of nonlinear functions of the data-generating process arc almost impossible to obtain in discrete time, but in continuous time Itos differentiation rule gives an explicit expression for such dynamics. Theoretical insights into the equilibrium structure of derivatives prices-for example, which slate variables affect derivatives prices and which do not-arc also more readily obtained in a continuous-time framework such as Cox, Ingersoll, and Ross (1985b). Therefore, each set of models oilers some valuable insights that are not contained in the other.

; 9.4 Pricing Path-Dependent Derivatives Via Monte Carlo Simulation

Consider a contract at date 0 that gives the holder the right but nol the obligation to sell one share of slock at date V for a price equal to the maximum of lhat stocks price over the period from 0 to 7. Such a contract, often called a lookback option, is clearly a put option since il gives the holder the option to sell at a particular price at maturity. However, in this case the strike price is stochastic and determined only al the maturity date, because the strike price depends on the path that ihe stock price takes from 0 to 7 , and not just on the terminal stock price P(T), such a contract is called a path-dependent option,

Path-dcpcndeni options have become increasingly popular as the hedging needs of investors become ever more complex. For example, many multinational corporations now expend great efforts to hedge against exchange-rate fluctuations since large portions of their accounts receivable and accounts payable arc denominated in foreign currencies. One of the most popular path-dependent options are foreign currency average rate or Asian options which gives the holder the right to buy foreign currency at a rate equal lo the average of the exchange rates over the life of the contract.2

Path-dependent options may be priced by the dynamic-hedging a>-proach of Section 9.2.1, but the resulting PDF is often intractable, flic risk-neutral pricing method offers a considerably simpler alternative in which the power of high-speed digital computers may be exploited. For example, consider the pricing of the option lo sell at the maximum. If l(t) denotes the date / stock price and (()) is die initial value of this put, we have

/7(0) = e~V:

Max l(l) - P(T)

ll< l<T

(9.4.1)

The term Asian routes Iron i ihe tact thai smelt options were lirsl actively wri Men on slocks blading on Asian exchanges. Because these exchanges are usually smaller than their Km орган tmi American counterpart, with relatively thin trading and low daily volume, prices on such xchanes are somewhat easier to manipulate. To minimize an options exposure to the risk if stock-price manipulation, a new option was created with the average of the stock prices over he options life playing the role ol the terminal siock price.

Max Id) n i i

r v: i/ivii

(9.1.2)

Max /(/)

/(()).

(9.4.3)

where F. is the expectations operator with respect to the i isk-neuli al probability distribution or equivalent martingale measure.

()bserve that in going from (9.4.2) In (9.4.3) we have used the fact thai die expected present value of /(7) discounted al die riskless rale r is /(()). I his holds because we have used the risk-neutral expectations operator K, and under die risk-neutral probabilities implicit in K* all assets must earn an expected return of r\ hence e F.*(7) = /(0).

To evaluate (9.4.3) via Monte Carlo simulation, wc simulate many sample paths of (/(/), find die maximum value for each sample path, or replication, and average die present discounted value of I he maxima over all the replications to yield an expected value over all replications, i.e., an estimate of (0). Two issues arise immediately: How do we simulate a continuous sample path, and how many replications do we need for a reasonably precise estimate of (0)?

9.4.1 Discrete \W:w/.i Continuous lime

by their very nature, digital computers are incapable of simulating truly continuous phenomena; bin as a practical matter they are often capable of providing excellent approximations. In particular, if we divide our lime interval [0. I into n discrete intervals each of length , and simulate prices

at each discrete date kh, Ii = 0.....the result will be an approximation

lo a continuous sample path which can be made successively more precise by allowing n lo grow and lo shrink so as lo keep 7 fixed.

For example, consider die case of geometric brownian motion (9.2.2) for which the risk-neutral dynamics are given bv

dP*(I) - rl {l)dl + ciP{l)dll(l). (9.4.4)

and consider simulating the following approximate sample path /*:

Il = /*(()) exp

X>v<>

;,( ) - M{rh.rT-h). (9.4.5)

Despite the fact that the simulated path / varies onlv al multiples of . die approximation may be made arbitrarily precise bv increasing n and therefore decreasing h-as n increases without bound, /* converges weakly to (9.4,4) (see Section 9.1,1 lor further discussion). Unfortunately, there are no general rules for how large n must be lo vield an adequate approximation-(boosing и lllitsl be done on a case-bv-case basis.

9.1.2 IImu Мину Simulations lo Perform

We can, however, provide some dear guidelines for choosing [he number of replications m to simulate. Recall lhat our Monte Carlo estimate of (()) involves a simple average across replications:

/7(0) = ,.!V V. -/-(0), y, = Max/ ,* (9.4.0)

III -f (!<*< *

where (P*k j is the /lb replication or sample path of the stock price process under the risk-neutral distribution which, in the case of (9.4.4), implies that / = -y. But since by construction the 1 /s are IID random variables with finite positive variance, the Central Limit Theorem implies that for large m:

y/lii (/7(0) - ))) ~ Af((). a-{ )), a;(n) = \av[rrT Yj ]. (9.4.7)

Therefore, for large m an approximate 95% confidence interval for 7/(0) may he readily constructed from (9.4.7):

Pr (())---A- < (()) < (()) +-~-\ = 0.95.

V s i у/т I

(9.4.8)

The choice of m thus depends directly on the desired accuracy of /7(0). If, for example, we require a /7(0) thai is within $0,001 of (()) with 95% confidence, m must be chosen so that:

1.9onrv(/i) / j 06 \-

--- < 0.001 => w > --- rj-(>i). (9.4.9)

x/w V 0.001 / 1

Typically Vaij )) ] is not known, but it can be readily estimated from the simulations in the obvious way:

1 l

var[i; i = -£< ; - y )\ y-Yy ,. (9.4.10)

Since the replications are IID by construction, estimators such as (9.4.10) will generally be very well-behaved, converging in probability lo their expectations rapidly and, when properly normalized, converging in distribution just as rapidly to their limiting distributions.

9.1. 1 Comparisons willi a Closed-Form Solution

In the special case of the option to sell al the maximum with a geometric Brownian motion price process, a closed-form solution for the option price

is given by Goldman, Sosin, and Gatto (1979):

P(0)

(()) = Р(())е-ГФ al

( al\\ ( (a4-o-2)7Y

(9.4.11)

where a = r - a/2.

Therefore, in this case we may compare the accuracy of the Monte Carlo estimator /7(0) with the theoretical value 7/(0). Table 9.6 provides such a comparison under the following assumptions (for simple returns):

Annual Riskfree Interest Rate = 5%

Annual Expected Stock Return = 15%

Annual Standard Deviation of Stock Return = 20%

Initial Stock Price P(0) = $40

Time to Maturity 7 = 1 Year. j

From the entries in Table 9.6, we see that large differences between dije continuous-time price 7/(0) = $4.7937 and the crude Monte Carlo estimator /7(0) can arise, even when rn and n are relatively large (the antithetic estimator is defined and discussed in the next section). For example, H(0) and /7(0) differ by 30 cents when n = 250, a nontrivial discrepancy given the typical sizes of options portfolios.

The difference between /7(0) and Я(0) arises from two sources: sampling variation in (0) and the discreteness of the simulated sample paths of prices. The former source of discrepancy is controlled by the number of replications m, while the latter source is controlled by the number of observations n in each simulated sample path. Increasing m will allow us to estimate E*[/7(0)] with arbitrary accuracy, but if n is fixed then E*[tf(0)] need not converge to the continuous-time price H(0). Does this discrepancy imply that Monte Carlo estimators are inferior to closed-form solutions when such solutions are available? Not necessarily. j

This difference highlights the importance of discretization in the pricing of path-dependent securities. Since we are selecting the maximum ovefc A exponentials of the (discrete) partial sum Xw=i r*> where к ranges from 0 to ii, as и increases the maximum is likely to increase as well.21 Heuristically,

-Although ii is probable thai the maximum of the partial sum will increase with n, it is not guaranteed. As we increase и in table M.ti. we generate a new independent random sequence I and ihere is always some chance that this new sequence with more terms will nevertheless yield smaller partial sums.

Table 9.6. Mimic C.inln estimation nj limhhmU option jniie.

Monlc Carlo cstiinalni nl I lie* price ul ,i onc-vcai Inok-haek put option with i otiliiuioiis-liuic t.oliluuui-Sostu-C.atto price C(ll=$-1.70:17. Farh row corresponds tu an iiHlejienilenl м-i ul siiiiiii.ilиms ol Kill,Will replications ol sample pallis ol length п. For the anlillielii - ,u iales simulations, each sequence ol III) random varialcs is useil twice-ilie original seiiieiiie anil its ncgame-yielding a total of 200.(100 sample paths, or IIHI.OIHI ncgalisely с orielaleil pairs ol pallis. SKI (0)1 and SKI/7(11) are ihe standard errors of IIW) and 0 . respei lively.

die maximum of the daily closing prices of 7over die year (n = 250 (railing days) imisl be lower tiian die maximum of die daily highs over dial same year (h- oo). Therefore, the continuous-lime price 77(0), which is closer

10 die maximum of the daily highs, will almost always exceed the simulation price 7/(0) which is discretized.

Which price is more relevant depends of course on the terms ol die particular contract. For example, average rale options on foreign exchange usually specify particular dales on which the exchange rale is measured, and

11 is almost always either a market closing rale (such as die corresponding spot rate of the 1MM futures closing) or a central bank fixing rale. In both cafes, the more relevant price would be die simulation price, since the- p.illt dependence is with respect to the discrete set of measured rales, and not an idealized continuous process.

i У.*/.-/ Coni/nilalionalEfficiency

II e two main concerns of any Monte Carlo simulation ate accuracy and сoinpiilalional cost, and in mosl cases there will be tradeoffs between the tw ). As we saw in Section 9.1.2, the standard error of die Monte Carlo estimator (()) is inversely proportional lo the square root of the number of replications in. hence a 50% reduction in the standard error requires four limes the number of replications, and so on. This type of Monte Carlo procedure is often described as civile Monlc Carlo (sec I laiiuiiersley

and 1 l.uidsc onib j 19(И for example), for obvious reasons. Therefore, a number of variance-reiliiclion techniques have been developed to improve the efficiency of simulation estimators. Although a thorough discussion of these techniques is beyond the scope of ibis text, we shall brielly review a few of llieni here.2

A simple technique for improving the performance of Monlc Carlo estimators is lo replace estimates by (heir population counterparts whenever possible, for this reduces sampling variation in the estimator. For example, when simulating risk-neutralized asset relurns, the sample mean of each replication will almost never be ecpial lo its population mean (the riskless rale), but we can correct this sampling variation easily by adding the difference between the riskless rate and the sample mean lo each observation of die- replication. If ibis is done for each replication, die result will be a set ol replications wilb no sampling error for die mean. The efficiency gain depends on the extent lo which sampling errors for die mean contributes to die overall sampling variation of the simulation. Inn in many cases the improvement can be dramatic.

A related technique is to exploit other forms of population information. For example, suppose we wish to estimate F./(-Y) and we find a random variable g(Y) such thai V?\g( >)] is close to l.l/(A) and E* ,tr( V)l is known (this is the population information to be exploited). F.*/(,\) might be the price of newly created path-dependent derivative which must be estimated, and K* [g())] the market price of an existing derivative with similar characteristics, hence a similar expectation. Ну expressing К* [/(Л)] as the sum of K*\g( Y)] and K*[/(.Y) - g(Y)\. the expectation to be estimated is decomposed into two terms where the lirsl term is known and the second term can be simulated with much smaller sampling variation. This technique is known as the control variale method-g()) is the control variate for /(.Y)-and its success depends on how close V.*[g())] is lo F,*(/ (A)].

Another form of population information lhat can be exploited is symmetry. If, for example, the population distribution is symmetric about its mean and this mean is known (as in the case of risk-neutralized asset returns), then mi/2 replications can yield in sample paths since each replication can

- Several tests provide excellent coverage of this material. I lamtnersley and I landscomh (HHil) is a classic, concise hut complete. Kalos and Whillock ( h)H(i) provide a more detailed and updated exposition of similar material. Fishman (IWti) is considerably more comprehensive and covers sevet al advanced topics not found in other Monte Carlo tests such as Markov chain sampling, Cihbs sampling, random lours, and simulated annealing. Iishuian (Il.lli) also contains many applications, explicit algorithms lor many ol the techniques covered, and FOKFRAN software (Irom an lip site) lor random number generation. Finally. Fang and Wang (I .I.l-l} present a compact introduction to a new approach to Monlc (i.u lo simulation based on purely deterministic sampling. Although il is still too early to tell hou this appio.u h compares lo the moil traditional methods. Fang and Wang (lO.ll) piosiilc some muiguing examples lb.il look quite promising.