Pricing currency options under double exponential jump diffusion in a Markov-modulated HJM economy

Abstract

Extending the framework of Amin and Jarrow (J Int Money Financ 10:310–329, 1991) and Bo et al. (Insur Math Econ 46:461–469, 2010), this study provides a theoretical exploration of currency options pricing under the presence of interest-rate regime shifts and exchange-rate asymmetric jumps. Evidence of interest-rate regime shifts inferred from UK and US zero coupon bond yields provides support for the regime-switching specifications which we reflect upon the domestic and foreign forward rates. Results of statistical tests conducted on JPY/USD and EUR/USD FX rates provide further support the rationale behind using a double exponential jump diffusion process within a Markov modulated Heath–Jarrow–Morton economy. Our numerical results suggest that, the pricing performance of our model is closely comparable to the Bo-Wang-Yang model for at-the-money options, yet yields improvements in percentage root mean errors for in-the-money options.

Appendices

Appendix 1

Let ${\mathbf{U}}$ be a diagonal matrix with vector elements ${\mathbf{\varsigma }} = (\varsigma_{1} ,\varsigma_{2} , \ldots ,\varsigma_{n} )$. Then, for any ${\mathbf{\varsigma }}$, the moment generation function of ${\varvec{\updelta}}(0,t)$ is given by

$$E\left[ {{{\text{exp}}} \left( {{\mathbf{\varsigma }} \cdot {\varvec{\updelta}}(0,t)} \right)\left| {X(0)} \right.} \right] = {{\text{exp}}} \left[ {({\varvec{\Psi}} + {\mathbf{U}})t} \right]X(0) \cdot {\mathbf{I}} = H({\varvec{\Psi}})$$

(5.1)

where ${\varvec{\Psi}}$ is a $n \times n$ transition-rate matrix, ${\mathbf{I}} = (1, \, 1, \ldots ,1) \in {\mathbb{R}}^{n}$, $\left( \cdot \right)$ represents an inner product, and $H\left( {\varvec{\Psi}} \right)$ is a moment generation function which completely characterize the probability density function.

Appendix 2

Suppose $\left\{ {\zeta_{1} ,\zeta_{2} , \ldots } \right\}$ is a sequence of i.i.d exponential random variables with parameters $\eta_{1}$ and $\eta_{2}$. Then for every $n \ge 1$, the tail probabilities are given by

$$\begin{aligned} & {\mathbf{\mathbb{P}}}\left( {{\mathcal{N}}(0,\sigma (t)^{2} ) + \sum\limits_{i = 1}^{n} {\zeta_{i}^{ + } \ge x} } \right) \\ & = \frac{{(\sigma (t)\eta_{1} )^{n} }}{{\sigma (t)\sqrt {2\pi } }}{{\text{exp}}} \left( {\frac{1}{2}(\sigma (t)\eta_{1} )^{n} } \right)\int\limits_{x}^{\infty } {{{\text{exp}}} \left( { - u\eta_{1} } \right)Hh_{n - 1} \left( { - \frac{u}{\sigma (t)} + \sigma (t)\eta_{1} } \right)du} , \\ \end{aligned}$$

(6.1)

$$\begin{aligned} & {\mathbf{\mathbb{P}}}\left( {{\mathcal{N}}(0,\sigma^{2} (t)) - \sum\limits_{i = 1}^{n} {\zeta_{i}^{ - } \ge x} } \right) \\ & = \frac{{(\sigma (t)\eta_{2} )^{n} }}{{\sigma (t)\sqrt {2\pi } }}{{\text{exp}}} \left( {\frac{1}{2}(\sigma (t)\eta_{2} )^{n} } \right)\int\limits_{x}^{\infty } {{{\text{exp}}} \left( { - u\eta_{2} } \right)Hh_{n - 1} \left( { - \frac{u}{\sigma (t)} + \sigma (t)\eta_{2} } \right)du,} \\ \end{aligned}$$

(6.2)

where $Hh_{n} (x) = \frac{1}{n!}\int_{x}^{\infty } {(u - x)^{n} } {{\text{exp}}} \left( {\frac{{ - u^{2} }}{2}} \right)du$, for $n = 0,1,2, \ldots$

Therefore, given (6.1) and (6.2), we get

$$\begin{aligned} & \varUpsilon \left( {\mu (t),\sigma (t), \, \lambda t, \, p_{E} , \, \eta_{1} , \, \eta_{2} ; \, \alpha } \right) = {\mathbf{\mathbb{P}}}\left( {\mu (t) + \sigma (t){\mathcal{N}}(0,1) + \sum\limits_{j = 1}^{n} {Z_{E,j} \ge \varpi } } \right) \\ & =\varPi (0){\mathbf{\mathbb{P}}}\left( {\mu (t) + \sigma (t){\mathcal{N}}(0,1) \ge \alpha } \right) + \\ & \sum\limits_{n = 1}^{\infty } \varPi (n;\lambda t)\sum\limits_{k = 1}^{n} {P(n,k){\mathbf{\mathbb{P}}}\left( {\mu (t) + \sigma (t){\mathcal{N}}(0,1) + \sum\limits_{j = 1}^{n} {\zeta_{j}^{ + } \ge \varpi } } \right)} \\ & + \sum\limits_{n = 1}^{\infty } \varPi (n;\lambda t)\sum\limits_{k = 1}^{n} {Q(n,k){\mathbf{\mathbb{P}}}\left( {\mu (t) + \sigma (t){\mathcal{N}}(0,1) - \sum\limits_{j = 1}^{n} {\zeta_{j}^{ - } \ge \varpi } } \right),} \\ \end{aligned}$$

(6.3)

where

$$\varPi \left( {n;\lambda t} \right) = {{\text{exp}}} \left( { - \lambda t} \right)\frac{{\left( {\lambda t} \right)^{n} }}{n!},$$

$$P(n,k) = \sum\limits_{i = k}^{n - 1} {\left( {\begin{array}{*{20}c} {n - k - 1} \\ {i - k} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)} \left( {\frac{{\eta_{1} }}{{\eta_{1} + \eta_{2} }}} \right)^{i - k} \left( {\frac{{\eta_{2} }}{{\eta_{1} + \eta_{2} }}} \right)^{n - i} p_{E}^{i} \left( {1 - p_{E}^{{}} } \right)^{n - i} ,$$

$$Q(n,k) = \sum\limits_{i = k}^{n - 1} {\left( {\begin{array}{*{20}c} {n - k - 1} \\ {i - k} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)} \left( {\frac{{\eta_{1} }}{{\eta_{1} + \eta_{2} }}} \right)^{n - i} \left( {\frac{{\eta_{2} }}{{\eta_{1} + \eta_{2} }}} \right)^{i - k} p_{E}^{n - i} \left( {1 - p_{E} } \right)^{i} .$$

Appendix 3 (Bo-Wang-Yang model (BWY), 2010)

The regime-switching jump diffusion model under the risk neutral measure is described by:

$$\frac{dS(t)}{S(t - )} = r^{D} (X(t)) - r^{F} (X(t))dt + \sigma_{J} (X(t))dW(t)\; + \left( {{{\text{exp}}} (Z_{J,n} ) - 1} \right)dN_{J} (t;\lambda (X(t))),$$

(7.1)

where $Z_{J,n}$ is a normally distributed with mean $\mu_{Z, \, J}$ and variance $\sigma_{Z,J}^{2}$.

Thus,

$$C_{BWY} (\left. t \right|{\mathcal{H}} (t )) = \int\limits_{t}^{T} {\sum\limits_{j = 1}^{2} {\sum\limits_{n = 0}^{\infty } {\varPi \left( {n;\lambda_{{}}^{*} \delta_{j}^{D} } \right)} C_{JD} \left( {\left. {n,\delta_{j}^{D} } \right|{\mathcal{H}} (t )} \right)} } \varphi \left( {\delta_{j}^{D} } \right)d\delta_{j}^{D} ,$$

where

$$\begin{aligned} & C_{JD} \left( {\left. {n,\delta_{j}^{D} } \right|{\mathcal{H}} (t )} \right) = \\ & S(t){{\text{exp}}} \left( { - r^{f} \delta_{j}^{D} } \right){\mathfrak{N}}\left( {d_{JD,1} (t, \, T)} \right) - K{{\text{exp}}} \left( { - r^{D} \delta_{j}^{D} } \right){\mathfrak{N}}\left( {d_{JD,2} (t, \, T)} \right), \\ \end{aligned}$$

$$d_{JD,2} (t, \, T) = \frac{{\ln \left( {\frac{S(t)}{K}\frac{{{{\text{exp}}} \left( { - r^{f} \delta_{j}^{D} } \right)}}{{{{\text{exp}}} \left( { - r^{D} \delta_{j}^{D} } \right)}}} \right) - \frac{1}{2}\left( {\sigma_{S} \delta_{j}^{D} + n\sigma_{J} } \right)}}{{\sqrt {\sigma_{S} \delta_{j}^{D} (t,T) + n\sigma_{J} } }},$$

$$d_{JD,1} (t, \, T) = d_{JD,2} (t, \, T) + \sqrt {\sigma_{S} \delta_{j}^{D} (t,T) + n\sigma_{J} } ,$$

where $\varPi \left( {n;\lambda_{{}}^{*} t} \right)$ is the probability density function for a Poisson process with intensity $\lambda_{{}}^{*} t$, and ${\mathfrak{N}}( \cdot )$ represents the cumulative normal distribution function.

Appendix 4

Further results on the first fourth moments, which include the mean, standard, skewness, and kurtosis of the sample data, the JD model, and the double exponential jump diffusion model, for the 100JPY/USD and the EUR/USD spot-FX rates are shown in Table 6.

Table 6 shows that the skewness and Kurtosis of the double exponential jump diffusion model most closely matches the sample data.