Option pricing

In the financial world, an option right is a right to choose between several possible trades at a time in the future that may be determined in advance or that may be subject to choice. An option is a contract in which an option right is sold. For example, consider a contract that gives the holder the right, but not the obligation, to exchange one million euros for one million American dollars at a given time $T$ in the future. Such a contract may be useful for a European company that will have to make a payment in American dollars at a known time. The contract allows the company to choose at time $T$ whether it will buy American dollars at the exchange rate $1:1$ or whether it will not do so; in the latter case the company may of course still buy American dollars directly in the market. The company's decision will depend on the actual exchange rate at time $T$. Because this exchange rate is not known at the time the contract is entered, it is not obvious on which principle the pricing of the contract can be based. An approach to this problem, which holds for options in general, was developed by F. Black, M. Scholes and R.C. Merton in the early 1970s [a3], [a9] and is now generally accepted. The Black–Scholes–Merton method is based on the observation that an institution that confers an option (say on the euro-dollar exchange rate) may modify the risk involved in the option by buying and selling dollars against euros during the life-time of the contract. Under appropriate assumptions it is in fact possible to eliminate risk completely, so that there is a unique price for the option that does not depend on the risk preferences of any of the parties involved in the contract.

The Black–Scholes–Merton option pricing methodology uses a fairly elaborate mathematical framework. The behaviour of the underlying variables is modelled by means of stochastic differential equations (cf. also Stochastic differential equation). In the original paper by Black and Scholes, [a3], stock prices are modelled by the geometric Brownian motion

$$dS_t=\mu S_tdt+\sigma S_tdw_t,$$

where $\mu$ and $\sigma$ are constants and $w_t$ is standard Brownian motion; later on, researchers have used a variety of other diffusion models (cf. also Diffusion equation) to describe the behaviour of financial indicators such as interest rates and exchange rates. In the general Black–Scholes–Merton framework one works with models in which there are several tradeable assets and in which a vector-valued Brownian motion enters. It is assumed that continuous trading is possible, so that portfolios may be formed of tradeable assets with continually adjusted weights (cf. also Portfolio optimization). In general, the weights may follow processes that are adapted to a filtration associated with the process of the underlying variables. Weight processes are usually subjected to integrability conditions and moreover constrained to be self-financing, which means that no funds are added or withdrawn; thus, any change in value of the portfolio is due to price changes of the assets. Consider the random variables (cf. also Random variable) that arise as portfolio values at time $T$ corresponding to such portfolio strategies that are followed during an interval $[0,T]$ and that start from a portfolio with some given value at time $0$. If any random variable with finite variance can be produced in this way, then the model under consideration is said to represent a complete market. Roughly speaking, markets are complete when the number of independent tradeable assets is at least one larger than the dimension of the vector of Brownian motions appearing in the model. In particular, in a complete market any option can be replicated, that is, reproduced by a suitable trading strategy. Under the assumption that the given model allows no arbitrage (i.e. no riskless profits), there can be only one initial portfolio value corresponding to a replicating portfolio for a given option. Again under the no-arbitrage assumption, this must then be the price of the option at time $0$.

A powerful tool in the pricing of options is the replacement of the probability measure in the given model by an equivalent martingale measure under which all price processes, after discounting, are martingales (cf. also Martingale). Under suitable hypotheses it can be shown that absence of arbitrage implies the existence of an equivalent martingale measure, and that at most one equivalent martingale measure can exist in a complete market. If a unique equivalent martingale measure exists, the price of an option can be computed as the expected value (with respect to this measure) of its discounted pay-off. The transformation to an equivalent martingale measure can often be simply achieved by a change of the drift term in the given stochastic differential equations (the Cameron–Martin–Girsanov theorem). For instance, the well-known Black–Scholes formula can be obtained in this way.

Options that have a fixed time of expiry are called European options. In the financial markets one also trades contracts in which the holder is free to choose the time at which the option is exercised. Such contracts are called American options. Even in a complete and arbitrage-free model, the pricing of such options cannot be based on an arbitrage argument alone. Usually, the price of an American option is defined by maximizing its value over all exercise strategies; the pricing problem then becomes an optimal stopping problem (cf. also Stopping time). For computational purposes, it is often useful to reformulate such problems as free boundary problems for a related partial differential equation (cf. also Differential equation, partial, free boundaries).

More information about option pricing can be found in, for instance, [a1], [a2], [a4], [a5], [a6], [a7], [a8], [a10], [a11], [a12], [a13], [a14].

References

[a1]	N.H. Bingham, R. Kiesel, "Risk-neutral valuation: The pricing and hedging of financial derivatives" , Springer (1998)
[a2]	T. Björk, "Arbitrage theory in continuous time" , Oxford Univ. Press (1998)
[a3]	F. Black, M. Scholes, "The pricing of options and corporate liabilities" J. Political Economy , 81 (1973) pp. 637–659
[a4]	R.J. Elliott, E. Kopp, "Mathematics of financial markets" , Springer (1999)
[a5]	I. Karatzas, S.E. Shreve, "Methods of mathematical finance" , Springer (1998)
[a6]	Y.-K. Kwok, "Mathematical models of financial derivatives" , Springer (1997)
[a7]	D. Lamberton, B. Lapeyre, "Introduction to stochastic calculus applied to finance" , Chapman and Hall (1996)
[a8]	D.G. Luenberger, "Investment science" , Oxford Univ. Press (1997)
[a9]	R.C. Merton, "Theory of rational option pricing" Bell J. Economics and Management Sci. , 4 (1973) pp. 141–183
[a10]	M. Musiela, M. Rutkowski, "Martingale methods in financial modeling. Theory and applications" , Springer (1997)
[a11]	L.T. Nielsen, "Pricing and hedging of derivative securities" , Oxford Univ. Press (1999)
[a12]	S.R. Pliska, "Introduction to mathematical finance. Discrete time models" , Blackwell (1997)
[a13]	A.N. Shiryaev, "Essentials of stochastic finance" , World Sci. (1999)
[a14]	P. Wilmott, "Derivatives. The theory and practice of financial engineering" , Wiley (1998)