Method of Lines for Valuation and Sensitivities of Bermudan Options

Abstract

In this paper, we present a computationally efficient technique based on the Method of Lines for the approximation of the Bermudan option values via the associated partial differential equations. The method of lines converts the Black Scholes partial differential equation to a system of ordinary differential equations. The solution of the system of ordinary differential equations so obtained only requires spatial discretization and avoids discretization in time. Additionally, the exact solution of the ordinary differential equations can be obtained efficiently using the exponential matrix operation, making the method computationally attractive and straightforward to implement. An essential advantage of the proposed approach is that the associated Greeks can be computed with minimal additional computations. We illustrate, through numerical experiments, the efficacy of the proposed method in pricing and computation of the sensitivities for a European call, cash-or-nothing, powered option, and Bermudan put option.

Appendix

1.1 Matrix Exponential

The matrix exponential is defined for $A\in {\mathbb {C}}^{n\times n}$ by,

$$\begin{aligned} e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!}+\cdots \cdot \end{aligned}$$

(28)

We shall now state an important theorem which yields the convergence of the matrix Taylor series (28).

Theorem 7.1

(Convergence of matrix Taylor series) Suppose f has a Taylor series expansion,

$$\begin{aligned} f(z) =\sum _{k=0}^{\infty }a_k (z-\alpha )^k, \quad \left( a_k =\frac{f^{(k)} (\alpha )}{k!}\right) \end{aligned}$$

(29)

with radius of convergence r. If $A\in {\mathbb {C}}^{n \times n}$ then f(A) is defined and is given by

$$\begin{aligned} f(A)=\sum _{k=0}^{\infty }a_k (A-\alpha I)^k \end{aligned}$$

(30)

if and only if each of the distinct eigenvalues $\lambda _1,\ldots ,\lambda _s$ of A satisfies one of the conditions

• 1. $|\lambda _i -\alpha |< r,$

• 2. $|\lambda _i -\alpha |=r,$

and the series for $f^{(n_i -1)}(\lambda )$ (where $n_i$ is the index of $\lambda _i$) is convergent at the point $\lambda =\lambda _i$, $i=1:s$.

Now, if we consider the power series (28), then the corresponding power series is given by,

$$\begin{aligned} f(z)=\sum _{k=0}^{\infty }\frac{z^k}{k!} \end{aligned}$$

(31)

whose radius of convergence is,

$$\begin{aligned} r=\frac{1}{\lim _{n\rightarrow \infty }\frac{a_{n+1}}{a_n}} \end{aligned}$$

where, $a_n = \frac{1}{n!}$ (since the limit in the denominator exists in this case). Hence, this readily yields

$$\begin{aligned} r =\frac{1}{\lim _{n\rightarrow \infty } \frac{n!}{(n+1)!}}=\infty \end{aligned}$$

Therefore, on applying the theorem above we know that the series (28) is defined and further by standard results in analysis (as in Gillespie, 1955), we can differentiate the series term by term to obtain $\frac{d}{dt}e^{At}=Ae^{At}=e^{At}A.$

Another representation of a matrix exponential is,

$$\begin{aligned} e^A =\lim _{s\rightarrow \infty } ( I+A/s)^s \end{aligned}$$

(32)

The formula above is the limit of the first-order Taylor expansion of A/s raised to the power of $s\in {\mathbb {Z}}$. In a more general setting, we can take the limit as $r\rightarrow \infty$ or $s\rightarrow \infty$ of r terms of the Taylor expansion of A/s raised to the power of s, thereby generalising both (28) and (32). Results show that this general formula also yields $e^A$ and provides an error bound for finite r and s. We shall not delve into these details but refer the reader to Gillespie (1955) for a detailed analysis of the same.

1.2 Derivations of the Greeks

Theta ($\Theta$): Since the matrices do not involve the parameter t, a direct differentiation of the function $U(\tau )$ in (15) with respect to t yields the following formula for $\Theta$,

$$\begin{aligned} \frac{\partial U(\tau )}{\partial t}&=-\zeta e^{\tau \zeta }U_0 -{\zeta }^{-1}\zeta e^{\tau \zeta }F\\&=-\zeta e^{\tau \zeta }U_0 - e^{\tau \zeta }F \end{aligned}$$

Vega ($\nu$): The calculation of this is a bit involved since the coefficients of the matrices A and F involve the parameter $\sigma$. We first use the identity $\zeta {\zeta }^{-1}=I$ and differentiate it partially with respect to $\sigma$ and utilise the chain rule to obtain,

$$\begin{aligned} \frac{\partial \zeta }{\partial \sigma }{\zeta }^{-1}+\zeta \frac{\partial \zeta ^{-1}}{\partial \sigma }&=0\\ \implies \zeta \frac{\partial \zeta ^{-1}}{\partial \sigma }&=-\frac{\partial \zeta }{\partial \sigma }{\zeta }^{-1}\\ \implies \frac{\partial \zeta ^{-1}}{\partial \sigma }&=-{\zeta }^{-1}\frac{\partial \zeta }{\partial \sigma }{\zeta }^{-1} \end{aligned}$$

Now, a simple application of chain rule to (15) , combined with the formula obtained and the straight-forward result $\frac{\partial \zeta }{\partial \sigma }=A'$ yields the formula (24) for $\nu$.

Rho ($\rho$): An almost identical formulation as that of $\nu$ yields the following formula,

$$\begin{aligned} \frac{\partial \zeta ^{-1}}{\partial r}=-{\zeta }^{-1}\frac{\partial \zeta }{\partial r}{\zeta }^{-1} \end{aligned}$$

Using this and the result $\frac{\partial \zeta }{\partial r}=B'+C'$, one readily obtains the formula (25) for $\rho$.

1.3 Accuracy and Stability of MOL

It can be seen using a direct application of Taylor’s expansion that $U(\tau ,x)$ satisfies the following equation,

$$\begin{aligned} U(\tau ,x-h)&=U(\tau ,x)-h U_x(\tau ,x)+\frac{h^2}{2}U_{xx}(\tau ,x)\\&\quad -\frac{h^3}{3!}U_{xxx}(\tau ,x_i)+\frac{h^4}{4!}U_{xxxx}(\tau ,x)+{\mathcal {O}}(h^5) \end{aligned}$$

which gives,

$$\begin{aligned} \frac{U(\tau ,x -h)}{h^2}&=\frac{U(\tau ,x)}{h^2} -\frac{U_x(t,x)}{h}+\frac{U_{xx}(\tau ,x)}{2}\\&\quad -\frac{h}{3!}U_{xxx}(\tau ,x)+\frac{h^2}{4!}U_{xxxx}(\tau ,x)+{\mathcal {O}}(h^3). \end{aligned}$$

Similarly,

$$\begin{aligned} U(\tau ,x +h)&=U(\tau ,x)+h U_x(t,x)+\frac{h^2}{2}U_{xx}(\tau ,x)\\&\quad +\frac{h^3}{3!}U_{xxx}(\tau ,x)+\frac{h^4}{4!}U_{xxxx}(\tau ,x)+{\mathcal {O}}(h^5) \end{aligned}$$

which leads to

$$\begin{aligned} \frac{U(\tau ,x +h)}{h^2}&=\frac{U(\tau ,x)}{h^2} +\frac{U_x(t,x)}{h}+ \frac{U_{xx}(\tau ,x)}{2}\\&\quad +\frac{h}{3!}U_{xxx}(\tau ,x)+\frac{h^2}{4!}U_{xxxx}(\tau ,x)+{\mathcal {O}}(h^3). \end{aligned}$$

Upon adding these two equations, one obtains the expression for $D^2 U(\tau )$ in (17).

Using these and the fact that $U(\tau ,x)$ denotes the exact solution of (16), one obtains the local accuracy of the method as follows (Higham, 2004),

$$\begin{aligned} R= \frac{\partial U(\tau ,x)}{\partial \tau }-\frac{1}{2}\sigma ^2 {x}^2 D^2 U(\tau ,x) -r x D U(\tau ,x) +r U(\tau ,x) \end{aligned}$$

(33)

which, on application of (17) and (18) becomes,

$$\begin{aligned} R&=\frac{\partial U(\tau ,x)}{\partial \tau }-\frac{1}{2}\sigma ^2 {x}^2U_{xx}(\tau ,x) -\sigma ^2 {x}^2\frac{h^2}{24}U_{xxxx}(\tau ,x)+{\mathcal {O}}(h^3)\\&\quad -rx U_x(\tau ,x)-\frac{rx}{3!}h^2U_{xxx}(\tau ,x)+{\mathcal {O}}(h^3)+rU(\tau ,x) \end{aligned}$$

Since, $U(\tau ,x)$ is a solution to the Black–Scholes equation (3), the above expression simplifies to,

$$\begin{aligned} R=-\sigma ^2 {x}^2\frac{h^2}{24}U_{xxxx}(\tau ,x)+{\mathcal {O}}(h^3) -\frac{rx}{3!}h^2U_{xxx}(\tau ,x) +{\mathcal {O}}(h^3) \end{aligned}$$

which gives the desired local accuracy of the method as ${\mathcal {O}}(h^2)$.

Recall that our system of differential equations described in Eq. (20) has a solution given by (15). This implies that the system is stable if $e^{t\zeta }$ does not blow up. To ensure this, it is enough if the real part of each of the Eigen values of the associated matrix $\zeta$ are negative, since F is a constant matrix in our case (Strang, 2006).

For general cases, though, F is not necessarily constant and is itself a function of time. In such a case, one could use the scheme discussed in Hochbruck and Ostermann (2010).