Asian Options

Compare the following two strategies: i the exporter buys a set of put options with strike X, maturing on each day of the month; or ii he buys a put option on the average exchange rate o

17 Asian Options

(i) Consider a company which is exporting goods continuously rather than in large chunks Equal payments are received daily in foreign currency and the exporter decides to hedge these for the next month by buying put options Compare the following two strategies: (i) the exporter

buys a set of put options with strike X, maturing on each day of the month; or (ii) he buys a

put option on the average exchange rate over the month with payoff max[0, X − Av] Clearly, the payoff of package (i) is greater than that of package (ii): simply imagine two

successive days on which the exchange rates are X − 10 and X + 10 The combined payoff

for two separate puts would be 10 while the payoff for the put on the average price is zero

An exporter is more likely to be interested in the average exchange rate rather than the rate on individual daily payments He would therefore execute the cheaper strategy (ii) above Options in which the underlying asset is an average price are known as Asian options They are of most interest in the foreign exchange markets, although they do appear elsewhere, e.g savings products whose upside return is related to the average of a stock index In the following,

we will continue to use the vernacular of equity derivatives

(ii) Average Price and Average Strike: There are two families of Asian options to consider:

rAverage price options with payoffs for call and put of

max[0, AvT − X] and max[0, X − AvT]

rAverage strike options with payoffs

max[0, ST − Av T] and max[0, AvT − S T] Average price options are more intuitively interesting and more common, although the two types are obviously closely related Both will be examined in this chapter

(iii) In-progress and Deferred Averaging: The underlying “asset” in an Asian option is A v T, the

average price up to maturity at time T This is not a tangible asset that can be delivered at

the expiry of an option, so these options are cash settled, i.e an amount of cash equal to the mathematical expression for the payoff is delivered at maturity

It very often happens that the averaging period does not run from “now” to time T Two

cases need to be considered:

rWe may be pricing an option that started at some timeτ in the past and the averaging may

already have started

rThe option may be only partly Asian, i.e the averaging does not start until some timeτ in

the future

(iv) Definition of Averages: The average of a set of prices is most simply defined as A N =

(N+ 1)−1"N

n=0S n If the averaging does not start until n = ν (deferred averaging of the last

subsection, then A N = (N − ν + 1)−1"N

n =ν S n Averaging could be calculated daily, weekly

or whatever is agreed This is the arithmetic average and is used for most option contracts The

various S nare lognormally distributed, but there is no simple way of describing the distribution

of A N Note that by convention, the average includes the price on the first day of averaging,

e.g it would include today’s price S0if averaging started now

An alternative type of average, defined by G N = {S0× S1× · · · × S N}1/(N+1), is called

the geometric average For deferred start averaging, G N = {S ν × S ν+1 × · · · × S N}1/(N−ν+1).

Taking logarithms of both sides gives

g N = lnG N

S0 = 1

N+ 1

N

n=0

ln S n = 1

N+ 1

N

n=0

x n

The x nare normally distributed (see Section 3.1) and we know that the sum of normal random

variables is itself normally distributed; the distribution of g Nis therefore normal

(v) Geometric vs Arithmetic Average: We have the unfortunate situation where options

in the market are all written on the arithmetic average while pricing is only easy for the geometric average Perhaps there is a simple bridge to get from one to the other?

On the right is a set of 20 daily prices of a commodity with a volatility of about 22%

The arithmetic and geometric averages are A20 = 103.95 and G20 = 103.92, which

are surprisingly close given the very different mathematical forms of the two averages Given the simplifying assumptions of option theory and the uncertainty surrounding volatility, one is tempted to say that these results are close enough to be taken as being the same

Could these two averages have come out close by accident? There is a mathematical

theorem which states that G N ≤ A N always; equality occurs if all the S nare identical The commodity prices in our list are close in size so the averages are close Now consider two series in which the numbers being averaged are much more variable, or equivalently stated, prices which are much more volatile:

1

20{1 + 2 + · · · + 20} = 10.5; {1 × 2 × · · · × 20}1

20 = 8.3

100 102 99 100 102 101 103 104 103 104 107 106 107 105 104 107 108 106 107 104

Even in this case, which is more extreme than price series generally encountered in finance, the difference is only about 25% It is frustrating to have the arithmetic and geometric results

so close, and we describe below how theoreticians have been prompted to devise schemes in which an arithmetic option is regarded as a geometric option plus a correction factor

(vi) Put–Call Parity: Before turning to various explicit models, it is worth pointing out that put–call

parity works for European Asian options – strange terminology, but meaning an average option with no payout before final maturity permitted For either a geometric or an arithmetic average price, we may write

Cav(T ) − Pav(T ) = Av N − X

Taking risk-neutral expectations and present valuing gives

Cav(0)− Pav(0)= e−rT {E[Av N]− X}

202

17.2 GEOMETRIC AVERAGE PRICE OPTIONS

which gives the price of an Asian put option in terms of the price of the corresponding call option Expressions for EAvN with different averaging periods can be calculated exactly for both geometric and arithmetic averages This result means that we can focus on call options,

as the put price follows immediately from the above formula

(i) Use of Black Scholes Model for Geometric Average Options: The notation of Chapter 3 is

used and extended for simple or deferred geometric averaging as follows:

r n= ln S n

S n−1 E[r n]= mδT var[r n]= σ2δT Underlying price: S n Geometric average price: G n

x n = lnS n

S0

g n = lnG n

G0

var[x n]= σ2T var[g n]= σ2

g T E[S n]= S0e(m+1σ2)T = F 0T E[G n]= G0e(m g+ 1σ2)T

g nis normally distributed so we can take over the whole barrage of the Black Scholes model

to price geometric average price options, using the following substitutions:

rS N → G N; x N → g N

rσ2→ σ2

rµrisk neutral= (r − q) → µ g = m g+1

2σ2

Remember, the term (r − q) only appears in the Black Scholes model as an input into the calcu-lation of the forward rate F 0T → S0e(r−q)T; so equation (17.2) is equivalent to the substitution

F 0T → S0e(m g+ 1σ2)T when using the Black Scholes model Alternatively, we can describe the substitutions in the Black Scholes model as

rS0→ S0; σ2→ σ2

g; q → q g = r − m g−1

2σ2

It just remains to work out expressions for m gandσ2

g The form of these depends on the precise averaging period being considered and will be given in the next three subsections

(ii) Simple Averaging: We first consider geometric averaging from now until maturity, i.e neither

deferred nor in progress Using previous definitions in this chapter:

G N

S0

= 1

S0

{S0× S1× · · · × S N} 1

N+1 = S0

S0

× S1

S0

× · · · × S N

S0

 1

N+1

Take logarithms of both sides:

g N = 1

N+ 1

N

n=0

Using the analysis and notation of Section 3.1(ii) we can further write

x n= ln S n

S0

= ln S1

S0

× S2

S1

× · · · × S n

S n−1 = r1+ r2+ · · · + r n

203

All the r i are independently and identically distributed with E[r i]= mδT and var[r i]= σ2δT When taking expectations or variances of x n we can therefore simply write each of the r i as r and put x n → nr.

E[g N]= 1

N+ 1

N

n=0

E[x n]= 1

N+ 1

N

n=0

E[r1+ r2+ · · · + r n]

N+ 1

N

n=0

E[nr ]= m δT

N+ 1

N

n=0

n

And similarly

var[g N]= 1

(N+ 1)2

N

n=0

var[x n]= 1

(N+ 1)2

N

n=0

var[nr ]= σ2δT

(N+ 1)2

N

n=0

n2

The following two standard results of elementary algebra

N

n=1

n =1

2N (N+ 1);

N

n=1

n2= 1

6N (N + 1)(2N + 1)

are used to give

m g T = E[g N]= 1

2m NδT = 1

2mT

σ2

g T = var[g N]= σ2NδT

3

(2N + 1)

(2N + 2) =

σ2T

3

(2N + 1)

It is interesting to compare these last two results with the analogous results for the logarithm

of the stock price x N [Section 3.1(ii)]:

rE[x N]= mT while E[g N]= 1

2mT It is no surprise that the expected growth of the average

is half the expected growth of the underlying

rlimN→∞var[g N]= σ2T /3 This is the “square root of three” rule of thumb for roughly

estimating the value of an Asian option from the Black Scholes model by dividing the volatility by√

3, which has long been used by traders

(iii) Deferred Start Averaging: The results of the last subsection need to be adapted if the averaging

period is not from “now” to the maturity of the option We assume that deferred start averaging beginsν time steps from now Equation (17.4) becomes

N − ν + 1

N

n =ν

x n

Using the same analysis as before, we can write (for n ≥ ν)

x n= ln S n

S0 = lnS ν

S0 ×S ν+1

S ν × · · · × S n

S n−1 = x ν + r ν+1 + r ν+2 + · · · + r n

204

17.2 GEOMETRIC AVERAGE PRICE OPTIONS

so that

E[g N]= 1

N − ν + 1E

# N

n =ν

{x ν + r ν+1 + r2+ · · · + r n}

$

N − ν + 1

N

n =ν+1

E[(n − ν)r]

Use E[x ν = E[νr] = νmδT and"N

n =ν+1 (n − ν) ="N −ν

n=1 n in the last equation to give

E[g N]= νmδT + mδT

N − ν + 1

1

2(N − ν)(N − ν + 1) = 1

2m δT (N + ν) (17.6) The corresponding expression for variance is given by

var[g N]= var[x ν + σ2δT

(N − ν + 1)2

N −ν

n=1

n2= σ2δT ν +1

6

(N − ν)(2N − 2ν + 1) (N − ν + 1)



(17.7)

In continuous time, with large N and setting N δT → T and νδT → τ, the last two equations

can be written

E[g N]= 1

2m(T + τ); var[g N]= σ2

2(N − ν + 1)

 (17.8)

(iv) In-progress Averaging: At some point in its life, every Asian option becomes an in-progress

deal The average then needs to be replaced by the average from now to maturity plus a non-stochastic past-average part The adaption is straightforward:

G N = {S −ν × S −ν+1 × · · · × S0× · · · × S N} 1

N +ν+1 = ¯G N +ν+1 ν {S0× S1× · · · × S N} 1

N +ν+1

where ¯G = {S −ν×S −ν+1 × · · · × S−1}1/νis the geometric mean of those past stock prices which

have already been achieved Using the methods of the last two subparagraphs, we have

N + ν + 1 g¯+

1

N + ν + 1

N

n=0

x n

Using the results of subsection (ii) above immediately gives

m g T = E[g N]= ν

N + ν + 1 ¯g+

N+ 1

N + ν + 1

1

σ2

g T = var[g N]=σ2T

6

(N + 1)(2N + 1)

In continuous time these are written

m g T =τ ¯g +

1

2mT

T + τ ; σ g2T = σ2T

3

T

T + τ

2

(17.11)

205

17.3 GEOMETRIC AVERAGE STRIKE OPTIONS

The payoff of a call option of this type is max[0, ST − G T]; but this is an option to exchange one lognormal asset for another, and can be priced by Margrabe’s formula [equations (12.1)] using the substitutions of equation (17.3):

CGAS= S0e−qT N[d1]− S0e−q g T N[d2] (17.12)

d1= 1

 g

√

T ln

e−qT

e−q g T +1

22

g T



; d2= d1−  g

√

T

2

g T = σ2T + σ2

g T − 2 cov[x T , g T]; q g = r − m g−1

2σ2

g

Expressions for m gandσ2

g corresponding to different types of averaging were derived in the last section Now we just need to derive an expression for the covariance term

(i) Deferred Start Averaging: Recall the results from Section 17.2(iii):

N − ν + 1

N

n =ν

x n; x n = r1+ r2+ · · · + r ν + · · · + r n

and

cov[r i , r j]= σ02δT i i = j

to give cov[x N , x n]= nσ2δT (n ≤ N) Then

cov[x N , g N]= 1

N − ν + 1cov

#

x N ,

N

n =ν

x n

$

= σ2δT

N − ν + 1

N

n =ν

n = σ2δT N + ν

2 This last may be writtenσ2(T + τ)/2 in continuous time, and using equation (17.8) for σ2

g

gives

2

g =σ2 3

T − τ T

(ii ) In-progress Averaging: Using the notation of Section 17.2(iv), cov ¯g, x N = 0 so that

cov[g N , x N]= 1

N + ν + 1cov

# N

n=0

x n , x N

$

= σ2δT

N − ν + 1

N

n=0

n = σ2δT N (N+ 1)

2(N + ν + 1)

Once again, the last expression can be written asσ2T2/2(T + τ) which yields the slightly

more complicated result

2

g = σ2 T2

3(T + τ)2 + τ

T + τ



SOLUTIONS

(i) The analysis of the last two sections on geometric Asian options is satisfyingly elegant; but Asian options encountered in the market are arithmetic, and there are no simple Black Scholes

206

17.4 ARITHMETIC AVERAGE OPTIONS: LOGNORMAL SOLUTIONS type solutions for these It was observed in Section 17.1(v) that arithmetic and geometric averages are surprisingly close in value, so that a natural approach is to seek an arithmetic solution expressed as a geometric solution plus a correction term

The arithmetic average A N has the following properties:

r A N is the sum of a set of correlated, lognormally distributed random variables S N, and does

not have a simply defined distribution

rAlthough the distribution of A N is ill-defined, exact expressions can be derived for the

individual moments, i.e E[ A λ N] with integerλ Expressions for these moments in terms of

observed or calculable parameters (volatility of the underlying stock, number of averaging points, risk-free rate, etc.) are given in Section A.13 of the Appendix

rIt has been observed in several fields of technology that the sum of lognormally distributed

variables can be approximated by a lognormal distribution, under a fairly wide range of conditions

These observations lead to various approximation methods There seems to be a bewildering array of these, but the most important ones are closely related We have included what we consider the most important approaches and a route map of the subject follows

1 Monte Carlo: The arithmetic average option problem is ideally suited for solution by

the Monte Carlo methods using the geometric average price as the control variate [see Section 10.4(iii)] These can achieve any degree of accuracy we please just by extend-ing calculation times They are therefore ideal tools for testextend-ing or calibratextend-ing some faster algorithm to be used for real-life situations

2 Exactly Lognormal Models: All methods explained in the next two sections exploit the

fact that the arithmetic average is at least approximately lognormal If we assume exact lognormality with the defining parameters m g andσ2

g as defined in the last section, we merely reproduce the geometric average results

rVorst’s method assumes the distribution of A nis exactly lognormal, but applies a

correc-tion term E[ A n]− E[G n] to the strike price

rThe simple modified geometric also assumes that the distribution is exactly lognormal;

it assumes that the variance is the same as for the geometric average, but it assumes that

the mean m aequals the exact mean of the arithmetic average

rLevy’s correction goes one step further than (4) by assuming that both the variance and

the mean of the lognormal distribution assumed for A nare equal to the calculated variance and mean of the arithmetic average Note that the mean and variance are now exactly correct, although the assumption of lognormality may be in error

3 Approximately Lognormal Models: In Section 17.5, we drop the assumption of exact

log-normality and merely assume the distribution of A n can be approximated by a lognormal

dis-tribution Correction terms to the results of the present section (particularly Levy’s method) are obtained in terms of an infinite but diminishing series of observable or calculable terms

4 Geometric Conditioning: In Section 17.6, we examine a very successful method due to

Curran, which makes no explicit assumptions about the form of the distribution of A N It

is more awkward to implement than a simple formula, but it is probably the recommended approach at present, giving very accurate answers over a wide range

207

(ii) Vorst’s Method: (Vorst, 1992) Let C G and C A be the values of a geometric and an arith-metic average price call option Given that the payoffs of these options at maturity are

C A (T, X) = max[0, A N − X] and C G (T, X) = max[0, G N − X], and also given the general result mentioned in Section 7.1 that G N ≤ A N , we have a lower bound for C A(0, X):

C G(0, X) ≤ C A(0, X) This is fairly obvious, but Vorst has also established an upper bound If G N ≤ A N then we can write

max[0, AN − X] − max[0, G N − X] =







0: A N < X; G N < X

A N − X: A N > X; G N < X

A N − G N: A N > X; G N > X (17.13) Note that a fourth possible combination on the right-hand side ( A N < X; G N > X) is not included because G N ≤ A N Two interesting results are derived from equation (17.13):

N

A - X max[0, A N - X]

X

X - dN

A N - X

G N

Figure 17.1 Vorst approximation

1 The equation can be summarized as

max[0, AN − X] − max[0, G N − X] ≤ A N − G N

Taking present values of risk-neutral expectations

gives

C A(0, X) ≤ CG(0, X) + e−rTEAN − G N

This gives an upper bound on the value of C A(0, X)

in terms of calculable quantities: C G(0, X) is the

subject of Section 17.2; the lead-in to equation

(17.3) shows that E[G N]→ S0e(m g+ 1σ2)T ; E[ A N]

is derived in Appendix A.13

2 Equation (17.13) can be manipulated to a slightly different form:

max[0, AN − X] =







A N − X: A N > X; G N < X

G N − (X − δ N): A N > X; G N > X

whereδ N = A N − G N This payoff is illustrated in Figure 17.1 Recalling thatδ N is always small, the stepped part of the payoff might be approximated by the diagonal dotted line, prompting the following (Vorst’s) approximation for an arithmetic average call option:

C A(0, X) ≈ CG(0, Xδ); X δ = X − Eδ N (17.14)

(iii) Simple Modified Geometric: Let us assume that A N is lognormally distributed with the

same volatility as G N, i.e.σ g We can write this as a N = ln(A N /S0)∼ N(m a T, σ2

g T ), where

a N = ln(A N /S0) It follows that

E[ A N]= S0e(m a+ 1σ2)T or m a T = lnE[ A N]

S0 −1

2σ2

Expressions forσ2

g under various averaging scenarios were derived in Section 17.2 The

corre-sponding expressions for E[ A N] are derived in Appendix A.13 The net effect of this approach is

208

17.5 ARITHMETIC AVERAGE OPTIONS: EDGEWORTH EXPANSION

to use the geometric average model but substitute the known risk-neutral drift of the arithmetic average

(iv) Levy Correction: (Levy, 1992) This is a logical step forward from the last sub section This time

we assume that a N ∼ N(m a T , σ2

a T ), i.e the distribution of the arithmetic mean is lognormal;

but this time we calculateσ2

a from first principles rather than just approximating it byσ2

g

Equation (A1.8) of the Appendix shows that E[ A λ N]= S λ

0e(λm a+ 1λ2σ2)Twhereλ is an integer.

Therefore we may write



m a+1

2σ2

a



T = ln E[A N]− ln S0



m a + σ2

a



T = 1 2



ln E[ A2

N]− ln S2

0



These are solved for m aandσ2

a using the expressions for the moments of A Ngiven by equations (A13.11)–(A13.13) The value of an arithmetic average price call option can then be written

CLevyA = e−rt {EA N N[d1]− X N[d2]} (17.16)

d1 = 1

σ a

√

T ln

EA n

2σ2

a T



; d2= d1− σ a

√

T

(v) Arithmetic Average Strike Options: Using Levy’s model, we assume that the arithmetic average

is lognormally distributed, so that the analysis is very similar to that given for geometric average strike options in Section 17.3 Again, the price of this option is given by Margrabe’s formula:

CAAS= S0e−qt N[d1]− S0e−q a T

d1 = 1

 a

√

T ln

e−qT

e−q a T +1

22

g T



; d2= d1−  a

√

T

2

a T = σ2T + σ2

a T − 2 cov[ln S T , ln A T]; q a = r − m a−1

2σ2

a

The term cov[ln S T , ln A N] can be calculated from equation (A1.23) which can be written as follows:

E[ A N S T]= E[A N ]E[S T]ecov[ln ST,ln AN ]T

A formula for each of the expected values is given in Appendix A.13

EXPANSION

(i) It is well known that a function can be expressed by a Maclaurin’s (Taylor’s) expansion as follows:

f (x + δx) = f (x) +∞

n=1

1

n!

∂ n f (x)

∂x n (δx)n

It is less well known that if a probability density function f ( A N) is approximated by another

distribution l( A N), then we may write

f ( A N)= l(A N)+∞

n=1

(−1)n

n!

∂ n l( A N)

∂ A n N

E n

This is called the Edgeworth expansion and is derived in Appendix A.14

209

We assume that the true distribution of A N has a true density function f ( A N) which is

close to but not identical to the lognormal distribution l( A N); the analytical form of the latter

is known so that the derivatives can be evaluated explicitly The terms E n are functions only

of the differences between the various cumulants under the true distribution of A N, and the corresponding cumulants under the lognormal distribution Furthermore, the cumulants

them-selves are functions only of the moments of A N which may be calculated explicitly for both the true distribution and for the lognormal distribution All terms on the right-hand side of the Edgeworth expansion are therefore calculable in principle

(ii) We will restrict ourselves to an investigation of the effects of higher moments up to the fourth term in the Edgeworth expansion:

f ( A N)= l(A N)−∂l(A N)

∂ A N

E1+ 1 2!

∂2l( A N)

∂ A2

N

E2− 1 3!

∂3l( A N)

∂ A3

N

E3+ 1 4!

∂4l( A N)

∂ A4

N

E4

(17.18) where

δκ n = κ f

n − κ1

n

E1= δκ1; E2= δκ2+ (δκ1)2; E3= δκ3+ 3δκ1δκ2+ (δκ1)3

E4= δκ4+ 4δκ1δκ3+ 3(δκ2)2+ 6(δκ1)2δκ2+ (δκ1)4

The cumulants may be obtained from

κ1= E[x] = µ; κ2= E(x − κ1)2

= σ2

κ3= E(x − κ1)3

; κ4= E(x − κ1)4

− 3κ2 2

The expectations of the powers of A N (moments) corresponding to κ f

n are given in Appendix A.13; the moments corresponding toκ1

n (i.e lognormal) are given by the standard

formula E[ A λ N]= S λ

0e(λm a+ 1λ2σ2)T which was encountered in connection with Levy’s method

in the last section

When we say that l( A N) is a lognormal distribution, this is clearly not enough to define the

distribution: we must, for example, specify the mean and variance Let us select l( A N) to have

mean and variance equal to the true mean and variance of A N, i.e set these parameters in the same way as for the Levy method Then by definition,κ f

1 = κ l

1andκ f

2 = κ l

2so that E1 = 0

and E2= 0

This is known as the Turnbull–Wakeham method (Turnbull and Wakeham, 1991) The only remaining inputs which have not been given explicitly are the two partial differentials But

l( A N) is just the lognormal probability density function

l( A N)= 1

A N

 2πσ2

a T exp

#

−1 2

ln( A N /S0)− m a T

σ a

√

T

2$

where m aandσ2

a are defined in Section 17.4(iv)

(iii) Using this last expansion and equation (A14.8), we can now write for the value of an arithmetic average

CTWA = CLevy

A + e−rT −1

3!

∂l(A N)

∂ A N

E3+ 1 4!

∂2l( A N)

∂ A2

N

E4



A N =X

(17.19)

210