Constant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation

Transcription

1 Constant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation Ying-Lin Hsu Department of Applied Mathematics National Chung Hsing University Co-authors: T. I. Lin and C. F. Lee Oct. 21, 2008 Ying-Lin Hsu (NCHU) Oct. 21, / 46

2 Outline Introduction Transition Probability Density Function Noncentral Chi-Square Distribution The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 β Approaches to 2 Some Computational Considerations Special Cases Concluding Remarks Ying-Lin Hsu (NCHU) Oct. 21, / 46

3 Introduction CEV model The CEV option pricing model is defined as ds = µsdt + σs β/2 dz, β < 2, where dz is a Wiener process and σ is a positive constant. Ying-Lin Hsu (NCHU) Oct. 21, / 46

4 Introduction The elasticity is β 2 since the return variance υ(s,t) = σ 2 S β 2 with respect to price S has the following relationship dυ(s,t)/ds υ(s,t)/s = β 2, which implies that dυ(s,t)/υ(s,t) = (β 2)dS/S. Ying-Lin Hsu (NCHU) Oct. 21, / 46

5 Introduction If β = 2, then the elasticity is zero and the stock prices are lognormally distributed as in the Black and Scholes model (1973). If β = 1, then the elasticity is -1. The model proposed by Cox and Ross (1976). Ying-Lin Hsu (NCHU) Oct. 21, / 46

6 Introduction We will focus on the case of β < 2 since many empirical evidences (see Campbell (1987), Glosten et al. (1993), Brandt and Kang (2004)) have shown that the relationship between the stock price and its return volatility is negative. The transition density for β > 2 is given by Emanuel and Macbeth (1982) Ying-Lin Hsu (NCHU) Oct. 21, / 46

7 Transition Probability Density Function Consider the constant elasticity of variance diffusion, ds = µ(s,t) + σ(s,t)dz, where µ(s,t) = rs as, and σ(s,t) = σs β/2, 0 β < 2. Then ds = (r a)sdt + σs β/2 dz. Ying-Lin Hsu (NCHU) Oct. 21, / 46

8 Transition Probability Density Function Let Y = Y (S,t) = S 2 β. By Ito s Lemma with we have dy = Y S = (2 β)s1 β, Y t = 0, 2 Y S 2 = (2 β)(1 β)s β, [ (r a)(2 β)y + 1 ] 2 σ2 (β 1)(β 2) dt + σ 2 (2 β) 2 Y dz. Ying-Lin Hsu (NCHU) Oct. 21, / 46

9 Transition Probability Density Function The Kolmogorov forward equation for Y becomes 2 P t = 1 [ σ 2 2 Y 2 (2 β)y P ] Y { [(r a)(2 β)y σ2 (β 1)(β 2) ] } P. Then f(s T S t,t > t) = f(y T y t,t > t) J where J = (2 β)s 1 β. Ying-Lin Hsu (NCHU) Oct. 21, / 46

10 Transition Probability Density Function By Feller s Lemma with a = 1 2 σ2 (2 β) 2, b = (r a)(2 β), h = 1 2 σ2 (β 2)(1 β), x = 1/T, x 0 = 1/t and t = τ = (T t), we have f(s T S t,t > t) = (2 β)k 1/(2 β) (xz 1 β ) 1/(2(2 β)) e x z I 1/(2 β) (2(xz) 1/2 ) where k = 2(r a) σ 2 (2 β)[e (r a)(2 β)τ 1], x = k S 2 β t e (r a)(2 β)τ, z = k S 2 β T. Ying-Lin Hsu (NCHU) Oct. 21, / 46

11 Transition Probability Density Function Cox (1975) obtained the following option pricing formula: C = S t e rτ e x x n G ( n /(2 β),k K 2 β) Γ(n + 1) n=0 Ke rτ n=0 e x x n+1/(2 β) G ( n + 1,k K 2 β) Γ ( n /(2 β) ), where G(m,ν) = (Γ(m)) 1 ν e u u m 1 du is the standard complementary gamma distribution function. Ying-Lin Hsu (NCHU) Oct. 21, / 46

12 Noncentral Chi-Square Distribution If Z 1,...,Z ν are standard normal random variables, and δ 1,...,δ ν are constants, then ν Y = (Z i + δ i ) 2 i=1 is the noncentral Chi-square distribution with ν degrees of freedom and noncentrality parameter λ = ν j=1 δ2 j, and is denoted as χ 2 ν (λ). When δ j = 0 for all j, then Y is distributed as the central Chi-square distribution with ν degrees of freedom, and is denoted as χ 2 ν. Ying-Lin Hsu (NCHU) Oct. 21, / 46

13 Noncentral Chi-Square Distribution The cumulative distribution function of χ 2 ν (λ) is F(x;ν,λ) = P(χ 2 ν (λ) x) = e λ/2 (λ/2) j j!2 ν/2 Γ(ν/2 + j) j=0 x 0 y ν/2+j 1 e y 2 dy, x > 0. Ying-Lin Hsu (NCHU) Oct. 21, / 46

14 Noncentral Chi-Square Distribution An alternative expression for F(x;ν,λ) is ( ) (λ/2) j e λ/2 F(x;ν,λ) = P(χ 2 ν+2j x). j! j=0 The complementary distribution function of χ 2 ν (λ) is Q(x;ν,λ) = 1 F(x;ν,λ). Ying-Lin Hsu (NCHU) Oct. 21, / 46

15 Noncentral Chi-Square Distribution The probability density function of χ 2 ν (λ) can be expressed as a mixture of central Chi-square probability density functions. p χ ν 2 (x) = e λ/2 (λ) j=0 = e (x+λ)/2 2 ν/2 ( 1 2 λ)j p j! χ 2 ν+2j (x) j=0 x ν/2+j 1 λ j Γ(ν/2 + j)2 2j j!. Ying-Lin Hsu (NCHU) Oct. 21, / 46

16 Noncentral Chi-Square Distribution An alternative expression for the probability density function of χ 2 ν (λ) is p χ 2 ν (λ)(x) = 1 2 ( x λ ) (ν 2)/4 exp { 1 2 (λ + x) } I (ν 2)/2 ( λx), x > 0, where I k is the modified Bessel function of the first kind of order k and is defined as ( ) 1 k ( I k (z) = 2 z z 2 /4 ) j j!γ(k + j + 1). j=0 Ying-Lin Hsu (NCHU) Oct. 21, / 46

17 The Noncentral Chi-Square Approach to Option Pricing Model Following Schroder (1989) with the transition probability density function, the option pricing formula under the CEV model is ( ) C = E max(0,s T K), τ = T t = e rτ f(s T S t,t > t)(s T K)dS T K = e rτ S T f(s T S t,t > t)ds T e rτ K K = C 1 C 2. K f(s T S t,t > t)ds T Ying-Lin Hsu (NCHU) Oct. 21, / 46

18 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Making the change of variable w = k S 2 β T, we have Let y = k K 2 β. ds T = (2 β) 1 k 1/(2 β) w (β 1)/(2 β) dw. Ying-Lin Hsu (NCHU) Oct. 21, / 46

19 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 C 1 = e rτ y = e rτ (x/k ) 1/(2 β) = e rτ S t e (r a)τ = e aτ S t y e x w (x/w) 1/(4 2β) I 1 (2 xw)(w/k ) 1/(2 β) dw 2 β y y e x w (x/w) 1/(4 2β) I 1 (2 xw)dw 2 β e x w (w/x) 1/(4 2β) I 1 (2 xw)dw 2 β e x w (w/x) 1/(4 2β) I 1 (2 xw)dw, 2 β Ying-Lin Hsu (NCHU) Oct. 21, / 46

20 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 C 2 = Ke rτ y = Ke rτ y x 1 4 2β w (1 2β+2β 2)/(4 2β) e x w I 1 (2 xw)dw 2 β e x w (x/w) 1/(4 β) I 1 (2 xw)dw. 2 β Ying-Lin Hsu (NCHU) Oct. 21, / 46

21 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Recall that the probability density function of the noncentral Chi-square distribution with noncentrality λ and degree of freedom ν is p χ 2 ν (λ)(x) = 1 2 (x/λ)(ν 2)/4 I (ν 2)/2 ( λx)e (λ+x)/2 = P(x;ν,λ). Let Q(x;ν,λ) = x p χ 2 ν (λ)(y)dy. Then letting w = 2w and x = 2x. Ying-Lin Hsu (NCHU) Oct. 21, / 46

22 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 C 1 ( = S t e aτ e (x+w)/2 w y x ( = S t e aτ 2y e (x +w )/2 ) 1/(4 2β) ( ) I 1 2 xw dw 2 β w x = S t e aτ Q(2y; ν, x ), ( = S t e aτ Q 2y; ) 2 β, 2x, ) 1/(4 2β) I 1 2 β ( 2 x w ) 1 2 dw Ying-Lin Hsu (NCHU) Oct. 21, / 46

23 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 C 2 = Ke rτ y = Ke rτ 2y ( e x w x ) 1/(4 2β) ( ) I 1 2 xw dw w 2 β e (x +w )/2 ( = Q 2y; 2 2 ) 2 β, 2x ( x w ) 1/(4 2β) I 1 2 β ( 2 x w ) 1 2 dw Ying-Lin Hsu (NCHU) Oct. 21, / 46

24 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Thus, C = S t e aτ Q(2y; β, 2x) Ke rτ Q(2y; β, 2x). It is noted that 2 2/(2 β) can be negative for β < 2. Thus further work is needed. Ying-Lin Hsu (NCHU) Oct. 21, / 46

25 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Using the monotone convergence theorem and the integration by parts, we have = = = = y y y P(2y;2ν,2k)dk e z k (z/k) ν 1 kz ν 1 e z z n+ν 1 Γ(n + ν) g(n + ν,z)g(n + 1,y) n=0 n=0 y n=0 e k k n Γ(n + 1) dk g(n + ν 1,z) g(i,y). i=1 (zk) n n!γ(n + ν 1 + 1) dk Ying-Lin Hsu (NCHU) Oct. 21, / 46

26 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 G(n, y) = = e k k n 1 dk Γ(n) y k n 1 y Γ(n) de k = yn 1 e y + Γ(n) n y i 1 e y = Γ(i) = i=1 n g(i,y). i=1 y k n 2 e k Γ(n 1) dk Ying-Lin Hsu (NCHU) Oct. 21, / 46

27 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Next, applying the monotone convergence theorem, we have = = = = = Q(z;ν,k) 1 ( y 2 k z z n=0 n=0 n=0 ) (ν 2)/4 I ν 2 2 ( ky ) e (k+y)/2 dy 1 ( ( ) y (ν 2)/4 1 (ν 2)/2 ky 2 k) 2 ) n ( e k/2 kn 1 2 Γ (n + 1) z k/2 (k/2)n e Γ (n + 1) z n=0 ( 1 2 (1/2) (ν+2n)/2 e y/2 y ν+2n 2 1 ) (ν+2n)/2 Γ ( ν+2n 2 Γ ( ν+2n 2 k/2 (k/2)n e Q(z;ν + 2n,0), Γ (n + 1) (ky/4) n n!γ ( ν+2n )e (k+y)/2 dy 2 )dy ) e y/2 y ν+2n 2 1 dy Ying-Lin Hsu (NCHU) Oct. 21, / 46

28 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 where Q(z;ν + 2n,0) = = (1/2) (ν+2n)/2 z Γ ( ν+2n 2 1 z/2 Γ ( ν+2n 2 = G(n + ν/2,z/2). ) e y/2 y ν+2n )e y y ν+2n 2 1 dy 2 1 dy Ying-Lin Hsu (NCHU) Oct. 21, / 46

29 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Furthermore Q(z;ν,k) = = g (n + 1,k/2) G(n + ν/2,z/2) n=0 ( g (n,k/2) G n + ν 2 ),z/2. 2 n=0 Hence Q(2z; 2ν, 2k) = g (n,k) G(n + ν 1,z). n=0 Ying-Lin Hsu (NCHU) Oct. 21, / 46

30 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Again Q(2z; 2ν, 2k) = g (1,k) G(ν,z) + g (2,k) G(ν + 1,z) + g (3,k) G(ν + 2,z) + = g (1,k) [G(ν 1,z) + g (ν,z)] + g (2,k) [G(ν 1,z) + g (ν + 1,z)] +g (3,k) [G(ν 1,z) + g (ν,z) + g (ν + 1,z) + g (ν + 2,z)] + = G(ν 1,z) + g (ν,z) + g (ν + 1,z) [1 g (1,k)] +g (ν + 2,z) [1 g (1,k) g (2,k)] + = G(ν 1,z) + g (ν + n,z) g (ν + 1,z) [g (1,k)] n=0 g (ν + 2,z) [g (1,k) + g (2,k)] + = 1 g (ν + 1,z) [g (1,k)] g (ν + 2,z) [g (1,k) + g (2,k)]. Ying-Lin Hsu (NCHU) Oct. 21, / 46

31 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 We conclude that n Q(2z;2ν,2k) = 1 g (n + ν,z) g (i,k). n=1 i=1 And y P(2z;2ν,2k)dk = 1 Q(2z;2(ν 1),2y). Ying-Lin Hsu (NCHU) Oct. 21, / 46

32 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 Thus C 2 = Ke rτ P = Ke rτ Q = Ke rτ (1 Q(2x; y ( 2x; ( 2y; β, 2x ) 2 β, 2w ) 2 2 β, 2y) ). dw Ying-Lin Hsu (NCHU) Oct. 21, / 46

33 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 We obtain C = S t e aτ Q(2y; 2 + 2, 2x) Ke rτ 2 β ( 1 Q(2x; ) 2 2 β, 2y), where y = k K 2 β, x = k S 2 β t e (r a)(2 β)τ, k = 2(r a)/ ( σ 2 (2 β)(e (r a)(2 β)τ 1) ) and a is the continuous proportional dividend rate. Ying-Lin Hsu (NCHU) Oct. 21, / 46

34 The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C 1 and C 2 When β > 2 (see, Emanuel and MacBeth (1982), Chen and Lee (1993)), the call option formula is as follows: ( C = S t e aτ 2 Q(2x;, 2y) Ke rτ 1 Q(2y; ) β 2 β 2, 2x). Ying-Lin Hsu (NCHU) Oct. 21, / 46

35 The Noncentral Chi-Square Approach to Option Pricing Model β Approaches to 2 We consider that the noncentral Chi-square distribution will approach log-normal as β tends to 2. Since when either λ or ν approaches to infinity, the standardized variable χ 2 ν (λ) (ν + λ) 2(ν + 2λ) tends to N(0,1) as either ν or λ. Ying-Lin Hsu (NCHU) Oct. 21, / 46

36 The Noncentral Chi-Square Approach to Option Pricing Model β Approaches to 2 χ 2 ν (λ) (ν + λ) lim β 2 2(ν + 2λ) 2k S 2 β T (ν + λ) = lim β 2 2(ν + 2λ) 2r S 2 β T (1 β)σ 2 (e r τ(2 β) 1) 2r S 2 β t e r τ(2 β) = lim β 2 σ 2 (e r τ(2 β) 1) σ 2 (e r τ(2 β) 1)/(2 β) (1 β)σ 2 (e r τ(2 β) 1) + 4r S 2 β t e r τ(2 β) = ln S T [ln S t + (r σ 2 /2)τ] σ, τ where r = r a. Ying-Lin Hsu (NCHU) Oct. 21, / 46

37 The Noncentral Chi-Square Approach to Option Pricing Model β Approaches to 2 Thus, ln S T ln S t N(ln S t + (r a σ 2 /2)τ,σ 2 τ) as β 2. Similarly, it also holds when β 2 +. And we also obtain the same result for β > 2. Ying-Lin Hsu (NCHU) Oct. 21, / 46

38 The Noncentral Chi-Square Approach to Option Pricing Model Some Computational Considerations First initialize the following four variables (with n = 1) ga = e z z ν = g(1 + ν,z), Γ(1 + ν) gb = e k = g(1,k), Sg = gb, R = 1 (ga)(sg). Ying-Lin Hsu (NCHU) Oct. 21, / 46

39 The Noncentral Chi-Square Approach to Option Pricing Model Some Computational Considerations Then repeat the following loop beginning with n = 2 and increase increment n by one after each iteration. The loop is terminated when the contribution to the sum, R, is declining and is very small. ( ) z ga = ga = g(n + ν,z), n + ν 1 ( ) k gb = gb = g(n,k), n 1 Sg = Sg + gb = g(1,k) + g(n,k) R = R (ga)(sg) = the nth partial sum. Ying-Lin Hsu (NCHU) Oct. 21, / 46

40 The Noncentral Chi-Square Approach to Option Pricing Model Some Computational Considerations As each iteration, ga equals g(n + ν,z), gb equals g(n,k) and Sg equals g(1,k) + + g(n,k). The computation is easily done. Ying-Lin Hsu (NCHU) Oct. 21, / 46

41 The Noncentral Chi-Square Approach to Option Pricing Model Some Computational Considerations Using the approximation Q(z;ν,k) = Pr(χ 2 > z) ( χ 2 ) z ν + k = Pr ν + k > ( ) h = Pr χ 2 > ν + k ( ) z h ν + k ( 1 hp[1 h + 0.5(2 h)mp] z ν+k = Φ h 2P(1 + mp) ) h, where h = (ν + k)(ν + 3k)(ν + 2k) 2, P = (ν + 2k)/(ν + k) 2 and m = (h 1)(1 3h). Ying-Lin Hsu (NCHU) Oct. 21, / 46

42 The Noncentral Chi-Square Approach to Option Pricing Model Special Cases Q(z,1,k) = z f Y (y)dy = 1 F Y (z) = 1 P(Y z) = 1 P ( (z + δ) 2 z ) = 1 P( z + δ z) = 1 P( δ z z δ + z) = 1 [N ( z δ) N ( δ z)] = N ( z k) + 1 N ( z k) = N ( k z) + N ( k z). Ying-Lin Hsu (NCHU) Oct. 21, / 46

43 The Noncentral Chi-Square Approach to Option Pricing Model Special Cases Q(z,3,k) = Q(z,1,k) j=0 = Q(z,1,k) + 2 2π 1 k = Q(z,1,k) + (k/4) j z j+1 2 j!γ ( 1 2 j=0 1 k 2π j=0 e (k+z)/2 + j + 1) ( (kz) 1/2 ) 2j+1 (2j + 1)! = Q(z,1,k) + 1 [ 1 e kz 1 k 2π e (k+z)/2 ( kz) 2j+1 (2j + 1)! e (k+z)/2 2π e kz ] e (z+k)/2 = Q(z,1,k) + 1 [ 1 e ( k z) 2 /2 1 ] e ( k+ z) 2 /2 k 2π 2π = Q(z,1,k) + 1 [N ( k z) N ( k + ] z). k Ying-Lin Hsu (NCHU) Oct. 21, / 46

44 The Noncentral Chi-Square Approach to Option Pricing Model Special Cases Q(z;5,k) [ = Q(z;1,k) + k 1/2 N ( k z) N ( k + ] z) +(k 1 z k 3/2 )N ( k z) + (k 1 z + k 3/2 )N ( k + z) = Q(z;1,k) + (k 1/2 k 3/2 + k 1 z)n ( k z) (k 1/2 1 k 1 z)n ( k + z) = Q(z;1,k) +k 3 2 [ (k 1 + kz)n ( k z) (k 1 kz)n ( k + ] z). Ying-Lin Hsu (NCHU) Oct. 21, / 46

45 Concluding Remarks The option pricing formula under the CEV model is quite complex because it involves the cumulative distribution function of the noncentral Chi-square distribution Q(z; ν, k). Some computational considerations are given in the article which will facilitate the computation of the CEV option pricing formula. We can use the function pchisq(q, df, ncp) in free software R to calculate the cumulative distribution function of the noncentral Chi-square distribution, where q is the quantile, df is the degree of freedom which non-integer values are allowed and ncp is the noncentrality parameter. Ying-Lin Hsu (NCHU) Oct. 21, / 46

46 Concluding Remarks THE END. Thanks For Your Attentions! Ying-Lin Hsu (NCHU) Oct. 21, / 46