Foreign exchange option pricing in the currency cycle with jump risks

Abstract

This paper examines regime switching behavior and the nature of jumps in foreign exchange rates, as well as their implications in currency option pricing. Considering the characteristics of long swing as well as the short term jumps in exchange rates, we adopt the regime-switching model with jump risks to capture the movement of exchange rates in the developed and emerging countries. Our results show that ‘high-variance’ and ‘low-variance’ describes most of our sample currencies’ trajectories. The regime-switching model with jump risks is proven to capture better exchange rate changes than the regime-switching model (RSM) and the Black–Scholes model (BSM). In addition, our results show that the currency option pricing model when considering regimes of high-variance or low-variance states as well as the jump nature of exchange rates, is better than the traditional BSM and RSM.

Appendix: Change of measure in Esscher transform and currency option pricing under the RSMJ model

Suppose the exchange rate follows the RSMJ with Eq. (19), and there is no risk premium occurring during the transition of the states, h _B and h _J are the parameters of the Brownian motion and the systematic jump risks in the Esscher transform, respectively. $h_{B} = \left[ {\begin{array}{*{20}c} {h_{B1} } & 0 \\ 0 & {h_{B2} } \\ \end{array} } \right]$; where h _B1 and h _B2 refer to the h _B under the states 1 and 2, respectively. Because the jump risk is non-diversifiable, the exchange rate dynamics under the physical probability measure can be transferred to that under the risk neutral probability measure by the Esscher transform in this paper.

Let

$$A = \left\{ {\left( {\mu_{{q_{t} }} - \frac{1}{2}\sigma_{{q_{t} }}^{2} } \right)s + \sigma_{{q_{t} }} W(s)} \right\}\quad {\text{and}}\;B = \sum\limits_{m = 1}^{N(s)} {\ln Y_{m} } .$$

(27)

The Randon–Nikodym derivatives of the Brownian motion and the systematic jumps risk term, η _B and η _J are derived as follows:

$$\eta_{B} = \left[ {\begin{array}{*{20}c} {\exp \left\{ { - \frac{{h_{B1}^{2} \sigma_{1}^{2} s}}{2} + h_{B1} \sigma_{1} W(s)} \right\}} & 0 \\ 0 & {\exp \left\{ { - \frac{{h_{B2}^{2} \sigma_{2}^{2} s}}{2} + h_{B2} \sigma_{2} W(s)} \right\}} \\ \end{array} } \right],$$

where

$$\eta_{{Bq_{t} }} = \frac{{e^{{h_{{Bq_{t} }} A}} }}{{E(e^{{h_{{Bq_{t} }} A}} )}} = \frac{{\exp \left\{ {h_{{Bq_{t} }} (\mu_{{q_{t} }} - \frac{1}{2}\sigma_{{q_{t} }}^{2} )s + h_{{Bq_{t} }} \sigma_{{q_{t} }} W(s)} \right\}}}{{\exp \left\{ {h_{{Bq_{t} }} (\mu_{{q_{t} }} - \frac{1}{2}\sigma_{{q_{t} }}^{2} )s + \frac{{h_{{Bq_{t} }}^{2} \sigma_{{q_{t} }}^{2} s}}{2}} \right\}}} = \exp \left\{ { - \frac{{h_{{Bq_{t} }}^{2} \sigma_{{q_{t} }}^{2} s}}{2} + h_{{Bq_{t} }} \sigma_{{q_{t} }} W(s)} \right\}$$

and

$$\eta_{J} = \prod\limits_{m = 0}^{N(s)} {Y_{m}^{{h_{J} }} } \cdot e^{{ - \lambda s\xi^{{(h_{J} )}} }} .$$

Because the Brownian motion term and the systematic jumps risk term are independent from each other, the Brownian motion under the risk neutral measure can be transferred from that under the physical measure by change of measure:

$$dQ(W^{Q} (s)) = dP(W(s)) \cdot \eta_{{Bq_{t} }} = \frac{1}{{\sqrt {2\pi s} }}\exp \left\{ { - \frac{{\left[ {W(s) - h_{{Bq_{t} }} \sigma_{{q_{t} }} s} \right]^{2} }}{2s}} \right\}.$$

(28)

Next, we deal with the jump terms in the Esscher transform, which can be separated into two parts: jump sizes and jump frequencies. Given the number of jumps, n, the jump terms under the risk neutral measure can be rewritten from those under the physical measure by change of measure:

$$\begin{aligned} dQ(N^{Q} (s) & = n,\ln Y_{1}^{Q} , \ldots ,\ln Y_{n}^{Q} ) \\ & = dP(N^{Q} (s) = n)dP(\ln Y_{1}^{Q} , \ldots ,\ln Y_{n}^{Q} ) \cdot \prod\limits_{m = 1}^{N(s)} {Y_{m}^{{h_{J} }} } \cdot e^{{ - \lambda s\xi^{{(h_{J} )}} }} \\ & = dP(N^{Q} (s) = n)\left\{ {\prod\limits_{m = 1}^{n} {\left[ {f(\ln Y_{m} )\exp (h_{J} \ln Y_{m} )} \right]} } \right\}\left\{ {E\left[ {\exp (h_{J} \ln Y_{m} )} \right]} \right\}^{ - n} \left\{ {E\left[ {\exp (h_{J} \ln Y_{m} )} \right]} \right\}^{n} e^{{ - \lambda s\xi^{{(h_{J} )}} }} \\ \end{aligned}$$

(29)

According to the assumption of independent jump sizes, the individual jump size under the risk neutral measure is distributed as follows:

$$f^{Q} (\ln Y_{m} ) = \frac{1}{{\sqrt {2\pi \sigma_{y}^{2} } }}\exp \left\{ { - \frac{{\left[ {\ln Y_{m} - (\mu_{y} + h_{J} \sigma_{y}^{2} )} \right]^{2} }}{{2\sigma_{y}^{2} }}} \right\}.$$

(30)

Next, the probability of the number of jumps n under the risk neutral measure is evaluated as follows:

$$dQ(N^{Q} (s) = n) = \frac{{e^{{ - \lambda s\left( {\xi^{{(h_{J} )}} + 1} \right)}} \left[ {\lambda s\left( {\xi^{{(h_{J} )}} + 1} \right)} \right]^{n} }}{n!}.$$

(31)

Therefore, the jump sizes under the risk neutral measure follow a normal distribution with $\ln Y_{m}^{Q} \sim N(\mu_{y} + h_{J} \sigma_{y}^{2} ,\sigma_{y}^{2} )$, and the number of jumps under the risk neutral measure follows a Poisson process with an arrival rate $\lambda s\left( {\xi^{{(h_{J} )}} + 1} \right)$.

The Esscher transform parameters have to satisfy the martingale condition, which is derived under the RSMJ dynamics of the exchange rate with the risk neutral measure:

$$\mu_{{q_{t} }} - r + h_{{Bq_{t} }} \sigma_{{q_{t} }}^{2} + \lambda \left( {\xi^{{(h_{J} + 1)}} - \xi^{{(h_{J} )}} - \xi^{(1)} + \xi^{(0)} } \right) = 0.$$

(32)

From the infinite solutions of the Esscher parameters, we can find one condition to satisfy the martingale condition:

$h_{B} = \left[ {\begin{array}{*{20}c} {\frac{{r - \mu_{1} }}{{\sigma_{1}^{2} }}} & 0 \\ 0 & {\frac{{r - \mu_{2} }}{{\sigma_{2}^{2} }}} \\ \end{array} } \right]$ and h _J  = 0.

Given the jump size and the jump frequency under the risk neutral measure is:

$\ln Y_{m}^{Q} \sim N(\mu_{y} ,\sigma_{y}^{2} )$ and $N^{Q} (s)\sim Poi(\lambda s)$.

Under the risk neutral measure (also called the Merton measure, 1976), the dynamic process of exchange rates under the RSMJ can be solved in the following form:

$$\begin{aligned} S(T) & = \left( {S\left( 0 \right)\exp \left\{ {\int\limits_{0}^{T} {(r - r_{f} - \xi^{(1)} + \xi^{(0)} )dt} - \frac{1}{2}\int\limits_{0}^{T} {\sigma_{{q_{t} }}^{2} dt} + \int\limits_{0}^{T} {\sigma_{{q_{t} }} dW^{Q} \left( t \right)} + \sum\limits_{m = 1}^{n} {\ln Y_{m}^{Q} } } \right\}} \right) \\ & \mathop = \limits^{dist} S(0)\exp \left\{ {\left( {r - r_{f} - \xi^{(1)} + \xi^{(0)} } \right)T - \frac{1}{2}\theta_{{k_{1} }}^{2} + \theta_{{k_{1} }} Z + \sum\limits_{m = 1}^{n} {\ln Y_{m}^{Q} } } \right\}, \\ \end{aligned}$$

(33)

where Z is an identically independent standard normal distribution and the weighted variance under the state q _i is$\theta_{{k_{1} }}^{2} = k_{1} \sigma_{1}^{2} + (T - k_{1} )\sigma_{2}^{2}$.$\sum\nolimits_{m = 1}^{n} {\ln Y_{m}^{Q} }$ and $\theta_{{k_{1} }} Z_{n} + \sum\nolimits_{m = 1}^{n} {\ln Y_{m}^{Q} }$ also follow normal distributions:$\sum\nolimits_{m = 1}^{n} {\ln Y_{m}^{Q} } \sim N\left( {n\mu_{y} ,n\sigma_{y}^{2} } \right)$ and $\theta_{{k_{1} }} Z + \sum\nolimits_{m = 1}^{n} {\ln Y_{m}^{Q} } \sim N\left( {n\mu_{y} ,\theta_{{k_{1} }}^{2} + n\sigma_{y}^{2} } \right)$.Therefore, we can price a European-style call option, $C_{RSMJ,T}^{{}} (0)$ with strike price K, the local and foreign risk-free interest rates, r and r _f , and maturity date T as follows:

$$\begin{aligned} C_{RSMIJ} (0) & = e^{ - rT} E_{N(T)}^{Q} E_{D}^{Q} E_{{q_{0} }}^{Q} E^{Q} \left[ {\hbox{max} \left\{ {S(T) - K,0} \right\}\left| {N^{Q} (T) = n,q_{0} = i,D = k_{1} } \right.} \right] \\ & = E_{N(T)}^{Q} E_{D}^{Q} E_{{q_{0} }}^{Q} \left\{ {S(0)e^{ - rT} E^{Q} \left[ {e^{{\ln \frac{S(T)}{S(0)}}} I_{{\left\{ {\ln \frac{S(T)}{S(0)} > \ln \frac{K}{S(0)}} \right\}}} \left| {N^{Q} (T) = n,q_{0} = i,D = k_{1} } \right.} \right]} \right\} \\ & - Ke^{ - rT} E_{N(T)}^{Q} E_{D}^{Q} E_{{q_{0} }}^{Q} \left\{ {E^{Q} \left[ {I_{{\left\{ {\ln \frac{S(T)}{S(0)} > \ln \frac{K}{S(0)}} \right\}}} \left| {N^{Q} (T) = n,q_{0} = i,D = k_{1} } \right.} \right]} \right\} \\ & = E_{N(T)}^{Q} E_{D}^{Q} E_{{q_{0} }}^{Q} \left\{ {A - B} \right\}. \\ \end{aligned}$$

(34)

The term A of Eq. (34) can be represented as:

$$A = S(0)e^{ - rT} E^{Q} \left[ {e^{{\ln \frac{S(T)}{S(0)}}} I_{{\left\{ {\ln \frac{S(T)}{S(0)} > \ln \frac{K}{S(0)}} \right\}}} \left| {N^{Q} (T) = n,q_{0} = i,D = k_{1} } \right.} \right] = S(0)e^{{ - r_{f} T + n\left( {\mu_{y} + \frac{{\sigma_{y}^{2} }}{2}} \right) - \lambda (\zeta^{(1)} - 1)T}} N(d_{{1k_{1} }} )$$

where $N( \cdot )$ is the cumulative distribution function for a standard normal random variable,

$$d_{{1k_{1} }} = \frac{{\ln \frac{S(0)}{K} + (r - r_{f} - \lambda (\zeta^{(1)} - 1))T + \frac{1}{2}\theta_{{k_{1} }}^{2} + n(\mu_{y} + \sigma_{y}^{2} )}}{{\sqrt {\theta_{{k_{1} }}^{2} + n\sigma_{y}^{2} } }}.$$

The term B can be shown as:

$$Ke^{ - rT} E^{Q} \left[ {I_{{\left\{ {\ln \frac{S(T)}{S(0)} > \ln \frac{K}{S(0)}} \right\}}} \left| {N^{Q} (T) = n,q_{0} = i,D = k_{1} } \right.} \right] = Ke^{ - rT} N(d_{{2k_{1} }} ),$$

where

$$d_{{2k_{1} }} = \frac{{\ln \frac{S(0)}{K} + (r - r_{f} - \lambda (\zeta^{(1)} - 1))T - \frac{1}{2}\theta_{{k_{1} }}^{2} + n\mu_{y} }}{{\sqrt {\theta_{{k_{1} }}^{2} + n\sigma_{y}^{2} } }} = d_{{1k_{1} }} - \sqrt {\theta_{{k_{1} }}^{2} + n\sigma_{y}^{2} } .$$

Given the steady-state probability of the initial market state, π _i ; the probability that over the T periods, k ₁ periods are assigned to the state 1 conditional on the initial market state i, $\gamma_{{T,k_{1} \left| {q_{0} = i} \right.}}$, we can further rewrite the pricing equation as:

$$C_{RSMJ,T} (0) = \sum\limits_{n = 0}^{\infty } {\sum\limits_{{k_{1} = 0}}^{T} {\sum\limits_{i = 1}^{2} {\pi_{i} \cdot \gamma_{{T,k_{1} \left| {q_{0} = i} \right.}} \frac{{e^{ - \lambda *} \left( {\lambda_{{}}^{*} } \right)^{n} }}{n!}\left[ {S(0)e^{{ - r_{f} T + n\left( {\mu_{y} + \frac{{\sigma_{y}^{2} }}{2}} \right) - \lambda (\zeta^{(1)} - 1)T}} N(d_{{1,k_{1} }} ) - Ke^{ - rT} N(d_{{2,k_{1} }} )} \right]} } } ,$$

(35)

where $\lambda^{ * } = \lambda T$,

$$d_{{1,k_{1} }} = \frac{{\ln \frac{S(0)}{K} + (r - r_{f} - \lambda (\zeta^{(1)} - 1))T + \frac{1}{2}\theta_{{k_{1} }}^{2} + n(\mu_{y} + \sigma_{y}^{2} )}}{{\sqrt {\theta_{{k_{1} }}^{2} + n\sigma_{y}^{2} } }},$$

$d_{{2,k_{1} }} = d_{{1,k_{1} }} - \sqrt {\theta_{{k_{1} }}^{2} + n\sigma_{y}^{2} }$, and N(x) is the cumulative distribution function for a standard normal random variable with upper integral limit x.