Option Pricing: Classic Results
We recall here the basics of the most classic result of option pricing, perhaps the most famous result in mathematical finance: the Black–Scholes theory for the pricing of “European options” in a perfect market, infinitely divisible and liquid, with no “f
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Option Pricing: Classic Results
2.1 Introduction Several authors have proposed a nonstochastic version of the famous Black–Scholes theory. McEneaney [112] may have been the first to replace the stochastic framework with a robust control approach. He derives the so-called stop-loss strategy for bounded variation trajectories, as we do here. He also recovers the Black–Scholes theory, but this is done at the price of artificially modifying the portfolio model with no other justification than recovering the Itˆo calculus and the Black–Scholes partial differential equation (PDE). Cox, Ross, and Rubinstein [57] introduced a nonstochastic approach to the theory of option pricing in a discrete-time setting. We will discuss their approach as compared to the “interval market model” in the next chapter. Their discrete model clearly involves no claim of being realistic for any finite time step. Its only objective is to converge, as the time step vanishes, to a continuous random walk, to recover either the Black–Scholes theory or another one with possible price jumps, depending on how the market model behaves in that limiting process. This approach has been generalized and extended by Kolokoltsov [95,97], as will be explained in Part IV of the book. The crucial point, usually attributed to Robert Merton, in the Black–Scholes theory of option pricing [46] is that of finding a portfolio together with a selffinanced trading strategy that “replicates” (ensures the same return as) the option to be priced. Hence, if no “arbitrage” (riskless profitable trading) is to exist in that market, the price of the option should be equal to that of the replicating portfolio. What is requested is that the portfolio and strategy constructed replicate the option, i.e., yield the same payment to the owner for all possible outcomes of the underlying stock’s value. As has been stressed by several authors, this statement is not in terms of probabilities, and therefore the precise (probabilistic) model adopted
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 2, © Springer Science+Business Media New York 2013
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2 Option Pricing: Classic Results
for a stock’s price should be irrelevant. As a matter of fact, it is known that if one adopts the classic Samuelson model, dS = μ dt + σ dB, S
(2.1)
with B(t) a Wiener process, then the famous Black–Scholes equation and formula do not contain μ . Explaining this fact has been a concern of many an article or textbook. In our formulation, μ just does not appear in the problem statement. We will further argue that the volatility σ appears only as a characteristic of the set of allowable histories S(·), not as a probabilistic entity. We first show an elementary theory that emphasizes this point and let us discuss a zero-volatility, yet stochastic – in the sense that S(·) is a priori unknown and can thus be thought of as stochastic or, in Aubin’s words, tychastic: see part V – model. This model le
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