376 9. Derivative Pricing Models

Table 9.5. Option prices on assets with negatively autocorrelaled returns.

( .oiup.u isou ot lU.uk-St holes call option iи ices on a liyjtothelieal $40 slock under an at ittmlclic lliowtttatt motion versus a trending ()t nstcin-Uhlcnbeck process lor log-prices, assuming a standard deviation olloi daily t ontinuousty com pounded returns, and a daily continuously compounded riskfree rate of log( 1.0M/3ti I. As autocorrelation becomes larger in absolute value, option pi it es im t rase.

Лn Empirical Illustration

To gauge the empirical relevance ol this adjustment lor autocorrelation, Tahlc 9.5 reports a comparison ol Blark-S< holes prices under arithmetic Brownian motion and under the trending Ornstein-Uhlenbeck process for various holding periods, sirikc prices, and daily autocorrelations from -5 to -45% lot a hypothetical $40 stock. The unconditional standard deviation ol daily returns is held fixed at 2% per day. The Black-Scholes price is calculated according to (9.2. Hi), selling a equal to the unconditional standard deviation. The trending O-U prices are calculated by solving (9.3.57) and (9.3.58) Гогст given r and the return autocorrelations p\(r) of-0.05, -0.10, -0.20, -0.30, -0,40. and -0.45, and using these values of a in the Black-Scholes formula (9.2.16), where т - I.

9.3. Implementing Parametric Option Pricing Modeb

ЙгйЯ*

377%:;?..

The first panel of Table 9.5 shows that even extreme autocorrelation in daily returns does not affect short-maturity in-the-money call option prices very much. For example, a daily autocorrelation of -45% has no impact on the $30 7-day call; the price under the trending O-U process is identical to the standard Black-Scholes price of $10.028. But even for such a short maturity, differences become more pronounced as the strike price increases; the at-the-money call is worth $0.863 in the absence of autocorrelation, but increases to $1.368 with an autocorrelation of -45%.

However, as the time to maturity increases, the remaining panels of Table 9.5 show that the impact of autocorrelation also increases. With a - 10% daily autocorrelation, an at-the-money 1-year call is $7.234 and rises lo $10.343 with a daily autocorrelation of-45%, compared to the standard Black-Scholes price of $6.908. This is not surprising since the sensitivity of the Black-Scholes formula to a-the options vega-is an increasing function of the time-to-maturity (see Section 9.2.1). From (9.2.28), we see that for shorter-maturity options, changes in a have very little impact on the call price, but longer-maturity options will be more sensitive.

In general, the effects of asset return predictability on the price of derivatives depends intimately on the precise nature of the predictability. For example, the importance of autocorrelation for option prices hinges critically on the degree of autocorrelation for a given return horizon r and, of course, on the data-generating process which determines how rapidly this autocorrelation decays with т. For this reason, Lo and Wang (1995) introduce several new stochastic processes that are capable of matching more complex patterns of autocorrelation and predictability than the trending O-U process.

9.3.5 Implied Volatility Estimators

Suppose the current market price of a one-year European call option on a nondividend-paying stock is $7.382. Suppose further that its strike price is $35, the current stock price is $40, and the annual simple riskfree interest rate is 5%. If the Black-Scholes model holds, then the volatility a implied by the values given above can only take on one value-0.200-because, the Black-Scholes formula (9.2.16) yields a one-to-one relation between the options price and a, holding all other parameters fixed. Therefore, theop-tion described above is said to have an implied volatility of 0.200 or 20%] So common is this notion of implied volatility that options traders often qvjote prices not in dollars but in units of volatility, e.g., The one-year European call with $35.000 strike is trading at 20%. j

Because implied volatilities are linked directly to current market prijees (via the Black-Scholes formula), some investment professionals have argued that they are better estimators of volatility than r*\irniiltTt*bii*er1(rtibrH1tit1fii1, (Ыл such h1. frftfritr-rl vM*fmfi лг tAiiii mri v/tifi fltrWttrM HltcHUM

pirically because stock prices do not follow a lognormal diffusion, we may be able lo specify an alternate price process thai lils die dala belter, in which case die implied paramclci (s) of options on the same slock may indeed be numerically identical. Alternatively, if the Black-Scholes model fails empirically because in practice it is impossible lo trade continuously due to transactions costs and other institutional constraints, then markets are never dynamically complete, options are never redundant securities, and we should never expect implied parameters of options on the same stock lo be numerically identical for any option-pricing formula. In this case, the degree lo which implied volatilities disagree may be an indication of how redundant options really are.

fhe fact lhat options traders quote prices in terms of Black-Scholes implied volatilities has no direct bearing on (heir usefulness from a pricing point of view, but is a remarkable testament lo die popularity of the Black-Scholes formula as a convenient heuristic. Quoting prices in terms of licks rather than dollars has no far-reaching economic implications simply because there is a well-known one-to-one mapping between ticks and dollars. Moreover, just because options traders quote prices in terms of Black-Scholes implied volatilities, this does not imply thai they are using the Black-Scholes model to set their prices. Implied volatilities do convey information, bin ibis information is identical lo the information contained in die market prices on which the implied volatilities are based.

). 1.6 Stochastic-Volatility Models

Several empirical studies have shown that the geometric Brownian motion (9.2.2) is nol an appropriate model for certain security prices, for example, Beckers (1983), Black (1970), Blallberg and Conceits (1974), Christie (1982), fama (1905), l.o and MacKinlay (1988, 1990c), and Mandelbrot (1963, 1971) have documented important departures from (9.2.2) for US stock relurns: skewness, excess kurtosis, serial correlation, and time-varying volatilities. Although each of these empirical regularities has implications for option pricing, it is the last one thai has received the most attention in the recent derivatives literature,2 partly because volatility plays such a central role in die Black-Scholes/Merlon formulation and in industry practice.

If, in the geometric Brownian motion model (9.2.2) the о is a known deterministic function of time cr{t), (hen the Black-Scholes formula still applies but with n replaced by the integral j1 a{s)ds over the options lile. However, if a is stochastic, the situation becomes more complex. For

-.Sec. lor example, Aniiii and Ni (НИГ.1). Hall ami Коша (НИМ). Ileikei.s (IWO). Cos

(I.I7: ), Coldcllhere, О-.l). Ileslim (1<1ЗД. Holm....... Ilalen......I St Imei/ei (IWZ). Hull

ami White (I.IH ).Johnson and Shainio (1*187). S,on (I.IH7). and Wiggins ( И1К7).

siiuty they are based on current prices which presumably have expectations of die future impounded in them.

\lowcvcr, such an argument overlooks the lad that an implied volatility is intimately related to a specific parametric option-pricing model-typically the Black:Scholes model-which, in turn, is intimately related to a particular set of dynamics for the underlying stock price (geometric Brownian motion in tl с Black-Scholes case). Herein lies the problem with implied volatilities: If th: Black-Scholes formula holds, then the parameter a can be recovered with ntt error by inverting the Black-Scholes formula for anyone options price (each of which yields the same numerical value for ет); if the Black-Scholes formula does not hold, then the implied volatility is difficult to interpret sinci; it is obtained by inverting the Black-Scholes formula. Therefore, using the mplied volatility of one option to obtain a more accurate forecast of vola ility lo be used in pricing other options is either unnecessary or logically incojnsistent.

Го sec this more clearly, consider the argument that implied volatilities are better forecasts of future volatility because changing market conditions cause volatilities vary through time stochastically, and historical volatilities cannot adjust to changing market conditions as rapidly. The folly of this argument lies in the fact that stochastic volatility contradicts the assumptions required by the Black-Scholes model-if volatilities do change stochastically through lime, the Black-Scholes formula is no longer the correct pricing formula and an implied volatility derived from the Black-Scholes formula provides no new information.

Of course, in this case the historical volatility estimator is equally useless, since il need nol enter into the correct stochastic volatility option-pricing formula (in fact it docs not, as shown by Hull and While [1987], Wiggins [ 1987] and others-see Section 9.3.0). The correct approach is to use a historical estimator of the unknown parameters entering into the pricing formula-in the Black-Scholes case, the parameter a is related to the historical volatility estimator of continuously compounded returns, but under other assumptions for the slock price dynamics, historical volatility need not play such a central role.

This raises an interesting issue regarding the validity of the Black-Scholes formula. If the Black-Scholes formula is indeed correct, then the implied volatilities of any set of options on the.same stock must be numerically identical. Ol course, in practice they never are; thus the assumptions of the Black-Scholes model cannot literally be true. This should nol come as a complete .surprise; after all die assumptions of the Black-Scholes model imply thai options are redundant securities, which eliminates die need for organized options markets altogether.

The difficulty lies in determining which of the many Black-Scholes assumptions are violated. If, for example, the Black-Scholes model fails em-

.Mi)

l>. Derivative Pricing Models

example,.suppose dial die Гшккимеша! assets dynamics are given liv:

dl = /I Ill I 1-я / < ! (9.3.64)

* : da = u{a)dl + H(a)dlln. (9.3.05)

where crl) and /!( ) are ai hi lrar\ hind ions ol volatility a, and /fy and ll are standard Brownian motions with instantaneous correlation dllpdli, = p ill. In this case, it may not he possible to determine the price of ;m option In arbitrage arguments alone, lor die simple reason that there may not exist a dynamic sell-Imam ing porllolio strategy involving stocks and riskless bonds thai ran perfectly replicate the options payoff.

I leiu istically, stochastic volatility introduces a second source of uncertainty into the replicating portfolio and if this uncertainty (lln) is not per-lectly correlated with the uncertainty inherent in the stock price process (/( ), the replicating portfolio will not be able to span the possible outcomes thill an option may realize at maturity (see I larrison and Kreps [ 1979] and Dtiffic and I luang 19851 for a more rigorous discussion). Of course, if a were the price ol a traded asset, then under relatively weak regularity conditions there would exisi a dynamic self-financing portfolio strategy consisting of slocks, bonds, and the volatility asset that could perfectly replicate the option.

In the absence of this additional hedging security, the only available method lor pricing options in ihe presence of stochastic volatility of the lorm (9.3.1)5) is to appeal lo a dynamic equilibrium model. Perhaps the simplest approach is to assert lhat the risk associated with stochastic volatility is not priced in equilibrium. This is die approach taken by 1 lull and White (1987) for ihe case where volatility follows a geometric Brownian motion

ill - Ц Ill I-\-a Id Щ, (9.3.61))

da- = vtadl + $a~dli . (9.3.67)

By assuming ihat volatility is uncorrelated with aggregate consumption, they show that equilibrium option prices are given by the expectation of the Black-Scholes formula, where the expectation is taken with respect lo the average volatility over the options life.

Using the dynamic equilibrium models of Carman (1976b) and Cox, Ingersoll, and Ross (1985b), Wiggins (1987) derives the equilibrium price ol volatility risk in an economy where agents possess logarithmic utility functions, yielding an equilibrium condition-in the form of a PDE with certain boundary conditions-for ihe instantaneous expected return of the option price. Other derivative-pricing models with stochastic volatility take similar approaches, 11 it- differences coining from the type of equilibrium model employed or the choice of preferences dial agents exhibit.

9.J. Implementing Parametric Option Pricing Models Parameter Estimation

One of the most challenging aspects of stochastic-volatility models, is the fa that realizations of the volatility process are unobservable yet option-pricing formulas are invariably functions of the parameters of the process driving о To date, there has been relatively little attention devoted to this important issue for continuous-time processes like (9.3.64)-(9.3.65) primarily because of the difficulties inherent in estimating continuous-time models with discretely sampled data. However, a great deal of attention has been devoted to a related discrete-time model: the autoregressive conditional heteroskedasticity (ARCH) process of Engle (1982) and its many variants (see Chapter 12 and Bollerslev, Chou, and Kroner [1992]).

Although originally motivated by issues other than option pricing, ARCH models does capture the spirit of some of the corresponding continuous-time models. Recent studies by Nelson and Foster (1994), Nelson and Ramaswamy (1990), and Nelson (1991,1992,1996) provide some important links between the two. In particular, Nelson (1996) and Nelson and Fostjer (1994) derive the continuous-record asymptotics for several discrete-time ARCH processes, some of which converge to the continuous-time processes of Hull and While (1987) and Wiggins (1987). The empirical properties of these estimators have yet to be explored but will no doubt be the subject future research.

Discrete-Time Models \

Another approach is to begin with a discrete-time dynamic equilibriufn model for option prices in which the fundamental assets price dynamics are governed by an ARCH model. Although it is typically impossible to price securities by arbitrage in discrete time, continuous-time versions must appeal lo equilibrium arguments as well in the case of stochastic volatili; hence there is little loss of generality in leaving the continuous-time framework altogether in this case. This is the approach taken by Amin and Ng (1993), who derive option-pricing formulas for a variety of price dynamics- stochastic volatility, stochastic consumption growth variance, stochastic interest rates, and systematic jumps-by applying the discrete-time dynamic equilibrium models ofBrennan (1979) and Rubinstein (1976).

Discrete-time models are also generally easier to implement empirically since virtually all historical data are sampled discretely, financial transactions are typically recorded at discrete intervals, parameter estimation and hypothesis testing involve discrete data records, and forecasts are produced at discrete horizons. For these reasons, there may be an advantage in modeling stochastic volatility via AKCH and pricing derivatives within a discrete-time equilibrium model.

However, continuous-time models do offer other insights that are harder to come by within a discrete-time framework. For example, the dynamics