Derivative Pricing Models

ЧИР*

Is .1 is

Now In us further itsiiirt thixai biliage portfolio lo lie completely riskless in ilie sense thai its return is iioiisiochastic on (), 7. This inav he guaranteed bv choosing /j, = I and If, = 1 so lhat

a I I /; = 0. V, e [0, V.

implying thai

,(/)

(0.2.12)

(0.2.13)

lor every / e 0, /j. where /;*(/) and /*(/) are the number of shares of the slock and the option, respectively, held in the self-financing zero-risk portfolio. Note thai unless i)(,{l)/dP is constant through lime, both /;*(,) and nJt) must be lime-varying lo ensure thai this portfolio is riskless al all limes. Such a portfolio is said lo be perfectly hedged and -dG(t)/i)P is known as the hedge ratio.

Imposing (9.2.(i) and (0.2.13) loi all / e 0, 7 yields a dynamic portfolio strategy that is riskless and requires no net investment. Hut then surely its nonslochastic return ill must be zero, otherwise there would exist a costless, riskless dynamic portfolio strategy that yields a positive return. Therefore, lo avoid arbitrage, it must be the case lhat

( - r)l*(t) + ( (/) - r)l(l)

(0.2.14)

where a time argument has been added lo Цк(-) to emphasize die fact that il varies through lime as well.

Surprisingly, this simple no-arbilragc condition reduces the possible choices of G lo just one expression which is a second-order linear parabolic ID К for G:

i ., ., n-c. не, ix;

-o-r--- + >/ - +--rC.

I il/- HI iU

(0.2.15)

subject lo 11 u- following two boundary conditions: G(/(7h 7) = Max/(/)-.V,0, and (.(О. /) = 0. The unique solution is, of course, the Black-Scholes formula:

G(/(M. I)

d. =

/<0ф(./,1 - \V Ф(г/..)

logl/m/.V) + (r+ iff-)(7-/) oJTt

log(/(/)/.V) + (с- т,а-)(Т~1)

(0.2. Hi) (0.2.17)

(0.2. IS)

9.2. Л lirief Iteview of Derivative Ihicing Methods

where Ф(-) is the standard normal cumulative distribution function, and 7 -/is ihe time-to-maturity of the option. , :

The importance of assumptions (Al) and (A3) should now be appar-... ent: it is the combination of Brojvjjian motion (with its continuous sample paths) and the ability to trade continuously that enables us to construct! a perfectly hedged portfolio. If either of these assumptions failed, there; would be occasions when the return to the arbitrage portfolio is nonzero and stochastic, i.e., risky. If so, the arbitrage argument no longer applies.

Mertons derivation of the Black-Scholes formula also showcases the power of Itos Lemma, which gives us the dynamics (9.2.8) of the option price G that led to the PDE (9.2.15). In a discrete-time setting, obtaining the dynamics of a nonlinear function of even the simplest linear stochastic process is generally hopeless, and yet this is precisely what is required to construct a perfectly hedged portfolio.

More importantly, the existence of a self-financing perfectly hedged portfolio of options, stocks, and bonds implies that the option may be synthetically replicated by a self-financing dynamic trading strategy consisting of only stocks and bonds. The initial cost of setting up this replicadng portfolio of stocks and bonds must then equal the options price to rule out arbitrage because the replicating portfolio is self-financing and duplicates the options payoff at maturity. The hedge ratio (9.2.13) provides the recipe for implementing the replicating strategy.

Option Sensitivities

The sensitivities of G to its other arguments also play crucial roles in trading and managing portfolios of options, so much so that the partial derivatives12 of G with respect to its arguments have been assigned specific Greek letters in the parlance of investment professionals and are now known collectively as option sensitivities or, less formally, as the Greeks :13

Delta A =

Gamma Г

Tbeta (-) =

3G IF

92G ЭС

(9.2.19) (9.2.20) (9.2.21)

Tlic lerm partial derivatives in this context refers, of course, to instantaneous rates of chillier. This is unfortunate coincidence of terminology is usually not a source of confusion, hut readers should beware.

M()f course, vej a is not a Greek letter and V is simply a script V.

Vega

ас 1c7

(9.2.22)

(9.2.23)

For the Black-Scholes formula (9.2.10), these option sensitivities can he evaluated explicitly:

A s Ф(г/,)

<PUh)

Г =

Poy/r-t Pa

2JT-1 R s (7-.)Х<г(7->Ф(<* ) V = Ру/Т1фШ

</>(4) - Хгв (Т- Ф()

(9.2.24) (9.2.25)

(9.2.26)

(9.2.27) (9.2.28)

where ф(-) is the standard normal probability density function. We shall have occasion to consider these measures again once wc have developed methods for estimating option prices in Section 9.3.3.

9.2.2 The Martingale Approach

Once Black and Scholes (1973) andMerton (1973b) presented their option-pricing models, it quickly became apparent that their approach could be used to price a variety of other securities whose payoffs depend on the prices of other securities: Find some dynamic, costless self-financing portfolio strategy that can replicate the payoff of the derivative, and impose the no-arbitrage condition. This condition usually reduces to a PDE like (9.2.15), subject to boundary conditions that are determined by the specific terms of the derivative security.

It is an interesting fact that pricing derivative securities along these lines docs not require any restrictions on agents preferences other than nonsatiation, i.e., agents prefer more to less, which rules out arbitrage opportunities. Therefore, the pricing formula for any derivative security that can be priced in this fashion must be identical for all preferences that do not admit arbitrage. In particular, the pricing formula must be the same regardless of agents risk tolerances, so that an economy composed of risk-neutral investors must yield the same option price as an economy composed of risk-averse investors. But under risk-neutrality, all assets must earn the

same expected rate of return which, under assumption (Л2), must equal the riskless rale r. This fundamental insight, due to ( ox and Ross (1976), simplifies the computation of option-pricing formulas enormously because in a risk-neutral economy the options price is simply the expected value of its payoff discounted al ihe riskless rate:

C{1) = e-,r-,>E,[Max[/,(7) - Л,0]]. (9.2.29)

However, the conditional expectation in (9.2.29) must be evaluated with respect to the random variable P*(T), not /(7 ), where / (/) is the terminal slock price adjusted for risk-neutrality.

Specifically, under assumption (A3), the conditional distribution of P(T) given /(0 is simply a lognormal distribution with E[log /(7 ) /()] = log/40 4- (Ц - y)(T-/) and Var[log/>(7) />(<)] = а*{Т-1). Under risk-neutrality, the expected rale of return for all assets must be r, and hence die conditional distribution of the risk-neutralized terminal stock price /*(/) is al.:o lognormal, but wilh E[\ogP{T) I P(t)] = log/(0 + (r - ~-){T-~t) and Varilog/VO I P(D] =a2(T-t).

For obvious reasons, this procedure is called the risk-neutral pricing method and under assumptions (Al) through (A4), the expectation in (9.2.29) may be evaluated explicitly as a function of the standard normal CDF and yields the Black-Scholes formula (9.2.16).

To emphasize the generality of the risk-neutral pricing method in valuing arbitrary payoff streams, (9.2.29) is often rewritten as

G(t) = r--r- E;[Max[/J(7) - Л,0]], (9.2.30)

where the asterisk in E* indicates that the expectation is to be taken with respect to an adjusted probability distribution, adjusted lo be consistent with risk-neutrality, ln a more formal selling, Harrison and Kreps (1979) have shown that the adjusted probability distribution is precisely the distribution under which the stock price follows a martingale; thus they call the adjusted distribution the equivalent martingale measure. Accordingly, the risk-neutral pricing method is also known as the martingale pricing technique. Wc shall exploit this procedure extensively in Section 9.4 where we propose to evaluate expectations like (9.2.30) by Monte Carlo simulation.

9.3 Implementing Parametric Option Pricing Models

Because there are so many different types of options and other derivative securities, it is virtually impossible lo describe a completely general method for implementing all derivative-pricing formulas. The particular features of each derivative security will often play a central role in how its pricing

formula is lo he applied mosl effectively. Bui there are several common aspects lo every implementation of a parametric option-pricing model-a model in which the price dynamics of ihe underlying security, called the fundamental asset, is specified up lo a finite number of parameters-and we shall lot us on these common aspects in this section.

To simplify terminology, unless otherwise stated we shall use the term option to mean any general derivative security, and the term stork to mean the derivative securitys underlying fundamental asset. Although there are certainly aspects ol some derivative securities thai differ dramatically from those ol standard equity options and cannot be described in analogous terms, they need not concern us al die current level of generality. After developing a coherent framework for implementing general pricing formulas, we shall turn to modifications tailored to particular derivative securities.

9. ?. / Parameter Estimation of Asset Price Dynamics

The lenn parametric in ibis sections title is meant to emphasize the reliance of a class of option-pricing formulas on the particular assumptions concerning the fundamental assets price dynamics. Although these rather strong assumptions often yield elegant and tractable expressions for the options price, they arc typically contradicted by the data, which does not bode well for the pricing formulas success. In fact, perhaps the most important aspect of a successful empirical implementation of any option-pricing model is correctly identifying the dynamics of the stock price, and uncertainly regarding these price dynamics will lead us to consider nonparametric alternatives in Chapter 12.

But for the moment, let us assert that the specific form of the stock price process /(/) is known up to a vector of unknown parameters в which lies in some parameter space 0, and thai it satisfies the following stochastic differential equation:

dlit) = aU\t:a)dl + h(P,t;f3)dn(t), . e [0, 7], (9.3.1)

where /{(/) is a standard Wiener process and 0 = [ n 0 ] is а (/<x 1) vector of unknown parameters. The functions a(P, t; a) and l>(P, t; (3) are called the drift and diffusion functions, respectively, or the coefficient functions collectively, for example, the lognormal diffusion assumed by Black and Scholes (1973) is given by die following coefficient functions:

(/./;< ) = цР (9.3.2)

HP. f.fi) = ctP (9.3.3)

In this case, die parameter vector в consists of only two elements, the constants a and в.ы j

In the more general case the functions a(P, t; a) and b(P, t; (3) must be restricted in some fashion so as to ensure the existence of a solution to the stochastic differential equation (9.3.1) (see Arnold [1974], for example). Also, for traciability we assume that the coefficient functions only depend on the most recent.price P(t); hence the solution to (9.3.1) is a Markov process. This assumption is not as restrictive as it seems, since non-Markov processes can often be re-expressed as a vector Markov process by expansion I of the stales, i.e., by increasing the number of state variables so that the collection of prices and state variables is a vector-Markov process. In practice, however, expanding the states can also create intractabilities that may be more difficult to overcome than the non-Markovian nature of the original price process.

For option-pricing purposes, what concerns us is estimating в, since pricing formulas for options on P(t) will invariably be functions of some or all of the parameters in в. In particular, suppose that an explicit expression for the option-pricing function exists and is given by GP(l), в) where other dependencies on observable constants such as strike price, time-to-maturity, and the interest rate have been suppressed for notational convenience.15 An estimator of the option price G may then be written as G = G(P(t), в), where в is some estimator of the parameter vector в. Naturally, the properties of G are closely related to the properties of Q, so that imprecise estimators of the parameter vector will yield imprecise option prices and vice versa. To quantify this relation between the precision of G and of в, we must first consider the procedure for obtaining 0.

Maximum Likelihood Estimation

The most direct method for obtaining 8 is to estimate it from historical data. Suppose we have a sequence of n+l historical observations of Pt) sampled at non-stochastic dates to < l\ < < t which are not necessarily equally spaced. This is an important feature of financial time series since markets are generally closed on weekends and holidays, yielding irregular sampling intervals. Since P(t) is a continuous-time Markov process by assumption, irregular sampling poses no conceptual problems; the joint density function / of the sample is given by the following product:

№......IV, в) = /,(/> ; 0)П/( I (9.3.4)

Nine ill.il .ill hough the drill anil diffusion lunriinns depend on distinct parameter vectors и and p. these two vectors may contain some parameters in common.

lr>Kven iff,cannot be obtained in closed-form, is a necessary input for numerical solutions of (, and must still be estimated. ,