A Derivative of Digital Objects andEstimation of Default Risks
in Electronic Commerce
Kanta Matsuura
Institute of Industrial Science, University of Tokyo,
Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japankanta@iis.u-tokyo.ac.jp

Abstract. In electronic commerce, traded digital objects are likely asso-

ciated with several numerical values as well as their prices. These values
may change unpredictably over time and bring risks both to the providers
and to the consumers of the application. One possible strategy for hedging the risks is to introduce derivatives regarding the uncertain values.
This paper shows a theoretical pricing equation of the derivatives when
the underlying digital objects have systematic default or revocation risks.
We can make use of this pricing to estimate the risks.

1 Introduction
With the help of applied cryptography, we are going to trade more and moredigital objects over an open network. Since digital objects can keep their original bit strings virtually forever, one may expect that there would be no riskof change. This is, unfortunately, not always the case. Digital objects can have
not only prices but also other important numerical values. For example, digital
certicates may have condence values or trust metrics 1]. Access-grant tickets may have priority numbers or QoS (Quality-of-Service) values 2] reserved.
Digitally-watermarked images 3] may have innocence values about their origins
in terms of copyright protection. Any product may be associated with some insurance contracts 4]. Reward points may be attached. Those additional values
and their e ectiveness may change unpredictably over time and cause risks. At
the worst case, the values get into defaults (e.g. the corresponding certicate isrevoked) and the holder may have a large nancial damage.
A popular way for hedging such stochastic risks is to introduce derivatives or
options written on underlying assets, typically regarding their prices. In nancial
theory, encouraged by the seminal paper by Black and Scholes 5], option-pricing
theories have been developed a lot. Most of them use assumptions including divisibility of the underlying assets, which is not trivial in the case of the digital
objects. Thus we are motivated to study option pricing with models and assumptions suitable for digital objects. The rest of this paper consists of modeling (Sect. 2), pricing (Sect. 3), discussion including an application (Sect. 4), andconclusions (Sect. 5).

Object Provider

6

Object?
6

Object?

Customer

6

RewardPayment

Object Server?

Report6

RefundPayment
Customert t t?

ObjectCustomerFig. 1. A network commerce architecture where boxes with wider lines indicate thatthe entities inside are more trusted.

2 Objects with Default/Revocation Risks
Our model is based on an architecture illustrated in Fig. 1.

(Object Provider) Copyright management and related technical maintenanceare dicult and non-trivial tasks. So are management and maintenance ofnetwork-security infrastructure (e.g. public-key infrastructure). These tasks
may require sucient trustworthiness and reliability. We need specializedentities. They would be happier if the objects they provide are distributedand circulated more frequently in larger amounts it would improve theirreputation and/or make attached advertisement more protable.
(Object Server) Selling digital objects to untrusted customers through poor
communication channels is dicult task, too. We need specialized entities.
(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.
In a network life, we would want to pay for digital objects in electronic cash.
Such digital payment systems could be more ecient if the monetary value ofeach coin is less granular 6]. Therefore, if we want to allow as wide variety ofelectronic cash systems as possible, highly discrete (i.e. very sparse) prices would
be helpful. So we rstly assume an object whose price is xed. For notational
simplicity, we assign this xed price as the unit of network currency. Also forsimplicity, we assume each share of the object has a single value. This value isrepresented by a stochastic random variable H (t). We assume that its dynamics
is given bydH = (t H (t))Hdt +  (t H (t))HdW(1)where (t H (t)) and (t H (t)) are adapted processes and W is a Wiener processunder the objective measure.
We assume a value-proportional tradability the holder of a share of theobject can sell it at a value-proportional price Sp dened byVSp (t) =(2)H (t)whenever he wishes as long as no default occurs. V is the nominal value of theshare, which is equal to H (t0 ) where t0 is the time the share is issued.
The default/revocation is assumed to happen according to a Poisson processwith intensity . Once the Poisson event occurs, the value-proportional tradability is ruined but the holder can refund his share for the original unit price.
Di erent from the conventional nancial market, we do not assume the divisibility of the object. We do not assume that we can go short for it, either. Alsodi erent from the conventional nancial market, we assume that the Poissonprocess represents a systematic risk which can fully appear in the risk premium.
As for other issues, we assume a typical liquid and arbitrage-free market 7].
The derivative we study here is a European call option written not on theprice but on the value as follows. We assign the issuing time of the option as thetime origin (t=0) for notational convenience.

Denition 1 (A European Call). A European call option on the object is aderivative which provides a right to buy one share of the object with a reserved
value K at a particular time Tm 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 strikevalue, and Tm is called the maturity.
Let C (t) = c(t H (t)) be the price process of the call option. As a continuoustime model, we place the following mathematical assumptions.

{ The function c(t h) is a C 12 -mapping in the domain R+ R++, and c(t 0) =0 for all t 2 R+ . R++ is the set of positive real numbers and R+ is the setof non-negative real numbers.
{ The price process of the riskless asset is described by dB(t) = rf B(t)dt,
where the short rate rf is a deterministic constant.

3 Pricing
By establishing a riskless portfolio composed of one share of the digital object
and adjusted amount of options, we can reach the following pricing theorem.
Due to the space limitation, the full proof 8] is not given here.

Theorem 1 (Boundary Value Problem for Pricing). The only pricing
function of the form C (t) = c(t H (t)) is obtained whenc(t h) =c^(t 1=h) for h > 00 for h = 0and c^(t g) is the solution of the boundary value problem
2 2g c^gg + (rf ; )gc^g ; (rf + )^c + c^t = 0 c^(Tm  g) = maxf0 Kg ; 1g

2
in the domain 0 Tm]  R++ .

4 Discussion
4.1 Jump ProcessesWe derived Theorem 1 by using a systematic risk assumption. This is di erent
from the conventional nance 9]. The conventional nonsystematic-risk assumption is an extreme assumption and there have been a lot of arguments about it
11]. In fact, jumps observed in stock prices are reported to be systematic acrossthe market portfolio 12]. Heuristically speaking, the more similarly network entities look at the default/revocation risk, the better model our choice would give.
Our choice could go better with the recent trend in the public-key infrastructure
toward a single-directory system 13], 14].

4.2 Application
According to Theorem 1, the option price depends on , Tm , rf , h, K , and therisk of default/revocation . This suggests that the market data (C , , Tm , rf ,h, K ) may help us with an indirect measurement of the risk . This needs aninverse estimation, which may be too heavy. However, if what you want to do isjust to see whether  exceeds a certain value, say, 0 , then you may be able touse a more practical strategy. That is, in the region where C is locally monotoneincreasing/decreasing with respect to , the following procedure without repeat
is worth a try.
1. By using recent market data, estimate the short rate rf and the volatility .
2. Set  = 0 .
3. Solve the boundary value problem in Theorem 1.
4. Compare the result with the current option price data.
5. By using a tool for statistical test, examine whether you can say the computed price is higher(monotone-decreasing case)/lower(monotone-increasingcase) than the observed price with sucient probability.
6. If the answer is Yes, think of it as an alarm.

5 Concluding Remarks
We described an E-commerce architecture and a simple model of digital objects
each object has an abstracted value as well as a xed price. The value can change
stochastically and can be ruined at the worst case. The object is not divisible andwe cannot go short for it. A European call option written not on the price buton the value was introduced. A PDE for pricing the option was derived. In thediscussion, applications of the pricing were studied: to estimate the probabilityof revocation and to detect an alarm of high probability.

References
1. Reiter, M. K., Stubblebine, S. G.: Resilient Authentication Using Path Independence. IEEE Trans. Comput. 47 (1998) 1351{1362
2. Xiao, X., Ni, L. M.: Internet QoS: A Big Picture. IEEE Network. 13 (1999) 8{183. Katzenbeisser, S., Petitcolas, F. (eds.): Information Hiding Techniques for
Steganography and Digital Watermarking. Artech House Publishers, Boston London (2000)
4. Reiter, M. K., Stubblebine, S. G.: Authentication Metric Analysis and Design.
ACM Trans. Info. & Sys. Security 2 (1999) 138{1585. Black, F., Scholes, M.: The Pricing of Options and Corporate Liabilities. J. PoliticalEcon. 81 (1973) 637{654
6. Eng, T., Okamoto, T.: Single-Term Divisible Electronic Coins. In: De Santis, Alfredo (ed.): Advances in Cryptology | EUROCRYPT'94. Lecture Notes in Computer Science, Vol. 950. Springer-Verlag, Berlin Heidelberg New York (1995) 306{319
7. Bjork, T.: Arbitrage Theory in Continuous Time. Oxford University Press, NewYork (1998)
8. Matsuura, K.: Security Tokens and Their Derivatives. Technical Reports, Centrefor Communication Systems Research, University of Cambridge (2001)
http://www.ccsr.cam.ac.uk/techreports/tr29/index.html

9. Merton, R. C.: Option Pricing When Underlying Stock Returns are Discontinuous.
J. Financial Econ. 3 (1976) 125{144
10. Sharpe, W. F.: Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. J. Finance 19 (1964) 425{442
11. Colwell, D. B., Elliott, R. J.: Discontinuous Asset Prices and Non-Attainable Contingent Claims. Math. Finance 3 (1993) 295{308
12. Jarrow, R. A., Rosenfeld, E. R.: Jump Risks and the Intertemporal Capital AssetPricing Model. J. Business 57 (1984) 337{351
13. Buldas, A., Laud, P., Lipmaa, H.: Accountable Certicate Management Using Undeniable Attestations. In: Proc. 7th ACM Conf. on Comp. & Comm. Security,Athens (2000) 9{1814. Gassko, I., Gemmell, P. S., MacKenzie, P.: Ecient and Fresh Certication. In:Imai, H., Zheng, Y. (eds.): Public Key Cryptography | PKC 2000. Lecture Notes
in Computer Science, Vol. 1751. Springer-Verlag, Berlin Heidelberg New York(2000) 342{353