The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Option Pricing Theory

  • Jonathan E. IngersollJr.
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1707

Abstract

Financial contracting is as old as human history. Deeds for the sale of land have been discovered that date to before 2800 bc. The Code of Hammurabi (c1800 bc) regulated, among other things, the terms of credit. Contingent contracting was also common. Under the Code crop failure due to storm or drought served to cancel that year’s interest on a land loan. The trading of the first options is probably equally ancient.

Financial contracting is as old as human history. Deeds for the sale of land have been discovered that date to before 2800 bc. The Code of Hammurabi (c1800 bc) regulated, among other things, the terms of credit. Contingent contracting was also common. Under the Code crop failure due to storm or drought served to cancel that year’s interest on a land loan. The trading of the first options is probably equally ancient.

Although options have certainly been traded for centuries, it is only in recent years that they have reached any degree of importance. In 1973 the Chicago Board of Trade founded The Chicago Board Options Exchange to create a centralized market for trading call options on listed stock. The American, Pacific, and Philadelphia Stock Exchanges followed suit within a few years. In 1977 the trading of puts on these exchanges began.

By the early 1980s puts and calls could be traded on over 400 listed stocks, and options were available on many other financial instruments such as Treasury bonds and bills, foreign currencies and futures contracts. The volume of trade had grown as well. In terms of the number of shares controlled, option volume often exceeded that on the New York Stock Exchange.

Curiously the recent revolution in option pricing theory also dates to 1973 with the publication by Fischer Black and Myron Scholes of their classic paper on option valuation. In the past decade and a half, the valuation of options or various other contingent contracts has been one of the primary areas of research among financial economists.

Option contracts are examples of derivative securities; that is securities whose values depend on those of other securities or assets. For example, a call option on a share of stock gives the owner the right to purchase a share of that stock at a set price. The value of this right obviously depends on the price per share of the stock on which the option is based.

Terminology

Before discussing the academic study of options, it is useful to consider some terminology. The two most common types of option contracts are puts and calls. A call is an option to buy, and a put is an option to sell. Puts and calls are contracts between two investors. The purchaser of the option is the party to whom the contract gives certain rights or ‘options’. The call’s owner is said to have a long position. The creator or writer of the call has certain financial obligations if the owner chooses to exercise the option. The writer of an option is said to have a short position.

The owner of a call has the right, but not the obligation, to buy a fixed amount (usually 100 shares for exchange listed stock options) of a particular asset on or before a given date, the maturity or expiration date, upon payment of a stated fee. This fee is called the exercise price, striking price, or contract price. The owner of the call does not receive any dividends paid by the common stock or have any other rights of ownership until the option is exercised. The owner of the put has the right to sell on similar terms.

When purchasing the option, the amount that the long party pays to the short party is called the premium. If the stock price is above the striking price then the difference is the call’s intrinsic value, i.e., for a call with a striking price of X on a stock with price S, the intrinsic value is Max(SX, 0). For a put the intrinsic value is the exercise price less the stock price when the former is larger, i.e., Max(XS, 0). An option’s intrinsic value is sometimes called the whenexercised value. An option’s intrinsic value does not measure its market value. Typically an option sells for more than its intrinsic value.

When options are first written the striking price is usually set near the currently prevailing stock price. The option is then said to be at-the-money. As the stock price changes, the option will become in-the-money or out-of-the-money. A call option is in-the-money when the stock price is above the striking price and out-of-the money when the stock price is below the striking price.

The options just described are American options. They can be exercised at any time on or before the expiration date. Options that can only be exercised at maturity are called European options. Actually this is a misnomer. While American options are traded on exchanges in the United States and Canada (and Europe), European contracts are not traded on that continent.

A warrant is similar to a call option. The primary difference is that a warrant is issued by a corporation against its own stock. When a warrant is exercised, the corporation issues new shares to the owner of the warrant. Warrants typically have maturities of several years or longer. There have even been a few perpetual warrants issued. When they are issued, warrants are usually substantially out-of-the-money.

A rights issue, like a warrant, is granted by a corporation against new stock. Usually a rights issue expires in a few weeks to a few months after it is issued. When rights are issued, they are typically substantially in-the-money.

Many other financial contracts contain implicit or explicit options. Convertible bonds, for example, give the owner the right to swap the bonds for shares of stock. This option is like a warrant. Instead of paying a cash exercise price, the bondholder relinquishes the right to the future interest and principal payments. A callable bond includes the company’s right to ‘repurchase’ a bond at a set call price. Much of the development in option pricing subsequent to the Black–Scholes option pricing model has been in the application of the model to these and other situations.

Preliminary Considerations

Call options are the most common and one of the simplest types of derivative assets so this discussion will be illustrated primarily with calls. Most of the general principles apply with only minor changes to any derivative asset.

A call option with an exercise price of X on a share of stock with a current price per share of S is worth SX if exercised, for it enables its owner to purchase for X something worth S. To avoid any possibility of arbitrage a call option must sell for at least this difference. In addition because a call has limited liability (that is the owner cannot be forced to exercise when it is not advantageous to do so), it must be worth at least zero. Thus, C(S, τ) ≥ Max (SX, 0), where C(S, τ) is the market price of a call with time to maturity of τ.

As a general rule this inequality will be strict, and the call will be worth more ‘alive’ than when exercised. One exception is at the time a call matures. Then the owner has only two choices–execise the option or let it expire. At this point the preceding relation must hold as an equality, C(S, 0) = Max (SX, 0). It is this functional relation between the value of the call at maturity and the stock price prevailing at that time that makes the call a derivative asset and allows its price to be determined as a function of the prevailing stock price.

Some general restrictions on option values can be derived with no assumptions beyond the absence of arbitrage opportunities. For example, a call with a low exercise price must be worth at least as much as an otherwise identical call with a high exercise price. The intuition is simple. The owner of the call with the low striking price could exercise whenever the owner of the other call did and would always have a lower cost of doing so. Two important restrictions of this type are Stoll’s (1969) put–call parity relation and the proof that a call option on stock which pays no dividend should not be exercised prior to maturity.

The put–call parity relation holds for European puts and calls on stocks not paying dividends. It is
$$ P\left(S,\tau \right)+S=C\left(S,\tau \right)+X/{\left(1+r\right)}^{\tau }. $$
(1)
To prove this relation consider two portfolios. The first holds one share of stock and a put. The second holds a call and a zero coupon bond with a face value of X maturing on the options’ expiration date. If the stock price is ST at the expiration of the option, then the first portfolio is worth Max(XST, 0) + ST = Max(ST, X). The second is worth Max(STX, 0) + X = Max(ST, X) These values are the same, and neither portfolio makes any interim disbursements. Therefore, absence of arbitrage implies that the current value of the two portfolios must be equal. Equation (1) expresses the equality of these two portfolios’ current values. One importance of this relation is that once either the put or call pricing problem has been solved, the answer to the other is also known.
To prove the optimality of holding a call option until maturity consider the following two portfolios. The first holds just one share of stock. The second holds the call and a zero coupon bond with a face value of X. At expiration, the first portfolio is worth ST. The second is worth Max (ST, X). As the former value is never larger, the current value of the first portfolio cannot be greater than that of the second, or
$$ C\left(S,\tau \right)\ge S-X/\left(1+{r}^{\tau}\right)>S-X. $$
(2)
This proves that an option is worth more alive than when exercised. An investor who no longer wishes to hold a call could realize more by selling the option than exercising it.

These two relations do not exhaust the general statements that can be made about option prices. Other propositions, also depending only on the absence of arbitrage, have been proved by Merton (1973) and Cox and Ross (1976b). To go beyond general propositions of this type and derive a precise value for an option, further assumptions must be made.

There were many attempts at a consistent and self-contained model of option valuation. All of these models made assumptions about the distribution of the stock’s return (a lognormal distribution was the usual choice) and the absence of market frictions such as taxes, transactions costs, and short sales constraints. Most of the models included unspecified parameters which had to be measured to use the formulae.

This area of research was revolutionized with the 1973 publication of the Black–Scholes option pricing model deriving a formula depending on only five directly observable variables, the stock’s price (S), the exercise price (X), the time to maturity (τ), the risk-free rate of interest (r), and the variance of changes in the logarithm of the stock price (σ2).

Option Models Prior to Black–Scholes

Option pricing theory did not begin with the Black–Scholes model. Many economists had tackled this problem previously. While some of the attempts are flawed by current standards, later developments almost certainly would not have come about without the earlier works. There is room here only to highlight some of the more important steps leading to the Black–Scholes model.

The earliest model of option pricing was probably developed by Louis Bachelier (1900). In examining stock price fluctuations he was led to some aspects of the mathematical theory of Brownian motion five years prior to Einstein’s classic paper of 1905. Postulating an absolute Brownian motion without drift and with a variance of σ2 per unit time for the stock price process, he determined that the expected value of the call option at maturity should be
$$ C=S\cdot \varPhi \left(\frac{S-X}{\sigma \sqrt{\tau }}\right)-X\cdot \varPhi \left(\frac{S-X}{\sigma \sqrt{\tau }}\right)+\sigma \sqrt{\tau}\cdot \phi \left(\frac{S-X}{\sigma \sqrt{\tau }}\right) $$
(3)
where Φ (⋅) and ϕ(⋅) are the standard cumulative normal and normal density functions. In keeping with an assumption of a zero expected price change for the stock, he did not discount this expectation to find a present value. This model was rediscovered more than fifty years later by Kruizenga (1956).

By contemporary standards this model must have been very advanced. The model is only lacking in two primary areas. The use of absolute Brownian motion allows the stock price to become negative – a condition at odds with the assumption of limited liability. The assumption of a mean expected price change of zero ignores a positive time value for money, the different risk characteristics of options and the underlying stock, and risk aversion. Despite these shortcomings, the formula is actually quite good at predicting the prices of short-term calls. It fails at long maturities, however, by requiring the option price to grow proportionally to the square root of maturity.

Most of the developments in option pricing for the next half century or more were ad hoc econometric models. Typical of this type is the model of Kassouf (1969) who estimated call prices with the formula
$$ C=X\left({\left[{\left(S/X\right)}^{\gamma }+1\right]}^{1/\gamma }-1\right],1\le \gamma <\infty . $$
(4)
This formula does bound the call price above by the stock price and below by its intrinsic value, Max(SX, 0). It also gives correct maturity values for calls when the parameter γ is set to ∞. Kassouf fit his model by estimating the parameter γ using time to maturity, dividend yield, and other variables.
Major new developments in option pricing began in the 1960s. Sprenkle (1961) assumed a lognormal distribution for the stock price with a constant mean and variance (although not specifically a diffusion) and allowed for a positive drift in the stock’s price. His equation for a call value can be written as
$$ C= {\displaystyle \begin{array}{l}{\mathrm{e}}^{\alpha \tau}S\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(\alpha +\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right]\hfill \\ {}-\left(1-\pi \right)\times X\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(\alpha -\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right].\hfill \end{array}} $$
(5)
The parameter π was an adjustment for the market ‘price for leverage’. Sprenkle did not discount this expectation to determine the option value. (Note that if π is set to zero, (5) gives the expected terminal value for the option.)
Boness’s (1964) model was very similar. He also assumed a stationary lognormal distribution for stock returns, and recognized the importance of risk premiums. For tractability he assumed that ‘[i]nvestors are indifferent to risk’. He used this last assumption to justify discounting the expected final option value by α, the expected rate of return on the stock. His final model was
$$ {\displaystyle \begin{array}{l}C=S\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(\alpha +\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right]\\ {}-{\mathrm{e}}^{-\alpha \tau}X\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(\alpha -\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right].\hfill \end{array}} $$
(6)
This equation is identical in form to the Black–Scholes formula described below. Its only difference is its use of α, the expected rate of return on the stock, rather than the risk-free rate of interest. If Boness had carried his assumption that investors are indifferent to risk to its logical conclusion that α = r, he would have derived the Black–Scholes equation. Of course, his derivation would still have been based on the assumption of risk neutrality.
Samuelson (1965) recognized that the expected rates of return on the option and stock would generally be different due to their different risk characteristics. He posited a higher (constant) expected rate of return for the option, β, although recognizing that a ‘deeper theory would deduce the value of [the expected rate of return]’. He also realized that this assumption would mean that it might be optimal to exercise a call option prior to its maturity but was unable to solve for the optimal exercise policy except in the case of perpetual calls. His model for a European call was
$$ {\displaystyle \begin{array}{l}C={\mathrm{e}}^{\left(\alpha -\beta \right)\tau }S\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(\alpha +\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right]\\ {}-{\mathrm{e}}^{-\beta \tau}X\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(\alpha -\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right].\hfill \end{array}} $$
(7)
Boness’s equation above is a special case of this model for α = β.

Samuelson and Merton (1969) examined option pricing in a simple equilibrium model of portfolio choice that allowed them to determine the stock’s and option’s expected rates of return endogenously. They verified that the option problem could be stated in ‘utili-probability’ terms in a function form identical to the problem statement in terms of the true probabilities. When stated in this fashion, the adjusted expected rates of return on the stock and option were the same. This approach anticipated the development of the risk-neutral or preference-free method of valuing options that is now accepted as a matter of course.

The Black–Scholes Option Pricing Model

The Black–Scholes option pricing model is based on the principle that there should be no arbitrage opportunities available in the market. The following simple model, due to Cox et al. (1979), can be used to illustrate the principle behind the Black–Scholes model.

Assume that over a single period the stock price can change in only one of two ways. From its current level S, the stock price can increase to hS or fall to kS. Let C(S,n) denote the value of a call option on the stock when the stock price is S and there are n of these ‘steps’ remaining before the option matures.

Consider a portfolio that is short one call option and long N shares of stock. This portfolio is currently worth NSC(S,n). After one period this portfolio will be worth either NhSC(hS, n − 1) or NkSC(kS,n − 1). Suppose N is chosen so that these last two quantities are equal; i.e.,
$$ N=\frac{C\left( hS,n-1\right)-C\left( kS,n-1\right)}{\left(h-k\right)S} $$
(8)
then after one period the portfolio will be worth
$$ \frac{kC\left( hS,n-1\right)- hC\left( kS,n-1\right)}{\left(h-k\right)} $$
(9)
with certainty. To avoid an arbitrage opportunity the current value of the portfolio must be equal to this value discounted at (1 + R) where R is the risk-free rate of interest (not annualized) over the time of a single step in the stock price. That is,
$$ {\displaystyle \begin{array}{ll}\hfill & C\left(S,n\right)\\ {}& =\frac{1}{1+R}\left[\frac{1+R-k}{h-k}C\left( hS,n-1\right)+\frac{h-1-R}{h-k}C\left( kS,n-1\right)\right].\hfill \end{array}} $$
(10)
This equation relates the value of a n step call option to the value of a n − 1 step call. At the time it matures, the value of a call with an exercise price of X is C(S, 0) = Max (SX, 0). As this functional form is known, (10) can be used to derive the value of a one-period call for different stock prices. Given these values, (10) can be used again to derive the value of a two-period call. The value of any call can be computed by using (10) recursively.
The resulting formula for a n step call is
$$ {\displaystyle \begin{array}{ll}C\left(S,n\right)=& {\left(1+R\right)}^{-n}\hfill \\ {}& \times \sum \limits_{i=1}^n\frac{n!}{i!\left(n-i\right)!}{q}^i{\left(1-q\right)}^{n-i}\left(S{h}^i{k}^{n-1}-X\right)\hfill \end{array}} $$
(11)
where q ≡ (1 + Rk) (hk) and I is the smallest integer for which ShikniX.
The fraction and the next two terms involving q in the summation can be recognized as the probability of i successes in n trials with a success probability of q from a binomial distribution. Thus the formula in (11) can be rewritten as
$$ C\left(S,n\right)={\left(1+R\right)}^{-n}{E}^{\ast}\left[\operatorname{Max}\left({S}_n-X,0\right)\right] $$
(12)
where Sn is the random stock price after n steps and E*[⋅] denotes the expectation using the artificial probabilities q and 1 − q for the up and down steps. Similarly equation (10) can be expressed as
$$ {\displaystyle \begin{array}{ll}C\left(S,n\right)& =\frac{1}{1+R}\left[ qC\left( hS,n-1\right)+\left(1-q\right)C\left( kS,n-1\right)\right]\hfill \\ {}& =\frac{1}{1+R}{E}^{\ast}\left[C\left(S,n-1\right)\right].\hfill \end{array}} $$
(13)
Again an ‘artificial’ expectation has been taken. It should be noted that q is not the actual probability that the stock price will change from S to hS – in fact this true probability has not be used here at all.

In deriving their model Black and Scholes did not assume that the stock price followed this binomial step process. They used instead a geometric or lognormal Brownian motion process. Geometric Brownian motion can be constructed as the limit of this type of binomial process as the step sizes h − 1 and k − 1 shrink to zero while the number of steps per unit time goes to infinity.

Taking these limits in (10) gives the Black–Scholes partial differential equation
$$ \frac{1}{2}{\sigma}^2{S}^2{C}_{SS}+ rS{C}_s- rC+{C}_t=0 $$
(14)
where r is the continuously-compounded (annualized) rate of interest on a risk-free asset, σ2 is the variance of changes in the logarithm of the stock price per unit time and subscripts on C denote partial differentiation. Applying the limits to (11) yields the Black–Scholes call option pricing formula
$$ {\displaystyle \begin{array}{ll}C\left(S,\tau \right)=& S\cdot \varPhi \left[\frac{\ln \left(S/X\right)+\left(r+\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right] \hfill \\ {}& -{\mathrm{e}}^{- r\tau}X\cdot \varPhi \left[\frac{\ln \left(S/X\right)-\left(r+\frac{1}{2}{\sigma}^2\right)\tau }{\sigma \sqrt{\tau }}\right].\hfill \end{array}} $$
(15)
where Φ (⋅) is the standard cumulative normal distribution function and τ ≡ Tt is the time until maturity. (Black and Scholes derived this differential equation and its solution working directly with the continuous time diffusion and not by taking limits.)

The Black–Scholes formula is identical to Samuelson’s with α = β = r and to Boness’s with α = r. In fact the most remarkable feature about the model is that the resulting formula does not depend on the stock’s or the option’s expected rates of return or any measure of the market’s risk aversion. Only five variables determine the option’s price: S, τ, r, X and. σ2 Except for the variance, each of these variables is known, and the variance can be measured with a high degree of certainty.

The absence of the expected rates of return or any measure of risk aversion from the Black–Scholes model was at first troubling. This puzzle was explained by Cox and Ross (1976a) and Merton (1976) who introduced the risk neutral or martingale representation. This idea was later developed more formally by Harrison and Kreps (1979) and others.

The fact that a hedging argument can be used to derive (10), which does not include explicitly expected rates of return, investor preferences, or probabilities means that given the stock price and the interest rate, the value of the option cannot depend directly on these either. To solve for the option price, then, we need only find the equilibrium solution in some world where returns, preferences, and probabilities are consistent with the actual stock price process and interest rate. The solution obtained will then be generally applicable.

The most convenient choice of equilibrium is often an economy with risk neutral investors. In such an economy all expected rates of return must be equal to the risk-free rate. If the stock price has a lognormal distribution, then Boness’s model applies with α = r.

In the risk neutral economy the Black–Scholes formula has an interpretation identical to that in (12). The cumulative normal in the second term in (14) is the risk neutral ‘probability’ that the option will mature in-the-money. Thus, the second term is the discount factor multiplied by the ‘expected’ exercise payment. The first term is the discounted value of the expectation of the stock’s price at expiration conditional on ST > X.

Extensions of the Black–Scholes Model

The derivation of the Black–Scholes model rests on six assumptions: (i) There are no transactions costs, taxes or restrictions on short sales. (ii) The risk-free rate of interest is constant. (iii) The stock pays no dividends. (iv) The stock price evolution is geometric Brownian motion. (v) The market is open continuously for trading. (vi) The option is European.

Subsequent modifications of the basic model have shown that it is quite robust with respect to relaxations of these assumptions. Thorpe (1973) examined the short sale constraint. Leland (1985) allowed for transactions costs. Ingersoll (1976) and Scholes (1976) considered the effects of differing tax rates on capital gains and dividends. Merton (1973) generalized the model to allow for dividends and a stochastic interest rate. He also proved that assumption (vi) was not necessary if the stock did not pay dividends. Cox and Ross (1976a) and Merton (1973) utilized alternative stochastic processes. Cox and Ross (1976a) and Merton (1976) considered the option problem when the stock’s price evolution did not have a continuous sample path. Rubinstein (1976) and Brennan (1979) obtained the Black–Scholes solution with discrete-time trading by imposing conditions on the utility function of the representative investor.

Other types of options have also been valued using the same methods or extensions of them. Some examples are European puts by Black and Scholes (1973), ‘down-and-out’ options by Merton (1973), commodity options by Black (1976) and interest rate options by Cox et al. (1985b). To solve these or similar problems, the Black–Scholes partial differential equation (14) is used.

While (10) and, therefore (14), were developed to price call options, the characteristics of the call are captured entirely by the condition at maturity C(S, 0) = Max(SX, 0). Thus, this equation is a general one that can be used to price calls, puts, or any other derivative asset whose value depends on just the price of the primitive asset.

To solve this equation for other problems the appropriate boundary condition is required
$$ C\left(S,T\right)=H(S). $$
(16)
H (⋅) specifies a contractual or otherwise known payment at the derivative asset’s maturity. If the derivative asset’s value arises solely from this payment at maturity, then the formal solution to (14) with boundary condition (16) is
$$ C\left(S,t\right)={\mathrm{e}}^{-r\left(T-t\right)}{E}^{\ast}\left[H(S)\right]. $$
(17)
For some contracts a portion or all of the value may be due to payments that are received at random times prior to maturity. In this case (17) does not measure the full value. For example, a down-and-out option is a call contract that is cancelled if and when the stock price falls below the ‘knock-out’ price. At this point a partial rebate is usually given. Let K and R denote the knock-out price and rebate. Then the conditions imposed to value this option are
$$ {\displaystyle \begin{array}{l}C\left(K,u\right)=R\forall u<T\\ {}C\left(S,T\right)=\operatorname{Max}\left(S-X,0\right)\\ {} \mathrm{if}\;S(u)>K\mathrm{for}\;t<u<T.\end{array}} $$
(18)
The value of the down-and-out option is
$$ {\displaystyle \begin{array}{l}C\left(S,t\right)=R{E}^{\ast}\left[{\mathrm{e}}^{-r\left(U-t\right)}I\left(U\le T\right)\right]\\ {}+{\mathrm{e}}^{-r\left(T-t\right)}{E}^{\ast}\left[\operatorname{Max}\left({S}_T-X,0\right)I\left(U>T\right)\right].\hfill \end{array}} $$
(19)
Here U is a random variable that takes on the value u if the first time that the stock price drops to K is u. I (⋅) is an indicator function with the value one if its argument is true and zero otherwise. The first expectation is taken over the random variable U. This term measures the value contributed by the receipt of the rebate. The second expectation is taken over both random variables U and ST. This term measures the value contributed by the right to exercise if it was not cancelled.

The pricing of the American put has a similar feature. The payment received upon exercise, XS, is known (conditional on the stock price at that time) but its timing is not. In addition, unlike the timing of the rebate in the previous problem, the timing of the exercise is not contractually stated. It is chosen by the put’s owner.

Suppose that the put owner chooses a rule for exercising. This rule will generate a random time U at which the option is exercised. The random variable U must be a Markov time; that is, whether or not exercise occurs at a particular time can depend on information known at that time but cannot in any way anticipate the future. For a given rule U, the put’s value is
$$ {E}^{\ast}\left[{\mathrm{e}}^{-r\left(U-t\right)}\left(X-{S}_U\right)\right]. $$
(20)
As the owner of the put has the choice, the rule chosen will be that which maximizes the value of the option
$$ P\left(S,t\right)=\underset{U}{\sup }{E}^{\ast}\left[{\mathrm{e}}^{-r\left(U-t\right)}\left(X-{S}_U\right)\right]. $$
(21)
In principle the American put could be valued by solving (20) for all exercise rules and choosing that one which maximized the value. Samuelson (1965) conjectured and Merton (1973) proved that in such problems the value and the optimal exercise rule could be determined simultaneously by imposing the ‘high contact’ condition.
The partial differential equation (14) is solved subject to the maturity condition P(S, T) = Max(XS, 0) and
$$ P\left[K(t),t\right]=X-K(t) $$
(22a)
$$ {\left.\frac{\partial P\left(S,t\right)}{\partial S}\right|}_{S=K(t)}=-1. $$
(22b)
K(t) denotes the optimal exercise policy; that is if the stock price falls to K(t) at time t, then the put is exercised. Equation (22a) is the standard condition at exercise. Equation (22b) is the high contact condition.

The high contact requirement assures that for the optimal policy the slope of the pricing function, P(⋅) is equal to the slope of the payoff function (−1 in the relevant region of exercise). This is just the usual tangency condition at an optimum.

No analytical solution to the American put problem has yet been derived. Brennan and Schwartz (1977), Parkinson (1977) and others have described numerical techniques for these problems and other contracts for which there are no analytical solutions.

Applications of Option Pricing to Valuing Corporate Securities

After deriving their call option formula Black and Scholes make an observation that may be one of the most important in the field of finance. They argue that the same methods can be used to value other contingent claims, in particular the components of a firm’s capital structure. This observation has led to an enormous amount of research. Option pricing techniques have been applied to a wide variety of financial instruments and contracts including corporate bonds, futures, variable rate mortgages, insurance, investment timing advice, and the tax code.

For the simplest problems the call formula can be applied directly. Consider a firm with assets whose value, V, evolves according to a geometric Brownian process. The firm’s capital structure consists of common stock and single issue of zero coupon bonds with an aggregate face value of B which mature at time T. At that time the firm will be liquidated.

If VTB, then the bondholders can be paid and the equity will be worth VTB. If VT < B, the assets will be insufficient to pay the bondholders, and there will be nothing left for the shareholders. Thus, the payoff to the common shares is Max(VTB, 0). This is just like a call option so currently the equity must be worth C(V, Tt; B). By the Modigliani–Miller irrelevancy theorem, the value of the debt and equity must sum to V so the debt is worth
$$ D\left(V,T-t;B\right)=V-C\left(V,T-t;B\right). $$
(23)

This same valuation applies even if the firm is not to be liquidated. To repay the bondholders, the firm must raise B dollars. Selling assets to do this is the same as a liquidation. The only other way to raise this money is by a new offering of securities. To raise B dollars the firm will have to offer a security that is worth B. If the firm’s assets are not worth at least B, this cannot be done. If they are, then again by the Modigliani–Miller theorem the original equity will be worth VTB.

A zero coupon convertible bond can be priced similarly. Suppose there are N shares of common outstanding and the convertibles can be exchanged for n shares in aggregate. If all the bondholders convert, then they will own the fraction γn/(N + n) of the equity. Clearly the bondholders will convert if γVT > B. Otherwise they will receive B, unless the firm is insolvent, in which case they will get just VT. Thus, the bondholders will receive
$$ {\displaystyle \begin{array}{ll}\operatorname{Max}\left[\gamma V,\operatorname{Min}\left(V,B\right)\right]=& \operatorname{Max}\left(\gamma V-B,0\right)\hfill \\ {}& +\left[V-\operatorname{Max}\left(V-B,0\right)\right].\hfill \end{array}} $$
(24)
This is the payoff to an option plus an ordinary zero coupon bond so the convertible’s value must be C(γV, Tt; B) + D(V, Tt; B). If the convertible is also callable, as most are, then methods used to determine the optimal exercise policy for and the value of an American put must be used. This problem has been solved by Ingersoll (1977).

Most corporate securities receive periodic coupons or dividends. While a default-free coupon bond can be valued as a portfolio of zero coupon bonds, this method will not work when there is default risk because the omission of one coupon puts the whole bond in default. These securities can be priced as a series of options, however.

Consider a company with common stock on which it is not paying dividends and a single issue of coupon bonds with aggregate periodic coupons of c, at times T1,…,Tn, and an aggregate par value of B, repaid at Tn. Once the next to last coupon is paid only a single payment remains B + c. Therefore, just after the next to last payment the bond can be treated like a zero coupon bond. Its value at that time is
$$ {D}_{n-1}V,{T}_{n-1}=D\left(V,{T}_n-{T}_{n-1};B+c\right). $$
Between times Tn−2 and Tn−1 the company makes no payments to the holders of its securities so the standard Black–Scholes equation (14) applies. The solution for the bond’s value at time Tn−2 is
$$ {\displaystyle \begin{array}{ll}\hfill & {D}_{n-2}\left(V,{T}_{n-2}\right)\\ {}& ={\mathrm{e}}^{-r\left({T}_{n-1}-{T}_{n-2}\right)}{E}^{\ast}\left[D\left({V}_{T_{n-1}},{T}_{n-1}\right)\right]\hfill \end{array}} $$
(25)
as given in (17). The price at earlier times can be determined by a recursive application of (25). Geske (1977) addresses this compound option problem.
Another way to price claims with coupons or dividends is to approximate the sequence of payments as continuous flows. The general problem is to value a particular claim, F(S, t), when the price evolution of the firm’s value is
$$ \mathrm{d}V=\left[\alpha V-\Delta \left(V,t\right)\right]\mathrm{d}t+\sigma V\mathrm{d}\omega . $$
(26)
Δ(V,t) is the total flow of all disbursements (dividends, coupons, etc.) paid by the firm and dω is the increment to a Wiener process.
The equilibrium price process for the claim is
$$ \mathrm{d}F\left(V,t\right)=\left[\beta \left(V,t\right)F-\delta \left(V,t\right)\right]\mathrm{d}t+\left({F}_v/F\right)\sigma V\mathrm{d}\omega $$
(27)
where β (⋅) is the (endogenous) expected rate of return on the derivative asset and δ (⋅) is the portion of the total disbursement received by the owners of the derivative asset.
Ito’s Lemma is used to determine the expected rate of price appreciation which is equated to the rate of capital gains required in equilibrium to earn β.
$$ {\displaystyle \begin{array}{l}\frac{1}{2} {\sigma}^2{V}^2{F}_{vv}+\left[\alpha V-\Delta \left(V,t\right)\right]{F}_V+{F}_t\\ {}=\beta \left(V,t\right)F\left(V,t\right)-\delta \left(V,t\right).\hfill \end{array}} $$
(28)
The equivalent risk neutral processes replace α and β by r, the risk-free rate. Thus, the general valuation equation is
$$ \frac{1}{2}\ {\sigma}^2{V}^2{F}_{VV}+\left[ rV-\Delta \left(V,t\right)\right]{F}_V- rF+{F}_t-\delta \left(V,t\right)=0. $$
(29)
Equation (29) is the fundamental valuation equation for the financial claims against a firm. It can be used for any situation when the standard Black–Scholes conditions hold and the value of the claim to be priced depends solely on time and the value of the assets of the firm. The basic requirement for this second condition is that there be no other sources of uncertainty beyond that affecting the value of the assets. Thus, the interest rate cannot be stochastic, the dividend policy must be a known function of the firm value and time, the firm cannot alter its investment or financing policies in unanticipated ways.

If this second requirement is not met, then the value of the claim being priced will depend on other variables as well – variables that measure the overall state of the economy. Cox et al. (1985a) have developed a theoretical context in which all these pricing problems can be handled. The basic Black–Scholes method is still valid, but the pricing equation will include these additional state variables.

Other Applications of Option Pricing

In recent years option pricing techniques have been used in a great variety of situations. PBGC insurance and the effects of ERISA on corporate pension plans have been considered as have FDIC insurance and the implicit insurance in government loan guarantees. The asymmetries of the tax code and their effects on corporations and investors have been analysed. Option pricing methods have been used to value market timing advice and to examine the efficiency of dynamic portfolio strategies such as contingent immunization. More on the applications of option pricing and extensive bibliographies can be found in the survey articles by Mason and Merton (1985) and Smith (1976) and in the texts by Cox and Rubinstein (1985) and Ingersoll (1987).

It should be clear that the realm of applications goes far beyond the more obvious corporate securities. A bibliography of the published papers alone would be extensive, and working papers are continually added. Option pricing theory has become an important element in our understanding of financial contracting and a practical tool in widespread applications.

See Also

Bibliography

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Jonathan E. IngersollJr.
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