security and Economics . . . . . . . . . . . . . . . . 44
5.4.1 Charging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4.2 Metering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4.3 Payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4.4 Punishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4.5 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Setok and Finance/Economics . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Setok and Information/Network Security . . . . . . . . . . . . . . . . . . 49

2

1 Introduction
Applied cryptography1] opens a door to a market of digital products in a network society.
With the help of advertisement attached, the products do not necessarily have positive
prices they can be free of charge. In this case, the recipients may no longer think of themas products. So we will use the phrase digital objects, instead of digital products, in thefollowing.
Since digital data in general can keep their original bit strings virtually forever, one
may expect that there would be no risk of change in qualities of digital objects. This is,unfortunately, not the case.
In addition to the prices, digital objects likely have other numerical values. For example, digital images and multimedia contents may have popularity indices. Their qualities
may be controlled by copyright protection mechanisms2], 3], and hence they may havequality labels and usage-restriction numbers. Multi-purpose tokens may have reward
points which would expire someday. Any object may be associated with an insurance
contract stating how much will be paid in the case of a signicant damage to the object
4]. If the use of the objects needs access-grant tickets, they may have priority numbers
or QoS (Quality-of-Service) values reserved 5]{8]. The functions mentioned above are
typically achieved by using digital signatures whose trustworthiness is resorted to digital certicates, and the certicates themselves may be combined with condence values
9]{12]. These additional values may change unpredictably over time and cause risks.
Let us see a simple example. Suppose that you buy a digital ticket issued and digitally
signed by an issuer. The ticket can be used for a remote-access service. On purchase, youof course verify the digital signature. Signature verication procedure includes vericationof a certicate associated with the public key of the signer. It fortunately succeeds. On
the next day, you make an attempt to use the ticket. The remote-access server veries theissuer's signature on your ticket. Unfortunately, the public-key certicate of the issuer isturned out to have been revoked due to an unexpected event, and therefore the verication
fails. You can no longer use your ticket1 .
This risk may be recognized as a new type of credit risk13], 14] specic to network
economics based on information-security technologies. We need appropriate risk management theories, and rstly theories need models we have to identify what are primaryfeatures of the digital objects distributed in the society. The rst report of this identication and possible application of it was given in 15]. However, it was unfortunately in
its infancy. The purpose of this paper is to show the full framework on this identication, and then to develop a basic theory of derivatives in the model identied. From theengineering point of view, we also explore applications of the framework. Dierent fromthe corresponding theory and application in 15], this paper enriches their implications
signicantly: more generalizations, deeper interpretations, surveys of related areas, and acomprehensive discussion on the relation between information-security technologies andeconomics.
Specically, the paper is organized as follows. First, in Section 2, we model uncertain
digital objects as a security token, which will be abbreviated into a word coinage setok
in contrast to an existing word stock. Written on a tradable setok, a European call option
This is a simple example because, in general, we can have a trust metric computed from a set ofcerticates. A revocation does not always imply a complete default of the certicate set.
1

3

is dened and priced in Section 3, where discrete-time models are rstly studied. The
last part of Section 3 deals with a continuous-time model. By using a well-known lemmain stochastic calculus, an option-pricing formula is derived. A numerical analysis of theformula is provided as well. We then explore an application of option pricing in Section 4:
inverse estimation of compromise probability. After a survey of related works in Section 5,
Section 6 concludes the paper with indicating future directions. The survey includes a
relation between this work and credit derivatives. A comprehensive discussion on therelation between information/network security and economics is included there as well.

2 Security Token
2.1 Network Society

We start with our basic architecture of the network commerce, which is illustrated inFig. 1.

Object Provider

6

object?
6

object?

Customerpayment

6

rewardObject Server?

report6

refundpayment
Customert t t?

objectCustomer

Figure 1: Basic architecture of the network commerce. Boxes with wider lines indicate thatthe entities inside are more trusted. The object provider and server are able to keep in touchwith the up-to-date market information. Some customers are able to keep in touch with the
up-to-date market information but the others are not. Due to better reputation and/or advertisement prots, the provider would be happier if the objects are distributed and circulated more
frequently in larger amounts. The refund may depend on the price and/or values of the object.
The payment from a customer may be regarded as a deposit, depending on the situation thisarchitecture can model a rental system as well. Basically, each payment is made by digital cash.

4The observation to have this architecture is as follows:(Object Provider) For example, copyright management and related technical maintenance are not easy and trivial tasks with respect to digital objects. Management
and maintenance of network-security infrastructure (e.g. public-key infrastructure)
are not, either. These tasks may require sucient trustworthiness and reliability. We need specialized entities which are eligible for them. Typically, they are
trusted organizations or licensed rms. Object providers would be happier if the objects they provide are distributed and circulated more frequently in larger amounts
it would improve their reputation and/or make attached advertisement more profitable. They would have a motivation to give rewards for active usage of the objects.
They are able to keep in touch with the up-to-date market information.
(Customer) We do not trust individuals in terms of (i) their own behaviour, (ii) theirnancial situation, and (iii) resources (for communication and computation) available to them. Some customers are able to keep in touch with the up-to-date market
information but the others are not. In terms of trading amount, the former would
be the major contributors. Since we allow untrusted individuals, the system canbe ubiquitous: many people can enjoy transactions in various situations including
mobile settings.
(Object Server) Selling digital objects to distrusted customers is another dicult andnon-trivial task. We need specialized entities which can do it and have a goodconnection with object providers. Typically, they are trusted organizations or rms
they can be less trusted in comparison with object providers but they must be moretrusted than customers. Due to the rewards from object providers as well as theirbasic business reasons, object servers would like to enhance their trading activities
with customers. One may think that it is easy to sell more because copying digital
data in general is so easy. However, as already implied, this is not true for digital
objects if the object providers work well, they are the only entities who can increase
the number of the objects either by copying or by creating a new version. Thus
object servers would have a motivation to re-circulate the objects. They may beable to get the objects back from their customers in exchange for some refund. The
refund may depend on the price and/or values of the object. Object servers are able
to keep in touch with the up-to-date market information.
The observation above is very important in view of forthcoming derivative studies
because how to specify observability often constitutes a core part of credit-derivative
pricing16].

2.2 Setok

Based on the basic architecture, we model uncertain digital objects as follows.

De nition 2.1 (Setok) A security token or setok is a digital material which nominally contains the following four attributes:
5 contents which may include MAC (Message Authentication Code), digital signa-tures, or other security-related control sequences if necessary, a non-negative explicit price (denoted by S) which is paid when the setok is purchased by a customer, a set of non-negative explicit values (denoted by V1 , V2 ,   , Vm where m is referred
to as the dimension of the explicit values) which represent some qualities of thecontents in a way that larger values of each element imply better qualities regarding
the feature represented by the element when the setok is purchased, and a timestamp which indicates when the setok is purchased,
and is associated with
 a non-negative implicit price (denoted by S ) and a set of non-negative implicit values (denoted by V1 , V2 ,   , Vn where n is referred

to as the dimension of the implicit values)in the following way.
 The explicit price is speci ed as the occurrence of a price-interpretation pro-

cess Y (t)=y (t S (t)) i.e. the speci c numerical value y (t0  S (t0)) is written as
the explicit price of the setok which is purchased at time t = t0 . Each occurrenceof the price-interpretation process is called the up-to-date price at time t. The
price-interpretation process is a non-negative process and also called the up-to-dateprice process. y(t s) is called a price-interpretation function and monotone
increasing with respect to s. Customers are unable to change the explicit price.
 The explicit values are speci ed as the occurrences of value-interpretation processes H1 (t)=h1(t, V1(t), V2(t),   , Vn(t)), H2(t)=h2(t, V1(t), V2(t),   , Vn(t)),
  , Hm (t)=hm (t, V1 (t), V2 (t),   , Vn (t)) i.e. the speci c numerical value hi (t0 ,V1(t0 ), V2 (t0),   , Vn(t0)) is written as the i-th explicit value of the setok which ispurchased at time t = t0 (i = 1, 2,   , m). Each occurrence hi (t, V1 (t), V2 (t),  , Vn (t)) is called the i-th up-to-date value at time t. The value-interpretation
processes are non-negative processes, and also called the up-to-date value processes. h1 (t, v1, v2,   , vn ), h2 (t, v1 , v2 ,   , vn),   , hm (t, v1, v2,   , vn )are called value-interpretation functions. Customers are unable to change theexplicit values.

When we say \security token", we consider three notions at the same time:
1. Information security such as condentiality, authenticity, integrity, non-repudiation,
and availability.
2. Ocial pieces of writing, e.g. bonds and stocks.
3. What gives the owner the right to certain property.
6Denition 2.1 accepts not only purely nancial digital materials but also digital commodities as setoks we have not specied the contents.
Customers are distrusted. Depending on the payment scheme, customers may be evenanonymous when they buy setoks. So each payment must be settled on site in exchange ofthe corresponding pieces of the setok. This should be done in a secure way no customer
can exploit a setok without payment, and no server can exploit a payment without sending
the setok. Servers are trusted but we do not want to allow customers to lay frame-up
accusation against servers. Thus setoks are transmitted to customers in pieces e.g. \threepieces" are possible but \two and a half pieces" are impossible. A piece of setok will bereferred to as a share of the setok.
A setok in the market is denoted by (S  Y  V1, V2 ,   , Vn H1, H2,   , Hm ) or
sometimes shorthandly by (S , Y  V , H , n, m). Likewise, a share of the setok alreadypurchased and held by someone is denoted by (S V1, V2,   , Vm  t0 ) or sometimes
shorthandly by (S V , m t0).
The price-interpretation function can model the eect of taxes, transaction costs, regulatory issues, and so on. The value-interpretation functions can model the eect of
security policies, regulatory issues, editorial policies, transmission delay, and so on. Suppose that we make an electronic version of a stock in a way that each share of the setok
has the explicit values which tell the rm's information evaluated somehow. The \evaluation" or \interpretation" may include an editorial process or aggregation procedures.
It is impractical to write every history of the rm on a setok, even though it is a digitalmaterial.
It should be noted that we have not mentioned any increasing or decreasing propertiesof the value-interpretation functions. A tricky but eciently implementable example isa one-way hash function. Even when we have implicit values which are extremely large
in size (number of bits), hashed values can be in a practical size. This example might
remind us of a gamble the explicit values, which are completely unpredictable in advance,
represent yet some qualities of the material in a way that larger values of each element
imply better qualities. The implicit values do not necessarily represent such qualitiesthey cause uctuation.
The \up-to-date" processes, Y (t) and H (t), are observable in the market and hence
are adapted processes. Note that some of the implicit processes S (t), V1 (t), V2 (t),   , Vn (t) may be not adapted
processes even if{ the explicit value is one-dimensional (i.e. m = 1),{ both of the price/value interpretation functions are easy to compute, andstrictly monotone increasing functions with respect to some implicit price or
implicit values, and{ the other implicit price/value processes are adapted processes.
For not strictly monotone increasing functions, we may easily accept the note above. The
following is a trivial example:

7

Example 2.1 Let us consider a setok (S Y  V H n 1) with a price-interpretation func-tion y (t s) = s and a value-interpretation function h1 (t, v1 , v2,   , vn)= h0 where h0
is a deterministic constant. y is strictly monotone increasing with respect to s but h1 is
not strictly monotone increasing with respect to any vj . The implicit price process is anadapted process but the implicit value processes are not.

However, at a rst glance, it would be more dicult or counter-intuitive, especiallyfor those who are unfamiliar with information-security engineering, to see that strictly
monotone increasing interpretation functions do not always guarantee the adaptation.
We demonstrate it by using a one-way hash function.

Example 2.2 Let us consider a setok (S Y  V H 2 1) with a price-interpretation function y (t s) = s and a value-interpretation function

h1 (t v1 v2 ) = h(v2 ) + p  v1
where h() : N ! Z p is a one-way hash function, N = f1 2 3   g is the set of positive
integers, Z p=f0 1 2     p;1g, and p is a large positive integer (constant). Let us suppose
V1(t) 2 f0 1g, V2 (t) 2 N for any t 2 T . Then y is strictly monotone increasing withrespect to s, and h1 is strictly monotone increasing with respect to v1 . The implicit priceprocess is an adapted process. V1 is also an adapted process because H1 (t)  p implies
V1(t) = 1 and H1(t) < p implies V1(t) = 0. Nevertheless, V2 is not an adapted process.

Remark 2.1 Regarding stochastic variables, we follow the conventions in notation:
 Stochastic variables often appear with suppression in the following part of the papere.g. for readability reasons, we would write S instead of S (t) nor S (t)!] where! 2 and is the universe of the probability space considered.
 (1) An occurrence of a stochastic variable, and (2) corresponding arguments in functions describing other processes by the use of the stochastic variable, are usually
written in small letters.

2.3 Price and Value

Recall that sucient information to determine the implicit price and values is not necessarily obtained through market observation, although the up-to-date price and values
are observable in the market. Let us consider in more detail about the relationship or
correlation among the implicit/up-to-date price and values.
The explicit values represent some qualities of the setok. They depend on the implicit
values. The bridge between them is the value-interpretation functions. Changes in theimplicit values may be relaxed through the interpretation. They may be exaggerated,too.
8Let us suppose a value-interpretation function hi which is monotone increasing withrespect to the j -th implicit value. In this case, intuitively, the implicit value Vj also represents some sort of quality of the material. Let us suppose that some sort of compromise
has just reduced the j -th implicit value to be zero. Hopefully best eorts are made to
make the up-to-date values reect this emergency well enough. The eorts may include,for example, implementing the underlying directory system with real-time revocation as
well as periodical/regular update2 . From the viewpoint of setok, an ideal situation isdened as follows.

De nition 2.2 (Value Response to Compromise) Let a setok (S Y  V H n m) haveone or more value-interpretation functions which are monotone increasing with respect toone or more implicit values. Let fVj1 , Vj2 ,   , Vjs g be the set of all such implicit value
processes.
Then, the setok is said to be compromised if and only if Vj1 (t)=Vj2 (t)=  =Vjs (t) =0. This setok is said to be compromise-responsive in value if and only if the following
condition is satis ed.
 For any hi which is monotone increasing with respect to one or more implicit values
Vj1 , Vj2 ,   , Vjs , a compromise Vj1 (t)=Vj2 (t)=  =Vjs (t) = 0 implies Hi(t) = 0.
When Hi(t) becomes zero, the setok is said to be revoked with respect to the i-th

up-to-date value.

Do you accept a positive price for a \compromised" material? The answer may be
not unique and depend on the relationship between the contents and the implicit/explicit
values. We shall dene a special, but easier to accept, situation.

De nition 2.3 (Price Response to Compromise) Let a setok (S Y  V H n m) haveone or more value-interpretation functions which are monotone increasing with respect toone or more implicit values. Let fVj1 , Vj2 ,   , Vjs g be the set of all such implicit value
processes.
This setok is said to be compromise-responsive in price if and only if the compromise Vj1 (t)=Vj2 (t)=  =Vjs (t) = 0 implies Y (t) = 0.

2.4 Resale of Setoks

Object servers have a motivation to re-circulate the objects they have sold. They may
be able to get the objects back from their customers in exchange for some refund, which
would in turn motivate the customers to return the objects. This refund may depend on
the price and/or values of the object. Formally, we dene refundability and tradability.
2

In this context, the compromise is a primary disaster and the revocation is a response to it.

9De nition 2.4 (Refundability) A share of setok is said to be T -refundable if andonly if the following two conditions are satis ed:
 The explicit price is positive.
 The holder can sell it whenever he wants during a set of time intervals T , i.e. at
any time t 2 T , at the price of the explicit price.

is called a refundable period and allowed to be composed of open and closed timeintervals all of the forms TL  TU ], TL TU ), (TL  TU ], and (TL  TU ) (and set of them)
are available. The refundable period can be either deterministic or stochastic. Whenthe context does not need the refundable period, we can just say \refundable" instead of\T -refundable".
In particular, a share of setok is said to be 1-refundable if and only if it is t0 1)refundable where t0 is the timestamp on it.

TDe nition 2.5 (Strict Refundability)
A share of setok is said to be strictly T refundable if and only if the following three conditions are satis ed:
 It is T -refundable.
 The refundable period T is deterministic.
 The holder cannot sell it at any price when it is out of the refundable period T .

When the context does not need the refundable period, we can just say \strictly refundable"
instead of \strictly T -refundable".
In particular, a strictly -refundable setok is said to be unrefundable where  is theempty set.

De nition 2.6 At time t, a strictly T -refundable setok with non-empty refundable period
T is said to be still refundable if and only if there exists a time t such that t > t and00

t 2 T where t is the current time.
0

There are several important things to be pointed out. Firstly, note that we havedened the refundability with respect to a share of setok. Even for the same setok, shares
sold at dierent time could have dierent refundable periods the refundability of (S V ,m t0) and that of (S V , m t1 ) (t1 6= t0) are not necessarily the same. This allows
a dynamic change of management policies of a setok. For example, let us suppose asetok which has been managed so far in a way that all the shares are strictly refundable
and the refundable period is very long. This policy gives an assurance to customers,
but could cause too much nancial load (e.g. reserve funds) on the server and/or too
much administrative load (e.g. security-parameter directory) on the provider. If needed,
the policy can be changed in a way that shares will be sold with a shorter refundable
10

period, or even sold with no refundability, from now on. However, due to the former
strict refundability, the refundable periods of the shares already sold cannot be changed
accordingly. If their refundable periods were stochastic, the corresponding change wouldbe (not mandatory but) possible. Thus, in our framework, we can accommodate a wide
variety of situations by specifying which kind of refundability is used.
It should be also noted that the refund of a setok is possible only at its explicit priceS . Changes in values are not considered. This can model a rental system with deposit, forexample. However, depending on the applications, a more exible resale may be allowed.
This is easier if the explicit value is one-dimensional.

De nition 2.7 (Single-Valued Setok) A setok is said to be single-valued if and only

if it has one-dimensional explicit value. In the case of a single-valued setok, we often omit
the subscript \1" when we denote the explicit value, the corresponding value-interpretation
function, and the corresponding value-interpretation process. Hence we may writeV = H (t0) = h(t0  V1(t0) V2(t0 )     Vn(t0 )):De nition 2.8 (Tradability) A share of single-valued setok is said to be T -tradable
if and only if the following two conditions are satis ed:
 The explicit value V is positive.

 Whenever he wishes during a set of time intervals T , i.e. at any time t 2 T , thevalue-interpretation process is positive and the holder of the setok can sell it. This
resale is possible only at the value-proportional price Sp de ned bySp = h(t V (t) V V(t)     V (t)) y (t S (t)) :12nis called a tradable period and allowed to be composed of open and closed timeintervals all of the forms TL TU ], TL  TU ), (TL  TU ], and (TL TU ) (and a set of them)
are available. The tradable period can be either deterministic or stochastic.
In particular, a setok is said to be 1-tradable if and only if it is t0  1)-tradable
where t0 is the timestamp on it.

TLemma 2.1 The value-proportional price Sp is a non-negative adapted process.
P roof: Trivial from Denition 2.8.

Q:E:D:It should be noted here that Sp can have a zero occurrence there can be a trade at a
price of zero.
11One may wonder why we restrict the trading to the value-proportional price there
might be a case in which the mapping from the space of (time, implicit values, explicit
value, implicit price) to the space of trading price takes a more complicated form. Our
answer to this question is that we model the complexity in terms of value-interpretation
functions.

De nition 2.9 (Strict Tradability) A setok is said to be strictly T -tradable if andonly if the following three conditions are satis ed:
 The setok is T -tradable.
 The tradable period T is deterministic.
 The holder of it cannot sell it at any price when it is out of the tradable period T .

In particular, a strictly -tradable setok is said to be untradable where  is the emptyset.

De nition 2.10 A strictly T -tradable setok with non-empty tradable period T is said tobe still tradable if and only if there exists a time t such that t > t and t 2 T where t000

is the current time.

2.5 Divisibility

Tradability and refundability do not cover all the exibility concerns about setoks. Suppose a situation in which I need a single-valued setok (S Y  V H 1 1) with an explicit
value of 100. And suppose that I am so unlucky that the current market board says
H (t) = 95. Shall I buy two shares of the setok? Do I have to reconsider my purchaseplan? This annoyance depends on the divisibility of the setok.

De nition 2.11 (Online Divisibility) A setok (S Y  V H n m) is said to be onlinedivisible if and only if the following condition is satis ed.
 Whenever the occurrence of the price-interpretation process is positive, anyone canpurchase an arbitrary fraction of the setok with keeping proportional explicit
values i.e. at an arbitrary order price Sc > 0, he can buy the setok at the explicit
price Sc and explicit values

Sc h (t  V (t ) V (t )     V (t )) (i = 1 2     m)
n 0
Y (t0 ) i 0 1 0 2 0
assigned where t0 is the timestamp on it.
12It should be noted here that we make an order by specifying the order price, not by
specifying the explicit values. This can decrease communication overhead.
We may face a similar annoyance when we are going to sell a share of setok.

De nition 2.12 (Oine Divisibility) A share of setok (S V1 V2     Vm t0 ) which has
a positive explicit price S is said to be oine-divisible if and only if the holder of it
can divide it into two pieces, (S1  V11 V21      Vm1  t0 ) and (S2  V12  V22      Vm2  t0 ), in a
price-proportional manner, i.e.
 S1 > 0 S2 > 0S1 + S2 = SandiVji = S Vj  (i = 1 2 j = 1 2     m):
S

One may wonder why we have dene not only the online divisibility but also theo"ine divisibility. This distinction is needed to enrich the variety of exibilities. In
the setok world, entities or agents live under uncertainty. Even if all the exogenous
stochastic dynamics are given, their optimal strategies depend on the divisibilities in adierent manner roughly speaking, the online divisibility contributes to the exibilityof purchase whereas the o"ine divisibility contributes to the exibility of resale andsubsequent investments. So the online divisibility is dened with respect to a setok
whereas the o"ine divisibility is dened with respect to a share of setok.
One may also wonder why we have paid so much attention to divisibility in a digitalworld, by saying \It is very easy to divide digital data, isn't it? That should be a virtueof computers and computer networks!" Unfortunately, this is not the case in general. Our
experiences in applied cryptography and security protocols tell us that accountability israther dicult or expensive at best.
Looking back at stocks, such a piece of paper could not be physically divided. Nevertheless, the conventional nancial theory works well with the divisibility assumption.
Why? A rigorous answer might be dicult to place here but we can roughly say that the existence of standardized market with reliable communication network devoted
to nance,and the newspapers and other media which report the daily market occurrences andphysically timestamp on them
contribute a lot. This counter-intuitive dierence between paper stocks and digital setoks
arise not from whether they themselves are digital or not but from their assumptions on
the available infrastructures and the trustworthiness of the entities involved Figure 1 is
so important in that sense.
13

3 Call Option on a Simple Setok
3.1 Simple Settings

In a network life, we would want to pay for digital products in electronic cash. As reviewed
later in 5.4.3, electronic cash systems could be more ecient if the monetary value of eachcash or coin is less granular 17]. Some systems have only a few kinds of xed-value coins.
Therefore, if we want to allow as wide variety of electronic cash systems as possible, axed price would be helpful. This section assumes a setok whose price-interpretation
process is an identity process3 .
We also assume here a strictly T -tradable setok where T 6= . We have dened
tradability only for single-valued setoks. Hence we consider a single-valued setok in thissection. In the derivative theory, tradability and divisibility is important with respect tothe completeness and eciency of the market. Since a rigorous discussion would takea long time, we recommend those who are interested to consult good literatures such as19]{25].
We do not assume any divisibility of the setok. This is because we prefer settings lessrestrictive against security protocol design and implementation.
The nal important decision is how to regard the setok as a \project" during thetime interval we keep it. Our framework considers the contents, which suggests that thepossession of a setok might yield something. Again, this section chooses the simplest
story: nothing happens.
With some more specications made, we have the following assumptions.

Assumption 3.1 In Section 3, a single-valued setok (S Y  V H n 1) with the following
properties is studied.

1. The price-interpretation process is an identity process, i.e. Y (t) = 1 for all t.
2. The setok is not online divisible.
3. The setok is not oine divisible

4. No one can go short for the setok.
5. Each share of the setok is strictly T -tradable.
6. The tradable period T begins at the time of purchase and is composed of a single timeinterval of a xed positive length T . We will make it explicit by saying \T -tradable",
where T is not boldfaced.
7. The possession of the setok has no meaning as a project and hence, in a nancial
term, yields no dividends.
In the conventional nance, continuous-price models were studied rst, and then the biases induced
by discreteness were appended 18]. In the case of the setok, we have decided to take a dierent approach
because the discreteness would be much more signicant.
3

148. All the up-to-date and implicit value processes are positive and nite, i.e. 0 <H (t) < 1 and 0 < Vj (t) < 1 (j =1, 2,   , n) for any t 2 .

In the real world, one of the simplest derivatives is European call options written on
a stock price. However, we have assumed a xed-price setok. We are unable to write any
option on the setok price. What we can do is with respect to the values.

De nition 3.1 (A European Call) A European call option on the setok, purchased attime t = t0, is de ned as a derivative which provides a right to buy one share of the setok
with a reserved explicit value K at a particular time Tm < t0 + T in the future for its
xed price, 1, regardless of the up-to-date value H (Tm ) at t = Tm . The reserved value K
is called the strike value or the exercise value, and Tm is called the exercise date or
the maturity date, or just simply the maturity.

It should be noted that there is no obligation for the holder of the option to exercise
the right on the exercise date. Obviously, he will exercise it if H (Tm) < K and he won't
if H (Tm )  K .
Our nal setting statement is assumptions on the market.

Assumption 3.2 (Ideal Market) We assume an ideal market which satis es the following conditions.

(a) There are no transaction costs of trading both in time and in money: any transaction
can be completed immediately, free of charge.

(b) The market is completely liquid, i.e. it is always possible to buy and/or sell unlimited

quantities. In particular, it is possible to borrow unlimited amounts from the bank(by selling bonds short). The short rate rf is a deterministic constant.

(c) There is no bid-ask spread, i.e. the selling price is equal to the buying price.
(d) The market is free of arbitrage.
The four items above are common for both the setok market and the option market.
In addition, the option market is assumed to have the following properties as well.

(e) The option can be bought and sold on a market at any fraction.
(f) Anyone can go short for the option.
We seem to have supposed that investors interested in derivatives are diligent enoughto be connected both with the setok network and with the common nancial network. Or
otherwise, due to the economic benet from options, expensive cryptographic tools are
deployed to have such an ideal high-quality communication for setok-option transaction
15

even over insecure channels. Obviously, we consider dierent architectures for setoks andfor options. Let us have a more close look at the dierence and the common features byreferring to the items in Assumption 3.1 and in Assumption 3.2.
The common features, (a), (b), and (c) in Assumption 3.2, are for simplicity. The
feature (a) for setoks is supported by secure timestamps non-negligible delay may occur
in the architecture of the setok world, but no one can abuse the delay for cheating. The
feature (d) in Assumption 3.2 is the most common in nancial theory.
With respect to setoks, no divisibility is assumed (2. and 3. in Assumption 3.1).
Going short is not allowed, either (4. in Assumption 3.1). This is due to the architecture described in 2.1 at the beginning of Section 2. By contrast, with respect to options,divisibility is assumed ((e) in Assumption 3.2) and going short is allowed ((f) in Assumption 3.2). This implies that participation in the setok-option market has the samerequirements as in the conventional option market: security and reliability of the resources
are good enough, and participants are trusted enough.

3.2 Pricing in Discrete-Time Models
3.2.1 Single-Period Binomial Model

This subsection is the rst attempt to show option pricing in the setok world. We have
Assumption 3.1. To get a simple but good and instructive start, we furthermore assume
that the implicit value process V is one-dimensional, i.e. n = 1, and that the valueinterpretation function h(t v) is a deterministic function h(v) such that h(v) > 0 for anyv  0. Furthermore, for a readability reason, we assign the time unity so that Tm = 1.
The simplest model used here is a single-period binomial model described in Fig. 2.
At present (t = 0), the up-to-date value is H (0) = H0 = h(V0). At the maturity, there are
two possible states: H (Tm)up] = h(u  V0) and H (Tm)down] = h(d  V0) where d and u arepositive constants such that 0 d < 1 < u. The former, the state after upward change,
occurs with probability pu. Hence the latter, the state after downward change, occurs
with probability 1 ; pu. You do not know these probabilities. Please nd a reasonable
price of the European call option. This is the problem to be solved below.
The option has a payo of
Cu = maxf0hK(uV; h) (uV0)g(1)0
in the case of the upward change. That is, if h(uV0) < K , the holder of the option
exercises it he buys one share of the setok at the price 1, with the strike value K . Thanks
to Denition 2.8, Assumption 3.1, and Assumption 3.2, the holder can immediately resale
this share for the value-proportional priceK 1= K 
Sp = h(uV(2)
h(uV0)
0)and achieve a positive gain ofh(uV0) :Sp ; 1 = K h;(uV
(3)
0)
16

3 u  V0pu

Implicit Value 


V0 
QQ



QQ
Q1Q; pu
QQ
Qs d  V0

u

u

t=0

t=1

-

Figure 2: A single-period binomial model for pricing the European call option. The setok priceis kept to be 1 all the time. At the beginning of the period (t = 0), the up-to-date value ofthe setok is h(V0 ). At the end of the period (t = Tm = 1), i.e. the maturity of the option,either the \upward" or the \downward" state has occurred: the former is with the option payo
Cu and the up-to-date value of the setok h(uV0 ) while the latter is with Cd and h(dV0 ), where
0  d < 1 < u. Although this illustration shows that the upward-change probability is pu
and the downward-change probability is 1 ; pu , the pricing does not require these probabilities.
They appear here just to tell you that the two states are the only possible states at t = 1. It is
assumed that h(uV0 ) 6= h(dV0 ), which di erentiates the two states in the market as well as in
the implicit world.

He is not obliged to exercise the option but he does. This is what is called the `greedy'
assumption in economics: anyone able to obtain anything of value for free will not hesitate
to do so. On the contrary, if h(uV0)  K , the holder of the option does not exercise it
hence gets nothing. Eqn. (1) is a mathematical representation of this. Likewise, we ndthat the option has a payo of
Cd = maxf0hK(dV; )h(dV0)g(4)0
in the case of the downward change.
By using the no-arbitrage condition, it is easy to see the following lemma.

Lemma 3.1 In the binomial single-period model described in Fig. 2, the following musthold:

P roofh(V0)max fh(uV0) h(dV0)g:

1 + rf :If h(V0)= max fh(uV0) h(dV0)g > 1 + rf , then you can achieve an arbitrary large
amount of gain M fh(V0 )= max fh(uV0) h(dV0)g ; (1 + rf )g by the following steps withprobability 1.
1. At time t = 0, go short for the bank and sell the riskless Tm -bond to get M dollars.
17

2. Immediately after that, i.e. also at t = 0, buy M shares of the setok by using all
the M dollars.
3. At time t = Tm , sell all the setok for the value-proportional price. This is possible
because the setok is strictly T -tradable and T > Tm .
4. The value-proportional price is at least h(V0 )= maxfh(uV0) h(dV0)g. Hence what
you obtain in total is at least Mh(V0 )= maxfh(uV0) h(dV0)g.
5. At the same time, you pay (1 + rf )M dollars for the bond.
This is not permissible. Therefore,h(V0)max fh(uV0) h(dV0)g 1 + rf :Q:E:D:

Our task is to nd a riskless portfolio composed of one share of setok and M European
call options. Let C be the price of one share of the option (at t = 0). Then, our initialinvestment is given by
1 + MC:(5)
In order to achieve risk-freeness, the portfolio must have exactly the same payo atthe maturity t = Tm = 1 regardless of the state (upward or downward). That is,h(V0)  1 + MC = h(V0)  1 + MC :(6)udh(uV0)h(dV0 )
Note that the rst term of each side in Eqn. (6) represents the payo resulting from one
share of the setok. This is possible because the setok is strictly T -tradable and T > Tm.
We are afraid that it might be unclear whether the possibility means obligation ornot the holder of the setok can sell it at t = Tm < T for the value-proportional pricebut is he obliged to do so? He can wait for a more protable situation as far as the setok
is still tradable. Why we assume that he sells the setok at t = Tm? A naive answer is:that is because the set of time indices is f0 1g in the space considered, and thus we have
only today (t=0) and tomorrow (t=1). Another excuse can be found in the no-project
assumption (statement 7. in Assumption 3.1). This informally tells us that we can have astock-like look at the setok in terms of wealth/asset evaluation the evaluation is based on
the price for which the holder can sell it now (if he would do so). More detailed discussion
in this direction would need an equilibrium theory and be beyond the scope of the rst
stage of our setok framework. Anyway, in the context of our current option-pricing theory,
it is sucient that we understand the riskless portfolio strategy is possible if there exists a number M which satises
Eqn. (6) and is allowed in the setok/option market assumed here.
18

The setok values are exogenously given.
Note also that we cannot sell the setok during the time interval t 2 (0 1). If you wereable to know the value-interpretation process time-continuously and found H (t ) < h(V0 )at some time t 2 (0 1), then you would be tempted to sell the setok at t , before thematurity of the option. This is not what the single-period model considers. We can get
no information, we can buy nothing, and we can sell nothing on the way.
After a manipulation on Eqn. (6) with the help of h(uV0) 6= h(dV0 ), we have
(
)
1
1
h
(
V
0)
(7)
M = C ; C h(uV ) ; h(dV )du00Because of the assumption of the ideal option market (Assumption 3.2), the portfolio is
feasible regardless of the actual occurrence of M given by Eqn. (7) any M 2 R is allowed.
Thus we have established a riskless portfolio. In order to achieve no arbitrage, theportfolio must have the rate of return exactly as low as the short rate rf . Therefore,
looking at the initial investment (Eqn. (5)), we notice that
V0 )  1 + MC
(1 + rf )(1 + MC ) = hh((uV(8)u0)
must hold. Let us insert Eqn. (7) into Eqn. (8). Then, after some manipulations, we
obtain; p)CdC = pCu +1 (1(9)+ rf
where p is dened byf(h(dV0 )g 1 ; (1 + rf ) fh(V0 )g 1
:
(10)
p=
fh(dV0 )g 1 ; fh(uV0 )g 1
The story above can be summarized as the following theorem.
0

0

0

;

;

;

;

Theorem 3.1 (Option-Pricing Formula (Single-Period Model)) In the binomial
single-period model described in Fig. 2, let us consider a European call option de ned by
De nition 3.1 under Assumption 3.1 and Assumption 3.2. Choose the option's maturity
Tm as a time unit and assume that the short rate of interest is a constant rf for the timeunit. Let the price process and the strike value of the option be C (t) and K , respectively.
Then, the following pricing formula holds.
; p)CdC (0) = pCu +1 (1
+ rf
where
Cu = maxf0hK(uV; h) (uV0)g0

Cd = maxf0hK(dV; )h(dV0)g0f(h(dV0 )g 1 ; (1 + rf ) fh(V0 )g 1
p=:fh(dV0 )g 1 ; fh(uV0 )g 1
;

;

;

19

;

We are happy because the pricing does not need pu. This feature can be interpretedby the following statements:
 Pricing Theorem 3.1 does not need the objective measure. Instead, it looks simple
and sound with the risk-neutral measure,which coincides with the well-known feature of the basic nancial option pricing theories.
This will be soon revisited in 3.2.2.
We are also happy to see that
 no divisibility assumption on the setok is needed,
and that the setok is allowed to have a nite lifetime in tradability.
Both go well with the basic architecture described in Fig. 1. From the engineering pointof view, such a less restrictive situation most likely allows cheaper and more ecient
protocols which are suitable for general customers.

3.2.2 Hedging Probability and Martingale Measure

Before proceeding, let us further investigate the meaning of p in the pricing formula(Theorem 3.1).

Lemma 3.2 (Hedging Probability) If h(V0 )= minfh(uV0) h(dV0)g  1 + rf , the parameter p given in Theorem 3.1 satis es 0
hedging probability.

P roofp

1. In this case, we refer to p as the:We have assumed that h(v) > 0 for any v  0 and that h(uV0) 6= h(dV0).
When h(uV0) > h(dV0), we have h(V0)=h(dV0)  1 + rf because of the assumption
h(V0)= minfh(uV0) h(dV0)g  1 + rf . h(uV0) > h(dV0) of course implies 1=h(dV0) >1=h(uV0). Consequently,h(V0 )
f(h(dV0 )g 1 ; (1 + rf ) fh(V0 )g 1h(dVn0 ) ; (1 + rf ) o
=
 0:
p=
fh(dV0 )g 1 ; fh(uV0 )g 1
h(V0 ) h(dV1 ) ; h(uV1 )
;

;

;

;

0

0

By using h(uV0) > h(dV0) and Lemma 3.1, we have h(V0)=h(uV0)remember 1=h(dV0) > 1=h(uV0). Hence

1 + rf . We

11
(1 + rf ) ; hh((uVV00))(1+rf ) fh(V0 )g ; f(h(uV0 )go  0:n=1;p=fh(dV0 )g 1 ; fh(uV0 )g 1
h(V0 ) h(dV1 0 ) ; h(uV1 0)
;

;

;

;

20

When h(uV0) < h(dV0), we have h(V0 )=h(uV0)  1 + rf because of the assumption
h(V0)= minfh(uV0) h(dV0)g  1 + rf . h(uV0) < h(dV0) of course implies 1=h(dV0) <1=h(uV0). Consequently,(1 + rf ) ; hh((uVV00))o  0:n1;p=h(V0 ) h(dV1 0 ) ; h(uV1 0)
By using h(uV0) < h(dV0) and Lemma 3.1, we have h(V0)=h(dV0) 1 + rf . We
remember 1=h(dV0) < 1=h(uV0). Henceh(V0 )h(dVn0 ) ; (1 + rf ) op= 0:h(V0) h(dV1 0 ) ; h(uV1 0)
Q:E:D:

Proposition 3.1 (A Martingale Measure) Let us discount the option price process
C (t) by the (riskless) short rate rf , and de ne a process C (t) byC (0) = C (0)C (1) = (1 + rf ) 1C (1):;Then,
1. The hedging probability p provides a probability measure if we assign p as the probability of the upward change and 1 ; p as the downward one.
2. Under this measure, the discounted price process C (t) has the martingale property
such thathiC (0) = E0 C (1)where E0 denotes the expectation operator conditioned by the available information
at t = 0.
P roof:

Let us denote the state after the upward change by \up", and that by the downwardchange by \down".
The rst statement is trivial from the denition given by Eqn. (10) and Lemma 3.2.
To prove the second statement, please note that the no-arbitrage requirement gives
C (1)up] = Cu and C (1)down] = Cd  i.e. the option price at the maturity must be equal
to the payo. Therefore, we havehi
E0 C (1) = pC (1)up] + (1 ; p)C (1)down]= p(1 + rf ) 1C (1)up] + (1 ; p)(1 + rf ) 1 C (1)down]= (1 + rf ) 1 fpCu + (1 ; p)Cdg
Theorem 3.1 says that this equals to C (0), which is by denition equal to C (0). Q:E:D:
;

;

;

21

3.2.3 Multiple-Period Binomial Model
By repeating the binomial process, we can easily extend the single-period model to a
multiple-period model (Fig. 3). Let the maturity Tm be 1 as in the previous subsection
and divide the time interval 0 1] into a large number of periods, say, N periods: 0 1=N ),1=N 2=N ),   , (N ; 1)=N 1). We must pay attention to three things:(i) The short rate is dened as a rate of interest for a time unit. Hence we use rf =N foreach period.

3 u3V0




2

u
V
0 Q

3
QQ


QQ


QQs u2dV
uV0 
3

Q
3 0

QQ



Implicit ValueQQ
QQs udV 
V0 
QQ3 0 QQQ

QQ
QQ

QQ

QQs ud2Vs dV0 Q
Q
3 0
QQ


QQ


QQs d2V 
0 Q
QQQQQQs d3V
0

t

t=0

t

t

t = 1=3

t = 2=3tt=1-

Figure 3: A multiple-period binomial model for pricing the European call option. For drawing
convenience, the illustration has only N = 3 periods. The setok price is kept to be 1 all thetime. At the beginning of the period (t = 0), the up-to-date value of the setok is h(V0 ). At
the end of the period (t = Tm = 1), i.e. the maturity of the option, we have N possible states
denoted by the number j of upward changes which have occurred. We also denote each state
at time t = i=N by the number j of upward changes which have occurred. We suppress thetransition probability, which was denoted by pu and 1 ;pu in the case of the single-period model,
because they are not used in the pricing. We assume that H (i=N )j ] 6= H (i=N )k] (j 6= k) for
any i 2 f1 2     Ng.

22

(ii) At each edge of the periods, i.e. t = i=N (i = 1 2     N ), the state is determinedby how many upward changes have occurred. Let j be the number of upward changes
which have occurred during the rst i periods. Then the up-to-date value of the setok at
t = i=N is given by
h uj di j V0 :(11)
In particular, the payo of the option, which must be equal to the option price at thematurity, is given by
j N j
(12)
C (1)j ] = maxf0hK(u;j dNh(uj Vd ) V0 )g :
;

;

;

0

(iii) In general, the hedging probability p depends on the state at the beginning of theperiod. For the period i=N (i + 1)=N ), 
p = p Ni j ]
 r11
fh(d  uj di j V0 )g ; 1 + Nf fh(uj di j V0 )g=fh(d  uj di j V0 )g 1 ; fh(u  uj di j V0 )g 1
 r1
1
fh(uj di j +1 V0 )g ; 1 + Nf fh(uj di j V0 )g(13)=fh(uj di j +1 V0 )g 1 ; fh(uj +1 di j V0 )g 1
if j upward changes have occurred during the rst i periods. Thus p is an adapted process.
Due to the property (iii), it is in general cumbersome to explicitly write the pricing
formula. Instead, we had better rstly show the backward algorithm for computation:
;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

Algorithm 1: 
1. Stand at t = (N ; 1)=N and compute C NN 1 j ] (j = 0 1     N ; 1) by using(or carefully speaking, \interpreting") Theorem 3.1 with the cares (i), (ii), and (iii)
listed above. Thus you have  N ; 1
p NN 1 j ]C (1)j + 1] + f1 ; p NN 1 j ]gC (1)j ]C N j ] =1 + rNf 
for j = 0 1     N ; 1, where C (1)j ] (j = 0 1     N ) and p NN 1 j ] (j =0 1     N ; 1) are given by Eqn. (12) and Eqn. (13), respectively.
 
2. Go back to t = (N ; 2)=N and note that C NN 1 j ] represents what youobtain ifN2
you sell one share of option at the state j at t = (N ; 1)=N . Compute C N j ](j = 0 1     N ; 1) by using(or carefully
speaking, \interpreting") Theorem 3.1.
N
1
In place of the payos, use C N j ]. Thus you have
 
 
n
  o  
N ; 2
p NN 2 j ]C NN 1 j + 1] + 1 ; p NN 2 j ] C NN 1 j ]C N j ] =1 + rf
;

;

;

;

;

;

;

;

;

;

;

N

N 1

 
where C N j ] (j = 0 1     N ; 1) are given by the previous step. p NN 2 j ](j = 0 1     N ; 2) are given by Eqn. (13).
;

;

23

3. Repeat the above procedure until you reach t = 0 and obtainp(0)0]gC (1=N )0] :C (0) = p(0)0]C (1=N )1] +1 f+1 ;rfN
Next, let us explore a situation which gives an easy-to-write formula. Our concern
is the dependence of the hedging probability on the up-to-date value of the setok at the
beginning of each period. We want to avoid this dependence by assigning a specic formof value-interpretation function h.
Let us considerh(v) = avb(14)
where a and b are positive constants. Then, at each period and each state, we can use
the same hedging probability given byd b ; 1 + rNf
p= d b;u b :(15)
By using Algorithm 1, we have;;;a(u d V0) g 
C (1)j ] = maxf0aK(u;j dN j V )b(16)

C N N; 1 j ] = pC (1)j + 1]1 ++ (1rf ; p)C (1)j ] 

(17)

j N j
;

;



N ; 2r  1b0N

N ; 1 
pC N j + 1] + (1 ; p)C N j ]C N j ] = 1 + N r  2no
= 1+ fp2C (1)j + 2] + 2p(1 ; p)C (1)j + 1] + (1 ; p)2C (1)j ]  (18)fN ; 1;;  , and nally

N

!

 N
r
f NX N jp (1 ; p)N j C (1)j ]C (0) = 1 + Njj =0 r  NX
N N!
j N j V )b g
f0
j (1 ; p)N j maxf0 K ; a(u d= 1+ Np:j dN j V0 )b
j
a
(
u
j =0


;

;

;

;

;

;

(19)

Eqn. (19) can be easily proved by mathematical induction outlined above. Intuitively, thefollowing statements would be helpful.
!
N There arej paths between the start (t = 0) and each nal state j at the maturity(t = 1).
 Every path must experience exactly N changes.
 r 1
 Each upward change makes the payo be multiplied by p 1 + Nf.
;24 r 1
 Each downward change makes the payo be multiplied by (1 ; p) 1 + Nf.
;

 So the contribution of each path to the state j with the payo C (1)j ] is

 r  N1+ f
pj (1 ; p)N j C (1)j ];;N

Thus, although it seems an articial example, we have obtained a simple closed-form
pricing formula. We would like to summarize it.

Theorem 3.2 (Option-Pricing Formula (Multiple-Period Model))In thebinomial multiple-period model described in Fig. 3, let us consider a European call option
de ned by De nition 3.1 written on a setok which has the value-interpretation function
h(v) = avb (a b : positive constants). We have Assumption 3.1 and Assumption 3.2 aswell as the following two assumptions:
1. The short rate of interest is a constant rNf during each period of length 1=N .
2. The maturity is Tm = 1 (chosen as the time unit).
Let the price of the option now (t = 0) and the strike value of the option be C (0) and K ,respectively. Then, the following pricing formula holds.
 r  NX
N N!
j N j bfj (1 ; p)N j maxf0 K ; a(u d V0 ) gp
C (0) = 1 + Na(uj dN j V0)bj =0 j
;

;

;

;

where





d b ; 1 + rNf
p= d b;u b :If we further assume K > aV0b dN , which is the condition for the option to have non;;;zero probability for a positive payo at the maturity, then the formula can be expressed ina more (economically) instructive way:!
 r  N X
 nn N! pj (1 ; p)N j rff NX N j
C (0) = 1 + N
K
p (1 ; p)N j
j dN j V0 )b ; 1 + N
j
j
a
(
u
j =0
j =0
;

0

;

;

0

;

;

where

3
2 
K
6 ln aV0bdN 77
n = 64:b ln ud 5
0

Although Theorem 3.2 mentions merely about the option price at t = 0, Algorithm 1
gives us whole the price process at t 2 0 1=N 2=N     1]. It should be also noted thatthe multiple-period pricing formula, Eqn. (19), holds even if the transition probabilities(pu for the upward change and 1 ; pu for the downward change) change period by period25

as far as both the probabilities are positive. Suppose that you are a great person: security
minister of a large country. You are happy if the setok market is stable in terms of the
up-to-date values. The value-interpretation function is xed as the form of Eqn. (14).
You have made best political eort to achieve better stability. Your eort may contributeto
1. the reduction of the volatility factors of the value-interpretation function, i.e. a
and b,
2. the reduction of the volatility factor of the implicit value process, i.e. ju ; dj, or
3. the adaptive control of the transition probability for instance, pu > 1 ; pu when
H (t) is lower than a certain desirable value H0 and pu < 1 ; pu when H (t) > H0.
The rst two may be detected and you could be proud of it if you watch the setok and theoption price processes, whereas the last one cannot. There can be an administrative

strategy which stabilizes the underlying setok with no inuence on the option
price.

Before ending this subsection, we would like to place again the question on the tradeof the setok. This was discussed in the single-period model: \The holder of the setok can
wait for a more protable situation as far as the setok is still tradable. Why we assume
that he sells the setok at t = Tm ?" In the case of Algorithm 1 for deriving the pricing
formula in the multiple-period model, the evaluation regarding the nal period may beeasy to accept after the discussion in the single-period model. So the remaining questions
would be:
 The holder of the setok can wait for a more protable situation as far as the setok
is still tradable (and in fact, it is tradable under the assumption Tm < T ). Why weassume that he sells the setok at each t = i=N (i = 1 2     N ; 1) and reshape his
portfolio period by period?
 Is each period long enough for the investors to operate such a dynamic portfolio
strategy?
Regarding the rst question, we may have the same discussion as in the single-period
model as far as we look at each state of each period as if it were the beginning of thesingle-period model. Detailed discussion in total may lead us to equilibrium theory, which
is out of scope of this paper. The important thing is the fact that each of the dynamicallyconstructed portfolios
 is feasible and satises the riskless property and the no-arbitrage condition.
Regarding the second question, recall the ideal market assumption (Assumption 3.2),
in particular the statement (a). There is no transaction costs both in time and in money.
From the engineering point of view, this would be very demanding. But we have assumed
it. From the users' point of view, there are two excuses. One is the \diligent investor"
statement which have followed Assumption 3.2. The other is a setok-specic scenario:
26

users of the content tend to dislike both (i) an old information and (ii) huge and possibly
expensive storage of setoks. For example, once an entity has veried a digital signature,
he deletes the certicate used4 .

3.3 Pricing in a Continuous-Time Model

3.3.1 Model Description

The previous subsection 3.2 has investigated the models which are discrete both in timeand in values. What happens if we consider not discrete but somewhat continuous models?
Specically, we want to model a situation in which we can have trades whenever we wish,
 we can always observe the market, and the implicit values can take any non-negative real values.
We consider the following model.

Assumption 3.3 (Continuous-Time Model) In Section 3, we are investigating setok(S Y  V H n 1) under Assumption 3.1, and the European call option on it. Let C (t) =c(t H (t)) be the price process of the option.
As a simple continuous-time model, we further assume the followings.
 The function c(t h) is a C 12 -mapping.
 The dynamics of the (observable) up-to-date value process H is given by

dH = (t H (t))Hdt + (t H (t))HdW
where (t H (t)) and  (t H (t)) are adapted processes and W is a Wiener process
under the objective measure.
 De ne G(t) = fH (t)g 1 and corresponding occurrence as g = 1=h. We sometimes;

regard c as a function of t and g . To avoid confusion, we write c^(t g) = c(t 1=g ),where we assume the function c^ is also a C 12 -mapping.

 The price process of the riskless asset is described by the dynamics

dB (t) = rf B (t)dt
where the short rate rf is a deterministic constant.

This is a story of light-weight security. Digital forensics may require more for future dispute settlements 26], 27]. Technologies available for such purposes | secure audit log | are discussed in 28]{30],
for instance.
427

In general, the process H driven by the assumed dynamics

dH = Hdt + HdW

(20)

in Assumption 3.3 is said to be the geometric Brownian motion with drift if  and 
are deterministic constants. In this case, the absolute change in H over any nite timeinterval is log-normally distributed. The geometric Brownian motion with drift appear ina lot of random variables in the real society. We estimate  and  by using the observed
market data they are exogenously given parameters.
In fact, Assumption 3.3 is mathematically quite similar to the well-known BlackScholes model31] for pricing options on stocks. However, it is worth noting that Assumption 3.1 says the setok is neither online nor o"ine divisible. In addition, we cannot go
short for it. These properties are dierent from those of stocks.

3.3.2 Pricing

We wish to derive a pricing formula for the European call option purchased at t = 0 andmatured t = Tm , where Tm is not necessarily equal to 1 but smaller than T . Firstly, let
us dene an adapted process G(t) by

G(t) = g(t H (t))

(21)

where

g(t h) = h1 :(22)
By using It^o's Lemma and Eqn. (22), the dynamics of this process is given by
dG = gt + Hgh + 12 2 H 2ghh dt + HghdW
1 2H 2 2  dt ; H dW+= 0 ; H
H 2 !2H3
H2
2= ;  +  dt ;  dW
(23)
H H
H
where we denote partial derivatives by subscripts:@g = g  @g = g  @ 2 g = g :(24)@t t @h h @h2 hh
In the rest of this paper, we may use this type of notation without clearly stating it as
far as the context is clear.
Secondly, let us consider a portfolio composed of one setok and M options. The
portfolio is dynamically changed, over and over again. As in the discrete-time multipleperiod model, let us think of any innitesimal time interval of length dt. Each dynamics isgiven in the form of SDE. Let F be the monetary value (in terms of the initial investment
at the beginning of the innitesimal time interval) of this portfolio:
F = 1 + MC:
28

(25)

We sell the setok for the value-proportional price and immediately buy one setok withthe up-to-date value for the xed up-to-date price 1 at the end of the time interval.
By assumption, these procedures take no time and are allowed even in the case of theinnitesimal time interval. Thus, over the time interval, the \resale and buy" brings
G + dG ; 1 = dG
(26)
GG
and tells us that the dynamics of F is given by(27)
dF = dG
G + MdC
We have to be careful about the fact that stochastic dierentials are the expectedvalue at the beginning of the time interval. So a more instructive notation for
Eqn. (27) would be] + M (t)E dC ]
Et dF ] = EGt (dG
(28)
t
t)
where Et denotes the expectation operator conditioned by the information available attime t. We apply It^o's Lemma for dC , and insert it as well as Eqn. (23) into Eqn. (27).
Then we obtain5
(
!
)
2

2
2
dF = M ct + Hch + 2 H chh ;  +  dt + (MHch ; ) dW
(
!
)
2

2
2
2
= M c^t ; Gc^g +  Gc^g + 2 G c^gg ;  +  dt ;  (MGc^g + 1) dW: (29)Thanks to the divisibility of the option, by choosing
M = ; G1c^ (30)gwe can make the portfolio risk-free, i.e. force the diusion dW -term to be zero. Note that
M can change over time. This is a dynamic replication of the risk-free asset.
Since any risk-free asset must have rf as the rate of return, we have
!
2

2
2
(31)
M c^t ; Gc^g +  Gc^g + 2 G c^gg ;  + 2 = rf F:
It should be noted that F = F (t) on the right-hand side of Eqn. (31).
Finally, we insert Eqn. (25) and Eqn. (30) into Eqn. (31). The resultant relation must
hold for any occurrence of the adapted process G. So let us use g instead of G to obtain
the partial dierential equation (PDE)2 g2c^ + r (gc^ ; c^) + c^ = 0:(32)
t
2 gg f g
We solve this PDE under the boundary conditionc^(Tm  g) = maxf0 Kg ; 1g
(33)
5

It is elementary to see that ch = ;G2 c^g and chh = 2G3 c^g + G4 c^gg .

29

which says that the option price at the maturity must be equal to the payo at that time6.
The following theorem gives the summary and the notational remark.

Theorem 3.3 (Boundary Value Problem for Option Pricing) Consider the Euro-pean call option de ned by De nition 3.1 written on the setok under Assumption 3.3. The
maturity of the option is Tm and the strike value is K . H (t) is the up-to-date value processof the setok.
Then the only pricing function of the form C (t) = c(t H (t)) consistent with the noarbitrage condition is obtained when c(t h) = c^(t 1=h) and c^(t g) is the solution of theboundary value problem

2 g2c^ + r (gc^ ; c^) + c^ = 0
t
2 gg f gc^(Tm  g) = maxf0 Kg ; 1gin the domain 0 Tm ]R+.

In general, it is dicult to obtain an analytical closed-form solution for the boundary
value problem in Theorem 3.3. However, we do not have to be disappointed. We can use
numerical approach to obtain approximate solutions. The form of the PDE considered is
not really strange.
We have to mention an extremely simple case: if  and  are deterministic constants,
a closed-form solution is easily obtained as follows.

Theorem 3.4 (Option-Pricing Formula (Continuous-Time Model)) Let the up-

to-date value process follow the geometric Brownian motion with drift, i.e.  and  bedeterministic constants. Then Theorem 3.3 yields the pricing formula C (t) = c(t H (t)),
wherec(t h) = K N d1(t h)] ; exp f;rf (Tm ; t)g N d2(t h)]hwhere N is the cumulative distribution function for the standard normal distribution, i.e.
!
Zd
2
1
x
N d] = p
exp ; 2 dx
2
;1

and

!
)
(  
2
1

K
d1(t h) = p
ln h + rf + 2 (Tm ; t)  Tm ; tqd2(t h) = d1(t h) ;  Tm ; t:6This must hold for any state at the maturity.

30

Let us see on which parameters the option price in Theorem 3.4 depends. It depends
on the diusion , the maturity date Tm , the short rate rf , the up-to-date value h, and ofcourse the strike value K . By contrast, it does not depend on the drift  and the lengthof the tradability period T .

Remark 3.1

 In the continuous-time model, the maturity Tm is not necessarily equal

to the time unit.

 Theorem 3.3 holds even if  and  are time-variant and stochastic as far as they

are adapted processes.

 Typically, numerical analysis can be used to solve the boundary value problem.
 The PDE (Eqn. (32)) holds as long as we consider a European-type option and thepricing function is of the form C (t) = c(t H (t)) consistent with the no-arbitrage,
regardless of the payo at the maturity. Therefore, in the case of another Europeantype option, all we have to do is to replace the boundary condition (Eqn. (33)) with
an appropriate condition which describes the payo.

3.3.3 Analysis of the Option-Pricing Formula
In order to investigate the characteristics of the option-pricing formula, we plot the option
prices given by Theorem 3.4 with changing the up-to-date value H (0) 2 80 120]. Because
the price at t > 0 equals to the current (t = 0) price of an option with maturity Tm ; t,
we analyze C (0) only and denote it by C in the following. We choose one year as thetime unit. We interpret the short rate rf into the intuitive (yearly) rate r of interest forthe bank account by
r = exp(rf ) ; 1(34)
and we use percentage representation when we refer to r.
The basic parameter assignments are as follows.
Maturity: Tm = 1 year]
Volatility:  = 0:2
Exercise Value: K = 100
Short Rate: r = 0:5 %]
We are going to show four gures. Since smaller up-to-date values mean the option
holders are in better positions now, the curves are monotone decreasing in each gure.
Firstly, we investigate the eect of the maturity. The plots for Tm = 0:5, 1, and2 are given in Fig. 4. If the maturity is further away from now, things become more
uncertain. The uncertainty relaxes both chance and risk curves for larger Tm areless changing. Roughly speaking, larger Tm bring lower option prices for H (0) < K = 10031

the currently better position of the option holder is less evident under larger uncertainty.
Also roughly speaking, larger Tm bring higher option prices for H (0) > K = 100 thecurrently worse position of the option holder is also less evident under larger uncertainty.
Secondly, we investigate the eect of the volatility . The plots for  = 0:1, 0:2, and
0:4 are given in Fig. 5. If the volatility is larger, things would become more uncertain.
Therefore, the eect is quite similar to that of the maturity the uncertainty relaxes bothchance and risk, and curves for larger  are less changing.
Thirdly, we investigate the eect of the exercise value K . The plots for K = 80, 100,
and 120 are given in Fig. 6. It is obvious that larger exercise values mean currently better
positions of the option holder. Figure 6 represents this feature larger exercise values
bring higher option prices. It should be noted that the option price almost equals to thexed setok price, i.e. C ' 1 = Y (t), when the current position of the holder is neutral(i.e. when H (0) ' K ).
Finally, we investigate the eect of the short rate. The plots for r = 0:01, 0:5, 1:0,
and 10:0 %] are given in Fig. 7. In general, higher short rates make investments moreprotable. Figure 7 represents this feature larger short rates bring higher option prices.
However, realistic short rates show little dierence. Only the unrealistically high rate
(r = 10:0%) dominates evidently.

Volatility=0.2, K=100, r=0.5
3

2.5

From the most changing to the least:
Tm=0.5, Tm=1, Tm=2

Option Price

2

1.5

1

0.5

0
80

90

100
Up-To-Date Value

110

120

Figure 4: European call option price against up-to-date values of the setok whose price is xed
to be 1. The curves for the maturities Tm = 0:5, 1, and 2 are shown. Further maturities bringless changing curves by relaxing chances (for smaller up-to-date values) and risks (for larger
up-to-date values).

32

Tm=1, K=100, r=0.5
3

2.5

From the most changing to the least:
Volatility = 0.1, 0.2, 0.4

Option Price

2

1.5

1

0.5

0
80

90

100
Up-To-Date Value

110

120

Figure 5: European call option price against up-to-date values of the setok whose price is xed
to be 1. The curves for the volatilities  = 0:1, 0:2, and 0:4 are shown. Larger volatilities bringless changing curves by relaxing chances (for smaller up-to-date values) and risks (for larger
up-to-date values).

Tm=1, Volatility=0.2, r=0.5
3
From left-down to right-up:
K=80, K=100, K=120

2.5

Option Price

2

1.5

1

0.5

0
80

90

100
Up-To-Date Value

110

120

Figure 6: European call option price against up-to-date values of the setok whose price is xed tobe 1. The curves for the exercise values K = 80, 100, and 120 are shown. Higher exercise values
mean currently better positions of the option holders, and hence bring higher option prices.

33

Volatility=0.2, K=100, Tm=1
3
From left-down to right-up:
r=0.01, r=0.5, r=1 (similar)
r=10

2.5

Option Price

2

1.5

1

0.5

0
80

90

100
Up-To-Date Value

110

120

Figure 7: European call option price against up-to-date values of the setok whose price is xed
to be 1. The curves for the (yearly) short rates of interest r = 0:01, 0:5, 1:0 and 10:0 %] areshown. Higher short rates make investments more protable and give higher option prices.

4 Compromise and Revocation
4.1 Observation and Implication

Suppose again that you are a great person: security minister. The setok world of yourconcern works mostly well but there are possibility of revocation: implicit values can becompromised (i.e. discontinuously reduced to be zero), and the compromise can cause asudden and signicant ruin of the corresponding up-to-date values and/or price. Depending on the contents, this ruin can be viewed as a revocation. For example, a certicatewith zero condence would be no longer valid as itself, and no longer tradable. You want
to watch how often such a disaster is likely to happen. What can you do for that?
Of course, you can watch the market because your position is so important that you
have resources at least as good as the object providers' and servers'. You can analyse thestatistics. If the revocation frequency so far is not really high, you are probably happy.
But you may still concern about the setoks which have never experienced revocation but
may face it sometime in the future. You want to know the public opinion or publicfear. You may distribute a questionnaire which places a question: \How often do you
think the setok (S Y  V H n m) is likely to have revocation in value? Please specify it
in percentage." This may cost a lot and the result may be too subjective. It takes time,
too. The purpose of this section is to make an attempt to obtain less subjective opinion
by observing the setok/option market: we expect that some parameters related with thecompromise are implied by the market data including the option price. Thus we wish tohave an option-pricing theory under a risk of compromise.
34

4.2 Extended Settings

In this section, we study again a continuous-time model. As for the underlying setok,
we mostly follow Assumption 3.1. The fundamental dierence is the possibility of compromise and the response to it the setok considered is compromise-responsive in value.
Furthermore, we suppose that the primary meaning of compromise is a signicant damageon the eectiveness of the setok itself. Thus we change the assumptions on tradability
and project value:(Non-Strict Tradability) The tradability is assumed to be not strict here. Although
the setok is T -tradable, T is stochastic. Let H (0) > 0 and T (0) = T0 > 0. As
long as the up-to-date value H (t) is positive, T keeps the initial value T0. However,in the case of revocation, the tradability is ruined T (t) = 0 for any t such that
H (t) = 0. But the holders of the setok do not have to be completely discouraged.
In place of the tradability, a refundability arises.
(Project Opportunity) As long as the up-to-date value H (t) is larger than a threshold value H , each share of the setok gives the holder an opportunity of dividend
proportional to the length of time interval. We model this as a Poisson process.
The \no project" assumption which has been used so far in this paper is a special
case such that the intensity of the Poisson process is zero.
We specify this situation formally.

Assumption 4.1 (Setok with Revocation) In Section 4, a single-valued setok (S , Y
V , H , n, 1) with the following properties is studied.

1. The price-interpretation process is an identity process, i.e. Y (t) = 1 for all t.
2. The setok is not online divisible.

3. Any share of the setok is not oine divisible
4. No one can go short for the setok.

5. The setok is T -tradable and the length of the tradable period is given by T =T (t H (t)), where the function T (t h) : R+ R+ ! f0 T0 g is as follows.
(
T0 if h > 0
T (t h) =0 if h = 0:

T0 > 0 is a deterministic constant.
6. The setok is T -refundable and the length of the refundable period is given by T =R (t H (t)), where the function R (t h) : R+ R+ ! f0 T1 g is as follows.
R (t h) =(T1 > 0 is a deterministic constant.
35

0 if h > 0
T1 if h = 0:

7. As long as the up-to-date value H (t) is larger than a threshold value H , each shareof the setok gives the holder a dividend whose process is a Poisson process withintensity .
8. As long as no compromise has occurred, all the up-to-date and implicit value processes are positive and nite.
9. Compromise happens according to a Poisson process with intensity .
10. The setok is compromise-responsive in value.

The rst property Y (t) = 1 in Assumption 4.1 implies that the setok is not compromiseresponsive in price. The intensity of the dividend Poisson process represents a total eectof the use of the setok. For example, suppose the setok is a reusable access-grant ticket.
The holder feels like using it occasionally, say, with the probability of 1 dt during aninnitesimal time interval. Each use gives him a project value of 2 . However, the ticket
is valid only when the up-to-date value is large enough, say, no smaller than H . Thus,
during the innitesimal time interval (t t + dt], the dividend is dt = 1 2 dt if H (t)  H ,and 0 if H (t) < H .
According to Assumption 4.1, we change the assumption on the maturity Tm of theoption. In particular, we assume

Tm < minfT0 T1g:
(35)
We are interested in the price process of the European call option. The dierence from
Section 3 in each dynamics is the key for the pricing. The overview is as follows:
Compromise is incorporated into the assumption on the H dynamics of underlying
setok.
Non-strict tradability is incorporated both
 into the contribution of one share of the setok to the dynamics of a riskless
portfolio (dF )and into the contribution of the option to the dynamics of the riskless portfolio.
Project opportunity is incorporated only into the contribution of one share of the setok
to the dynamics of the riskless portfolio.
Now rstly we replace Assumption 3.3 with the following assumption which includes
the description of the compromise-responsive value process.

Assumption 4.2 (Continuous-Time Model with Revocation) In Section 4, we areinvestigating setok (S Y  V H n 1) under Assumption 4.1, and the European call option
on it. Let C (t) = c(t H (t)) be the price process of the option.
As an extended continuous-time model, we assume the followings.

36 The function c(t h) is a C 12 -mapping in the domain R+
all t 2 R+. R++ is the set of positive real numbers.

R++, and c(t 0) = 0 for The dynamics of the up-to-date value process H is given by

dH = (1 ; (t H (t))dt) f(t H (t))Hdt + (t H (t))HdW g + (t H (t))dt  (;H )where (t H (t)) and (t H (t)) are adapted processes and W is a Wiener process(under the objective measure). An adapted process (t H (t)) represents the intensityof the Poisson process. We regard the described revocation risk as a systematic risk(see 5.2).
 De ne G(t) = fH (t)g 1 and corresponding occurrence as g = 1=h. We sometimes;

look at c as a function of t and g . To avoid confusion, we write c^(t g) = c(t 1=g ),where we assume the function c^ is also a C 12 -mapping for 0 < g < 1.

 The price process of the riskless asset is described by the dynamics

dB (t) = rf B (t)dt
where the short rate rf is a deterministic constant.

c(t 0) = 0 suggests that the revocation makes the call option worthless: the holder canno longer sell the option. In addition, if the revocation happens, the payo at the maturity
is zero because of the non-strict tradability the revocation has ruined the tradability andbrought the refundability. Thus we notice that the dynamics of H is given by
dH = ( ; )Hdt + HdW:(36)

Next, we want to construct a riskless portfolio considering the non-strict tradability
and the project opportunity. This will be done in the next subsection 4.3.

4.3 Pricing

As usual, let us consider a riskless portfolio composed of one share of the setok and M
options. Let F be the monetary value (in terms of the initial investment at the beginningof the innitesimal time interval) of this portfolio. As long as no revocation happens, thedynamic strategy tells us to pay
F = 1 + MC(37)
at the beginning of the innitesimal time interval. After a revocation, we do not need toprice. We want to see dF . Unfortunately, the revocation will not allow g to remain niteg ! +1 as h ! +0. Therefore, without writing dG like Eqn. (23), we here consider
the meaning of Eqn. (27) or equivalently and more instructively Eqn. (28). What we aregoing to do is to write the expected gain conditioned by the information available at the
beginning of the innitesimal time interval (t t + dt].
Suppose the up-to-date value is currently large enough for the project: H (t)  H .
The refundability resulting from the revocation implies that the holder of the setok sells37

it for the xed up-to-date price Y (t) = 1. So the contribution from one share of the setok
to dF in this case (when H (t)  H ) isno(1 ; dt) ;( ; 2)dt ; dW + dt + dt  1 = ( + ;  + 2 )dt ; dW: (38)Unless compromise occurs (this probability is 1 ; dt), ;( ; 2 )dt ; dW describes thebasic contribution of the (tradability of the) setok given by Eqn. (23)7. Also, unless compromise occurs, dt describes the contribution of the project opportunity which is valid.
By contrast, if compromise occurs (this probability is dt), the holder will get a refund(=the xed price, 1) but buy nothing everything will then stop. Also, if compromise
occurs, the compromise-responsive value is ruined to be zero and hence smaller than H ,which means that there is no contribution of the project opportunity.
Suppose instead that H (t) < H . In this case, we do not have the contribution dt ofthe project opportunity. Therefore, the contribution from one share of the setok to dF isno(1 ; dt) ;( ; 2)dt ; dW + dt  1 = ( ;  + 2)dt ; dW:
(39)
Eqn. (38) and Eqn. (39) can be summarized asno+  u (H ; H ) ;  + 2 dt ; dW(40)

where we have used a step function u(h) : R! f0 1g dened by((if h  0) :u(h) = 10 (otherwise)(41)

As for the option, we incorporate the eect of the non-strict tradability, i.e. c(t 0) = 0,
into the dynamics:
!
)
(
2

2
2
MdC = M (1 ; dt) c^t ; Gc^g +  Gc^g + 2 G c^gg dt ; Gc^g dW+M dt  (;c^)
!
2

2
2
= M c^t ; Gc^g +  Gc^g + 2 G c^gg ; c^ dt ; MGc^g dW:(42)
The investigation above (Eqn. (40) and Eqn. (42)) results in
(
!
)
2

2
2
2
dF = M c^t ; Gc^g +  Gc^g + 2 G c^gg ; c^ + + u(H ; H ) ;  +  dt
; (MGc^g + 1) dW:(43)

Thanks to the divisibility of the option, by choosing
M = ; G1c^ g(44)

In Eqn. (23), each term is divided by H . This is cancelled out when we divide dG by G(= 1=H ) asin Eqn. (26) and in Eqn. (27).
738

we can make the portfolio risk-free. As usual, M can change over time.
Due to the no-arbitrage requirement, we have
!
2

2
2
M c^t ; Gc^g +  Gc^g + 2 G c^gg ; c^
+ + u(H ; H ) ;  + 2 = rf F (t)= rf (1 + MC ):(45)

Next, we insert Eqn. (44) into Eqn. (45). The resultant relation must hold for any
occurrence of the adapted process G as long as no compromise has occurred. So let ususe g instead of G(= 1=H ) to obtain the following PDE
2 g2c^ + nr ; ; u g 1 ; H o gc^ ; (r + )^c + c^ = 0:
(46)
f

g
f
t
2 gg
We solve this PDE under the boundary condition;c^(Tm  g) = maxf0 Kg ; 1g (0 < g < 1):(47)

The following theorem gives the summary and remarks.

Theorem 4.1 (Boundary Value Problem under a Revocation Risk) Let us have

Assumption 4.1 for the setok. Consider the European call option de ned by De nition 3.1
written on the setok under Assumption 4.2. The maturity of the option is Tm and thestrike value is K . H (t) is the up-to-date value process of the setok.
Then the only pricing function of the form C (t) = c(t H (t)) consistent with the noarbitrage condition is obtained when(h>0c(t h) = c^(t 01=h) forfor h = 0
and c^(t g) is the solution of the boundary value problem
2 g2c^ + nr ; ; u g 1 ; H o gc^ ; (r + )^c + c^ = 0
f

g
f
t
2 gg
;

c^(Tm  g) = maxf0 Kg ; 1gin the domain 0 Tm ]R++

. u(h) : R! f0 1g is a step function de ned by((if h  0) :u(h) = 10 (otherwise)

In general, it is dicult to obtain an analytical closed-form solution for the boundary
value problem in Theorem 4.1. However, we do not have to be disappointed. We can use
numerical approach to obtain approximate solutions. The form of the PDE considered is
not too strange.
394.4 Inverse Estimation

Return back to the role of security minister.
Let us see on which parameters the option price obtained from Theorem 4.1 would
depend. It does not depend on the drift , and the parameters regarding tradability/refundability periods, i.e. T0 and T1 . By contrast, it would depend on the diusion ,
the maturity date Tm, the short rate rf , the up-to-date value h, the project parameters( and H ), and of course the strike value K . In addition, it would depend on : the riskof compromise. This suggests that the market data may give you some information ona public opinion about the risk of compromise which has not yet occurred. You want toestimate by using the market data observed. This is an inverse problem.
Consider the following procedure for the inverse estimation:(Procedure 1)

1. By using recent market data excluding option prices, estimate the short rate rf andthe volatility  of the up-to-date value.
2. Guess the risk of compromise .
3. Solve the boundary value problem in Theorem 4.1.
4. Compare the result with recent option price data. If there are a lot of options (onthe same setok) with dierent maturities and/or dierent exercise values, use themas well. Compute the error of guess, e.g. in the least-square's sense. Some weighting
may be helpful.
5. If the error is small enough, quit.
6. If you do not satised with the error, change the guess and repeat. Typically, if thecomputed prices are too high, increase the guess. This is an intuitive observationfrom the fact that a revocation makes the option worthless. Once compromised,
one cannot expect any yield from exercising the option and immediately selling(refunding) the setok that would be just a waste of time.
The boundary value problem may take time to be solved. Procedure 1 in total may
be too heavy. But you may be more condent and feel more speedy than in the case ofquestionnaire.
What you want to do may be just to see whether exceeds a certain value, say, 0.
The intuitive observation tells us that the option price would be lower for larger . Thereis a possibility that the following more practical procedure without repeat will work well.

(Procedure 2)

1. By using recent market data excluding option prices, estimate the short rate rf andthe volatility  of the up-to-date value.
2. Set = 0.
3. Solve the boundary value problem in Theorem 4.1.
40

4. Compare the result with the current option price data. If there are a lot of options(on the same setok) with dierent maturities and/or dierent exercise values, usethem as well.
5. By using a tool for statistical test, examine whether you can say the computed prices
are higher than the observed prices with non-negligible probability.
6. If the answer is Yes, think of it as an alarm. It might be time for you to demonstrate
your administrative talent as a minister.

4.5 Eect of Derivatives

This framework is virtual in a sense that there has not been established such a setok
world. If we really want to have an option market on the setoks, we must be careful
about the possible change caused by the introduction of derivatives. Even in the existingnance, the eects of derivatives are on-going theoretical and empirical research topics.
There are dierent opinions.
There is a common public and regulatory perception that derivative securities may
increase volatility and can have a destabilizing eect on the underlying market. Basically,
they are afraid that poorly informed speculators could have a destabilizing eect. Academic reports on this side are few. A theoretical example is 32] and an empirical example
is 33].
Among academic people, however, the opposite opinion has been dominating so far:
if derivatives promote information dissemination and collection, introduction of themwould reduce volatility. The majority of studies have been in this direction. Examplesof theoretical approaches include 34]{37]. Empirical supports include 38]{41]. There
are also empirical studies which show volatility decreases when the derivatives get more
popular in trading quantity 42], 43]. If this is the case for setoks as well, we are happy
to introduce derivatives.
Diculties of this discussion arise, for example, from the followings44].
 The very rst introduction to a particular market is only once.
 There could be a lot of correlation among other assets.
 Regulatory situation can change.
Anyway, it is worth continuing studies.

5 Related Works

5.1 Foreign Derivatives and International Issues

The setok market may seem similar to the foreign exchange market when we consider
tradable single-valued setoks in particular, the ratio of the price to the value can have
similar properties to those of foreign exchange rates. The theories of currency and foreign
derivatives 45], 46] use dierent short rates in dierent countries, which is the major
41issue of the theory. Foreign currencies are regarded as tradable and divisible assets. This
assumption makes the theory easier.
By contrast, our setok framework has been domestic so far the key point is not in
how to deal with the short rate(s) but in how to model the tradability, refundability, anddivisibility. Of course, international settings would be very interesting future work. If
we assumed that several dierent virtual currencies are available over the network, theresultant theory would be more interesting: virtually international economy.
As for international issues, there are empirical studies which show that there arecommon features as well as dierent features in derivative statistics and questionnaire
results among dierent countries 47]. Common features include Major participants in the option market are large rms.
 The highest motivation comes from hedging purposes rather than from speculating
and arbitrage.
 The hedging activities are mainly aimed at hedging anticipated transactions within
a year.
Dierent features include The highest concern is in the lack of information in some countries, but volatilities
in others.
 How popular derivatives are.
 How popular currency derivatives are.
The study of the derivative eect in the setok framework would, if it starts, have toconsider dierences over countries and regions. Note that even regarding conventional
nancial derivatives, recently mandated disclosures have just enabled us to obtain a large
number of samples to investigate rms' usage of derivatives 48].

5.2 Jump Processes

In the continuous-time model, we derived the PDEs for option pricing based on the noarbitrage requirement. As far as Section 3, there is nothing to be appended here. However,
the model with revocation in Section 4 needs more words here. In particular, about anassumption which is implied by Eqn. (45) in nancial words, Eqn. (45) implicitly assumes
that the revocation is a systematic risk. We mentioned it in Assumption 4.2 but haveplaced no discussion on it so far.
A related issue is found in the option-pricing theory which allows the underlying stock
to have jumps in its price process 19], 21], 49], 50] 8 . They resorted to the conventional
CAPM (Capital Asset Pricing Model) 52] by assuming that jump processes describe
nonsystematic or idiosyncratic risks, which implies that risks such as rms' defaults havea too wide variety of backgrounds with no good reason to be pre-distributed to appear in
In early days, a study using a sample of NYSE listed common stocks reported that jumps in common
stock returns leads relatively small deviations from the Black-Scholes formula 51], which suggests thatthe bias-elimination by 50] may be insignicant.
842

a global risk premium. This is an extreme assumption and there are a lot of arguments
about the systematic/nonsystematic jump processes 53]{56]. In fact, jumps observedin stock prices are reported to be systematic across the market portfolio 57], and thisphenomenon is more signicant in the foreign exchange market than in the stock market
58]. The feature of the foreign exchange and currency option markets can be partly
understood by the eect of news arrival 59], 60] and changes in monetary policies 61].
Most of the biases in American foreign currency options appeared consistent with thefact that the underlying spot currency rate follows a mixed jump diusion process 62].
The mixture of idiosyncratic and systematic jumps is studied in 63], 64]. A more recent
study on inecient index portfolios also supports systematic risks 65], 66]. However, it
should be noted that empirical studies are not so easy as one might think. For example,
the empirical study on the jump-diusion process of interest rates has not matured dueto the diculty of likelihood estimation 67], 68]. There are a lot of arguments about
CAPM as well 69]{71] 9 .
We have used Eqn. (45), which is based on the systematic case. Heuristically speaking,the more similarly network people or entities look at the revocation risk, the better model
our choice of Eqn. (45) would give. In other words, we hope that related information
is distributed more fairly with less hesitation, and revocation risks shall be more open
than conventional default risks. This is also an assumption, and we could have neither
empirical supports nor objections at present the setok world has not been establishedyet. What we can say now is based on a technological insight: our choice could go wellwith the recent trend in the public-key infrastructure toward a single-directory system72], 73].
If we can interpret popular information-security software and protocols into a utility
function, a conditional-expectation approach would be powerful. The corresponding discussion in nance is in 74], 75] for instance pricing formulae are derived in a way thatthe utility function of each investor is optimized. The resultant pricing formula can be
expressed by using the conditional expectation operator.

5.3 Credit Derivatives

We introduced a simple example of digital certicate revocation in Section 1. Intuitively,
the example reminds us of credit risks13], 14]. The rst issue to be considered in a
credit-risk study is how to dene credit risk and how to describe an appropriate model.
A typical understanding is: Credit risk is the risk that an obligor will default in performing its obligations76].
In our revocation example, the public-key infrastructure (i.e. the set of certicate
authorities involved) plays a role of an obligor.
How to interpret the word \default" is more dicult in our setok world. This isbecause not only the revocation of a certicate but also a change of security policies canmake a signature unveriable. Suppose that you have a set of certicates and the publickeys of the certicate authorities involved. Suppose also that the system uses a trust
These are nothing but examples there are really a vast number of studies on utility functions andthe CAPM.
943

metric approach: the signature verication software rstly computes a total trust value ofthe certicates from the input of trust values written on the certicates. If your security
policy allows the nal computed trust value, the verication succeeds. Thus not only achange of the certicates but also a change of your policy can cause a verication failure.
This is a matter of taxonomy and we believe that our framework can be used in either
case.
Recently, an inviting application of credit derivatives is emphasized77] in nance: tocreate synthetic assets (e.g. credit-linked notes). A portfolio including the option analyzed in Section 4 may be directed to this sort of application. Value/price interpretation
functions can contribute to synthetic assets.
An important line of research regarding credit derivatives is the use of transition of
ratings14]. Among a lot of studies, a pricing given in 78] is powerful and pretty much
related to our setok modeling. The pricing is based on a continuous-time dynamics forthe underlying asset together with rating-dependent coecients which cause incompleteness. In any case, given the rating and the other input parameters, we can compute theoccurrence of the variable described by the dynamics. In the setok world, value/priceinterpretation functions correspond to this computation procedure.

5.4 Information/Network Security and Economics

Total management of information-security is so hard because of human and social aspectsof it. For example, a lot of frauds are resulting from poor motivation for security management. A lot of fear is from nancial uncertainty. Even with sucient technologies (Iam not quite sure if such things really are available), most end users cannot understandthem. So the very starting point and the very nal defense would be interpreted in termsof money and literacy.
In fact, there have been a lot of studies in the interdisciplinary area between information/network security and economics. Let us start with a taxonomy of them: charging,metering, payment, punishment, and insurance. Details are given in the following subsections. Before proceeding to them, it should be noted that there would be more categories
if we had a wider view such as network and economics rather than information/network
security and economics.

5.4.1 Charging
In network engineering, Quality-of-Service (QoS) issues are not really new but still actively
providing research topics as well as testbeds and projects 6]{8]. This is because the mostcommon global network, the Internet, is a best-eort network. We may claim a lot of kindsof communication qualities. Most traditional components are bandwidth and reliability.
Dierent QoS levels may be eligible for dierent prices 5]. Competition among dierent
providers and taxation can make things more complicated 79]. Actual charging policies
are not straightforward. This issue is one of the main concerns in Internet economics 80].

Analyses and experiences, both in network engineering and in economics,
are required.
44

5.4.2 Metering
Cost sharing through usage-sensitive pricing or charging can be eective for managing
Internet growth 81]. The usage-sensitivity can be either direct or indirect. An Internet
Service Provider (ISP) may charge you on the basis of connection duration: how long you
keep the Internet connection via the ISP, for instance. This is an example of the directcase. A portal WWW site may charge a company whose advertisement is on the portal
page, on the basis of eect: how many visitors click the banner, for instance. This is an
example of the indirect case.
In order to implement usage-sensitive pricing, either direct or indirect, we need asecure metering scheme because just a simple record or log may be easily altered there isa risk of cheating. Ecient solutions for this security problem are not trivial, especially inthe case of real-time applications. There have been eorts to do better against this tradeo 82]{84]. Aiming at more general situation where secure audit log is needed, several
protocols have been developed with the help of chains of one-way hashed values 28]{30].
This research direction of \digital forensics" is important toward dispute settlement in
the network society.

This is an area where information-security engineering is applied to support
data used in nancial transactions.
5.4.3 Payment

When charged, we pay. But how? | We need technologies for payment over an open
and insecure network. There are a lot of payment systems. Some assume an underlying infrastructure of nancial transaction and settlement, while others assume no otherinfrastructures.
In the former case, we may use standard protocols which assume an underlying creditcard system. SET (Secure Electronic Transaction) 85] is a good example it provides
condentiality, reliability, entity authentication, and data integrity needed for the nancial
data transmission used in the credit-card system. It only secures data in transit over anotherwise insecure network.

This is also an area where information-security engineering is applied tosupport data used in nancial transactions.

In the latter case, we may use digital cash which is typically a digitally-signed string ofa certain format 85]. Actual implementations require more complexity due to the risk thatthe customer might spend the same digital coin more than once. Usual solutions encode
the customer's identity in the coin in a way that the identity is normally protected but
is revealed if the customer spends it twice. A typical trick for this is to have a challengeresponse protocol between the customer and the merchant when the coin is spent 86] if
the same coin is spent twice, then dierent challenges have to be answered, which gives
away the customer's identity. In terms of cryptographic primitives, blind signature isuseful in anonymous payment systems 87] anyone can check that a stripped signature
was formed by using the signer's private key, but the signer knows nothing about thecorrespondence between the elements of the set of stripped matters and the elements ofthe set of unstripped matters 88]. It should be noted that the security might rely muchon hardware such as smartcards 89], 90] depending on the system design.
45

The previous paragraph used a word \digital coin" which implies that each coin has
a xed monetary value. This is not necessarily true because the technical term has not
yet been well dened. What is worth noting here is that the system could be more ecient if the monetary value of each coin is less granular17].
In other words, very discrete digital cash could be more light-weight than almost analogue
solutions 10 . The eciency is more signicantly required in some applications: electronic
toll collection (ETC) in Intelligent Transportation Systems (ITS), for example 91].

This is an area where information-security engineering is used to constructapplication systems in nance/economics.
5.4.4 Punishment

A trivial use of punishment mechanism for information security is liability management. If
introduced in a proper way, liability can motivate people to manage information security
better. I would like to focus on other helpful aspects of punishment in the rest of thissubsection.
In a broader sense, information/network security concerns availability as well as condentiality, authenticity, and integrity. Typical threats to availability are Denial-of-Service(DoS) attacks an attacker may send plenty of connection requests to a target server inorder to exhaust the resource of the target. If the server is an online shopping mall,the attack may not only cause nancial damages but also do harm to the reputation of
the mall due to the resultant low availability. We dislike an extremely long queue in asupermarket.
To make matters worse, security mechanisms might enhance the DoS threat. Let
us suppose a secure handshake before starting a condential communication. In order to avoid impersonation by intruder-in-the-middle attacks, the handshake protocol
is equipped with entity authentication, typically by digital signatures. The signatures
shall be veried. Unfortunately, the verication cost is usually expensive in computation,and could be abused by DoS attackers they launch a lot of bogus requests. The recipientof these requests may be hanged up earlier than in the case of non-secured handshake,because the computational cost or damage per attack is more signicant due to the expensive verication. This kind of DoS attack is quite a bitter threat primitives for asecurity reduces the resistance against another security property. At the worst case, it
could encourage a poor implementation which omits the verication during a busy period.
Intrinsically, we could not make DoS impossible. But we can discourage the attackers
by making the attack cost the attacker a great deal. We are happier if the attacker
must consume his own computational resource before we consume the same order of ourresource. This can be done in several ways 92]{94]. A quite similar trick is proposed
to combat junk e-mails 95], 96]. The common idea is to make the attacks expensive.
In other words, attackers are punished by cost. They must invest quite a lot in theirresources to launch powerful attacks. Hopefully, the actual budget of them may not allowit.
We say \very discrete" and \almost analogue" because all the data distributed over a digital communication network are intrinsically discrete.
1046

In a similar line of study, there is a project called MarketNet 97] which limits thepower of attackers by their virtual budget available. It is assumed that a virtual currency
system including a virtual central bank works well for this purpose.

This is an example in which information-security engineering makes use of
a common feature of money: we like it and do not want to lose it probably
because lack of it may limit activities we want to do.
5.4.5 Insurance

A trivial eect of insurance for information security is people's motivation better management contributes to lower rate of disasters and thus lower risk premium. I would liketo overview another example usage of insurance.
Before we start public-key based secure communication, we must associate the publickey condently with its owner. Because single-authority systems do not scale well in alarge network, many systems resort to authentication by a path or chain of authorities
98]{102]. A single path is weak because it relies on the correctness of all the authorities
on the path. Multiple paths may increase assurance but the specic level or value ofthe assurance may be unclear. Therefore, trust metrics for measuring the assurance havebeen studied by a lot of researchers 9]{12].
Among them, if we can assume that any damage caused by broken paths can be atleast partly compensated by money, an insurance-based approach 104], 4] could absorb
miscellaneous uctuations in the widest range. That approach uses a directed graphmodel the nodes are public keys Kj (j = 1 2   ) and the edge K1 ! K2 exists in thegraph if the user has a certicate that assigns attributes (including the owner) to K2 andwhose signature can be veried by using K1. Each edge is labeled with the attributes
included in the certicate that the edge represents. Each edge K1 ! K2 also has anumeric value that represents an insurance amount if the private key corresponding to K2
is compromised and used maliciously, then the owner of K1 is liable for the stated amountof money. A natural and prudent trust metric in this model would be the minimum insured
amount of the name-to-key binding for the target key (Fig. 8). One of the advantages ofthis metric is its eciency the minimum amount can be computed by using well-knownecient maximum-ow algorithms.
We do not intend to discuss pros and cons about this metric here. Instead, we want topoint out that there could be similar but dierent usage of insurance. For example, let us
suppose a routing service for a network-security infrastructure which uses identity-based
cryptography. In this case, identity-to-address binding could have a similar problem tothat in public-key infrastructure, and thus we might think of the use of insurance.

47

K6

$50

Minimum Cut
- K1
- K2
$10

6

$10

?

K5

$30
$20

?
- K3

$30
$5

- K4

Figure 8: A trust model based on insurance. Each node represents a public key (of a particularowner) the user already have. An edge exists from the node i to the node j if the user has acerticate that assigns attributes to the key Kj and the owner's signature can be veried byusing Ki . Each edge has a numeric value that represents an insurance amount if the private
key corresponding to K2 is compromised and used maliciously, then the owner of K1 is liable for
10 dollars, for example. A trust metric is dened as the minimum insured amount, which canbe computed by using maximum-ow minimum-cut algorithms. In this gure, the minimum is$10 + $5 = $15.

6 Concluding Remarks
6.1 Summary

This framework is an attempt to use stochastic techniques for the future network risk
management. We have made an abstraction of uncertain digital objects in network commerce and dened the security token, which is abbreviated into a word coinage setok.
Each setok has its explicit price, explicit values, and timestamp on it as well as the maincontents. A number of properties of the setok were dened. They include value response
to compromise, price response to compromise, refundability, tradability, online divisibility,
and o"ine divisibility.
We then investigated a European call option written on the up-to-date value of asetok which has only one explicit value. This setok is called a single-valued setok. In
continuous-time as well as discrete-time models, we have derived several option-pricing
formulae. These formulae do not require any divisibility of the underlying setok. The
basic features of the option price were examined numerically. An intuitive interpretationof the results was given as well.
In search of applications, an inverse estimation of the compromise probability wasstudied. The result suggests that we may be able to estimate the public opinion about
the probability. Finally, the paper shows a wide variety of interdisciplinary surveys inrelated areas to enrich the implications of our work. These implications as well as theapplication are quite important for interdisciplinary researchers to try the next stage ofthis work.

48

6.2 Setok and Finance/Economics

As mentioned in Section 5, the foreign exchange market is related with this framework.
What lies in between is a virtually international and borderless market over the global
network. Our setok framework may be extended in that direction. However, we must be
careful about the eect of new derivatives in the market. An advanced social science will
be needed as well.
The setok framework has investigated only simplied situations so far. A lot of extensions shall be made (and in fact, we have already started some of them):
Untradable and unrefundable setoks: Studies of them may learn a lot from projectinvestment theory and real options, and vice versa the abstraction in our framework
may contribute to new challenges in the existing areas.
Eect of Short-Selling Restriction: We have derived option-pricing formulae witha restriction that short-selling of the setok is disallowed. In theory, it works. In
practice, however, we have to be careful. This is because our nancial experiences
tell that a short-selling restriction can reduce the informational eciency of themarket 105] 11 . In other words, some of the items in the ideal market assumption(Assumption 3.2) might be threatened. Since we could not have any empirical testat present, a reasonable simulation studies would be helpful.
Other exotic derivatives: In particular, futures are very attractive because futures have been most popular in foreign derivatives so far,and
 most futures are closed before and hence without actual delivery of underlying
assets.

6.3 Setok and Information/Network Security

Stochastic approaches are very powerful in studying uncertain social activities. This iswhat we are facing with in security system engineering. So one possible direction is to use
the stochastic technique achieved in the setok study for analyses of information-security
systems. An attempt in Section 4 is one of the simplest cases in this direction. Intrusion
detection 106]{114] is another good candidate. To put it in short, studies in this direction
are in search of virtual but reasonable sensors realized by distributed agents. Setok
markets are not necessarily actual economies. If agents make decisions by using a simple
set of policies and the sensing/alarm system works well as a whole, it could be regarded
as a good application of our setok framework to network security.
Another possible direction is from what the setok world needs. If some setok/option
properties could not be realized by the current state of the art in information-security

From January 3rd, 1994, the Stock Exchange of Hong Kong allowed 17 out of 33 constituent stocksof the Hang Seng Index (HSI) to be sold short subject to restrictions. These restrictions were lifted
on March 25th, 1996, and at the same time, 113 of the stocks including all the constituent stocks wereallowed to be sold short. These relaxations improved the informational eciency.
1149

engineering, there may be a good opportunity of creating new cryptographic protocols
and/or primitives. A exible timestamping protocol is a good candidate. For example,o"ine divisibility, which is dened but never used in this paper, would need a timestamp
which are secure but allows specied manipulation afterwards.
Finally, we have to point out that how nancial tools can help information security.
Total management of information-security is so hard because of human and social aspectsof it. In fact, a lot of frauds are resulting from poor motivation. A lot of fear is from
nancial uncertainty. Even with sucient technologies (I am not quite sure if they reallyare available), most end users cannot understand them. So the very starting point andthe very nal defense would be interpreted in terms of money and literacy. By using
security-economics approaches based on the setok framework, we are going to enhance
motivation and make things easier for people to understand. Financial tools will play acentral role there.

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