Closed-Form Formulae for European Options Under Three-Factor Models

Abstract

In this paper, we derive new closed-form valuations to European options under three-factor hybrid models that include stochastic interest rates and stochastic volatility and incorporate a nonzero covariance structure between factors. We make novel use of the empirically proven 3/2 stochastic volatility model with a time-dependent drift in which we are free to choose the moving reversion target. This model has been shown by many authors to empirically outperform other volatility models in maximising model fit. We also improve the valuation of European options under the Heston volatility and Cox, Ingersoll, Ross interest rate model, recently published in the literature, by replacing open-form infinite series with closed-form analytic expressions. For completeness, we also add a fuller covariance structure in this setting and detail closed-form valuations for options. The inclusion of nonzero covariances amongst the factors can significantly improve option pricing by allowing for a wider variety of market behaviour. The solutions are derived by firstly formulating the price of a European call option in terms of the corresponding characteristic function of the underlying price and then determining a partial differential equation for the characteristic function. By including empirically proven models into our analysis, the options formulae could provide more realistic prices for investors and practitioners.

Appendices

Appendix A: Series Solution (4.4)

The coefficients of the first few terms of $E= \sum _{n=0}^{\infty } \psi _n \tau ^n$ as in Eq. (4.4) with $c= {d-(a_7\rho \sigma -k)\over \sigma ^2}, \ a_1 = -\alpha +a_7\gamma \eta , \ a_5 = \gamma \chi a_7,\ a_2 = a_7(1-{\gamma ^2\over 2}) -{\phi ^2\gamma ^2\over 2}, \ \beta _4 = \gamma \eta a_7, \ a_7 = j\phi$, found using the mathematics package Maple (see [30]) are listed below:

$$\begin{aligned} \psi _1= & {} a_2 \\ \psi _2= & {} -{1\over 2(-1+g)}(-a_5cd+a_1a_2-a_1a_2g-a_1+a_1g-\alpha +\alpha g) \\ \psi _3= & {} {1\over 6(-1+g)^2}\left[ 2 \chi \eta c d-a_1 a_5 c d-2 a_1^2 a_2 g\right. \\&+\,a_1^2 a_2 g^2+4 \eta ^2 a_2 g-2 \eta ^2 a_2 g^2-2 \eta ^2 a_2^2 g+\eta ^2 a_2^2 g^2-a_5 c d^2 \\&+\,a_1^2 a_2-a_1^2+2 a_1^2 g-a_1^2 g^2+\eta ^2 a_2^2-2 \alpha ^2 g+\alpha ^2 g^2-2 \eta ^2 a_2\\&\left. +\,\alpha ^2+\chi ^2 c^2 d^2-2 \eta a_2 \chi c d-2 \chi \eta c d g+a_1 a_5 c d g-a_5 c d^2 g+2 \eta a_2 \chi c d g\right] \end{aligned}$$

$$\begin{aligned} \psi _4= & {} -{1\over 48(-1+g)^3}\left[ 6 \eta ^2 \alpha +6 \chi \eta c d^2-2 a_5 c d^3 g^2\right. \\&-\,18 \eta ^2 \alpha g+18 \eta ^2 \alpha g^2-6 \eta ^2 \alpha g^3-3 \alpha ^3-3 m^2 a_1 g \\&+\,3 m^2 a_1 g^2-m^2 a_1 g^3-3 m^2 \alpha g+3 m^2 \alpha g^2-m^2 \alpha g^3+3 a_1 \alpha ^2 g-3 a_1 \alpha ^2 g^2\\&+\,a_1 \alpha ^2 g^3-2 a_5 c d^3+9 \alpha ^3 g-9 \alpha ^3 g^2+3 \alpha ^3 g^3+m^2 a_1+m^2 \alpha -a_1 \alpha ^2+6 \chi ^2 c^2 d^3\\&+\,6 \chi ^2 c^2 d^3 g-6 \chi \eta c d^2 g^2-6 \eta a_2 \chi c d^2-8 a_5 c d^3 g+6 \eta a_2 \chi c d^2 g^2-18 \eta ^2 a_1 g\\&+\,18 \eta ^2 a_1 g^2-6 \eta ^2 a_1 g^3+6 \eta ^2 a_1+2 a_1^3 a_2+6 a_1^3 g-6 a_1^3 g^2+2 a_1^3 g^3-2 a_1^3 -20 a_1 \chi \eta c d g\\&+\,10 a_1 g^2 \chi \eta c d+12 a_5 c d \eta ^2 a_2 g-6 a_5 c d \eta ^2 a_2 g^2-6 a_5 c^2 d^2 \chi \eta g\\&-\,10 a_1 a_2 \chi \eta c d-12 \eta ^2 a_5 c d g+6 \eta ^2 a_5 c d g^2-6 a_5 c d \eta ^2 a_2+10 a_1 \chi \eta c d-2 a_1 g \chi ^2 c^2 d^2\\&+\,2 a_1 g^2 a_5 c d^2+6 a_5 c^2 d^2 \chi \eta +4 a_5 c d a_1^2 g-2 a_5 c d a_1^2 g^2\\&+\,6 \eta ^2 a_5 c d+16 a_1 g^3 \eta ^2 a_2-2 a_5 c d a_1^2-24 a_1 a_2^2 \eta ^2 g\\&+\,24 a_1 a_2^2 \eta ^2 g^2-8 a_1 a_2^2 g^3 \eta ^2+48 a_1 \eta ^2 a_2 g-48 a_1 \eta ^2 a_2 g^2-2 a_1 a_5 c d^2\\&+\,2 a_1 \chi ^2 c^2 d^2+20 a_1 a_2 \chi \eta c d g-10 a_1 a_2 g^2 \chi \eta c d-6 a_1^3 a_2 g+6 a_1^3 a_2 g^2\\&\left. +\,8 a_1 a_2^2 \eta ^2-2 a_1^3 a_2 g^3-16 a_1 \eta ^2 a_2 \right] . \end{aligned}$$

Appendix B: Series Representations

The series for the functions W, P, Q and R as given in (4.9)–(4.11) with $c= {d-(a_7\rho \sigma -k)\over \sigma ^2}, \ a_1 = -\alpha +a_7\gamma \eta , \ a_5 = \gamma \chi a_7,\ a_2 = a_7(1-{\gamma ^2\over 2}) -{\phi ^2\gamma ^2\over 2}, \ \beta _4 = \gamma \eta a_7, \ a_7 = j\phi$, are given by

$$\begin{aligned} W(\tau )=\sum _{i=0}^{\infty } w_i \tau ^i, \ P(\tau )=\sum _{i=0}^{\infty } p_i \tau ^i,\ Q(\tau )=\sum _{i=0}^{\infty } q_i \tau ^i,\ R(\tau )=\sum _{i=0}^{\infty } r_i \tau ^i, \end{aligned}$$

where

$$\begin{aligned} w_i= & {} {1\over i!}[2g(-m+\alpha )d^i+g^2(m-\alpha )(2d)^i\\&+\,(\alpha +m)m^i -2g(m+\alpha )(m+d)^i+g^2(m+\alpha )(m+2d)^i];\\ w_0= & {} w_0+m-\alpha ;\\ p_i= & {} {1\over i!}\left\{ \left[ -4 a_1 g-\chi ^2 c^2 m-a_5 c m-2 \eta \chi c-2 a_2 m g+2 a_2 \alpha g\right. \right. \\&\left. +\,a_5 c \alpha +\chi ^2 c^2 \alpha -2 \eta \chi c g-a_5 c g m+a_5 c g \alpha -4 \alpha g\right] d^i\\&+\, \left[ 2 a_1 g^2-a_2 \alpha g^2-(1/2) \chi ^2 c^2 \alpha +(1/2) \chi ^2 c^2 m+a_2 m g^2\right. \\&\left. +\,2 \eta \chi c g+a_5 c g m-a_5 c g \alpha +2 \alpha g^2\right] (2d)^i\\&+\,\left[ a_2 m+a_2 \alpha +(1/2) \chi ^2 c^2 m+a_5 c m-2 \eta \chi c\right. \\&\left. +\,a_5 c \alpha +(1/2) \chi ^2 c^2 \alpha -2 a_1-2 \alpha \right] m^i\\&+\,\left[ 4 a_1 g-\chi ^2 c^2 m-a_5 c m+2 \eta \chi c-2 a_2 m g-2 a_2 \alpha g-a_5 c \alpha \right. \\&\left. -\,\chi ^2 c^2 \alpha +2 \eta \chi c g-a_5 c g m-a_5 c g \alpha +4 \alpha g\right] (m+d)^i \\&+\,\left. \left[ -2 a_1 g^2+(1/2) \chi ^2 c^2 m+a_2 m g^2+a_2 \alpha g^2+(1/2) \chi ^2 c^2 \alpha \right. \right. \\&\left. \left. -\,2 \eta \chi c g+a_5 c g m+a_5 c g \alpha -2 \alpha g^2\right] (m+2d)^i \right\} ; \\ p_0= & {} p_0+\left[ 2 \alpha +a_5 c m+(1/2) \chi ^2 c^2 m+a_2 m+2 a_1\right. \\&\left. -\,a_5 c \alpha -(1/2) \chi ^2 c^2 \alpha +2 \eta \chi c-a_2 \alpha \right] ; \\ q_i= & {} {1\over i!}\left\{ \left[ -4 \eta ^2 g-2 a_1 m g+2 a_1 \alpha g-\eta \chi c m g\right. \right. \\&\left. \left. +\,\eta \chi c \alpha g-\eta \chi c m+\eta \chi c \alpha \right] d^i \right. \\&+\,\left[ -g \left( -a_1 m g+a_1 \alpha g-2 \eta ^2 g\right. \right. \\&\left. \left. +\,\eta \chi c \alpha -\eta \chi c m\right) \right] (2d)^i \\&+\,\left[ a_1 m+a_1 \alpha +\eta \chi c \alpha +\eta \chi c m-2 \eta ^2\right] m^i \\&+\,\left[ 4 \eta ^2 g-\eta \chi c \alpha -\eta \chi c m-\eta \chi c \alpha g-\eta \chi c m g-2 a_1 \alpha g-2 a_1 m g\right] (m+d)^i \\&+ \,\left. \left[ g\left( -2\eta ^2 g+a_1 m g+a_1\alpha g+\eta \chi c \alpha + \eta \chi c m\right) \right] (m+2d)^i \right\} ;\\ q_0= & {} q_0+\left[ \eta \chi c m-\eta \chi c \alpha +2 \eta ^2+a_1 m-a_1 \alpha \right] ; \\ r_i= & {} {\eta ^2\over 2}w_i. \end{aligned}$$

Appendix C: A Note on the 3/2 Stochastic Volatility Model with a Full Correlation Structure

With $\delta =1, \ p={3\over 2}$ and letting $k=a, \theta = A/a$, so that the volatility dynamics is

$$\begin{aligned} \mathrm{d}v= v(A-a v)\mathrm{d}t + \sigma v^{3/2} \mathrm{d}Z_2 +\chi \sqrt{r} \mathrm{d}Z_3, \end{aligned}$$

we need to solve (4.2) for the corresponding characteristic function. If we let

$$\begin{aligned} f(\phi ; \tau ,T, y,v,r) = {\mathrm{e}}^{ j\phi y} N(\tau , r, v) \end{aligned}$$

then on substituting into (4.2), we find that N needs to satisfy

$$\begin{aligned} {\partial N\over \partial \tau }= & {} \left( {\sigma ^2 v^{3}+\chi ^2 r\over 2}\right) {\partial ^2 N\over \partial v^2}+ {\eta ^2 r\over 2} {\partial ^2 N\over \partial r^2} + \left\{ \alpha \beta + r\left[ \gamma \eta a_7 -\left( \alpha + B(\tau ) \eta ^2\right) \right] \right\} {\partial N\over \partial r} \nonumber \\&+ \,\chi \eta r {\partial ^2 N\over \partial r \partial v} + \left[ v^2(-a+\rho \sigma a_7)+Av+r(-\chi \eta B(\tau )+\gamma \chi a_7)\right] {\partial N\over \partial v} \nonumber \\&+ \,\left\{ v\left[ -{\phi ^2\over 2}-{a_7\over 2}\right] + r \left[ -{\gamma ^2\phi ^2\over 2} + a_7 \left( 1-\gamma \eta B(\tau ) -{\gamma ^2\over 2}\right) \right] \right\} N, \end{aligned}$$

(C.1)

where $a_7 = j\phi$ and subject to $\tau =0, N= 1.$ With $\chi \ne 0$, (C.1) does not admit solutions of the form $F(r,\tau )G(v,\tau )$ or any Lie symmetries apart from the usual superposition and scaling symmetries of linear PDEs.