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

Abstract

Even though interest rates fluctuate randomly in the marketplace, many option-pricing models 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.

1. 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.

2. 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 and is expressed by , where . Also, is the Hull-White interest rate model, is mean reversion rate of interest, is volatility of interest rate, and is average direction of interest rate movement. Using notation as the expectation with respect to the risk-neutral measure, we have the no-arbitrage price of a European option with a payoff function given by From Feynman-Kac formula (cf. [10]), the solution of satisfies the following PDE: where is the terminal condition and is the identity operator.

2.1. A Review of Bond Pricing Formula

Under the Hull-White interest rate model for the short rate, , given by , the no-arbitrage price at time of a zero-coupon bond maturing at time is as follows: with the final condition .

Then, by using Feynman-Kac formula (cf. [10]), the solution of satisfies the following equation: and by writing the form of the solution as , where , and are given by In Section 3, we will define , , and as , , and , respectively.

2.2. A Review of the Mellin Transforms

To derive a closed solution of , we use the Mellin transform. For a locally Lebesgue integrable function , the Mellin transform , is defined by and if and such that exists, the inverse of the Mellin transform is expressed by

3. The Derivation of the Formula of European Option Price

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 , the Mellin transform of the payoff function does not exist. Therefore, a somewhat modified form of is needed to guarantee the existence of the integral, and we define the sequence of the payoff function such that as follows: If we define the call option price with the payoff function 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 .

If we define as the Mellin transform of , then the inverse of the Mellin transform is given by By substituting (12) into the PDE mentioned above, the PDE is transformed by where and the terminal condition is given by .

To simplify PDE (13), let us assume can be expressed by the form and then must satisfy with the final condition .

Now, to solve PDE (15), we set . Here, and by substituting this functional form of the solution into PDE (15), we have two ordinary differential equations (ODEs) with respect to and as follows: with and . ODEs (16) yield where .

Therefore, from the solution of and (14), we obtain the solution of as follows: where From the definition of , , and , the bond price stated in Section 2.1 is

Finally, we have the price of the European call option: and to compute the above integral (21), we define the following equation: where . Then leads to the following equation: To compute the integral of (23), we use the following lemma.

Lemma 1. Let and be complex numbers satisfying . Then, holds.

Proof. Let and . Then , where . If , then becomes

However, to apply Lemma 1 to (23), the following lemma is also required.

Lemma 2. For and given above, holds.

Proof. From the definition of and , is satisfied. However, since we can prove , where .

In (23), if we set then, from Lemmas 1 and 2, yields the following equation: 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 as follows: where is the symbol of the Mellin convolution of and and 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 , 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

Theorem 3. Under the payoff function , the formula of European call option with Hull-White interest rate is given by where is the price of zero-coupon bond mentioned in Section 2.1, with , and is the normal cumulative distribution function defined by .

Proof. Most of all, let . By applying change of variable , of (32) leads to where . From the definition of , , and mentioned before, we have Therefore, the solution of is where .
Also, by setting and using the similar way stated above, of (32) is transformed into Similarly,
Therefore, the solution of is Finally, by combining (32), (36), and (39), we obtain the following result: The proof is completed.

3.2. The Case of Put Option

Corollary 4. If is European vanilla put option with Hull-White interest rate, then the formula leads to the following relation:

Proof. In case of European put option, one has the payoff function . Unlike the case of call option, for the payoff of the put option, the Mellin transform exists and the value is given by . Therefore, for the function , we can apply the same Mellin transform method used to derive the formula of call option and then we obtain the formula of European vanilla put option with Hull-White interest rate stated above.

4. Concluding Remarks

In this paper, we have investigated that a closed formula for the European option price under Hull-White interest rate can be derived by applying the method of Mellin transforms. The formula is simply given by the normal cumulative distribution function. Also, one can apply the Mellin transform methods to other types of options except the ones that we did already.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions to improve the paper. The research of Ji-Hun Yoon was supported by BK21 PLUS SNU Mathematical Sciences Division.