Asian Options

17 Asian Options 17.1 INTRODUCTION (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: r Average price options with payoffs for call and put of max[0, Av T − X ] and max[0, X − Av T ] r Average strike options with payoffs max[0, S T − Av T ] and max[0, Av T − 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 Av 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: r We may be pricing an option that started at some time τ in the past and the averaging may already have started. r The option may be only partly Asian, i.e. the averaging does not start until some time τ 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=0 S n . If the averaging does not start until n = ν (deferred averaging of the last 17 Asian Options 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 n are 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 S 0 if averaging started now. An alternative type of average, defined by G N ={S 0 × S 1 ×···×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 = ln G N S 0 = 1 N + 1 N  n=0 ln S n = 1 N + 1 N  n=0 x n The x n are normally distributed (see Section 3.1) and we know that the sum of normal random variables is itself normally distributed; the distribution of g N is 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 A 20 = 103.95 and G 20 = 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 n are 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 C av (T ) − P av (T ) = Av N − X Taking risk-neutral expectations and present valuing gives C av (0) − P av (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 EAv N  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. 17.2 GEOMETRIC AVERAGE PRICE OPTIONS (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 ] = σ 2 δT Underlying price: S n Geometric average price: G n x n = ln S n S 0 g n = ln G n G 0 E[x n ] = mT E[g n ] = m g T var[x n ] = σ 2 T var[g n ] = σ 2 g T E[S n ] = S 0 e (m+ 1 2 σ 2 )T = F 0T E[G n ] = G 0 e (m g + 1 2 σ 2 g )T g n is 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: r S N → G N ; x N → g N r σ 2 → σ 2 g (17.1) r µ risk neutral = (r − q) → µ g = m g + 1 2 σ 2 g (17.2) 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 → S 0 e (r−q)T ; so equation (17.2) is equivalent to the substitution F 0T → S 0 e (m g + 1 2 σ 2 g )T when using the Black Scholes model. Alternatively, we can describe the substitutions in the Black Scholes model as r S 0 → S 0 ; σ 2 → σ 2 g ; q → q g = r − m g − 1 2 σ 2 g (17.3) It just remains to work out expressions for m g and σ 2 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 S 0 = 1 S 0 {S 0 × S 1 ×···×S N } 1 N +1 =  S 0 S 0 × S 1 S 0 ×···× S N S 0  1 N +1 Take logarithms of both sides: g N = 1 N + 1 N  n=0 x n (17.4) Using the analysis and notation of Section 3.1(ii) we can further write x n = ln S n S 0 = ln S 1 S 0 × S 2 S 1 ×···× S n S n−1 = r 1 + r 2 +···+r n 203 17 Asian Options All the r i are independently and identically distributed with E[r i ] = mδT and var[r i ] = σ 2 δT . 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[r 1 + r 2 +···+r n ] = 1 N + 1 N  n=0 E[nr] = mδT 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] = σ 2 δT (N + 1) 2 N  n=0 n 2 The following two standard results of elementary algebra N  n=1 n = 1 2 N (N + 1); N  n=1 n 2 = 1 6 N (N + 1)(2N + 1) are used to give m g T = E[g N ] = 1 2 mNδT = 1 2 mT σ 2 g T = var[g N ] = σ 2 N δT 3 (2N + 1) (2N + 2) = σ 2 T 3 (2N + 1) (2N + 2) (17.5) 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)]: r E[x N ] = mT while E[g N ] = 1 2 mT. It is no surprise that the expected growth of the average is half the expected growth of the underlying. r lim N→∞ var[g N ] = σ 2 T /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 begins ν time steps from now. Equation (17.4) becomes g N = 1 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 S 0 = ln S ν S 0 × 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 + 1 E  N  n=ν { x ν + r ν+1 + r 2 +···+r n }  = E[x ν ] + 1 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 2 mδT (N + ν) (17.6) The corresponding expression for variance is given by var[g N ] = var[x ν ] + σ 2 δT (N − ν + 1) 2 N −ν  n=1 n 2 = σ 2 δT  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 2 m(T + τ); var[g N ] = σ 2 3  T + 2τ − T − τ 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 ×···×S 0 ×···× S N } 1 N +ν+1 = ¯ G ν N +ν+1 {S 0 × S 1 ×···×S N } 1 N +ν+1 where ¯ G ={S −ν×S −ν+1 ×···× S −1 } 1/ν is the geometric mean of those past stock prices which have already been achieved. Using the methods of the last two subparagraphs, we have g N = ν 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 2 mT (17.9) σ 2 g T = var[g N ] = σ 2 T 6 (N + 1)(2N + 1) (N + ν + 1) 2 (17.10) In continuous time these are written m g T = τ ¯ g + 1 2 mT T + τ ; σ 2 g T = σ 2 T 3  T T + τ  2 (17.11) 205 17 Asian Options 17.3 GEOMETRIC AVERAGE STRIKE OPTIONS The payoff of a call option of this type is max[0, S T − 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): C GAS = S 0 e −qT N[d 1 ] − S 0 e −q g T N[d 2 ] (17.12) d 1 = 1  g √ T  ln e −qT e −q g T + 1 2  2 g T  ; d 2 = d 1 −  g √ T  2 g T = σ 2 T + σ 2 g T − 2cov[x T , g T ]; q g = r − m g − 1 2 σ 2 g Expressions for m g and σ 2 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): g N = 1 N − ν + 1 N  n=ν x n ; x n = r 1 + r 2 +···+r ν +···+r n and cov[r i , r j ] =  σ 2 δTi= j 0 i = j to give cov[x N , x n ] = nσ 2 δT (n ≤ N ). Then cov[x N , g N ] = 1 N − ν + 1 cov  x N , N  n=ν x n  = σ 2 δT N − ν + 1 N  n=ν n = σ 2 δT N + ν 2 This last may be written σ 2 (T + τ )/2 in continuous time, and using equation (17.8) for σ 2 g gives  2 g = σ 2 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 + 1 cov  N  n=0 x n , x N  = σ 2 δT N − ν + 1 N  n=0 n = σ 2 δT N (N + 1) 2(N + ν + 1) Once again, the last expression can be written as σ 2 T 2 /2(T + τ ) which yields the slightly more complicated result  2 g = σ 2  T 2 3(T + τ ) 2 + τ T + τ  17.4 ARITHMETIC AVERAGE OPTIONS: LOGNORMAL 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. r Although the distribution of A N is ill-defined, exact expressions can be derived for the individual moments, i.e. E[ A λ N ] with integer λ. 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. r It 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 testing or calibrating 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 and σ 2 g as defined in the last section, we merely reproduce the geometric average results. r Vorst’s method assumes the distribution of A n is exactly lognormal, but applies a correc- tion term E[A n ] − E[G n ] to the strike price. r The 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 a equals the exact mean of the arithmetic average. r Levy’s correction goes one step further than (4) by assuming that both the variance and the mean of the lognormal distribution assumed for A n are 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 17 Asian Options (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, A N − 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)isnot 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 - d N A N - X G N Figure 17.1 Vorst approximation 1. The equation can be summarized as max[0, A N − X ] − max[0, G N − X ] ≤ A N − G N Taking present values of risk-neutral expectations gives C A (0, X) ≤ C G (0, X) + e −rT EA N − 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 ] → S 0 e (m g + 1 2 σ 2 g )T ;E[A N ] is derived in Appendix A.13. 2. Equation (17.13) can be manipulated to a slightly different form: max[0, A N − X ] =    0: A N < X; G N < X A N − X: A N > X; G N < X G N − (X − δ N ): A N > X; G N > X where δ N = A N − G N . This payoff is illustrated in Figure 17.1. Recalling that δ 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) ≈ C G (0, X δ ); X δ = X − Eδ 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 /S 0 ) ∼ N (m a T,σ 2 g T ), where a N = ln(A N /S 0 ). It follows that E[A N ] = S 0 e (m a + 1 2 σ 2 g )T or m a T = ln E[A N ] S 0 − 1 2 σ 2 g (17.15) Expressions for σ 2 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,σ 2 a T ), i.e. the distribution of the arithmetic mean is lognormal; but this time we calculate σ 2 a from first principles rather than just approximating it by σ 2 g . Equation (A1.8) of the Appendix shows that E[ A λ N ] = S λ 0 e (λm a + 1 2 λ 2 σ 2 a )T where λ is an integer. Therefore we may write  m a + 1 2 σ 2 a  T = ln E[A N ] − ln S 0  m a + σ 2 a  T = 1 2  ln E[ A 2 N ] − ln S 2 0  These are solved for m a and σ 2 a using the expressions for the moments of A N given by equations (A13.11)–(A13.13). The value of an arithmetic average price call option can then be written C Levy A = e −rt {EA N N[d 1 ] − X N[d 2 ]} (17.16) d 1 = 1 σ a √ T  ln EA n  X + 1 2 σ 2 a T  ; d 2 = d 1 − 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: C AAS = S 0 e −qt N[d 1 ] − S 0 e −q a T N[d 2 ] (17.17) d 1 = 1  a √ T  ln e −qT e −q a T + 1 2  2 g T  ; d 2 = d 1 −  a √ T  2 a T = σ 2 T + σ 2 a T − 2 cov[ln S T , ln A T ]; q a = r − m a − 1 2 σ 2 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 ]e cov[ln S T ,ln A N ]T A formula for each of the expected values is given in Appendix A.13. 17.5 ARITHMETIC AVERAGE OPTIONS: EDGEWORTH 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 17 Asian Options 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 E 1 + 1 2! ∂ 2 l(A N ) ∂ A 2 N E 2 − 1 3! ∂ 3 l(A N ) ∂ A 3 N E 3 + 1 4! ∂ 4 l(A N ) ∂ A 4 N E 4 (17.18) where δκ n = κ f n − κ 1 n E 1 = δκ 1 ; E 2 = δκ 2 + (δκ 1 ) 2 ; E 3 = δκ 3 + 3δκ 1 δκ 2 + (δκ 1 ) 3 E 4 = δκ 4 + 4δκ 1 δκ 3 + 3(δκ 2 ) 2 + 6(δκ 1 ) 2 δκ 2 + (δκ 1 ) 4 The cumulants may be obtained from κ 1 = E[x] = µ; κ 2 = E  (x − κ 1 ) 2  = σ 2 κ 3 = E  (x − κ 1 ) 3  ; κ 4 = E  (x − κ 1 ) 4  − 3κ 2 2 The expectations of the powers of A N (moments) corresponding to κ f n are given in Appendix A.13; the moments corresponding to κ 1 n (i.e. lognormal) are given by the standard formula E[ A λ N ] = S λ 0 e (λm a + 1 2 λ 2 σ 2 a )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 )tohave 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 1 and κ f 2 = κ l 2 so that E 1 = 0 and E 2 = 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  2πσ 2 a T exp  − 1 2  ln(A N /S 0 ) − m a T σ a √ T  2  where m a and σ 2 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 C TW A = C Levy A + e −rT  − 1 3! ∂l(A N ) ∂ A N E 3 + 1 4! ∂ 2 l(A N ) ∂ A 2 N E 4  A N =X (17.19) 210 [...]... could equally have been described as an expectation over all x and y (and over all p and q and r as well for that matter!) This is written Ex [u(x)] = +∞ +∞ dx −∞ u(x) f (x, y) dy −∞ 211 (17.22) 17 Asian Options Comparing the last two equations gives the general relationship f (x | y) = f (x, y) f (y) (17.23) (ii) Conditioning is now applied with A N in place of x and G N in place of y: E A N [max[0,... and must therefore make a simplifying approximation The simplest approximation is X C1 = − E A N [max[0, X − A N ] | G N ]l(G N ) dG N 0 (17.27) X ≈− max[0, E AN X − A N | G N ]l(G N ) dG N 0 213 17 Asian Options In general, the effect of switching the max[·, ·] and E[·] operators is equivalent to switching between the value of a European put option and its asymptotes, as E[max [0, X - ST]] shown in... 24.10 24.24 23.58 24.02 24.10 19.38 18.02 19.00 19.34 19.50 19.26 19.35 19.37 15.44 14.26 15.06 15.40 15.57 15.63 15.49 15.47 Note: Results taken from Levy and Turnbull (1992) and Curran (1992) 215 17 Asian Options A Monte Carlo method has been used to obtain the “right answer” which is quoted with an indication of its accuracy (1 standard deviation) The numbers really speak for themselves but a few comments... = ln Sn ; S0 gn = ln Gn ; S0 zx = xn − mtn ; √ σ tn zg = gN − m g T √ σg T where tn is the time from now to the point when the stock price is Sn The correlation coefficient 212 17.6 ARITHMETIC AVERAGE OPTIONS: GEOMETRIC CONDITIONING between z x and z g may now be written cov[xn , g N ] √ √ σ tn σ g T We will write this correlation as ρn for short The covariance term may be explicitly derived by using... different from normal due to a high level of skewness and kurtosis But the computation significantly increases in complexity as none of the terms of equation (17.18) now drop out 17.6 ARITHMETIC AVERAGE OPTIONS: GEOMETRIC CONDITIONING This is a different approach from that taken in the last two sections, and gives better results over a wider range In fact, the answers come out so close to the Monte Carlo... papers quoted in the footnote In-progress results are less interesting in the sense that they carry the dead weight of the “average-until-now” and therefore have less optionality than deferred average options; results for in-progress averaging are therefore excluded Furthermore we only quote results for the rather extreme volatility of 50% For a volatility of 30%, errors in the answers obtained by the... methods are of the order of one-third of the errors obtained at 50%; at a volatility of 10%, the answers obtained by the different methods are effectively all the same Prices are given for two different options in Table 17.1 Both are on a non-dividend-paying stock with a volatility of 50%; the interest rate is 9% and averaging takes place at weekly intervals over 1 year The “Simple Averaging” option runs... starting now; the “Deferred Averaging” option is similar but has the averaging start in 20 weeks and then run for 52 weeks, i.e it is a 72-week option Table 17.1 Comparison of models for arithmetic average options Simple Averaging Strike Price Stock Price = 100 Deferred Averaging Strike Price 90 Monte Carlo (±0.03) Geometric Vorst Modified geometric Levy T&W (elementary) T&W (higher moments) Geometric conditioning...17.6 ARITHMETIC AVERAGE OPTIONS: GEOMETRIC CONDITIONING (iv) In the last subsection, we defined the approximating distribution l(A N ) as having mean and variance equal to the true mean and variance of A N (always assuming of... designed as an improvement on Levy Turnbull and Wakeham is much improved if we match skewness and kurtosis [see Section 17.5(iv)], although it is still inadequate for in-the-money deferred average call options 216 . ARITHMETIC AVERAGE OPTIONS: LOGNORMAL SOLUTIONS (i) The analysis of the last two sections on geometric Asian options is satisfyingly elegant; but Asian options encountered. families of Asian options to consider: r Average price options with payoffs for call and put of max[0, Av T − X ] and max[0, X − Av T ] r Average strike options