Analytic Method for Pricing Vulnerable External Barrier Options

Abstract

External barrier options are financial securities that have two assets for stochastic variables, where the payoff depends on one underlying asset and the barrier depends on another state variable such that it determines whether the option is knocked in or out. In this study, considering the financial derivatives subject to the default risks of the option writer in over-the-counter markets since the global financial crisis of 2007–2008, we study vulnerable external barrier option prices by utilizing multivariate Mellin transforms and the method of images and then examine the behaviors and sensitivities of the vulnerable external barrier option prices in terms of the model parameters. Based on the results obtained, our study has two main contributions. First, by using multivariate Mellin transform approaches, we can find an explicit-form pricing formula for the option prices more effectively and easily, resolving the complexity of calculation of the option prices by using probabilistic or other methods. Second, we verify that our closed-form solution has been accurately and efficiently obtained by comparing the closed-form solution with the Monte Carlo simulation solution.

Appendices

Appendix A. Basic Properties of Triple Mellin Transforms

This appendix summarizes the basic properties of triple Mellin transforms, which are useful for the transformation of PDEs with the given terminal and boundary conditions (refer to Thakur et al. (2016)).

Definition Appendix A.1

The triple Mellin transform is defined as

$$\begin{aligned} \mathcal {M}_{xyz} \, \left[ f(x,y,z); p, q, r \right] \triangleq F(p,q,r)=\int _{0}^{\infty }\int _{0}^{\infty } \int _{0}^{\infty } f(x,y,z) \, x^{p-1}y^{q-1}z^{r-1} \mathrm {d} x \, \mathrm {d} y \, \mathrm {d} z, \end{aligned}$$

(A.1)

where f(x, y, z) is a locally Lebesgue integrable function, and $p,q,r \in \mathbb {C}$. Furthermore, if $a< \mathrm {Re}(p), \mathrm {Re}(q)<b$, and if $c_1$ and $c_2$ are such that $a<c_1<b$ and $a<c_2<b$, respectively, then the inverse of the triple Mellin transform is given as

$$\begin{aligned} f(x,y,z)=&\mathcal {M}_{xyz}^{-1} [ F(p,q,r)]\nonumber \\ =&\frac{1}{(2\pi i)^2}\int _{c_1- i \infty }^{c_1 + i\infty }\int _{c_2- i \infty }^{c_2 + i\infty } \int _{c_{3}-i\infty }^{c_{3}+i\infty }F(p,q,r) x^{-p}y^{-q} z^{-r} \mathrm {d} p \, \mathrm {d}q \, \mathrm {d}r. \end{aligned}$$

(A.2)

We then have the following properties for the derivatives. For $l,m,n \in \{1,2, \cdots \}$,

$$\begin{aligned} \mathcal {M}_{xyz}\left[ \frac{\partial ^l}{\partial x^l}f(x,y,z); p,q,r\right]&=(-1)^l \frac{\Gamma (p)}{\Gamma (p-l)}F(p-l,q,r), \end{aligned}$$

(A.3)

$$\begin{aligned} \mathcal {M}_{xyz}\left[ \frac{\partial ^m}{\partial y^m}f(x,y,z); p,q,r\right]&=(-1)^m \frac{\Gamma (q)}{\Gamma (q-m)}F(p,q-m,r), \end{aligned}$$

(A.4)

$$\begin{aligned} \mathcal {M}_{xyz}\left[ \frac{\partial ^n}{\partial y^n}f(x,y,z); p,q,r\right]&=(-1)^n \frac{\Gamma (r)}{\Gamma (r-n)}F(p,q,r-n), \end{aligned}$$

(A.5)

$$\begin{aligned} \mathcal {M}_{xyz}\left[ \left( x\frac{\partial }{\partial x}\right) ^l f(x,y,z); p,q,r\right]&=(-p)^l F(p,q,r), \end{aligned}$$

(A.6)

$$\begin{aligned} \mathcal {M}_{xyz}\left[ \left( y\frac{\partial }{\partial y}\right) ^m f(x,y,z); p,q,r\right]&=(-q)^m F(p,q,r), \end{aligned}$$

(A.7)

$$\begin{aligned} \mathcal {M}_{xyz}\left[ \left( z\frac{\partial }{\partial z}\right) ^n f(x,y,z); p,q,r\right]&=(-r)^n F(p,q,r), \end{aligned}$$

(A.8)

where $\Gamma (z)$ is a gamma function defined by $\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t} \mathrm {d}t$ for $z\in \mathbb {C}$. Using $m=n=2$ in (A.5) and (A.6), we can obtain the following formula:

$$\begin{aligned} \mathcal {M}_{xyz}\left[ x^2\frac{\partial ^2}{\partial x^2} f(x,y,z); p,q,r\right]&=\left( p^2+p\right) F(p,q,r), \end{aligned}$$

(A.9)

$$\begin{aligned} \mathcal {M}_{xyz}\left[ y^2\frac{\partial ^2}{\partial y^2} f(x,y,z); p,q,r\right]&=\left( q^2+q\right) F(p,q,r). \end{aligned}$$

(A.10)

Appendix B. Multivariate Normal Distribution

For a random variable $X=[X_1, \cdots , X_n]^T\, (n=1,2, \cdots )$ is said to have an n-dimensional Gaussian distribution with expectation $\mathbb {E}[X]=\mu \in \mathbb {R}^n$ and covariance matrix $\Sigma \in \{A\in \mathbb {R}^{n\times n} : A=A^T\, \mathrm {and}\,x^T A x >0,\; \forall x\in \mathbb {R}^n -\{0\}\}$ if its probability density function is given as

$$\begin{aligned} f(x)= \dfrac{1}{(2\pi )^{n/2} |\Sigma |^{1/2}} \exp \left( -\dfrac{1}{2} (x-\mu )^T \Sigma ^{-1} (x-\mu )\right) \quad \quad \mathrm {for} \; x\in \mathbb {R}^n. \end{aligned}$$

(B.1)

The joint CDF is defined as

$$\begin{aligned} \mathcal {N}_n (x_1, \cdots , x_n)=\underbrace{\int _{-\infty }^{x_1} \cdots \int _{-\infty }^{x_n}}_{n-\mathrm {times}} f(y_1, \cdots , y_n) \, \mathrm {d}y_n \cdots \mathrm {d}y_1. \end{aligned}$$

(B.2)

For the other properties of a multi-variable normal distribution, we refer to Hogg et al. (2005). Specifically, for $n=3$, the standard normal CDF is presented by

$$\begin{aligned} \mathcal {N}_{3}(x_1,x_2,x_3) = \int _{-\infty }^{x_1} \int _{-\infty }^{x_2} \int _{-\infty }^{x_3} \exp \left( -\dfrac{p(x,y,z)}{2\det {\Sigma }} \right) \mathrm {d}x \, \mathrm {d}y \, \mathrm {d}z, \end{aligned}$$

where

$$\begin{aligned}&p(x,y,z)=(1-\rho _{23}^2) x^2+(1-\rho _{13})^2 y^2 + (1-\rho _{12}^2)z^2 - 2(\hat{\rho }_{12} xy + \hat{\rho }_{13} xz + \hat{\rho }_{23} yz), \\&\hat{\rho }_{12} = \rho _{12}-\rho _{13}\rho _{23}, \; \hat{\rho }_{13} = \rho _{13}-\rho _{12}\rho _{23}, \; \hat{\rho }_{23} = \rho _{23}-\rho _{12}\rho _{13}. \end{aligned}$$