# Options

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1747

## Abstract

An option is a security whose owner has a right to buy (sell) it at a specified price on a specified date (or, with an American-type option, on or before the specified date). Trading of options on common stock began in 1973 and has since spread to other commodities. Option pricing theory provides a unified theory for the pricing of corporate liabilities. Of its more recent extensions, perhaps the most significant is its application in the evaluation of operating or ‘real’ options in the capital budgeting decision problem.

### Keywords

Arbitrage Bachelier, L. Black, F. Capital budgeting Deposit insurance Forward contracts Futures contracts Government loan guarantees Ito’s lemma Merton, R. C. Option pricing theory Options Pension fund insurance Probability Scholes, M.A ‘European-type call (put) option’ is a security that gives its owner the right to buy (sell) a specified quantity of a financial or real asset at a specified price, the ‘exercise price’, on a specified date, the ‘expiration date’. An American-type option provides that its owner can exercise the option on or before the expiration date. If an option is not exercised on or before the expiration date, it expires and becomes worthless.

Options and forward or futures contracts are fundamentally different securities. Both provide for the purchase (or sale) of the underlying asset at a future date. A long position in a forward contract obliges its holder to make an unconditional purchase of the asset at the forward price. In contrast, the holder of a call option can choose whether or not to purchase the asset at the exercise price. Thus, a forward contract can have a negative value whereas an option contract never can.

The first organized market for trading options was the Chicago Board Options Exchange (CBOE) which began trading options on common stocks in 1973. The initial success of the CBOE was followed by an expansion in markets to include options on fixed-income securities, currencies, stock and bond indices, and a variety of commodities. Although these markets represent an increasingly larger component of total financial market trading, options are still relatively specialized financial securities. Option pricing theory has, nevertheless, become one of the cornerstones of financial economic theory.

This central role for options analysis derives from the fact that option-like structures pervade virtually every part of the field. Black and Scholes (1973) provide an early example: shares of stock in a firm financed in part by debt have a payoff structure which is equivalent to a call option on the firm’s assets where the exercise price is the face value of the debt and the expiration date is the maturity date of the debt. Option pricing theory can thus be used to price levered equity and, therefore, corporate debt with default risk.

Identification of similar isomorphic relations between options and other financial instruments has led to pricing models for seniority, call provisions and sinking fund arrangements on debt; bonds convertible into stock, commodities, or different currencies; floor and ceiling arrangements on interest rates; stock and debt warrants; rights and stand-by agreements. In short, option pricing theory provides a unified theory for the pricing of corporate liabilities.

The option-pricing methodology has been applied to the evaluation of noncorporate financial arrangements including government loan guarantees, pension fund insurance and deposit insurance. It has also been used to evaluate a variety of employee compensation packages including stock options, guaranteed wage floors, and even tenure for university faculty.

Perhaps the most significant among the more recent extensions of option analysis is its application in the evaluation of operating or ‘real’ options in the capital budgeting decision problem. For example, a production facility which can use various inputs and produce various outputs provides the firm with operating options that it would not have with a specialized facility which uses a fixed set of inputs and produces a single type of output. Option-pricing theory provides the means of valuing these production options for comparison with the larger initial cost or lower operating efficiency of the more flexible facility. Similarly, the choice among technologies with various mixes of fixed and variable costs can be treated as evaluating the various options to change production levels, including abandonment of the project. Research and development projects can be evaluated by viewing them as options to enter new markets, expand market share or reduce production costs.

As these examples suggest, option analysis is especially well suited to the task of evaluating the ‘flexibility’ components of projects. These, corporate strategists often claim, are precisely the components whose values are not properly measured by traditional capital-budgeting techniques. Hence, option-pricing theory holds for the promise of providing quantitative assessments for capital budgeting projects that heretofore were largely evaluated qualitatively. Survey articles by Smith (1976) and Mason and Merton (1985) provide detailed discussion of these developments in option analysis along the extensive bibliographies.

The lineage of modern option pricing theory began in 1900 with the Sorbonne thesis, ‘Theory of Speculation’, by the French mathematician Louis Bachelier. The work is rather remarkable because, in analysis the problem of option pricing, Bachelier derives much of the mathematics of probability diffusions; this, five years before Einstein’s famous discovery of the theory of Brownian motion. Although, from today’s perspective, the economics and mathematics of Bachelier’s work are flawed, the connection of his research with the subsequent path of attempts to describe an equilibrium theory of option pricing is unmistakable. It was not, however, until nearly 75 years later with the publication of the seminal Black and Scholes article (1973), that the field reached a sense of closure on the subject and the explosion in research on option pricing applications began.

As with Bachelier and later researchers, Black and Scholes assume that the dynamics for the price of the asset underlying the option can be described by a diffusion process with a continuous sample path. The breakthrough nature of the Black–Scholes analysis derives from their fundamental insight that the dynamics trading strategy in the underlying asset and a default-free bond can be used to hedge against the risk of either a long or short position in the option. Having derived such a strategy, Black and Scholes determine the equilibrium option price from the equilibrium condition that portfolios with no risk must have the same returns as a default-free bond. Using the mathematics of Ito stochastic integrals, Merton (1973, 1977) formally proves that with continuous trading, the Black–Scholes dynamic portfolio will hedge all the risk of an option position held until price exercise or expiration, and therefore, that the Black–Scholes option price is necessary to rule out arbitrage.

Along the lines of the derivation for general contingent claims pricing in Merton (1977), a sketch of the arbitrage proof for the Black–Scholes price of a European call option on a nondividend-paying stock in a constant interest rate environment is as follows.

*V*(

*t*), can be described by a diffusion process with a stochastic differential equation representation given by:

*α*is the instantaneous expected return on the stock;

*σ*

^{2}is the instantaneous variance per unit time of the return, which is a function of

*V*and

*t*; dz is a standard Wiener process. Let

*F*[

*V, t*] satisfy the linear partial

*r*is the interest rate. Let

*F*be such that it satisfies the boundary conditions:

*F*on these boundaries are identical to the payoff structure on a European call option with exercise price

*E*and expiration date

*T*. From standard mathematics, the solution to (2) and (3) exists and is unique.

Consider the continuous-time portfolio strategy which allocates the fraction *w*(*t*) ≡ *F*_{1}[*V*, *t*] *V*(*t*)/*P*(*t*) to the stock and 1−*w*(*t*) to the bond, where *P*(*t*) is the value of the portfolio at time *t*. Other than the initial investment in the portfolio at there are no contributions or from the portfolio until it is liquidated at *t* = *T*.

*t*depends only on the first derivative of the solution to (2)–(3) and the current values of the stock and the portfolio. It follows from the prescribed allocation

*w*(

*t*) that the dynamics for the value of the portfolio can be written as:

*F*is twice-continuously differentiable. Hence, we can use Ito’s Lemma to express the stochastic process for

*F*as:

*F*is evaluated at

*V*=

*V*(

*t*) at each point in time

*t*. But,

*F*satisfies (2). Hence, we can rewrite (5) as:

*Q*(

*t*) to be the difference between the value of the portfolio and the value of the function

*F*[

*V*,

*t*] evaluated at

*V*=

*V*(

*t*). From (4) and (6), we have that d

*Q*=

*rQ*d

*t*which is a nonstochastic differential equation with solution

*Q*(

*t*)

*= Q*(0)exp[

*rt*] and

*Q*(0) =

*P*(

*0*)−

*F*[

*V*(0), 0]. Hence, if the initial investment in the portfolio is chosen so that

*P*(0) =

*F*[

*V*(0), 0] then

*Q*(t) = 0 and

*P*(

*t*) =

*F*[

*V*(

*t*),

*t*] for all

*t*.

Thus, we have constructed a dynamic portfolio strategy in the stock and a default-free bond that exactly replicates the payoff structure of a call option on the stock. The solution of (2) and (3) for F and its first derivative *F*_{1} provides the ‘blueprint’ for that construction. The standard no-arbitrage condition for equilibrium prices holds that two securities with identical payoff structures must have the same price. It follows, therefore, that the equilibrium price of the call option at time *t* must equal the Black–Scholes price, *F*[*V*(*t*), *t*].

The extraordinary impact of the Black–Scholes analysis on financial economic research and practice can in large part be explained by three critical elements: (1) the relatively weak assumptions for its valid application; (2) the variables and parameters required as inputs are either directly observable or relatively easy to estimate, and there is computational ease in solving for the price; (3) the generality of the methodology in adapting it to the pricing of other options and option-like securities.

Although framed in an arbitrage type of analysis, the derivation does not depend on the existence of an option on the stock. Hence, the Black–Scholes trading strategy and price function provide the means and the cost for an investor to create synthetically an option when such an option is not available as a traded security. The findings that the equilibrium option price is a twice continuously differentiable function of the stock price and that its dynamics follow an Ito process are derived results, not assumptions.

The striking feature of (2) and (3) is not the variables and parameters that are needed for determining the option price but rather, those not needed. Specifically, determination of the option price and the replicating portfolio strategy does not require estimates of either the expected return on the stock, *α* or investor risk preferences and endowments. In contrast to most equilibrium models, the pricing of the option does not depend on price and joint distributional information for all available securities. The only such information required is about the underlying stock and default-free bond. Indeed, the only variable or parameter required in the Black–Scholes pricing function that is not directly observable is the variance rate function, *σ*^{2}. This observation has stimulated a considerable research effort on variance-rate estimation in both the academic and practising financial communities.

With some notable exceptions, equations (2) and (3) cannot be solved analytically for a closed-form solution. However, powerful computational methods have been developed to provide high-speed numerical solutions of these equations for both the option price and its first derivative.

As in the original Black and Scholes article, the derivation here focuses on the pricing of a European call option. Their methodology is, however, easily applied to the pricing of other securities with payoff structures contingent on the price of the underlying stock. Consider, for example, the determination of the equilibrium price for a European put option with exercise price E and expiration date *T*. Suppose that in the original derivation we change the boundary conditions specified for *F* in (3) so as to match the payoff structure of the put option on these boundaries. That is, we now require that *F* satisfy *F* ≤ *E* ; *F*[0, *t*] = *E* exp [–*r*(*T* – *t*)]; *F*[*V*, *T*] = max [0, *E* – *V*]. Once *F* and its derivative are specified, the development of the replicating portfolio proceeds in identical fashion to show that *P*(*t*) = *F*[*V*(*t*), *t*]. With the revised boundary conditions, the portfolio payoff structure will match that of the put option at exercise or expiration. Thus, *F*[*V*(*t*), *t*] is the equilibrium put option price.

As shown in Merton (1977), the same procedure can be used to determine the equilibrium price for a security with a general contingent payoff structure, *G*[*V*(*T*)], by changing the boundary conditions in (3) so that *F*[*V*, *T*] = *G*[*V*]. A particularly important application of this procedure is in the determination of pure state- contingent prices.

*π*[

*V*,

*t*;

*E*,

*T*] denote the solution of (2) subject to the boundary conditions:

*δ*(

*x*) is the Dirac delta function with the properties that

*δ*(0) is infinite in such a way that

*π*[

*V*,

*t*;

*E*,

*T*]d

*E*is the price of a security which pays $1 if

*V*(

*T*) =

*E*at time

*T*and $0, otherwise.

*F*[

*V*,

*T*] =

*G*[

*V*] can be written as:

*π*} are derived, the equilibrium price of any contingent payoff structure can be determined by mere summation or quadrature.

*T*where each ‘unit’ contains a long position in an option with exercise price

*E*−

*ε*; a long position in an option with exercise price

*E*+

*ε*; and a short position in two options with exercise price

*E*. If one takes a position in 1/

*ε*

^{2}units of this portfolio, the payoff structure at time

*T*with

*V*(

*T*) =

*V*is given by:

*ε*→ 0 is

*δ*(

*E−V*) which is the payoff structure to a pure contingent-state security. If

*F*[

*V*, t; E, T] is the solution to (2) and (3), then it follows from (8) that:

For further discussion of options, see especially the January/March 1976 issue of the *Journal of Financial Economics;* the October 1978 issue of the *Journal of Business;* and the excellent book by Cox and Rubinstein (1985).

## See Also

### Bibliography

- Bachelier, L. 1900.
*Théorie de la speculation*. Paris: Gauthier-Villars. English translation in*The random character of stock market prices*, ed. P. Cootner, revised ed, 17–78. Cambridge, MA: MIT Press, 1967.Google Scholar - Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities.
*Journal of Political Economy*81: 637–659.CrossRefGoogle Scholar - Cox, J., and M. Rubinstein. 1985.
*Options markets*. Englewood Cliffs: Prentice-Hall.Google Scholar - Mason, S., and R.C. Merton. 1985. The role of contingent claims analysis in corporate finance. In
*Recent advances in corporate finance*, ed. E.I. Altman and M.G. Subrahmanyan, 7–54. Homewood: Richard D. Irwin.Google Scholar - Merton, R.C. 1973. Theory of rational option pricing.
*Bell Journal of Economics and Management Science*4 (Spring): 141–183.CrossRefGoogle Scholar - Merton, R.C. 1977. On the pricing of contingent claims and the Modigliani-Miller theorem.
*Journal of Financial Economics*5: 241–250.CrossRefGoogle Scholar - Smith, C.W. 1976. Option pricing: A review.
*Journal of Financial Economics*3 (1/2): 3–51.CrossRefGoogle Scholar