Mellin Transform Method for European Option Pricing with Hull-White Stochastic Interest Rate

Even though interest rates fluctuate randomly in themarketplace, many option-pricingmodels do not fully consider their stochastic nature owing to their generally limited impact on option prices. However, stochastic dynamics in stochastic interest rates may have a significant impact on option prices as we take account of issues of maturity, hedging, or stochastic volatility. In this paper, we derive a closed form solution for European options in Black-Scholes model with stochastic interest rate using Mellin transform techniques.


Introduction
In practice, random fluctuations of interest rate over time have a significant contribution to the change of an option price.Based on this observation, some work has been reported on the price formula of European options with stochastic interest rate.Most of all, Merton [1], Rabinovitch [2], and Amin and Jarrow [3] have proposed the formula of closed form European option pricing under the Gaussian interest rate by using relatively simple algebra.This method is also discussed in detail by Kim [4].Also, Fang [5] derived an exact pricing formula for European option under stochastic interest rate by applying martingale method.However, the closed formula for the prices of options has been studied usually by utilizing probabilistic techniques as the papers stated above.In this paper, we use analytic methods based on Mellin transforms as a better way to compute the option prices.
The Mellin transform is defined as an integral transform that may be considered as the multiplicative version of the two-sided Laplace transform.Many papers have shown that the Mellin transform technique would help us resolve the complexity of the calculation compared to the probabilistic approach.Panini and Srivastav [6] studied the pricing formula of a European vanilla option and a basket option using Mellin transforms.Panini and Srivastav [7] found also the pricing of perpetual American options with Mellin transforms.Frontczak and Schöbel [8] used Mellin transforms to value American call options on dividend-paying stocks.Also, Elshegmani and Ahmed [9] derived analytical solution for an arithmetic Asian option using Mellin transforms.
This paper is organized as follows.In Section 2, we formulate a European vanilla option with Hull-White interest rate and obtain a partial differential equation (PDE) for European call option under the stochastic interest rate.In Section 3, we apply Mellin transforms to derive a closed form solution of the option price with respect to a European call option and a European put option.In Section 4, we have concluding remarks.

Model Formulation
Let   be the value of the asset (stock) underlying the option, let   be the drift rate of the stock, and let  be the volatility of the underlying asset.Then, the dynamics of   is given by the SDE   =      +     , where   is the standard Brownian motion.Under a risk-neutral probability measure, the above given model is transformed into the SDEs: where  *  represents the standard Brownian motion under a risk-neutral world satisfying the following relation: and the correlation of  (3) From Feynman-Kac formula (cf.[10]), the solution of (, , ) satisfies the following PDE: where (, , ) = ℎ() is the terminal condition and  is the identity operator.

A Review of the Mellin Transforms.
To derive a closed solution of (, , ), we use the Mellin transform.For a locally Lebesgue integrable function (),  ∈ R + , the Mellin transform M((), ),  ∈ C, is defined by and if  < Re() <  and  such that  <  <  exists, the inverse of the Mellin transform is expressed by 3. The Derivation of the Formula of European Option Price If we define the call option price with the payoff function ℎ  () =   (, , ) as (, , ) satisfies the PDE given by (4).Then, if we find the solution   (, , ) by using the Mellin transform, we can obtain the formula of the option price (, , ) from (, , ) = lim  → ∞   (, , ).
Lemma 1.Let  and  be complex numbers satisfying Re() ≥ 0.Then, However, to apply Lemma 1 to (23), the following lemma is also required.Lemma 2. For  1 () and () given above, ( ) , then, from Lemmas 1 and 2, () yields the following equation: −(1/4)(ln ) 2 . (29) Now, we are trying to use relation to multiplicative convolution of Mellin transform and find   (, , ).The Mellin convolution of  and  is given by the inverse Mellin transform of f() ĝ() as follows: where () * () is the symbol of the Mellin convolution of  and  and  −1  is the symbol of the inverse Mellin transform.It is referred to in [11] with more details.
In (21) and ( 22), since  (,,) is the Mellin transform of () and   () is the Mellin transform of the payoff function ℎ  (), we have the following formula by using the Mellin convolution property mentioned above: Therefore, to find the European option price (, , ), if we take  → ∞ in both sides of (31), then

3. 1 .
The Case of Call Option.In this section, we derive the formula of European call option with the Hull-White interest rate using the Mellin transform.However, since the European call option has the payoff function (, , ) = ℎ() = ( − ) + , the Mellin transform of the payoff function does not exist.Therefore, a somewhat modified form of ℎ() = (−) + is needed to guarantee the existence of the integral, and we define the sequence of the payoff function ℎ  () such that lim  → ∞ ℎ  () = ℎ() as follows: