he reflected through ils mean lo produce a mirror-image wliich has die same statistical properties. This approach yields an added benefit: negative correlation among pairs ol replications, lithe summands of the Monte Carlo estimator are monotone functions of the replications they will also he negatively correlated, implying a smaller variance for the estimator.

This is a simple example of a more general technique known its the antithetic variates method in which correlation is induced across replications to reduce the variance of the sum. Л more formal motivation for this approach comes from the following theorem: for any estimator which can he expressed as the sum of random variables, it is always possible to create a strict functional dependence between the summands which leaves the estimator unbiased but yields a variance that comes arbitrarily close to the minimum variance possible with these random variables (see Hammerslcy and Mauldon I950). ()l course, the challenge is to construct such a functional dependence, hut even if the optimal transformation is not apparent, substantial efficiency gains can be achieved by simpler kinds of dependence.

Variance reduction can also be accomplished by more sophisticated sampling methods. In slmtijicd sampling, the support of the basic random variable X being simulated is partitioned into a finite number of intervals and crude Monte (larlo simulations arc performed in each interval. If there is less variation in / (X) within intervals than across the intervals, the sampling variation of the estimator of К* /(X) will be reduced.

Importance sampling is a mote sophisticated version, sampling mote frequently in legions of the support where there is more variation in /(X)- where sampling is more important -instead of sampling at regular intervals. An even more sophisticated version of this method has recently been proposed by Fang and Wang (1994) in wliich replications are generated de-Icrniinisiically, not randomly, accordingly loan algorithm that is designed to minimize the sampling variation of the estimator directly. Il is still too early to say how this approach-called the number-theoretic method-compares to the more traditional Monte Carlo estimators, but il has already found its way into the financial community (see, for example, Paskov and Traub [1995]) and the preliminary findings seem encouraging.

Ли Illustration of Variance Reduction

To illustrate the potential [lower of variance-reduction techniques, we construct an antithetic-variates estimator of the price of the one-year lookback put option of Set lion 9.4.3. For each simulated price path /Д)1=( another can be obtained without further simulation by reversing the sign of each of the randomly generated НП standard normal variates on which the price path is based, yielding a second path /Д }д which is negatively correlated with the fust. If in sample paths of {/Дд are generated, the resulting

antithetic-variates estimator /7(0) is simply the average across all 2m paths

(0)

- 1 2m

/40)

(9.4.12)

where

Y. = Max P , Yj = Max />. 1 <><*< j* ()<*<

(l<*<n J 0<k<n

The relation between antithetic-variates and crude Monte Carlo can be more easily seen by rewriting (9.4.12) as

<°> = i ?-rTi t Y> +rrT7n £ pi-) -m (9-44

m4-f 2

(9.4.ll)

Equation (9.4.13) shows that /7(0) is based on a simple average of two averages, one based on the sample paths {/)j 0 and the other based on

1 Д )*=<> nc :iCl lnal l,ese lwo averages are negatively correlated leads lo a reduction in variance.

Equation (9.4.14) combines the two sums of (9.4.13) into one, with the averages of the antithetic pairs as the summands. This sum is particularly easy to analyze because the summands are IID-the correlation is confined within each summand, not across the summands-hence the variance of the sum is simply the sum of the variances. An expression for the variance of /7(0) then follows readily

Var[/7(0)] = e~

-Var m

-1 /1

, Var[y; ]4--Cov[y> m \2 2

oHn)

-d+P)

(9.4.15)

(9.4.16)

where a*(n)=Varle-rTYjn] = Var[e-rTYjn] and psCorr[e-rTYjn, e~rTYja].

Equation (9.4.15) shows that the variance of /7(0) can be estimated by the product of e~rl j m and the sample variance of the IID sequence {(Yj + Yjn)/2). There is no need to account for the correlation between antithetic pairs because this is implicitly accounted for in the sample variance of[(Yj +Yjn)/2).

Equation (9.4.16) provides additional insight into the variance reduction that antithetic variates affords. The reduction in variance comes from

two sources: a doubling ol ibc number ol replications from in lo 2 w, and the factor l+p which should be less than one if the correlation between the antithetic variates is negative. Note that even if the cot relation is positive, the variance of (()) will still be lower than the crude Monte (lat lo estimator (()) unless there is perfect correlation, i.e., p-\. Also, while we have doubled the number of replications, we have done so in a computationally trivial way: changing signs. Since the compulations involved in pseudorandom number generation are typically more demanding than mete sign changes, this is another advantage of aiililhclit -variates simulations.

A comparison of the crude Monte Carlo estimator (()) to the antithetic-variates estimator (()) is provided in fable 9.0. for most of the .simulations, the ratio of the standard error of (()) lo the standard error of (()) is 0.0000/0.0105=0.400, a reduction of about 00%. In comparison, a doubling of the number ol replications from in to 2m for the crude Mottle Carlo estimator would yield a ralio of 1/ч/2=0.707, only a 20% reduction. More formally, observe from (0.4.7) and (0.4. lb) that the ratio of the standard error of H (0) to the standard error of (0) is an estimator of,/(l-f p)/2, hence the ratio 0.0006/0.0105=0.400 implies a correlation of -08% between the antithetic pairs of the simulations in Table 0.6, a substantial value which is responsible for the dramatic reduction in variance of (0).

9.7.5 Extensions anil Limitations >

The Monte Carlo approach lo pricing path-dependent options is (pule general and may be applied lo virtually any European derivative security, for clxainplc, to price average-rale foreign currency options we would simulate price paths as above (perhaps using a different stochastic process more appropriate for exchange rates), compute the average for each replication, r-peal this many times, and compute the average across the replications.

Thus the power of the Cox-Ross risk-neutral pricing method is consider- able. However, there are several important limitations to this approach that s mtild be emphasized.

First, the Monte Carlo approach may only he applied to European options, options that cannot he exercised early. The early exercise feature of American options introduces the added complication of determining an optimal exercise policy, whit h must be done recursively using a dviiainic-programming-like analysis. In such cases, numerical solution of the corresponding IDE is currently die only available method for obtaining prices.

j Second, to apply the Cox-Ross technique lo a given derivative security, wk* must fust prove thai the security can be priced by arbitrage considerations alone. Recall that in the Hlack-St holes framework, the no-arbitrage condition was sufficient to completely determine the option price only because we were able to construct a dynamic portfolio of stocks, bonds, and

options that was riskless. In effect, this implies that the option is spanned by slocks and bonds or, more precisely, the options payoff al dale 7 can be perfectly replicated by a particular dynamic trading strategy involving only stocks and bonds. The no-arbitrage condition translates into the requirement that the option price must equal ihe cost ol the dynamic trading strategy that replicates the options payoff.

lint there are situations where the derivative .security cannot be replicated by any dynamic strategy involving existing securities. For example, if we assume that the diffusion parameter a in (0.2.2) is stochastic, then it may be shown lhat without further restrictions on a there exists no nondegenerate dynamic trading strategy involving stocks, bonds, and options that is riskless. 1 letn istically, because there are now two sources of uncertainty, the option is no longer spanned by a dynamic portfolio of stocks ami bonds (see Section 0.4.0 and Huang [ 1002 for further discussion).

Therefore, before we can apply die risk-neutral pricing method to a particular derivative security, we must fust check thai il is spanned by other traded assets. Since Goldman, Sosin, and Gatto (1070) demonstrate that the option to sell al the maximum is indeed spanned, we can apply the Cox-Ross method lo that case with the assurance that the resulting price is in fact the no-arbitrage price and that deviations from this price necessarily imply riskless profit opportunities. Hut il may be more difficult lo verify spanning for more complex path-dependent derivatives. In those cases, we may have to embed the security in a model of economic equilibrium, with.specific assumptions about agents preferences and their investment opportunity sets as, for example, in the stochastic-volatility model of Section 0.3.(i.

9.5 Conclusion

The pricing of derivative securities is one of the unqualified successes of modern economies, it has changed the way economists view dynamic models of securities prices, and it has had an enormous impact on the investment community. 1 he creation of ever more complex financial instruments has been an important stimulus for academic research and for the establishment of a bona fide financial engineering discipline. Recent innovations in derivative securities include: average rale options, more general lookback options, barrier options (also known as down and otil or birth and death options), compound options, dual-i ui rent v or dual-equity options, synthetic convertible bonds, spread-lock interest rale swaps, rainbow options, and many other exotic securities. In each of these cases, closed-form pricing formulas are available only for a very small set of processes for the underlying assets price, and a great deal of further research is needed to check whether such processes actually lit the data. Moreover, in many of

v. iJrrwulwr Pricing Models

these rases, analytical expressions lor hedging positions in these securities do not exist and must also he determined empirically.

There arc many unsettled issues in the statistical inference of continuous-time processes with discretely sampled data. Currently, the most pressing issue is the difficulty in obtaining consistent estimates of the parameters of lto processes with nonlinear drill and/or diffusion coefficients. For many lto processes of interest, we do nol have closed-form expressions for their transition densities and hence maximum likelihood estimation is not feasible. The GMM approach of I lansen and Scheinkman (1995) may be the most promising alternative, and empirical applications and Monte Carlo studies are sure to follow.

Another area of active current research involves developing better models of fundamental asset price dynamics. For example, casual empirical ol>-servalion suggests the presence of jump components in asset prices that are responsible for relatively large and sudden movements, but occur relatively infrequently and are therefore considerably more challenging to estimate precisely.Indeed, there is even some doubt as lo whether such jump processes can ever be identified from tliscretely sampled price data since the very act ol disc rete-sampling destroys the one clear distinction between diffusion processes and jump processes-the continuity of sample paths.

The difficulties in estimating parametric models of asset price dynamics have led to several attempts to capture the dynamics nanparametrically. For example, by placing restrictions on the drift coefficient of a diffusion process, Ait-Sahalia (1993) proposes a nonparametric estimator of its diffusion coefficient and applies this estimator lo the pricing of interest rate options, l.ongslalf < 1 *>*>£ ) proposes a lest of option-pricing models by focusing on ihe risk-neutral distribution implicit in option prices. And Hutchinson, l.o, and Poggio (МММ) attempt to price derivative securities via neural network models. Although it is still too early lo tell if these nonparametric and highly data-intensive methods will offer improvements over their parametric counterparts, the preliminary evidence is quite promising. In Chapter 12, we review some of these techniques and present an application to the pricing and hedging of derivative securities.

Closely related lo die issue of stock price dynamics are several open questions regarding die pricing of options in incomplete markets, markets in which the sources of uncertainty affecting the fundamental asset are not spanned by traded securities. For example, if the volatility of the fundamental assets price is stochastic, it is only under the most restrictive set of assumptions thai die price of an option on such an asset may be determined by arbitrage arguments. Since there is almost universal agreement

-% <, Im- <\.iuiili. Hall .mil Tomus (lW:t. 1КГ>). Мстит (I.l7l>h) develops an opiion-pi icing tin mill.i lor i omliiiuil ililliision/jiimp processes. See also Mellon (H)7lia) lor more general disi iission ol i lie imparl ol misspei ilving slock price dynamics on die pi icing ol options.

Problems

that volatilities do shift over time in random fashion, it is clear that issues regarding market incompleteness are central to the pricing of derivative securities.

In this chapter we have only touched upon a small set of issues that surround derivatives research, those that have received the least attention in the extant literature, with the hope that a wider group of academics and investment professionals will be encouraged to join in the fray and quicken the progress in this exciting area.

Problems-Chapter 9

9.1 Show that the continuous-time process p (t) of Section 9.1.1 converges in distribution to a normally distributed continuous-time process p(t) by calculating the the moment-generating function of p (t) and taking limijts.

9.2 Derive (9.3.30) and (9.3.31) explicitly by evaluating and inverting ihe Fisher information matrix in (9.3.7) for the maximum likelihood estimators ft and ex2 of the parameters of a geometric Brownian motion based pn regularly sampled data. j

9.3 Derive the maximum likelhood estimators Д, r>2, and у of the parameters of the trending Ornstein-Uhlenbeck process (9.3.46), and calculate their asymptotic distribution explicitly using (9.3.7). How do these three estimators differ in their asymptotic properties under standard asymptot cs and under continuous-record asymptotics?

9.4 You are currently managing a large pension fund and have invested most of it in IBM stock. Exactly one year from now, you will have to liquidate your entire IBM holdings, and you are concerned that it may be an inauspicious time to sell your position. CLM Financial Products Corporation hjas come to you with the following proposal: For a fee to be negotiated, they will agree to buy your entire IBM holdings exactly one year from now, but at a price per share equal to the maximum of the daily closing prices over the one-year period. What fee should you expect in your negotiations with CLM? Specifically:

9.4.1 Estimate the current (time 0) fair market price Я(0) of the option to sell at the maximum using Monte Carlo simulation. For simplicity, assume lhat IBMs stock price P(t) follows a geometric Brownian motion (9.2.2) so that

log - Л,1>(/2-/,).<Л 2-<*.)). (9.5.1)

and use daily relurns of IBM stock over the most recent five-year period to estimate the parameters ц and cr2 to calibrate your simulations. Assume