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Theory of Probability and its Applications

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1994

Volume 39, Issue 1, pp. 1-196


On Some Basic Concepts and Some Basic Stochastic Models Used in Finance

A. N. Shiryaev

Theory Probab. Appl. 39, pp. 1-13 (13 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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This paper can be viewed as an introduction to the papers of this issue devoted to some theoretical-probabilistic problems in financial mathematics.
In this paper, we describe key structures of finance theory (§ 1), present a brief historical bibliography (§ 2), and consider (§ 3) stochastic models of a stock exchange. In addition, the paper offers an insight into the problems of option pricing (§ 4).

Toward the Theory of Pricing of Options of Both European and American Types. I. Discrete time

A. N. Shiryaev, Yu. M. Kabanov, O. D. Kramkov, and A. V. Mel’nikov

Theory Probab. Appl. 39, pp. 14-60 (47 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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This paper consisting of two parts (I — discrete time, II — continuous time $[19]$) considers the main concepts, statements of problems, and results of financial mathematics in connection with options and option contract pricing as a kind of derivative securities. In § 1 it is assumed that the contracts are exercised in discrete $(B,S)$-market. There are two assets: riskless bank account $B = (B_n )_{n \geqq 0} $ and risky stock $S = (S_n )_{n \geqq 0} $, European as well as American options are examined. Special attention is paid to the “martingale” methods of option pricing and hedging strategies in particular for call options and put options.

Toward the Theory of Pricing of Options of Both European and American Types. II. Continuous Time

A. N. Shiryaev, Yu. M. Kabanov, D. O. Kramkov, and A. V. Mel’nikov

Theory Probab. Appl. 39, pp. 61-102 (42 pages) | Cited 13 times

Online Publication Date: July 17, 2006

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In the first part of the paper [29] the options pricing theory was developed under the assumption that a $(B,S)$-market is discrete (in space and in time). It is assumed in the present text that a $(B,S)$-market is operating continuously in time. The riskless bank account$B = (B_t )_{t \geqq 0} $ is evolving according to the “compound interests” formula (1.1), and a risky stock price $S = (S_t )_{t \geqq 0} $ is governed by geometric Brownian motion (1.4).
The “martingale” pricing theory is presented for fair (rational) option price, hedging strategies, and rational expiration times. The Black-Scholes formula for a standard European call option is derived. The paper considers a number of other particular examples of European as well as American options.

A New Look at Pricing of the ”Russian Option“

L. A. Shepp and A. N. Shiryaev

Theory Probab. Appl. 39, pp. 103-119 (17 pages) | Cited 25 times

Online Publication Date: July 17, 2006

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The “Russian option” was introduced and calculated with the help of the solution of the optimal stopping problem for a two-dimensional Markov process in [10]. This paper proposes a new derivation of the general results [10]. The key idea is to introduce the dual martingale measure which permits one to reduce the “two-dimensional” optimal stopping problem to a “one-dimensional” one. This approach simplifies the discussion and explain the simplicity of the answer found in [10].

Models for Option Prices

S. T. Rachev and L. Ruschendorf

Theory Probab. Appl. 39, pp. 120-152 (33 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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Cox, Ross, and Rubinstein [6] introduced a binomial option price model and derived the seminal Black–Scholes pricing formula. In this paper we characterize all possible stock price models that can be approximated by the binomial models and derive the corresponding approximations for the pricing formulas. We introduce two additional randomizations in the binomial price models seeking more general and more realistic limiting models. The first type of model is based on a randomization of the number of price changes, the second one on a randomization of the ups and downs in the price process.
As a result we also obtain price models with fat tails, higher peaks in the center, nonsymmetric etc., which are observed in typical asset return data. Following similar ideas as in [6] we also derive approximating option pricing formulas and discuss several examples.

On the Rational Pricing of the “Russian Option” for the Symmetrical Binomial Model of a $(B,S)$-Market

D. O. Kramkov and A. N. Shiryaev

Theory Probab. Appl. 39, pp. 153-162 (10 pages) | Cited 5 times

Online Publication Date: July 17, 2006

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We present in the binomial model of Cox, Rubinstein and Ross the closed form solution for the “Russian option”, i.e., the American type option with the reward sequence $f = (f_n )_{n \geqq 0} $ given by \[ f_n (\omega ) = \beta ^n \mathop {\max }\limits_{k \leqq n} S_k (\omega ), \] where $\beta $ is some discounting factor, $0 < \beta < 1$.
This option was introduced earlier by L. Sheep and A. N. Shiryaev [3], in the framework of the diffusion model of Black and Sholes.

Integral Option

D. O. Kramkov and E. Mordecki

Theory Probab. Appl. 39, pp. 162-172 (11 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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In the context of diffusion model of the $(B,S)$-market consisting of two assets: riskless bank account $B = (B_t )_{t \geqq 0} $ and risky stock $S = (S_t )_{t \geqq 0} $ described by (1.1) and (1.2) we consider the option of American type with payment function of “integral type” $f = (f_t )_{t \geqq 0} $: \[ f_t = e^{ - \lambda t} \left[ {\int_0^t {S^u du + s\psi _0 } } \right], \] The paper solves the problem of definition of the fair price of the integral option under consideration. The structure of the expiration time is also described.

Mean-Variance Hedging of Options on Stocks with Markov Volatilities

G. B. Di Masi, Yu. M. Kabanov, and W. J. Runggaldier

Theory Probab. Appl. 39, pp. 172-182 (11 pages) | Cited 30 times

Online Publication Date: July 17, 2006

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We consider the problem of hedging an European call option for a diffusion model where drift and volatility are functions of a Markov jump process. The market is thus incomplete implying that perfect hedging is not possible. To derive a hedging strategy, we follow the approach based on the idea of hedging under a mean-variance criterion as suggested by Föllmer, Sondermann, and Schweizer. This also leads to a generalization of the Black–Scholes formula for the corresponding option price which, for the simplest case when the jump process has only two states, is given by an explicit expression involving the distribution of the integrated telegraph signal (known also as the Kac process). In the Appendix we derive this distribution by simple considerations based on properties of the order statistics.

Large Financial Markets: Asymptotic Arbitrage and Contiguity

Yu. M. Kabanov and D. O. Kramkov

Theory Probab. Appl. 39, pp. 182-187 (6 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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We introduce a large financial market as a sequence of ordinary security market models (in continuous or discrete time). An important property of such markets is the absence of asymptotic arbitrage, i.e., a possibility to obtain “essential” nonrisk profits from “infinitesimally” small endowments. It is shown that this property is closely related to the contiguity of the equivalent martingale measures. To check the "no asymptotic arbitrage" property one can use the criteria of contiguity based on the Hellinger processes. We give an example of a large market with correlated asset prices where the absence of asymptotic arbitrage forces the returns from the assets to approach the security market line of the CAPM.

On the Russian Stock Exchange

M. V. Bondarenko and A. N. Vishnyakov

Theory Probab. Appl. 39, pp. 188-194 (7 pages)

Online Publication Date: July 17, 2006

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It seems reasonable to say that 1993 is a year of development for the Russian stock exchange. We observe a sharp increase in the amount of traded securities, and the appearance of a rather liquid asset — the voucher. As a consequence of the expanded credit market and increased turnover of currency transactions, futures and option contracts came into practice. The development of the security market structure made investment on the stock exchange possible. Despite of rich experience accumulated in developed western economies, its immediate implementation into the Russian stock exchange is impossible not taking into account the features peculiar to developing market relations of the transition period.
In this paper, we propose a brief overview of the distinguishing features of the Russian stock exchange. It should be pointed out that this review is not aimed at a development of the complete pattern for the Russian stock market. For instance, the analysis of the debt securities (short-term bonds, bills, notes, and others) as well as the analysis of the money market are not included in this overview. We consider mainly the markets for stocks, currency and some derivative instruments — future contracts.

News of Scientific Life: Actuarial and Financial Center for Scientific Investigation

Theory Probab. Appl. 39, pp. 195-196 (2 pages)

Online Publication Date: July 17, 2006

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