Explicit pricing formulas for vulnerable path-dependent options with early counterparty credit risk

Abstract

A path-dependent option is an exotic option, the value of which relies on the path of an asset, as well as the price of the underlying asset throughout all or part of the life of the option. Since the global financial crisis, in the over-the-counter markets, recognizing the importance of credit default risk that arises from the option’s trade has become necessary to consider an early counterparty credit risk while deriving the option price. In this article, we obtain the explicit-closed form formula for the vulnerable path-dependent options with the early counterparty default risk, by utilizing the method of double images and the Mellin transform approach. Furthermore, we investigate the pricing accuracy of the option price by comparing our closed-form solutions with Monte Carlo prices.

Appendix A: Benchmark solutions for VBOEDR and VLOEDR

Remark 2

(Appendix A.1.) (Benchmark solution for VBOEDR) The pricing formula of vulnerable down-and-out put option $\textrm{VBO}(t,x,v)$ is given by

$$\begin{aligned} \begin{aligned} \textrm{VBO}(t,x,v)&= Ke^{-r(T-t)}\bigg (\Phi _2(\Lambda _{x/B}^{+-}, \Lambda _{v/D^*}^{+-}; \rho ) - \Phi _2(\Lambda _{x/K}^{+-}, \Lambda _{v/D^*}^{+-}; \rho ) \bigg ) \\&\quad -x \bigg (\Phi _2(\Lambda _{x/B}^{++}, \Lambda _{v/D^*}^{+-} + \rho \sigma _X \sqrt{T-t} ; \rho ) \\&\quad - \Phi _2( \Lambda _{x/K}^{++} + \rho \sigma _X \sqrt{T-t} , \Lambda _{v/D^*}^{+-}+\ \rho \sigma _X \sqrt{T-t} ;\rho ) \bigg ) \\&\quad + K \delta v \bigg ( \Phi _2 ( \Lambda _{x/B}^{+-}+ \rho \sigma _V \sqrt{T-t}, \Lambda _{D^*/v}^{-+}; -\rho ) \\&\quad - \Phi _2(\Lambda _{x/K}^{+-}+\rho \sigma _V \sqrt{T-t}, \Lambda _{D^*/v}^{-+}; -\rho ) \bigg ) \\&\quad - \delta e^{\rho \sigma _X \sigma _V + r }xv \bigg ( \Phi _2( \Lambda _{x/B}^{++}+\rho \sigma _V \sqrt{T-t}, \Lambda _{D^*/v}^{-+} - \rho \sigma _X \sqrt{T-t} ; - \rho )\\&\quad - \Phi _2( \Lambda _{x/K}^{++}+\rho \sigma _V \sqrt{T-t}, \Lambda _{D^*/v}^{-+} - \rho \sigma _X \sqrt{T-t} ; - \rho ) \bigg ). \end{aligned} \end{aligned}$$

(5.1)

Remark 3

(Appendix A.2.) (Benchmark solution for VLOEDR) The price of the vulnerable lookback option $\textrm{VLO}=\textrm{VLO}(t,x,v,m)$ is given by

$$\begin{aligned} \begin{aligned} \textrm{VLO}&= me^{-r(T-t)} \Phi _2(\Lambda _{x/m}^{+-}, \Lambda _{D^*/v}^{--} ; -\rho ) - \delta xv \Phi _2(\Lambda _{x/m}^{++}, \Lambda _{D^*/v}^{--}-\rho \sigma _X \sqrt{T-t}; -\rho ) \\&\quad + \delta v(m \Phi _2(\Lambda _{m/x}^{--}-\rho \sigma _V \sqrt{T-t}, \Lambda _{v/D^*}^{++};-\rho ) \\&\quad - xe^{ \rho \sigma _X \sigma _V +r} \Phi _2(\Lambda _{m/x}^{-+}-\rho \sigma _V \sqrt{T-t}, \Lambda _{v/D^*}^{++} + \rho \sigma _X \sqrt{T-t}; \rho )) \\&\quad + \int \limits _1^\infty \left( \dfrac{x}{z}\right) ^{-(k_X-1)} m ^{k_X} \bigg [ e^{-r(T-t)} \Phi _2(\Lambda _{mz/x}^{+-}, \Lambda _{D^*/v}^{--}+\dfrac{2\rho }{\sigma _X \sqrt{T-t}} \ln \left( \dfrac{x}{mz} \right) ; -\rho ) \\&\quad - \delta v \Phi _2(\Lambda _{x/mz}^{--} -\rho \sigma _V \sqrt{T-t}, \Lambda _{v/D^*}^{++}- \dfrac{2 \rho }{\sigma _X \sqrt{T-t}} \ln \left( \dfrac{x}{z} \right) ; \rho ) \bigg ] {\textrm{d}} z. \end{aligned} \end{aligned}$$

(5.2)