Notes on the CEV model

Transcription

1 Notes on the CEV model Andrew Lesniewski Ellington Management Group 53 Forest Avenue Old Greenwich, CT 687 First draft of March 2, 24 This draft of November 6, 29 Statement of the problem We construct a family of Green s functions G X, for a forward following the CEV dynamics. Specifically, we consider the initial value problem: G X, = 2 b G X,, G X, = δ X, where δ X = δ X denotes Dirac s delta supported at X. This is actually the terminal value problem for the backward Kolmogorov equation written in terms of the time variable = T t. The function b has the form : We impose the natural boundary condition: b = β, β <. 2 G X,, as. 3 In addition, at =, we impose the following family of boundary conditions the Robin problem [2]: G X, + µg X, =. 4 = This reduces to the reflecting Neumann problem for µ =, and the absorbing Dirichlet problem for µ. It is these two boundary value problems that we consider in this manuscript. Taking the Laplace transform of G X,, G X, = Peter Carr informed me that the restriction β is not necessary e λ g λ, dλ, 5

2 Notes on the CEV model 2 we find that 2 2 2β g + λg =. 6 Simple algebra shows that g can be epressed as g λ, = 2ν u 2ν, 7 where ν =, i.e. ν >, 8 2 β and where u z satisfies Bessel s equation [4]: u + z u + ν2 u =. 9 Now see e.g. [5], any solution to Bessel s equation is a linear combination of Bessel s functions J ν z and Y ν z of the first and second kind, respectively: z 2 u z = A ν J ν z + B ν Y ν z. Recall we refer to [4] for details that J ν z is defined by the series and J ν z = k k z 2k+ν, k! Γ ν + k + 2 Y ν z = J ν z cos νπ J ν z sin νπ. 2 We shall determine the constants A ν and B ν so that all the conditions imposed on the solution are satisfied. Using and 2 as well as the well known properties of the gamma function we observe that, as, g λ, Γ ν + B ν λ Γ ν π 2 Dirichlet boundary condition ν ν Aν λ + cot νπ B ν λ ν ν + 2νλ 2 ν. Let us start with the Dirichlet problem. Since from 3, 3 g λ, = B ν λ Γ ν π ν ν, 4

3 Notes on the CEV model 3 we infer that the Dirichlet problem has a solution for all values of ν. Furthermore, we have B ν λ =, and thus g λ, = A ν λ 2ν J ν 2ν, 5 and we need to determine A ν λ. In order to do this recall first that the Hankel transform see e.g. [] H ν f of a function f is given by Its key property is that In particular, note that H ν f p = f = f J ν p d. 6 H ν f p J ν p p dp. 7 H ν δ X p = XJ ν px. 8 As a consequence, we get the following epression for A ν λ: A ν λ = 2ν 2 X ν 3 2ν 2 Jν X 2ν. 9 Using the identity [4]: e p2 J ν ap J ν bp p dp = 2 ep a2 + b 2 4 ab I ν, 2 2 we finally obtain the following eplicit representation for the Dirichlet Green s function G D X, : G D X, = X 4β /2 β 2 ep 2 β + X 2 β 2 β 2 2 I ν X β β Recall that the non-central χ 2 distribution with r degrees of freedom and the noncentrality parameter λ is given by the following probability density distribution: p ; r, λ = r 2/4 ep 2 λ + λ I r 2/2 λ, 22 2 and thus the Dirichlet Green s function can be written as G D X, = 4νX/ν 2 4ν 2 X /ν 4ν 2 /ν 2 p 2 ; 2ν + 2, 2. 23

4 Notes on the CEV model 4 Note that the total mass of G D X, is indeed less than one, meaning that there is a nonzero probability of absorption at zero. Using the series epansion [4]: we readily find that where I ν z = k z 2k+ν, 24 k! Γ ν + k + 2 G D X, dx = Γ ν Γ ν, Γ ν, = 2ν 2 /ν 2, 25 t ν e t dt 26 is the complementary incomplete gamma function [5]. The quantity Γ ν Γ 2ν 2 /ν ν, 2 is the probability of absorption at zero. For eample, in the square root process case, i.e. ν =, that probability equals ep 2 2. We shall now follow [3] in order to epress the option pricing function in terms of the cumulative noncentral χ 2 distribution function: χ 2 ; r, λ = The pricing function of a call struck at c is given by the integral max c, G D X, dx = c XG D X, dx c 27 p y; r, λ dy. 28 c G D X, dx. The first term on the right hand side of 29 is easy to calculate: after substituting we find that it evaluates to 4ν χ 2 2 c /ν 2 z = 4ν2 X /ν ; 2ν + 2, 4ν2 /ν 2. 3 In order to calculate the second term on the right hand side of 29, we shall first establish the following symmetry property of the noncentral χ 2 distribution: p y; r 2, λ dy + λ p ; r, µ dµ =. 32

5 Notes on the CEV model 5 Indeed, consider the function: From 24, ϕ λ = p y; r 2, λ dy + On the other hand, we verify readily that and thus for all positive λ, ϕ λ, as λ. λ p ; r, µ dµ. p ; r, λ = p ; r, λ + p ; r 2, λ, 2 λ p ; r, λ = p ; r, λ + p ; r + 2, λ, 2 d dλ ϕ λ = = =. Consequently, ϕ λ =, as claimed. Now, introducing the notation and using 32, we find that c G D X, dx = p y; r 2, λ dy p ; r, λ λ p y; r, λ dy p ; r, λ y λ = 4ν2 /ν 2 q = 4ν2 c /ν 2 = q q = = λ,, ν λ p z; 2ν + 2, λ dz z p λ; 2ν + 2, z dz λ p z; 2ν, q dz p z; 2ν, q dz. Putting everything together we obtain the following eplicit representation for the pricing function of a call struck at c: 4ν Vcall D, = χ 2 2 c /ν 2 ; 2ν + 2, 4ν2 /ν 2 4ν cχ 2 2 /ν 2 ; 2ν, 4ν2 c /ν 33 2.

6 Notes on the CEV model 6 From the put call parity, the pricing function of a put struck at c is: 4ν Vput D, = χ 2 2 c /ν 2 ; 2ν + 2, 4ν2 /ν 2 4ν c χ 2 2 /ν 2 ; 2ν, 4ν2 c /ν 2 3 Neumann boundary condition. 34 Let us now turn to the Neumann problem. Since, as, g λ, ν ν Aν λ + cot νπ B ν λ Γ ν + ν Γ ν B ν λ 2 π ν ν, 35 we conclude that the Neumann problem has a solution if /ν. Equivalently, the Neumann problem has a solution if β Assume first that ν <. The coefficients A ν and B ν must then obey the relation: and thus, as a consequence of 2 A ν λ = B ν λ cot νπ, 37 u z = B ν cot νπ J ν z + Y ν z = B ν sin νπ J ν z. 38 This implies that g λ, is given by g λ, = B ν λ 2ν J ν 2ν sin νπ. 39 The computation follows now the same outline as in the Dirichlet case. Using the technique of Hankel transforms, we find that the Neumann Green s function G N X, is given by G N X, = X 4β /2 β 2 ep 2 β + X 2 β 2 β 2 2 I ν X β β 2 2 4

7 Notes on the CEV model 7 or, equivalently, G N X, = 4νX/ν 2 4ν 2 X /ν p 2 ; 2ν + 2, 4ν 2 /ν 2. 4 This is a bona fide probability distribution of X; a straightforward calculation shows that by G N X, dx =. 42 Finally, the pricing functions with Neumann boundary conditions at zero are given 4ν Vcall N, = χ 2 2 c /ν 2 ; 2ν, 4ν2 /ν 2 4ν c χ 2 2 /ν 2 ; 2ν + 2, 4ν2 c /ν 2, 43 and 4ν Vput N, = χ 2 2 c /ν 2 ; 2ν, 4ν2 /ν 2 4ν cχ 2 2 /ν 2 ; 2ν + 2, 4ν2 c /ν References [] Davies, B.: Integral Transforms and Their Applications, Springer Verlag 22. [2] Guenther, R. B., and Fee, J. F.: Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publications 996. [3] Schroder, M.: Computing the constant elasticity of variance option pricing formula, J. Finance, 44, [4] Watson, G. N.: A Treatise on the Theory of Bessel Functions, Cambridge University Press 944. [5] Whittaker, E. T., and Watson, G. N.: A Course of Modern Analysis, Cambridge University Press 927.