Security Token and Its Derivative in Discrete-Time Models
Kanta MATSUURA
Institute of Industrial Science, University of Tokyo
Meguro-ku, Tokyo 153-8505, Japan

ABSTRACT
Aiming at modeling uncertain digital materials in the
network society, we de ne 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 main contents. The values are
uncertain and may cause risks. Several important
properties of the setok are de ned. Then, in order toprovide risk-hedging opportunities, a derivative written not on the price but on the value is introduced.
The derivative investigated is a simple Europeantype call option. We derive option-pricing formulae in a single-period model and in a multiple-period
model. These formulae do not require any divisibilityof the underlying setok.

Keywords: Electronic Commerce, Security Token,

Uncertainty, Derivative, Option Pricing, Binomial
Model.

1. INTRODUCTION
Applied cryptography triggers a market of digitalmaterials. They have their prices. In addition, they
likely have other numerical values. For example, digital certi cates may have con dence values [1]{[4].
Access-grant tickets may have priority numbers or
QoS (Quality-of-Service) values [5]{[8]. Digital images may have con dence values about their innocence in terms of illegal-copy regulation. Any product may be associated with some insurance contracts[9]. Reward points may be attached. Those additional values may change unpredictably over timeand cause risks.
A common tool for hedging risks is a nancial derivative. In the existing nance, a lot of theories havebeen developed. Likewise, we want to develop appropriate theories in the digital world. In the rst
place, theories need models. The purpose of this paper is to model the uncertain digital materials andintroduce a derivative written on them. This modeling is described in Section 2. Subsequently, by using
binomial processes, Section 3 demonstrates how to
price the derivatives based on the model. Finally
Section 4 concludes the paper.

Setok

2. MODELING

Let us start with typical entities in network commerce: Provider: Digital copyright management is neither easy nor trivial. Maintenance of security infrastructures (e.g. public-key infrastructure) is not, either. We need specialized entities which are eligible for them. Typically,
they are trusted organizations or licensed rms.
Providers would be happier if the digital materials they provide are circulated more frequently in larger amounts; it would improvetheir reputation and make attached advertisement more pro table. They would have a motivation to give rewards for active usage of thedigital materials.
 Customer: We do not trust individuals in termsof (i) behaviour, (ii) nancial situation, and(iii) resources (for communication and computation).
 Server: Selling digital materials to untrusted
customers is another dicult and non-trivial
task. We need specialized entities which can
do it and can have a good connection with
providers. Typically, they are trusted organizations or rms; they can be less trusted in comparison with providers but they must be moretrusted than customers. Due to the rewards
from providers as well as their basic businessreasons, servers would like to enhance theirtrading activities with customers. So servers
have a motivation to re-circulate the digitalmaterials. They may get the digital materials back from their customers in exchange forsome refund. The refund may depend on theprice and/or values of the material.
The observation above makes us model uncertain
digital materials as follows.
((De nition 1)) A security token or setok is a digitalmaterial which has the following four attributes:

 contents which may include MAC (message au-thentication code), digital signatures, or othersecurity-related control sequences if necessary, a non-negative explicit price (denoted by S)which is paid on purchase, a set of non-negative explicit values (denotedby V1 , V2 ,   , Vm where m is referred to as thedimension of the explicit values) which represents some qualities of the contents in a waythat larger values imply better qualities regarding the corresponding feature, and a timestamp which indicates when the setok isissued,and is associated with

 a non-negative implicit price (denoted by S )and a set of non-negative implicit values (denotedby V1 , V2 ,   , Vn where n is referred to as thedimension of implicit values)in the following way.

 The explicit price S is speci ed as the occur-

rence of a price-interpretation process Y (t) =y (t; S (t)); i.e. y (t0 ; S (t0 )) is written on the
setok as the explicit price of the setok which
is purchased at time t = t0 . This occurrenceis also called the up-to-date price. The priceinterpretation process is a non-negative process and also called the up-to-date price process. y = (t; s) is called a price-interpretation
function and monotone increasing with respect
to s. Once written, the explicit price is neverchanged.
 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)). These occurrences arealso called the up-to-date values. The valueinterpretation 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 valueinterpretation functions. Once written, the explicit values are never changed.
The up-to-date processes Y (t) and H (t) are observable in the market. A setok in the market is denotedby (S ; Y ; V1 , V2 ,   , Vn ; H1 , H2 ,   , Hm ) or sometimes shorthandly by (S , Y ; V , H , n, m). Likewise,a share of the setok already purchased and held bysomeone is denoted by (S; V1 , V2 ,   , Vm ; t0 ) or
sometimes shorthandly by (S; V , m; t0 ).
De nition 1 accepts not only purely nancial digitalmaterials but also digital commodities as setoks; wehave not speci ed the contents. For simplicity, therest of this paper studies a special case: a singlevalued setok de ned as follows.
((De nition 2)) 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 thesubscript \1".

Setok Properties

For single-valued setoks, we de ne several important
properties.
((De nition 3)) 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.
 At any time t 2 T , the value-interpretation
process is positive and the holder of the setok
can sell it. This resale is possible only at thevalue-proportional price Sp de ned bySp = h(t; V (Vt); yV ((t;t)S; (t)) ; V (t)) :
(1)
1
2
n
T is called a tradable period and allowed to be composed of open and closed time intervals; all of theforms [TL; TU ], [TL; TU ), (TL; TU ], and (TL; TU ) (anda set of them) are available. The tradable period canbe either deterministic or stochastic.
((De nition 4)) A T -tradable setok is said to bestrictly T -tradable if and only if the following two
conditions are satis ed:

 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 .

Since non-trivial security cares are needed to provide
or serve setoks, divisibilities of setoks are not trivial.
For example, it would be dicult for customers todivide digitally signed contents into two or more valid
pieces. We de ne two properties with respect to thisissue.
((De nition 5)) A setok (S , Y ; V , H , n, m) is said
to be online-divisible if and only if the following condition is satis ed.
 Whenever the occurrence of Y (t) is positive,
anyone can purchase 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 andexplicit values

Sc
Y (t0 ) hi (t0 ; V1 (t0 ); V2 (t0 );    ; Vn (t0 )) (2)assigned where t0 is the timestamp on it (i =1, 2,   , m).

((De nition 6)) A share of setok (S; V1 , V2 ,   , Vm ;t0 ) which has a positive explicit price S is said tobe oine-divisible if and only if the holder of it candivide 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-proportionalmanner, i.e.
 S1 > 0; S2 > 0S1 + S2 = S;(3)
iVji = SS Vj (i = 1; 2; j = 1; 2;    ; m): (4)OptionWhen we study pricing theories for options, we haveto specify market assumptions. Although we do notintend to stick to a single scenario, the rest of thispaper consider the following case.
((Assumption 1)) In the models below, a strictly T tradable single-valued setok (S , Y ; V , H , 1, 1) withthe 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 neither online nor oine divisible.
3. We cannot go short for the setok.
4. The tradable period T is composed of a single
time interval of a xed positive length T . We
will make it explicit by saying \T -tradable",
where T is not boldfaced.

5. The possession of the setok has no meaning as
a project.
6. All the up-to-date and implicit value processes
are positive and nite.
7. Value-interpretation function h(t; v) is a deterministic function h(v) such that h(v) > 0 for
any v  0.
((Assumption 2)) We assume an ideal market which
satis es the following conditions.
(a) Any transaction can be completed immediately,
free of charge.
(b) 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 riskless short rate is denoted by rf andassumed to be a deterministic constant.
(c) The selling price is equal to the buying price.
(d) The market is free of arbitrage.
(e) The option can be bought and sold on a market
at any fraction.
(f) Anyone can go short for the option.
The use of the constant price in Assumption 1 is because electronic cash systems are more ecient if themonetary value of each cash or coin is less granular[10]. That is, we want to allow as wide variety of
di erent electronic cash systems as possible.
Now we are ready for introducing an option written
on a setok.
((De nition 7)) A European call option on the singlevalued strictly T -tradable setok (S , Y ; V , H , 1, 1),purchased at time t = 0, is a derivative which provides a right to buy one share of the setok with a reserved explicit value K at a particular time Tm < T
in the future for its xed price, 1, regardless of the
up-to-date value H (Tm ) at 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.

3. OPTION PRICINGThis section is the rst attempt to show option pricing in the setok world. For a readability reason, we
assign the time unity so that Tm = 1 where Tm isthe maturity of the option considered.

Our task is to nd a riskless portfolio composed of
one share of setok and M options. The initial investment for this portfolio is 1 + MC . In order toachieve risk-freeness, the portfolio must have exactly
the same payo at the maturity t = Tm = 1 regardless of the state. That is,h(V0 )
h(V0 )
h(uV0 )  1 + MCu = h(dV0 )  1 + MCd : (9)3 u  V0pu (Implicit) Value


QQ

V0 
Q

Q1Q; pu
QQ
s dV

0

t

-

t

t=0

t=1

After a manipulation on Eq. (9) with the help ofh(uV0 ) 6= h(dV0 ), we have

M = Ch(;V0C)





1

1
h(uV ) ; h(dV ) :(10)

Figure 1: A single-period binomial model. At theend of the period (t = Tm = 1), either the \upward"or the \downward" state has occurred: the former
is with the up-to-date value h(uV0 ) while the latter
is with 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 option pricing does not require theseprobabilities. We assume that h(uV0 ) 6= h(dV0 ).

Due to Assumption 2, this portfolio is feasible.
In order to achieve no arbitrage, the portfolio musthave the rate of return exactly as low as the short
rate rf [11]. Therefore,

Single-Period Model

From Eq. (10) and Eq. (11), we obtain

The rst model used here is a single-period binomial
model described in Fig. 1. At present (t = 0), theup-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
are positive constants such that 0  d < 1 < u. The
former occurs with probability pu . Hence the latteroccurs with probability 1 ; pu . You do not know
these probabilities. Please nd a reasonable option
price C . This is the problem to be solved below.
Firstly, we point out that the option has a payo of
Cu = maxf0;hK(uV; h) (uV0 )g(5)
0
in the case of 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 the tradability, the holder can immediately resale this share for the value-proportional
price
K 1= K ;Sp = h(uV(6))
h(uV )
0and achieve a positive gain of0h(uV0 ) :Sp ; 1 = K h;(uV(7)0)
On the contrary, if h(uV0 )  K , the holder of theoption does not exercise it hence gets nothing. Likewise, the downward change gives a payo of
Cd = maxf0;hK(dV; h) (dV0 )g :
(8)
0

d

u

0

0

V0 )  1 + MC :(1 + rf )(1 + MC ) = hh((uVu)
0(1 ; p)Cd
C = pCu +1 +
r

(11)

(12)

f

where p is de ned by;1;1(1 + rf ) fh(V0 )g :p = f(h(dV0 )g ;;1
fh(dV0 )g ; fh(uV0 )g;1

(13)

The following theorem gives the summary.
((Theorem 1)) In the binomial single-period model
described in Fig. 1, let us consider a European call
option de ned by De nition 7 under Assumptions 1,
2 and the common frictionless no-arbitrage market
assumptions. Let the price process of the option be
C (t). Then, the following pricing formula holds.
(1 ; p)Cd
C (0) = pCu +1 +
r
f

(14)

where

Cu = C (1)[up] = maxf0;hK(uV; h) (uV0 )g
0
max
f
0
;
K
;
h(dV0 )g
Cd = C (1)[down] =h(dV0 );
1
(1 + rf ) fh(V0 )g;1 :p = f(h(dV0 )g ;fh(dV0 )g;1 ; fh(uV0 )g;1

(15)
(16)
(17)

3 u V


3


QQ

3 u2V0 Q(Implicit) Value 


uV 
3

Q
QQ


QQ
QQs u dV
3


2

0

QQ
QQs udV 
QQ
3

QQ

Q



V 
QQ
QQ
Q
0

0

0

0


QQ
s dV QQQ

QQs ud V
3



0

QQ
QQs d V 
QQ
Q
2

2

0

QQ
QQs d V
3

t

t=0

t

t = 1=3

0

t

0

t

t = 2=3

t=1-

Figure 2: A multiple-period binomial model. For the drawing convenience, the illustration has only N = 3
periods. Each state is denoted by the number j of upward changes which have occurred. It is assumed that
H (i=N )[j ] 6= H (i=N )[k] (j 6= k) for any i 2 f1; 2;    ; N g.

Multiple-Period Model

It is easy to extend the option pricing for dealing witha more realistic multiple-period model described inFig. 2; we can use the single-period pricing recursively from the nal period to the rst.
It is in general cumbersome to explicitly write thepricing formula in the multiple-period model. Thisis because the parameter p given by Eq. (17) in Theorem 1 depends on the state. So we had better rstly
show the backward algorithm for computation:((Algorithm 1))
1. The option price at the maturity is equal to thepayo .
2. Stand at t = (N ;1); =N and look at the nal period. Compute C NN;1 [j ] (j = 0; 1;    ; N ;1) by using Theorem 1.
3. Go back to t = (N ; 2)=N and compute the;option prices C NN;2 [j ] (j = 0; 1;    ; N ; 1)
by using
1. In place of the payo s,; Theoremuse C NN;1 [j ].
4. Repeat the above procedure until you reach t =0 and obtain C (0).
Let us explore a situation which gives an easy-towrite formula. Our concern is the dependence ofthe parameter p on the up-to-date value of the setok at the beginning of each period. We want toavoid this dependence by assigning a speci c form ofvalue-interpretation function h. An example which
achieves this independence is given in the following
theorem. Due to the space limitation, this paper
does not show the proof which is available from theauthor upon request.

((Theorem 2)) In the binomial multiple-period model
described in Fig. 2, let us consider a European call
option de ned by De nition 7 written on a setok
which has the value-interpretation function h(v) =avb (a; b : positive constants). The other assumptions are the same as in the single-period model, except that the short rate is a constant rf =N during
each period of length 1=N . Then, the following pricing formula holds.
;NC (0) = 1 + rNf


N 
X



N pj (1 ; p)N ;j maxf0; K ; a(uj dN ;j V0 )b ga(uj dN ;j V0 )bj =0 j

where

(18)

;



d;b ; 1 + rf
p = d;b ; u;bN :(19)

Although Theorem 2 mentions merely about the option price at t = 0, Algorithm 1 gives us whole theprice process at t 2 [0; 1=N; 2=N;    ; 1].

4. CONCLUSIONS
We have made an abstraction of uncertain digitalmaterials in the network society 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 main contents. Three important properties of the setok werede ned: tradability, online divisibility, and oine divisibility.
Written on the setok values, a simple European call
option is introduced. In multiple-period as well as
single-period models, we have derived option-pricing
formulae. These formulae do not require any divisibility of the underlying setok, which is easier for
information-security technologies to accept.
One might criticize this paper for using too simplied situations. However, what we really want to doby this paper is to trigger a new series of research. In
the conventional nance, encouraged by the seminal
paper by Black and Scholes [12], option-pricing theories have been developed a lot. This history has quite
a reputation although the Black-Scholes theory itself
had a lot of assumptions to make things too simple
and had a lot of biases. Likewise, we would be very
happy if a lot of studies emerge after this paper.

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