A Quick Algorithm for Pricing European Average OptionsAuthor(s): Stuart M. Turnbull and Lee MacDonald WakemanSource: The Journal of Financial and Quantitative Analysis, Vol. 26, No. 3 (Sep., 1991), pp. 377-389Published by: University of Washington School of Business AdministrationStable URL: http://www.jstor.org/stable/2331213Accessed: 27/05/2010 13:59

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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL 26, NO 3, SEPTEMBER 1991

A Quick Algorithm for Pricing European Average

Options

Stuart M. Turnbull and Lee Macdonald Wakeman*

Abstract

An algorithm is described that prices European average options. The algorithm is tested

against Monte Carlo estimates and is shown to be accurate. The speed of the algorithm is comparable to the Black-Scholes algorithm. A closed-form solution is derived for

European geometric average options.

I. Introduction

In the over-the-counter market, many options are written on average prices. For example, the payoff at maturity for a one-year average sterling call option is

the maximum of zero and the difference between the arithmetic daily exchange rate, averaged over some prespecified period and the exercise price. Thus, the

value of the option depends upon the average price and not the spot price. There are many possible uses for such options. For corporations that receive

or pay foreign currency claims on a regular basis, a foreign currency option based on the average of the exchange rate represents one way to reduce its

average foreign currency exposure. Similar comments also apply to interest

rate instruments. Average options, by their design, reduce the significance of

the closing price at the maturity of the option. This reduces the effects of any

possible abnormal price movements at the maturity of the option. Thus, average

options provide a way to ameliorate any possible price distortions that might arise because of a lack of depth in the market of the underlying asset.

The pricing and hedging of these options raises some interesting issues.

First, the value of the option depends upon the history of the asset price move?

ments over the averaging period. Thus, if the binomial tree approach is used

to price the option, it is necessary to keep track of 2n possible paths, where n

is the number of nodes. This becomes infeasible for large values of n unless

* School of Business, Queen's University, Kingston, Canada K7L 3N6 and MTK Global Capital Inc, 65 East 55th Street, New York, NY 10027. The paper was first written while Turnbull was visiting at the Australian Graduate School of Management. Helpful discussions with Ian Cooper, David Emanuel (of Emanuel and Macbeth Associates), Robert Kohn, Angelo Melino, and especially Edmond Levy are gratefully acknowledged. The authors also thank JFQA Managing Editor Jonathan Karpoff.

377

378 Journal of Financial and Quantitative Analysis

some approximation method is used. Second, if the asset price follows a log? normal distribution, the arithmetic average will not be log-normally distributed.

Standard option models, such as Black-Scholes, rely upon the assumption of

log-normality. Third, once the maturity of the option is less than the averag-

ing period, the hedging properties of the option can change quite dramatically. There is a practical need for a model that can quickly and accurately price such

options and determine the required hedge ratios such as delta, gamma, vega, and theta. Monte Carlo methods offer an obvious solution possibility, yet these

methods are relatively slow.

The objective of this paper is to describe a quick way to price European

average options. While it is very difficult to determine the probability distri?

bution for the average, all of its moments can be readily determined. Thus, an

Edgeworth series expansion can be used to approximate the distribution.

The basic model is described in Section II. The first part of this section

prices average options when the maturity of the option is greater than the aver-

aging period. The accuracy of the approximation is tested against Monte Carlo

estimates. The second part of this section addresses the case in which the ma?

turity of the option is less than or equal to the averaging period. Again, the

accuracy of the approximation is tested against Monte Carlo estimates. It is

demonstrated that when the maturity of the option is less than the averaging pe? riod, the price of an average value option can be greater than that of a standard

European option, contrary to what Kemna and Vorst (1990) assert. Some of the

characteristics of the delta ratio also are examined. A closed form solution for

European options written on the geometric average is described in Section III, and a comparison of arithmetic and geometric option prices is presented. It is

shown that, for some cases, the two sets of prices can be relatively close while, for other cases, there can be substantial differences. A summary is given in

Section IV.

II. The Option Model

Consider a call option of maturity T, written on the average of the past n

stock prices.1 It is initially assumed that the maturity of the option is greater than the averaging period (T > n).

A. Assumptions

Al. No transaction costs, no differential taxes, no borrowing or lending restric?

tions, and trading takes place continuously;

A2. The term structure of interest rates is flat and nonstochastic;

A3. It is assumed that the stock price S is described by a log-normal probability distribution,

(1) dS = Sadt + SOdZ,

1 The average is defined using stock prices at T ? n + 1,..., T.

Turnbull and Wakeman 379

where dZ is a standard Wiener process whose increments are uncorrelated; a is

the constant instantaneous mean; and 02 is the constant variance of the instan-

taneous rate of return.

Let A(t) denote the average at time t, which is defined by

(2) A(t) = [S(r) + S(r-1)+ +S(t-(n-l))]/n,

where S(t) is the stock price at time t, and it is assumed that the averaging is

done with daily stock prices. Weekends and holidays are ignored, though these

complications could easily be incorporated at the cost of additional notation.

Assumption A3 implies that the stock price is log-normally distributed, so that

(3) S(t) = S(t-l)exv[a-e2/2+0Y],

where Y ~ N(0,1). Thus, the average is the sum of log-normally distributed

random variables. Given that the log-normal distribution is not stable, the dis?

tribution of the average is not log-normal. Let C(A(t);T,K,n) denote the value of a European call option written on

the average. The option matures at time T; K is the exercise price; and n is the

number of prices included in the average. At maturity, the value of the option is

C(A(T),T,K,n) = max{A(7) -K,0}.

To value this option, the risk-neutral argument of Cox and Ross (1976) is used.

Under the equivalent distribution, relative prices follow a martingale and the

expected returns on the stock will be the risk-free rate of interest, r, implying that a = r in Equation (l).2 Given such an adjustment, the value of the option is

(4) C(A(t)',T,K,n) = exp[-r(T- t)]Et[A(T) -K\A(T) > K],

where Et is the expectation operator with respect to the equivalent probability

distribution, given the stock price at time t.

To evaluate the above expression, it is necessary to determine the prob?

ability density function for the average, A(T). In general, this is difficult to

do, given that A(T) is the sum of correlated log-normally distributed random

variables. To circumvent this difficulty, an Edgeworth series expansion will

be used. The true distribution will be approximated with an alternative distri?

bution. Given the work of Mitchell (1968), the alternative distribution will be

assumed to be log-normal.3 If f(y) denotes the true probability function and aiy) the approximating distribution, where aiy) is a log-normal probability density

function, then

/c, f/ , , , , c2 d2a(y) c3 d3a(y) c4 d4a(y) (5) fiy) =

^)+57-^--3T^r +

?!-^- + ^

2For foreign exchange options, the mean would be set such that a = r - r/, where r/ is the foreign exchange rate.

The question of convergence is discussed in detail in Mitchell (1968).

380 Journal of Financial and Quantitative Analysis

where c2 = X2(F)~Xi(A); c3 = X3(F)~X3(A); c4 = X+(F)~X+(A)+?>c22, Xj(F)[Xj(A)] is the 7th cumulant of the exact [approximating] distribution; and e(y) is a resid?

ual error term. If a random variable Y has a cumulative distribution function F, the first four cumulants are

Xx(F) = E(Y), X2(F) = E[Y-E(Y)]2,

Xi(F) = E[Y-E(Y)]\ X4(F) = E[Y - E(Y)f - 3E[Y - E(Y)]2,

where all expectations are with respect to the distribution F. (See Kendall

and Stuart (1977), pp. 67-73). The first two moments of the approximating distribution have been set equal to the first two moments of the exact distribution.

The moments of a random variable Y with respect to the a(y) distribution are

given by

cr2 o E(Ym) = expQL/m + -m2), m = 1,2,... .

To apply an Edgeworth expansion, it is necessary to determine the cumulants

of the distribution for the average, A(T). While it is difficult to determine the

probability density function for A(T), all of its moments can easily be determined

using a recursive relationship. Given the stock price at time t? 1, the stock price at time t can be written in the form,

(6) S(t) = S(t-l)Rt,

where Rt is the price relative, and from (3), it is log-normally distributed. Re~

peated use of (6) in Equation (2) implies that the average can be written in the

form,

(7) A(T) = S[T-(n-\W+Ri+i+Ri+iRi+2+ ? ? ?

+Ri+xRi+2 ? ? ? Ri+n-X]/n,

where / = 7' ? (w ? 1). Let Li+n = 1; and Li+j = 1 + Ri+jLi+j+x,j = 1, ? ? ? , n - 1, so that

(8) A(T) = S(i)Li+x/n.

But to value the option at t = 0, the stock S(i) is random. Let /?/ denote the

price relative from t = 0 to t = i, so that

(9) A(T) = S(0)RiLi+x/n.

From (9), the option will be exercised if

(10) Y = RiLi+l > k,

where k = nK/S(0). The mth moment of Y is given by

(11) E(Ym) = E(R?)E(L?+1),

Turnbull and Wakeman 381

given the Assumption A3 of independence. (See Boyle (1976).) The value of

E(Lf+\) can be determined from

E(L?+j) = E[(\+Ri+jLi+j+l)m],

j = l,...,n? 1. Thus, all the cumulants of the exact distribution can be calcu?

lated.

The Edgeworth series expansion is applied to (10), so that if *F = E(Y -

k)\Y>k), then

r 00 T =

J, {y~ (12) Y = (y-k)f(y)dy

J.

\ i\ , \j , C2 n\ c3da(k) c4d2a(k) (y - k)a(y)dy + -a(k) - - ? + - _ + *(*),

(see Jarrow and Rudd (1982), Equation (12)). The first term on the right-hand side of (12) can be writtten in the form

j; iy-k)aiy)dy = exp(jU + CT2/2)N(dx)-kN(d2),

where d\ = [?\n(k) + [X + <X2]/<X;d2 = di - <X. While this is similar to the

expression for Black-Scholes, it differs in an important way. In applying the

risk-neutral argument, the expected rate of return for the stock is set equal to the

risk-free rate of interest. However, this is not the case for the expected value of

the arithmetic averaging process, implying that the mean [X is not the risk-free

rate, due to the discrete nature of the average.

B. Accuracy

The accuracy of the approximation can be tested by pricing the option using the antithetic Monte Carlo method, as described by Boyle (1977) and Rubinstein

(1981). The accuracy of the approximation is examined in Table 1. In Part A, the instantaneous standard deviation of the rate of return is 20 percent per year and, in Part B, 30 percent per year. The maturity of the option is assumed to

be 120 days and the averaging period is 30 days. The reported Black-Scholes

values are derived using the standard deviation of the rate of return on the stock

and provide a measure of the effects of averaging. It is seen that, in all cases, the

error from using the approximation is very small. Similar results (not reported) are obtained using different averaging periods.

C. Pricing in the Averaging Period

Up to the present point, it has been assumed that the maturity of the option is greater than the averaging period. This assumption is now relaxed and it

is assumed that T < n. Given that the average is defined over n days, then

(n ? T) prices have been observed where, for convenience, T is treated as an

integer. Three important implications follow. First, the effect of averaging is to

reduce the variance of the terminal distribution, which will lower option prices.

382 Journal of Financial and Quantitative Analysis

TABLE 1

Accuracy: Maturity Greater than Averaging Period. The Approximation Is Calculated Using Equation (12).

Spot Price 100 Maturity 120 days Interest Rate 9 percent Averaging Period 30 days (Figures in parentheses are the standard errors for the Monte Carlo estimates)

Second, the effective exercise price is altered as maturity decreases. Consider a call option, which will be exercised if

A(T) [F + P]/n > K,

where F denotes the sum of future prices; and P denotes the sum of observed

prices. This can be written in the form,

F/n > K-P/n = EK,

where EK is defined to be the effective exercise price. The effective exercise

price changes as maturity decreases. The expected payoff at the maturity of a call option is

E[A(T)-K\A(T)>K] = E[F/n-EK\F/n>EK].

The Edgeworth series expansion can be applied to evaluate the above term.

Turnbull and Wakeman 383

The third implication is that the effective exercise price can be negative. If this happens, then it implies that a call option will be exercised for certain

at maturity. When the effective exercise price becomes negative, then even if

all future prices are zero, the average at maturity will still be greater than the

true exercise price. This also implies that a put option will be worthless.4 It

should be noted that this contradicts a result in Kemna and Vorst (1990) who

argue that the price of an average-value option will always be lower than that

of a standard European option. This is not correct if the maturity of the option is less than the averaging period.

D. Accuracy

The accuracy of the algorithm is tested against Monte Carlo estimates. To

calculate the average, it is assumed that all past prices have remained constant

at 100. The averaging period is assumed to be 120 days. Thus, if the maturity is 90 days, the first 30 past prices are assumed to be 100. The results are shown

in Table 2. In Part A, the volatility of the rate of return on the stock is 0.20

per year. It is seen that the approximation does an excellent job. In Part B, the volatility of the rate of return is 0.30 per year. Similar results are obtained,

though for deep in-the-money call options, the accuracy slightly deteriorates.

An interesting result is obtained for put options when the exercise price is

105. The put option price decreases and then increases as maturity decreases, and finishes in-the-money. Consider the option with one day left to maturity. The option will be exercised if

^-[5(1)+119* 100J/120 > 0,

where S(l) is the stock price at maturity. If K is 105, then the above can be

written

5.83-S(l)/120 > 0.

Given the spot price of the stock is 100, the probability of exercise is almost

unity. When the maturity is 120 days, no stock prices have been observed that

contribute to the average. Thus, the option is like an ordinary option and its

price declines as the variance of the terminal distribution declines, given that

maturity is decreasing. When the maturity is 90 days, however, the probability of the option being in-the-money at maturity has increased, given that the first

30 prices are 100. This effect will increase the price of the option. A priori, no

statement can be made about which effect will dominate.

The hedging properties of these options depend in part upon whether the

maturity of the option is greater or less than the averaging period. If the ma?

turity of the option is less than the averaging period, then the properties of the

4Suppose that at some point before maturity the effective exercise price becomes negative, so that EK = K-P/n < 0. At maturity, the value of a call option is [F + P]/n -K > 0, for all values of F, given that F > 0. Thus, a call option will be exercised for certain at maturity if the effective exercise price becomes negative. This is not the case, of course, for a standard European option. At maturity, the value of a put option is K - [F + P]/n = [K- P/n] - F/n < 0, as K - P/n < 0 and F > 0. The option is worthless. ?

384 Journal of Financial and Quantitative Analysis

option will depend upon the path of stock prices inside the averaging period, as

demonstrated in the last section.

In Table 3, the effects upon the option price and hedge ratio are examined

as the number of daily observations used to calculate the average is changed. The maturity of the option is 15 days. First, consider the case of 28 observations

used to calculate the average. For this case, the maturity of the option is clearly less than the averaging period. It is assumed that past prices have remained at

the spot rate when calculating the effective exercise price. For the call option with an exercise price of 100, there is a dramatic difference in the price and

hedge ratio when 28 observations are used for the average, compared to the

Turnbull and Wakeman 385

other two cases. The reason for this difference is that when 28 observations are

used to calculate the average, given the maturity is 15 days, 13 observations that

contribute to the average have already been observed. When 14 observations are

used to calculate the average, the maturity of the option is still greater than the

averaging period. For the call option with exercise price 95, there is very little

change in the price, as the option is deep in-the-money. However, the hedge ratio is reduced by nearly a half when 28 observations are used to calculate the

average. Thirteen observations that contribute to the average have already been

observed and, thus, the option is less sensitive to changes in the spot price.

TABLE 3

Increasing the Number of Observations in the Average Price and Hedge Ratios

!!!. Geometric Average

Consider a call option of maturity T written on the geometric average of

the past n stock prices. It is initially assumed that the maturity of the option is

greater than or equal to the averaging period (T>n). The geometric average is

defined by

(13) G(T) = [S(T - n + 1). S(T - n + 2)... .S(T)] \/n

Substituting (6) into the above expression gives, after simplification,

(14) G(T) = S(T-n)[RnT_n+l .Rnf-n+i--- ^r]1/n.

Given Assumption (A3), G(T) is log-normally distributed. Let exp(x) = G(T),

where x ~ N(mx, (J2). Using (6) and (3) then, under the risk-neutral distribution,

386 Journal of Financial and Quantitative Analysis

TABLE 4

Comparison of Arithmetic and Geometric Average Option Prices

Spot Price Maturity Interest Rate Averaging Period

100 120 days 9 percent 60 days

mx = ln[5(0)] + (r - 02/2)(T - n) + (r - 02/2) J j/n,

(15a) 7=1

and CTA2 = 02(T-n) + 02^J2/n2. 7=1

It is now assumed that the maturity of the option is less than the averaging period (T < n). Suppose that n0 prices have been observed, while the remaining n{(= n-n0) prices are random, where n\ is also the maturity of the option. The

geometric average can be written in the form,

G(n{) = [S(-n0 + l). S(-A20 + 2)...S(0). 5(1). S(n{)]

= G.S(0p/n[RTRn2~l-.Rnl]{/n>

1//1

where G = [S(-n0 + l)S(-n0 + 2).. .5(0)]1//7. For this case, Equation (15a) becomes

(15b)

mx = \n(G) + (nl/n)ln[S(O)] + (r-02/2)^j/n 7=1

and CTA2 = 02^j2/n2.

7=1

Turnbull and Wakeman 387

Theorem: European Geometric Average Options. a) The price of a European call option is given by

C(0) = B(T)[exp(mx + (T2/2)N(d)-KN(d-(Tx)],

where K is the exercise price; axd = - ln(K) + mx + a2;mx and (T2 are defined

by (15); and B(T) = exp(-rjT). b) The price of a European put option is given by

P(0) = B(T)[KN(-d + ax)-exp(mx + (T2/2)N(-d)].

Proof. The proof follows from using (4) and the theorem given in Smith

(1976). ?

TABLE 5

Pricing in the Averaging Period Arithmetic Price / Geometric Price

388 Journal of Financial and Quantitative Analysis

For the case of the maturity being greater than or equal to the averaging

period, the above expression can be written in a more familiar form. For a call

option,

C(0) = 5(0) exp(n)N(d) - KB(T)N(d - crx),

where Y] = -T02/2 - (r - 02/2)(n - l)/2 + a2/2; and (Jxd = ln[S(0)/KB(T)] +

r] + cr2. If n = 1, the above expression becomes the Black-Scholes formula for

a call option.

It is instructive to compare the prices of options written on arithmetic and

geometric averages. In Table 4, prices are compared for options that have a maturity of 120 days and an averaging period of 60 days. In Part A, the standard deviation is 0.20 per year. The results are very similar. In Part B, the

standard deviation is 0.30 per year. The increase in the standard deviation drives

a wedge between the two sets of prices, with the geometric average consistently

underpricing call options and overpricing puts. A similar result is seen in Table 5. The averaging period is set at 120 days and, when the maturity of the option is less than the averaging period, all past prices are assumed to be 100. It is

seen that, as the maturity of the option increases, the differences between the

two sets of prices increase.

It might be concluded from Tables 4 and 5 that, for "short" maturities and

"low" variance of return, the geometric average option provides a good estimate

for the price of an arithmetic option. If the maturity of the option is less than the averaging period, however, then care must be exercised as these options are

path dependent. Consider an option of 60-days maturity and an averaging period of 120

days, implying that 60 prices have been observed. Instead of assuming that all

Turnbull and Wakeman 389

past prices are fixed at 100, as was assumed in Table 5, it is assumed in Table

6 that, over the first 60-day period, prices go through a complete cycle, so that

the arithmetic average over the 60-day period is 100. It is seen that, even for a

standard deviation of 0.20 per year, substantial pricing differences can arise.

IV. Summary

An algorithm has been developed for pricing European arithmetic options. The algorithm was tested against Monte Carlo estimates and was found to be

accurate. The speed of the algorithm is similar to that of the Black-Scholes

algorithm. The algorithm can easily be extended to price average options when

nonconsecutive prices are used, and to price foreign currency average options. A closed form solution was derived for pricing European geometric options. From a comparison of arithmetic average and geometric average option prices, two conclusions can be drawn. First, if the maturity of the option is greater than the averaging period and if the standard deviation of the rate of return is

"small" and the option's maturity "short," then the results are similar.5 Second, if the maturity of the option is less than the averaging period, then there can be

substantial differences in the prices of the two types of options.

References

Boyle, P. P. "Rates of Return as Random Variables." Journal of Risk and Insurance, 43 (Dec. 1976), 694-711.

_"Options: A Monte Carlo Approach." Journal of Financial Economics, 4 (May 1977), 323-338.

Boyle, P. P, and D. Emanuel. "Options and the General Mean." Working Paper, Accounting Group, Univ. of Waterloo (July 1982).

Cox, J., and S. Ross. "The Valuation of Options for Alternative Stochastic Processes." Journal of Financial Economics, 3 (Jan./March 1976), 145-166.

Jarrow, R., and A. Rudd. "Approximate Option Valuation for Arbitrary Stochastic Processes." Journal of Financial Economics, 10 (Nov. 1982), 346-369.

Kemna, A. G. Z., and A. C. F. Vorst. "A Pricing Method for Options Based on Average Asset Values." Journal of Banking and Finance, 14 (March 1990), 113-129.

Kendall, M., and A. Stuart. The Advanced Theory of Statistics, Vol. 1, 4th ed. New York: Macmillan (1977).

Mitchell, R. L. "Permanence of the Log-Normal Distribution." Journal of the Optical Society of America, 58 (Sept. 1968), 1267-1272.

Rubinstein, R. Y. Simulation and the Monte Carlo Method. New York: John Wiley & Sons (1981).

Smith, C. "Option Pricing: A Review." Journal of Financial Economics, 3 (Jan./March 1976), 3-52.

5 This does suggest the geometric average option model could be used as a control variate in a Monte Carlo simulation to estimate arithmetic average option prices. See Kemna and Vorst (1990). However, such an approach would be a lot slower than using the algorithm described in this paper.