An Asymptotic Expansion Approach to Computing Greeks

Transcript

1 An Asymptotic Expansion Approach to Computing Greeks Ryosuke Matsuoka and Akihiko Takahashi Abstract We developed a new scheme for computing Greeks of derivatives by an asymptotic expansion approach. In particular, we derived analytical approximation formulae for Deltas and Vegas of plain vanilla and average call options under general Markovian processes of underlying asset prices. We also derived approximation formulae for Gammas of plain vanilla and average call options, and for Deltas of digital options under CEVConstant Elasticity of Variance processes of underlying assets prices. Moreover, we introduced a new variance reduction method of Monte Carlo simulations based on the asymptotic expansion scheme. Finally, several numerical examples under CEV processes confirmed the validity of our method. This paper presents the authors personal views, and these are not the official views of the Financial Services Agency, the Financial Research and Training Center, and of Tokio Marine & Nichido. Tokio Marine & Nichido Fire Insurance Co., Ltd. Graduate School of Economics, The University of Tokyo, Special Research Fellow at the Financial Research and Training Center, Financial Services Agency 7

2 1 Introduction We propose a new approximation formulae for computing Greeks, such as the Delta, the Gamma and the Vega that indicate risk indices, the first or second order derivatives of the value of an asset with respect to its parameters. In particular, we derive analytic approximation formulae of the Delta and the Vega of plain-vanilla and average options where the underlying assets prices follow general diffusion processes. We also show formulae of the Gamma of plain-vanilla and average options, and a formula of the Delta of digital options where the underlying assets prices follow CEV processes. Our method is based on the asymptotic expansion approach developed by Takahashi[1995,1999, Kunitomo and Takahashi[199,1,3a and Takahashi and Yoshida[4. We also introduce a new variance reduction method as an extension of Takahashi and Yoshida[5 to increase efficiency of Monte Carlo simulation in computation of Greeks. Moreover, we presented series of numerical examples where the underlying prices follow the constant elasticity of variance CEV processes and showed effectiveness of our method. Monitoring and controlling the risks of derivative securities are as important as pricing derivative securities in the practical world. In Black-ScholesBS model, option prices and their Greeks are obtained analytically. However, in the more realistic models, it is usually very hard to evaluate both of option prices and their Greeks analytically. Then, numerical methods are applied. We suppose that the underlying asset s price S t follows a stochastic differential equationsde; { ds t S t dt + σ S t dw t 1 S s >, where W t is a one-dimensional Brownian motion, r q. r and q denote a risk-free rate and a divident rate respectively which are assumed to be constants. Next, we consider a derivative security whose payoff function at maturity time T is given by φ such as where S T 1 T φ S T S T K + φ ST ST K + plain vanilla call option average call option, S t dt. The price of the derivative security can be represented by us E [ e rt φ S T e rt E [φ S T, where E [ denotes the expectation operator under the risk-neutral measure. To obtain the first order derivative of the price with respect to the underlying asset s price at the initial price s, a natural method is computing u s u s + u s, where is a sufficiently small positive number, and u s and u s + are generated by Monte Carlo simulations. However, its convergence speed is sometimes very slow due to the irregularity of φ. Moreover, convergence to the true value may not be achieved because its convergent value depends on. To overcome the problem, we may utilize a representation such as u s e rt E [ φ S T Y T, 73

3 where Y t satisfies the following SDE: { dy t Y t dt + σ S t Y t dw t Y 1. For the derivation, see Imamura, Uchida and Takahashi [5 for instance. Although this representation makes Monte Carlo simulations more efficient, using Monte Carlo simulation itself may be time-consuming, which cause many difficulties in the practical world such as trading business. Therefore, in the case that analytic formulae can not be obtained, ideally we need analytic approximation schemes which can generate values precise enough for practical pupose. The asymptotic expansion approach have been applied successfully to a broad class of Itô processes appearing in finance. Takahashi[1999 presented a third-order pricing formula for plain options and secondorder formulae for more complicated derivatives such as average options, basket options, and options with stochastic volatility in a general Markovian setting. Kunitomo and Takahashi[1 provided pricing formulae for bond optionsswap options and average options based on an interest rate model in the class of Heath-Jarrow-Morton[199 which is not necessarily Markovian. Takahashi[1995 also presented a second order scheme for average options on foreign exchange rates with stochastic interest rates in Heath-Jarrow- Morton framework. Moreover, Takahashi and Yoshida[4 extended the method to dynamic portfolio problems. Recently, Takahashi and Saito[3 successfully applied the method to American options, and Takahashi and Yoshida[5 proposed a new variance reduction scheme of Monte Carlo simulation with the asymptotic expansion. For details of mathematical validity based on the Malliavin calculus and of further applications, see Kunitomo and Takahashi[3a, 3b, 4, Takahashi and Uchida[4 and Takahashi and Yoshda[4,5. The organization of this paper is as follows. In section and 3, we derive approximation formulae for the Delta of plain-vanilla and average call options. In section 4, we introduce a variance reduction technique in computation of the Delta. Sections 5 and 6 treat approximations for computing the Vega of plain vanilla and average call options. In section 7, we propose a variance reduction technique in computation of the Vega. In section 8, when the underlying prices follow CEV processes, we show another derivation of approximation formulae for the Delta and the Vega, and derive approximation formulae for the Gamma of plain vanilla and average options as well as for the Delta of digital options. We present numerical examples in section 9. Finally, some concluding remarks are made in section 1. The Delta of Plain Vanilla Call Option First, we make the following assumption and definitions. Assumption 1. We assume that the underlying asset price S t follows the SDE: { ds t S t dt + ɛσ S t dw t S s > where W t is 1-dimensional Brownian Motion, and. In addition we assume that σ is a C class function and its derivative σ is bounded. We define the differentiation of S t by the initial value s. 74

4 Definition 1. We define Y t as a stocastic process following the SDE: { dy t Y t dt + ɛσ S t Y t dw t Y 1 Definition. We define the defined indicator function: { 1 if x a 1 {x a} if x < a Then, the payoff of Plain Vanilla Call Option is rewritten as S T K + S T K 1 {ST K}. Takahashi [1999 derived, the asymptotic expantion of S t : where S t A t + ɛa 1t + ɛ A t + oɛ, 3 da t A t dt, A s Then, we can easily obtain A t, A 1t and A t as da 1t A 1t dt + σ A t dw t, A 1 da t A t dt + σ A t A 1t dw t, A. A t s e t A 1t A t t t e t s σ A s dw s e t s σ A s A 1s dw s. Now we put X t S t A t /ɛ, g 1 A 1T and g A T /. Then, X T g 1 + ɛ g + oɛ. The asymptotic expantion of the Plain Vanilla Call Option e rt E [ S T K + is given by e rt E [ S T K + ɛ e rt E [ y + X T + ɛ e rt E [ y + g 1 + ɛ g + 1 {g1 +ɛ g + y}, where y A T K /ɛ. Under an appropriate assumption such as Assumption 6. in Kunitomo and Takahashi[3b, We obtain its approximated value as e rt E [ S T K + ɛ e rt E [ y + g 1 1 {g1 y} + ɛ e rt E [ g 1 {g1 y} [ +ɛ e rt E y + g 1 g 1 {g1 y} + o ɛ 75

5 where 1 { y} denotes the derivative of 1 { y}. Next, by the property of Brownian Motion, we know the distribution of g 1 follows a normal distribution N,, where [ E e T t σ A t dw t In addition, the conditional expectation E [g g 1 x is given by E [g g 1 x cx + f, where e T t σ A t dt. 4 c 1 s and f c. Accordingly, e rt E [ S T K + e T s σ A s σ A s e T v σ A v dv ds ɛ e rt y + x n [x;, dx + ɛ e rt cx + f n [x;, dx + o ɛ, y y 5 where n [x;, is the density function of N,, i.e. n [x;, 1 π e x. Now we define an approximated value of the call option as C, T ɛ e rt y + x n [x;, dx + ɛ e rt cx + f n [x;, dx y ɛ e rt y N y y + n [y;, + ɛ e rt f y n [y;,, 6 where N x is cumulative distribution function of N, 1. Next, consider its differentiation by s : s C, T ɛ e rt d N +ɛ e rt f ɛ e rt d N y + y d y s s 1 yd s + y s + 1 n [y;, s 1 + yd y s s y + f d + f y s y +ɛ e rt f s y + f d + c y yd s + y s n [y;, n [y;,, n [y;, 76

6 where d y s e T /ɛ. The derivatives of coefficients are calculated as follows: c { 1 s s 3 s + 1 s s s s s s s e T t σ A t dt e T t e t σ A t σ A t dt e T t σ A t σ A t dt } e T s σ A s σ A s e T v σ A v dv ds e T s σ A s σ A s e T v σ A v dv ds e T σ A s e T v σ A v dv ds e T σ A s σ A s e T v σ A v dv ds e T s σ A s σ A s e T v σ A v σ A v dv ds f c c c s s s s Although it looks quite complicated, simpler expressions of coefficients can be derived if we give σ specific forms such as CEV, which are shown in section 8. Then, we summarize the result obtained above. Theorem 1. The asymptotic expansion of the Delta of a plain vanilla call option of which payoff function with maturity time T is expressed as S T K + is given by where D, T + o ɛ, D, T ɛ e rt d N y +ɛ e rt f s y + f d + c y + 1 n [y;, s 1 + yd y s s n [y;,, 7 and the coefficents are given by y s e T K, ɛ 77

7 [ E e T t σ A t dw t e T t σ A t dt, and c 1 s e T s σ A s σ A s e T v σ A v dv ds 3 The Delta of Average Call Option f c. Utilizing the asymptotic expansion of S t, Takahashi [1999 derived S t Ãt + ɛã1t + ɛ Ãt + oɛ, 8 where S t 1 t t S u du, Ã t 1 Ãt t t A u du and so on. Now we put Xt St /ɛ, g 1 Ã1T and g ÃT /. Then, X T g 1 + ɛ g + oɛ. [ The asymptotic expantion of the average call option e rt E ST K + [ e rt E ST K + [ ɛ e rt E y + X T + is given by ɛ e rt E [ y + g 1 + ɛ g + 1 {g1 +ɛ g + y}, where y ÃT K /ɛ. We obtain its approximated value [ e rt E ST K + ɛ e rt E [ y + g 1 1 {g1 y} + ɛ e rt E [ g 1 {g1 y} [ +ɛ e rt E y + g 1 g 1 {g1 y} + o ɛ, where [ 1 T E T [ 1 T T E [ 1 T T E 1 T t s 1 e t s σ A s dw s dt e t s σ A s dt dw s e T s 1 σ A s dw s e T s 1 σ As ds. 9 78

8 In addition, the coefficients of the conditional expectation E [g g 1 x cx + f are given by 1 T [ t c T 3 e t s e T s 1 σ A s σ A s [ s e s v e T v 1 σ A v dv ds dt and f c. Accordingly, [ e rt E ST K + ɛ e rt y + x n [x;, dx + ɛ e rt y Now we define the approximated value as y cx + f n [x;, dx + o ɛ. C, T ɛ e rt y + x n [x;, dx + ɛ e rt cx + f n [x;, dx y ɛ e rt y N and consider its differentiation by s : s C, T ɛ e rt d N +ɛ e rt f ɛ e rt d N y + y d y y + n [y;, + ɛ e rt f y n [y;,, 1 y s s 1 yd s + y s + 1 n [y;, s 1 + yd y s s y + f d + f y s y +ɛ e rt f s y + f d + c y where d y s yd s + y s n [y;, n [y;,, e T 1 /T ɛ. The derivatives of coefficients are calculated as s 1 T s T T e T s 1 σ As ds e T s 1 e s σ A s σ A s ds e T s e T + e s σ A s σ A s ds, n [y;, 79

9 and c 1 T [ t { s s T 3 e t s e T s 1 σ A s σ A s [ s e s v e T v 1 σ A v dv ds dt} 1 T [ t 3 s T 3 e t s e T s 1 σ A s σ A s [ s e s v t + 1 T 3 [ s e s v t + 1 T 3 [ s e s v t + T 3 [ s e s v Then we sum up the result as a theorem. e T v 1 e t [ e T s 1 e T v 1 e t [ e T s 1 e T v 1 e t s [ e σ A v dv ds dt σ A s σ A v dv ds dt σ A s σ A s σ A v dv ds dt T s 1 σ A s σ A s e T v 1 σ A v σ A v dv ds dt, f s s c c s c s. Theorem. The asymptotic expantion of the Delta of an average call option, whose payoff function with maturity time T is expressed as ST K is given by where D, T ɛ e rt d N and the coefficents are given by + y +ɛ e rt f s y + f d + c y D, T + o ɛ, + 1 n [y;, s 1 + yd y s s y s e T 1 /T K, ɛ n [y;,, 11 8

10 [ 1 E T 1 T d e T 1 ɛt, t e t s σ A s dw s dt e T s 1 σ As ds, c 1 T 3 s t e s v [ e t s [ e e T v 1 T s 1 σ A s σ A s σ A v dv ds dt, and f c. 4 Variance Reduction Technique for Delta of Call Option Previous section showed the validity of the asymptotic expansion technique for the Delta, but we sometimes need more precise values, especially when ɛ is quite large or when the true value is quite small. For this purpose, we propose a variance reduction method, which is an extension of Takahashi and Yoshida[4. Although the value of Delta is obtained by calculating E [ e rt Y T 1 {st K}, it may be impossible to calculate it analytically. In the case, Monte Carlo simulation based on Euler-Maruyama Scheme is a standard method, which is called crude Monte Calro in this paper. However, crude Monte Carlo is usually quite time-consuming because of its low convergence speed, and hence, Takahashi and Yoshida [4 proposed the method based on an asymptotic approach to accelerate the simulation. The convergence speed depends on the variance of sample paths. The smaller the variance, the higher its convergence speed. Let X be a random variable. We take another random variable Y whose expectation is. Then, the random variable X Y has the same expectation as X, and its variance is given by V ar X Y E [X E [X Y E [X E [X E [X E [X Y + E [ Y V ar X + V ar Y Corr X, Y V ar X V ar Y V ar X V ar Y Corr X, Y V ar X V ar Y, where Corr X, Y denotes the correlation between X and Y. Let a >, and we use ay instead of Y. Then, V ar X ay is minimized when a Corr X, Y V arx V ary, and we obtain V ar X ay V ar X Corr X, Y V ar X 1 Corr X, Y V ar X. Thus, we have to find Y whose correlation with X is close to one. 81

11 We utilize the approximated Delta denoted by D, T. In the derivation of the theorem 1 or theorem, we obtain the approximated option price in an integral representation: C, T ɛ e rt y + x n [x;, dx + ɛ e rt cx + f n [x;, dx. y The candidate for a random variable Y is obtained by differentiating it with respect to s. Next, we prepare two lemmas for the ease of calculation. y Lemma 1. Let ϕ x be a C class function. Then 1 {x y} ϕ x dx ϕ y. 1 Proof. 1 {x y} ϕ x dx 1 {x y} ϕ x dx ϕ y Lemma. The differentiation of n [x;, with respect to s is Proof. s n [x;, s n [x;, 1 x s 1 n [x;, 13 s { 1 1/ exp 1 } x 1 π 1 1 3/ exp π s 1 x 1 1 1/ exp 1 s x 1 π 1 x 1 n [x;, s 1 x / exp 1 1 π x 1 x s 8

12 Using lemmas 1 and, the approximation of Delta is obtained as D, T ɛ e rt s ɛ e rt y y + x n [x;, dx + ɛ e rt cx + f n [x;, dx y s y d + 1 x y + x 1 n [x;, dx s +ɛ e rt d cy + f n [y;, c +ɛ e rt x + f n [x;, dx y s s +ɛ e rt 1 cx + f x y 1 n [x;, dx s ɛ e d rt + 1 x y + x 1 n [x;, dx s y +ɛ e rt d cy + f n [y;, c +ɛ e rt x + f n [x;, dx y s s +ɛ e rt 1 cx + f x 1 n [x;, dx. s y On the other hand, D, T is expressed as D, T ɛ e rt E[d + 1 g y + g s +ɛ d cy + f n [y;, + E [ϕ g 1, c s g 1 + f s + 1 cg 1 + f g1 1 s 1 {g1 y} where ϕx ɛ e rt d + 1 x y + x 1 14 s c +ɛ x + f + 1 cx + f x s s 1 1 s {x y} + ɛd cy + f n [y;,. Finally, we set Y ϕ g 1 D, T. We call this new method Hybrid Monte Carlo. 5 The Vega of Plain Vanilla Call Option In the previous section, we obtained an approximated value of Plain Vanilla Call Option price as C, T ɛ e rt y + x n [x;, dx + ɛ e rt cx + f n [x;, dx y ɛ e rt y N y + n [y;, + ɛ e rt f y n [y;,, y 83

13 and consider its differentiation by ɛ to obtain an approximation of the Vega. ɛ C, T ɛ e rt y N ɛ e rt y N y + n [y;, y + n [y;, y N + ɛ e rt f y n [y;, y y n [y;, + y n [y;, +ɛ e rt f y n [y;, ɛ e rt f y n [y;, + ɛ f y3 n [y;, e rt + ɛ f y + f y3 n [y;,. Then, we summarize the result as a theorem. Theorem 3. The asymptotic expantion of the Vega of Plain Vanilla Call option of which payoff function with maturity time T is expressed as S T K + is given by where and the coefficents are given by V, T + o ɛ, V, T e rt + ɛ f y + f y3 n [y;,, 15 y s e T K, ɛ [ E e T t σ A t dw t e T t σ A t dt, and c 1 s e T s σ A s σ A s e T v σ A v dv ds, 6 The Vega of Average Call Option f c. Repeating the similar argument used in the previous sections, we can obtain the approximated value of the Vega of Average Call Option. Hence, we only state the result as a theorem. 84

14 Theorem 4. The asymptotic expantion of the Vega of Average Call option of which payoff function with maturity time T is expressed as S T K +, is given by where and the coefficents are given by V, T + o ɛ, V, T e rt + ɛ f y + f y3 n [y;,, 16 y s e T 1 /T K, ɛ [ 1 E T 1 T d e T 1 ɛt, t e t s σ A s dw s dt e T s 1 σ As ds, c 1 T 3 s t e s v [ e t s [ e e T v 1 T s 1 σ A s σ A s σ A v dv ds dt, and f c. 7 Variance Reduction Technique for Vega of Call Option Previous section showed the validity of the asymptotic expansion technique for the Vega. However, we sometimes need more precise values, especially when ɛ is quite large or when the true value is quite small. For this purpose, we propose a variance reduction method. Although this method is quite similar to the case of Delta, we need some adjustments. We define the differentiation with respect to the volatility coefficient ɛ of S t satisfying the following SDE: { ds t S t dt + ɛσ S t dw t S s > 85

15 Definition 3. We define Z t as a stocastic process which satisfies the SDE: { dz t Z t dt + [σ S t + ɛσ S t Z t dw t Z 17 The value of Vega is obtained by calculating E [ e rt Z T 1 {st K}, but it is sometimes impossible to calculate it analytically. Although we utilize Monte Carlo simulations by using Euler-Maruyama Scheme, it is sometimes time-consuming. In a similar way as Variance Reduction Technique for Delta, let X be the sample path and we introduce a new random variable Y. According to Kunitomo and Takahashi [3b, the asymptotic expantion of S t is obtained by where S t A t + ɛa 1t + ɛ A t +, da t A t dt, A s The asympotic expansion of Z t is also given by da 1t A 1t dt + σ A t dw t, A 1 da t A t dt + σ A t A 1t dw t, A. Z t A 1t + ɛ A t +. Now we put X t S t A t /ɛ, g 1 A 1T and g A T /. Then, the asymptotic expantion of the Vega E [ e rt Z T 1 {st K} is given by E [ e rt Z T 1 {st K} e rt E [ g 1 + ɛ g + 1 {g1 +ɛ g + y}, where y A T K /ɛ. Thus, we obtain its approximated value as E [ e rt Z T 1 {st K} e rt E [ g 1 1 {g1 y} + ɛ e rt E [ g 1 {g1 y} [ +ɛ e rt E g 1 g 1 {g1 y} + o ɛ. Finally, the approximated value of Vega V, T is expressed as where V, T e rt E [ [ g 1 + ɛ g 1 {g1 y} + ɛ e rt E g 1 g 1 {g1 y} e rt E [ [ g 1 + ɛ E [g g 1 1 {g1 y} + ɛ e rt E g 1 E [g g 1 1 {g1 y} e rt E [ g 1 + ɛ c g1 + f [ 1 {g1 y} + ɛ e rt E g 1 c g 1 + f 1 {g1 y} E [ϕ g 1, ϕ x e rt x + ɛ c x + f 1 {x y} e rt y c y + f n [y;,. 18 Finally, we introduce a new random variable Y ϕ g 1 V, T, and implement the variance reduction technique described in section 4. We call this method Hybrid Monte Carlo as in section 4. 86

16 8 The Case of CEV processes 8.1 Another Deriviation of Approximated Value of Delta In this sections we assume the price processes of the underlying asset to be CEV processes: { ds t S t dt + ɛs γ t dw t, < γ 1 S s > 19 Then, we derive the approximation of the Delta in a more straightforward manner. We define Y t as Y t St s. Note that Y t the SDE: { dy t Y t dt + ɛ γ S γ 1 t Y t dw t, Y 1 and the Delta of Plain Vanilla Call Option is obtained by e rt E [ Y T 1 {st K}. 1 See Imamura, Takahashi and Uchida [5 for the derivation for instance. According to Kunitomo and Takahashi [3b, the asymptotic expantion of S t is given by S t A t + ɛa 1t + ɛ A t +, where A t s e t A 1t s γ t A t γ s γ 1 e t e 1 γs dw s t e t e γ s A 1s dw s. Similarly, the asymptotic expantion of Y t is obtained by Y t and B t satisfy SDEs: B t + ɛb 1t + ɛ B t +, where B t, B 1t db t B t dt, B 1 db 1t B 1t dt + γ A γ 1 t B t dw t, B 1 [ db t B t dt + γ γ 1 A γ t A 1t B t + γ A γ 1 t B 1t dw t, B Then, we can easily obtain B t, B 1t, B t as B t 1 s A t B 1t γ s A 1t B t γ 1 s A t. Now we put X t S t A t /ɛ, g 1 A 1T and g A T /, then Therefore, e rt E [ Y T 1 {st K} ɛ e rt E X T g 1 + ɛ g +. [ d + γ g 1 + ɛ γ 1 g + 1 s s {g1 +ɛ g + y}, 87

17 where y s e T K /ɛ and d y s e T /ɛ. We obtain its approximated value as e rt E [ Y T 1 {st K} [ ɛ e rt E [d + γs γ 1 g 1 1 {g1 y} + ɛ e rt E g 1 s {g1 y} +ɛ e rt E [d + γs g 1 g 1 {g1 y} + o ɛ, where 1 { y} means the derivation of 1 { y} in the sense of distribution. By the property of Brownian Motion, we know the distribution of g 1 N,, where [ E { s γ 1 γ e T t σ A t dw t e T t σ A t dt e T e γt if < γ < 1 s T et if γ 1. In addition, the conditional expectation E [g g 1 x cx + f,where c 1 s and f c. Accordingly, e rt E [ Y T 1 {st K} ɛ e rt d + γ x s y e T s σ A s σ A s e T v σ A v dv ds γ s 4γ 1 e 3T 1 e 1 γt 8 1 γ if < γ < 1 s 3 T e3t if γ 1, n [x;, dx + ɛ e rt γ 1 s +ɛ e rt d + γ s y cy + f n [ y;, + o ɛ. Thus we obtain the approximation as y ɛ e d rt N + γ s + ɛ y cx + f n [x;, dx d γ cy y + f + ɛ γ 1 f y n [y;,. s s where N x denotes a cumulative distribution function of standard normal distribution N, 1. Finally we see the agreement of the two approximated values derived in different manners. theorem 1, the approximated value of the Delta is given by By the D, T ɛ e rt d N y +ɛ e rt f s y + f d + c y + 1 n [y;, s 1 + yd y s s n [y;,. 88

18 In addition, the coefficients, c, f are monomials of s, and their differentiation with respect to s are Substituting them, we obtain s γ s, c s c s, D, T y ɛ e d rt N + γ n [y;, s γ 1 γ +ɛ e rt f y + f d + c y s s y ɛ e d rt N + γ n [y;, s +ɛ e rt γ 1 s f y + c y + f Hence, the two approximated values agree. Then we sum up the fact as a theorem. f γ 1 f. s s + yd γ y n [y;, s d γ y n [y;,. s Theorem 5. The asymptotic expantion of the Delta of Plain Vanilla Call option, whose payoff function with maturity time T is expressed as S T K +, is given by where D, T + o ɛ, D, T ɛ e rt d N and the coefficents are given by y + γ n [y;, s +ɛ e rt γ 1 s f y + c y + f y s e T K, ɛ [ E e T t σ A t dw t e T t σ A t dt { s γ 1 γ d γ y n [y;,, s e T e γt if < γ < 1 s T et if γ 1, c 1 s e T s σ A s σ A s e T v σ A v dv ds γ s 4γ 1 e 3T 1 e 1 γt 8 1 γ if < γ < 1 s 3 T e3t if γ 1, 89

19 and f c. Similarly, we gain the approximation formula for the Delta of the average option. Theorem 6. The asymptotic expantion of the Delta of Average Call option, whose payoff function with maturity time T is expressed as ST K, is given by D, T + o ɛ, where + D, T ɛ e rt d N y + γ n [y;, s +ɛ e rt γ 1 s f y + c y + f d γ y n [y;,, s 3 and the coefficents are given by y s e T 1 /T K, ɛ 1 T e T t 1 s γ T { e T 1 γ s T s e T T σ A t dt et 1 γ 1 d e T 1 ɛt, T + e γt γ1 γ1 γ T et 1 {e T T 3 + et 1 } } if < γ < 1, γ 1 if γ 1 if γ 1, 9

20 c and 1 T [ t T 3 e t s e T s 1 σ A s σ A s [ s e s v e T v 1 σ A v dv ds dt 1 + e T γ + e T 1 + γ +e 4γ 1T γ 7 + 1γ γ γ γ s 4γ 1 +e 1+γT 6 44γ + 96γ 64γ 3 5 T 3 +4e γt γ 3γ + 16γ 3 +e T γ γ + 48γ 3 e T 3 8γ + 89γ 11γ γ 4 /γ 3 16γ + 16γ /8 1 3γ + γ { } s e 3T + e T 1 T + e T 1 T + T T s 3 { 5 T e 3T T 3T e T T et 1 3} f c. if < γ < 1, γ 1 if γ 1 if γ 1, 8. The Gamma of Call Option The approximated value of Delta of Plain Vanilla Average Call Option is given by where D, T ɛ e rt d N D, T + o ɛ, y + γ n [y;, s +ɛ e rt γ 1 s f y + c y + f d γ y n [y;,. s Then, we can differentiate it with respect to s again. We get an approximated value of the Gamma. Theorem 7. The asymptotic expantion of the Gamma of Plain Vanilla Call option, whose payoff function with maturity time T is expressed as S T K +, is given by G, T + o ɛ, 91

21 where G, T ɛ e rt d γ d y γ s s + γ s + γ s y ɛ e d rt γ cy y + f + γ 1 s s γ +ɛ e rt + s y γ cy d + f + s γ 1 s f y + γ 1 f dn [y;,, s and the coefficents are given by n [y;, 4 γ f y + y d γ 1 y n [y;, s s c y + c y d + γ 1 f s s d γ s y y s e T K, ɛ [ E e T t σ A t dw t e T t σ A t dt { s γ 1 γ e T e γt if < γ < 1 s T et if γ 1, and c 1 s e T s σ A s σ A s e T v σ A v dv ds γ s 4γ 1 e 3T 1 e 1 γt 8 1 γ if < γ < 1 s 3 T e3t if γ 1, 8.3 The Delta of Digital Option f c. We consider the Digital Option whose payoff function at maturity time T is given by 1 {K1 S T K }. Before considering the Delta of Digital Option, we note that its price is represented by e rt E [ 1 {K1 S T K }. As above, X T g 1 + ɛ g +, where X t S t A t /ɛ, and e rt E [ 1 {K1 S T K } e rt E [ 1 { y1 X T y } e rt E [ 1 { y1 g 1 +ɛ g + y }, 9

22 where y 1 A T K 1 /ɛ and y A T K /ɛ. We obtain its approximated value as e rt E [ 1 {K1 S T K } e rt E [ [ 1 { y1 g 1 y } + ɛ e rt E g 1 { y1 g 1 y } + o ɛ e rt {N y1 N y } +ɛ cy 1 + f n [y 1 ;, ɛ cy + f n [y ;,. Its differentiation corresponds to the Delta of the Digital option. Theorem 8. The asymptotic expantion of the Delta of Digital option, whose payoff function of with maturity time T is expressed as 1 {K1 S T K }, is given by where D, T d γs y 1 +ɛ c y1 + c y 1 d + γ 1 s d γs y +ɛ and the coefficents are given by D, T + o ɛ, f + cy1 + f s γ s y 1d + y 1 γ s n [y 1 ;, c y + c y d + γ 1 f + cy + f γ y d s s s + y γ n [y ;,, s y 1 s e T K 1 ɛ, y s e T K, ɛ [ E e T t σ A t dw t e T t σ A t dt { s γ 1 γ e T e γt if < γ < 1 s T et if γ 1, 5 and c 1 s e T s σ A s σ A s e T v σ A v dv ds γ s 4γ 1 e 3T 1 e 1 γt 8 1 γ if < γ < 1 s 3 T e3t if γ 1, f c. 93

23 9 Numerical Examples The BSBlack-Sholes model is represented as { ds t S t dt + ɛs t dw t S s >, 6 and the CEV Constant Elasticity of Variance model is represented as { ds t S t dt + ɛs γ t dw t, < γ < 1 S s >. Hence, the volatility function of the models are expressed as σ S t S γ t < γ Approximation for the Delta of Plain Vanilla Option We apply the asymptotic expansion scheme to the BS and CEV processes. The coefficients are calculated as [ and E c 1 { s γ 1 γ s e T t σ A t dw t e T t σ A t dt e T e γt if < γ < 1 s T et if γ 1, e T s σ A s σ A s e T v σ A v dv ds γ s 4γ 1 e 3T 1 e 1 γt 8 1 γ if < γ < 1 s 3 T e3t if γ 1. The coefficients, c, and f are the function of s, T, γ and. In particular, paying attention to γ, we represent γ, c c γ. We can easily check the continuity at γ 1, i.e. γ 1 and c γ c 1,as γ 1. We apply the theorem 5 and obtain the approximated value of the Delta. We show some numerical examples figure 1. The initial value s is fixed to s 1. For the true values, we use the analytical values for BS model γ 1, and use the mean of 1,, paths generated by Euler-Maruyama scheme that divides one year to 365 intervals for CEV model γ < 1. The error ratio is calculated as approximate value true value /true value. We can observe that the accuracy of this approximation is valid. 94

24 9. Approximation for the Delta of Average Option We apply the asymptotic expansion scheme to the BS and CEV processes. The coefficients are calculated as 1 T e T t 1 s γ T { e T 1 γ s T s e T T σ A t dt et 1 γ 1 T + e γt γ1 γ1 γ T et 1 {e T T 3 + et 1 } } if < γ < 1, γ 1 if γ 1 if γ 1, and c 1 T [ t T 3 e t s e T s 1 σ A s σ A s [ s e s v e T v 1 σ A v dv ds dt 1 + e T γ + e T 1 + γ +e 4γ 1T γ 7 + 1γ γ γ γ s 4γ 1 +e 1+γT 6 44γ + 96γ 64γ 3 5 T 3 +4e γt γ 3γ + 16γ 3 +e T γ γ + 48γ 3 e T 3 8γ + 89γ 11γ γ 4 /γ 3 16γ + 16γ /8 1 3γ + γ { } s e 3T + e T 1 T + e T 1 T + T T s 3 { 5 T e 3T T 3T e T T et 1 3} if < γ < 1, γ 1 if γ 1 if γ 1. As with the Plain Vanilla case, the coefficients, c, and f are the function of s, T, γ and. We can easily check the continuity at γ 1, i.e. γ 1 and c γ c 1,as γ 1. We apply the theorem 6 and obtain the approximated value of the Delta. We show some numerical examples figure. The initial value s is fixed to s 1. For the true values, we use the analytical values for BS model γ 1, and use the mean of 1,, paths generated by Euler-Maruyama scheme that divides one year to 365 intervals for CEV model γ < 1. The error ratio is calculated as approximate value true value /true value. We can observe that the accuracy of this approximation is valid in general. 95

25 9.3 Computaion of Delta by the Variance Reduction Technique In this section, we show some numerical examples of variance reduction method. We use the BS and CEV models as the dynamics of stock prices. The result of Plain Vanilla Delta is exhibited in figure 3. In theorem 5, the Delta of Plain Vanilla Call Option is obtained by e rt E [ Y T 1 {st K}, and its asymptotic expansion is expressed as e rt E [ [ Y T 1 {st K} ɛ e rt E d + γ g 1 + ɛ γ 1 g + s s where y s e T K /ɛ and d y s 1 {g1 +ɛ g + y} e T /ɛ. Then, we obtained its approximated value D, T [ ɛ e rt E [d + γs γ 1 g 1 1 {g1 y} + ɛ e rt E g 1 s {g1 y} +ɛ e rt E [d + γs g 1 g 1 {g1 y} [ ɛ e rt E [d + γs γ 1 g 1 1 {g1 y} + ɛ e rt E E [g g 1 1 s {g1 y} +ɛ e rt E [d + γs g 1 E [g g 1 1 {g1 y} [ ɛ e rt E [d + γs γ 1 g 1 1 {g1 y} + ɛ e rt E c g s 1 + f 1 {g1 y} [ +ɛ e rt E d + γ c g 1 g s 1 + f 1 {g1 y} E [ϕ g 1, where ϕ x ɛ e rt d + γ x + γ 1 ɛ c x + f s s, 1 {x y} + ɛ e rt d γ s y c y + f n [y;,. Next, we define Y as Y ϕ g 1 D, T, and call this Y Attendant random variable of X generated by Crude Monte Carlo. In the simulation, the initial value s is fixed to s 1. For the true values, we use the analytical values for BS model γ 1, and use the mean of,, paths generated by Euler-Maruyama scheme that divides one year to 365 intervals for CEV model γ < 1. The errors 1, and 3 are defined by X true value /true value, Y/approximated value and X Y true value / true value, respectively. We generated 1 passes and took their average, and repeated it by 1 cases. We calculated the correlation between X and Y, and the average, standard deviation, maximum and minimum of 1 cases for each of six random variablesx, Y, X Y and errors 1,,3. The example of Average Delta is exhibited in figure 4. Items on the table are the same as above. We can observe that the standard deviaition as well as maximum and minimum values of the errors of HybridMCerror 3 are reduced for each case relative to those of CrudeMCerror

26 9.4 Approximation for the Vega of Plain Vanilla Option We apply the asymptotic expansion scheme to the BS and CEV processes. The coefficients are calculated as [ and E c 1 { s γ 1 γ s e T t σ A t dw t e T t σ A t dt e T e γt if < γ < 1 s T et if γ 1, e T s σ A s σ A s e T v σ A v dv ds γ s 4γ 1 e 3T 1 e 1 γt 8 1 γ if < γ < 1 s 3 T e3t if γ 1. We apply the theorem 3 and get the approximated value of the Delta. We show some numerical examples figure 5. The initial value s is fixed to s 1. For the true values, we use the analytical values for BS model γ 1, and use the mean of 1,, paths generated by Euler-Maruyama scheme that divides one year to 365 intervals for CEV model γ < 1. The error ratio is calculated as approximated value true value /true value. We can see that the accuracy of this approximation is generally valid. 97

27 9.5 Approximation for the Vega of Average Option We apply the asymptotic expansion scheme to the BS and CEV processes. The coefficients are calculated as 1 T e T t 1 s γ T { e T 1 γ s T s e T T σ A t dt et 1 γ 1 T + e γt γ1 γ1 γ T et 1 {e T T 3 + et 1 } } if < γ < 1, γ 1 if γ 1 if γ 1, and c 1 T [ t T 3 e t s e T s 1 σ A s σ A s [ s e s v e T v 1 σ A v dv ds dt 1 + e T γ + e T 1 + γ +e 4γ 1T γ 7 + 1γ γ γ γ s 4γ 1 +e 1+γT 6 44γ + 96γ 64γ 3 5 T 3 +4e γt γ 3γ + 16γ 3 +e T γ γ + 48γ 3 e T 3 8γ + 89γ 11γ γ 4 /γ 3 16γ + 16γ /8 1 3γ + γ { } s e 3T + e T 1 T + e T 1 T + T T s 3 { 5 T e 3T T 3T e T T et 1 3} if < γ < 1, γ 1 if γ 1 if γ 1. We apply the theorem 4 and obtained the approximated value of the Delta. We show some numerical examples figure 6. The initial value s is fixed to s 1. For the true values, we use the analytical values for BS model γ 1, and use the mean of 1,, paths generated by Euler-Maruyama scheme that divides one year to 365 intervals for CEV model γ < 1. The error ratio is calculated as approximated value true value /true value. As above, we can see that the accuracy of this approximation is valid. 98

28 9.6 Computation of Vega by the Variance Reduction Technique In this section, we show some numerical examples of variance reduction method. We use the BS and CEV models as the dynamics of stock prices. The example of Plain Vanilla Vega is exhibited in figure 7. The random variables X and Y correspond to Crude MC and Attendant, respectively. The initial value s is fixed to s 1. For the true values, we use the analytical values for BS model γ 1, and use the mean of,, paths generated by Euler-Maruyama scheme that divides one year to 365 intervals for CEV model γ < 1. The error 1, and 3 are defined byx true value /true value, Y/approximated value, and the error 3 is X Y true value /true value, respectively. We generated 1 passes and took their average, and repeated it by 1 cases. We calculated the correration between X and Y, and average, standard deviation, maximum and minimum of 1 cases for each of the six random variablesx, Y, X Y and errors 1,,3. The example of Average Vega is exhibited in figure 8. Items on the table are the same as above. We can observe that the standard deviaition as well as maximum and minimum values of the errors of HybridMCerror 3 are reduced for each case relative to those of CrudeMCerror 1. 1 Concluding Remarks We developed an asymptotic expansion method for computing Greeks. Because it takes almost no time to compute the approximated values, it would be useful for monitoring Greeks changing rapidly in volatile markets. We also introduced so called Hybrid Monte Carlo, a variance reduction technique based on combination of Monte Carlo simulation and the asymptotic expansion. We showed that our variance reduction technique can accelerate Monte Carlo simulations. Although we mainly treated the Delta and the Vega, our method can also be applied to computing the other Greeks such as Gamma; it is obtained by differentiating the approximated Delta by the initial price of the underlying asset. Clearly, the same method can be applied to term structure models and credit models such as Heath- Jarrow-Morton models, Market models and Duffie-Singleton models, which are topics in the next research. 99

29 European Call Delta r T σ γ K True Value Approximation error ratio 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1%.1 % % 1%.1 % % 1%.1 % % 1%.1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1%.1 % % 1%.1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % Figure 1: Delta of Plain Vanilla Option 1

30 Asian Call Delta r T σ γ K True Value Approximation error ratio 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 5%.5 % % 5%.5 % % 5%.5 % % 5%.5 % % 5% 1 % % 5% 1 % % 5% 1 % % 5% 1 % % 5% 1 3% % 5% 1 3% % 5% 1 3% % 5% 1 3% % 1%.5 % % 1%.5 % % 1%.5 % % 1%.5 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 5% 1 % % 5% 1 % % 5% 1 % % 5% 1 % % 5%.5 3% % 5%.5 3% % 5%.5 3% % 5%.5 3% % 5% 1 3% % 5% 1 3% % 5% 1 3% % 5% 1 3% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1%.1 3% % 1%.1 3% % 1%.1 3% % 1%.1 3% % 1%.5 3% % 1%.5 3% % 1%.5 3% % 1%.5 3% % Figure : Delta of Average Option

31 * +, -!!! "#!! "!!! $ % & '!!!! "#!! "!!!! $ % & '!!!! "#! " $ % & '!!!!! "#!!! "! $ % & '! "#!! "! $ % & '!!!!!! "# " $ % & ' Figure 3: Variance Reduction for Plain Vanilla Delta 1

32 * +,! "# "! $ % & '! "# "! $ % & '! "# "! $ % & '! "# "! $ % & '! "# "! $ % & '! "# "! $ % & ' Figure 4: Variance Reduction for Average Delta 13

33 European Call Vega r T σ γ K True Value Approximation error ratio 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1%.5 % % 1%.5 % % 1%.5 % % 1%.5 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % Figure 5: Vega of Plain Vanilla Option 14

34 Asian Call Vega r T σ γ K True Value Approximation error ratio 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 1% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1% 1 4% % 1%.5 % % 1%.5 % % 1%.5 % % 1%.5 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 3% % 1% 1 % % 1% 1 % % 1% 1 % % 1% 1 % % Figure 6: Vega of Average Option 15

35 * +,!"! # # # # $ % & ' #!"! # # # # $ % & ' # # # #!"! # # # # $ % & ' # # #!"! # # # # $ % & '!"! # # # # $ % & ' Figure 7: Variance Reduction for Plain Vanilla Vega 16

36 * +!!!! "#!! "!! $ % & '!!!! "# "!! $ % & '!!!!! "#!!!!!! "!!! $ % & '!!!! "#!! "! $ % & ' Figure 8: Variance Reduction for Average Vega 17

37 Reference [1 Imamura, S, Takahashi, A., and Uchida, Y. 5, On Risk Sensitivity Analysis of Options Based on Malliavin Calculus, Monetary and Economic Studies, Vol Bank of Japan in Japanese. [ Kunitomo, N. and Takahashi, A. 1, The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims, Mathematical Finance, Vol.11, [3 Kunitomo, N. and Takahashi, A. 3a, On Validity of the Asymptotic Expansion Approach in Contingent Claim Analysis, Annals of Applied Probability Vol.13, No.3, Aug. 3. [4 Kunitomo, N. and Takahashi, A. 3b, Foundation of Mathematical Finance-Application of Malliavin Calculus and Asymptotic Expansion Method-, Toyo-Keizai in Japanese. [5 Kunitomo, N. and Takahashi, A. 4, Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems, Stochastic Processes and Applications to Mathematical Finance, 195-3, World Scientific. [6 Takahashi, A. 1995, Essays on the Valuation Problems of Contingent Claims, Unpublished Ph.D. Dissertation, Haas School of Business, University of California, Berkeley. [7 Takahashi, A. 1999, An Asymptotic Expansion Approach to Pricing Contingent Claims, Asia-Pacific Financial Markets, Vol.6, [8 Takahashi, A. and Matsushima, S.4, Monte Carlo Simulation with an Asymptotic Expansion in HJM Framework, FSA Research Review 4, 8-13, Financial Services Agency in Japanese. [9 Takahashi, A. and Saito, T.3, An Asymptotic Expansion Approach to Pricing American Options, Monetary and Economic Studies, Vol., 35-87, Bank of Japan in Japanese. [1 Takahashi, A. and Uchida, Y.4, New Acceleration Schemes with the Asymptotic Expansion in Monte Carlo Simulation, Discussion Paper Series, CIRJE-F-98, Faculty of Economics, The University of Tokyo, Forthcoming in Advances in Mathematical Economics. [11 Takahashi, A. and Yoshida, N.5, Monte Carlo Simulation with Asymptotic Method, Discussion Paper Series, CIRJE-F-335, Faculty of Economics, The University of Tokyo, Forthcoming in The Journal of Japan Statistical Society. [1 Takahashi, A. and Yoshida, N.4, An Asymptotic Expansion Scheme for Optimal Investment Problems, Statistical Inference for Stochastic Processes, Vol.7-,