(1+2)-dimensional Black-Scholes equations with mixed boundary conditions

In this paper, we investigate (1+2)-dimensional Black-Scholes partial differential equations(PDE) with mixed boundary conditions. The main idea of our method is to transform the given PDE into the relatively simple ordinary differential equations(ODE) using double Mellin transforms. By using inverse double Mellin transforms, we derive the analytic representation of the solutions for the (1+2)-dimensional Black-Scholes equation with a mixed boundary condition. Moreover, we apply our method to European maximum-quanto lookback options and derive the pricing formula of this options.

Mixed boundary condition usually arises in the option pricing problem related to maximum process of underlying asset. Jeon et al. [5] derived an integral equation satisfying the Russian option using the Mellin transform. The Russian option satisfies (1+1)-dimensional Black-Scholes equations with mixed boudnary conditions. Dai et al. [2] obtained a pricing formula for European maximum-rate quanto lookback options by using joint probability density functions of the extreme and terminal values of the prices of the underlying assets. Since the European maximum-rate quanto lookback options can be formulated as (1+2)-dimensional Black-Scholes equation with mixed boundary conditions, the probabilistic method proposed by

JUNKEE JEON AND JEHAN OH
Dai et al. [2] is applicable for solving (1+2)-dimensional Black-Scholes equations with mixed boundary conditions. Our double Mellin transform technique also gives not only a pricing formula of European maximum-quanto lookback options but also an analytical representation for the solution of general (1+2)-dimensional Black-Scholes PDE with mixed boundary conditions. To the best of our knowledge, there have been no researches which derive an analytic representation of the solution of (1+2)-dimensional Black-Scholes equation with mixed boundary conditions. Since we analytically present the general solution of (1+2)-dimensional Black-Scholes equation with mixed boundary conditions, our approach can be applied to a variety of option pricing problems involving maximum or minimum process of underlying assets.
Since the Mellin transform can be regarded as a general Fourier transform, we can also adapt the double Fourier transform to solve the (1+2)-dimensional Black-Scholes equations with mixed boundary conditions. However, since the (1+2)dimensional Black-Scholes PDE operator is degenerate, we should consider the change of variables to apply the Fourier transform. Also, after applying the Fourier transform, we have to restore to the original variables. This procedure makes a lot of complexity. In contrast to the Fourier transform, we can directly obtain the simple ordinary differential equation(ODE) without any change of variables when we apply the Mellin transform to the problem. For this reason, when we extend the problem to whole domain, the Mellin transform is simpler and easier than the Fourier method.
We briefly introduce our approach to the proofs of our main results; Theorem 3.2 and Theorem 4.1. In order to extend a solution of the PDE with mixed boundary condition, we shall consider the associated PDE with Dirichlet boundary condition. We then extend the solution of the associated PDE to the domain {(t, x, y) | 0 ≤ t < T, 0 < x < ∞, 0 < y < ∞} by using the double Mellin transform. In the process, assuming a priori that the double Mellin transform of a solution exists, we find a solution and verify that it actually solves the problem. We also derive the extended PDE with Dirichlet boundary condition and deduce the extended PDE with mixed boundary condition. Using Theorem 3.2, we finally provide the pricing formula of European maximum-quanto lookback options in Theorem 4.1.
The rest of the paper is organized as follows. In the next section we review Mellin transform and set up notation. In Section 3 we state and prove the main results. Finally, in the last section we derive the price of the European maximum exchange rate quanto lookback option.

Preliminaries.
2.1. The review of Mellin transform. Here, we summarize the definition and basic properties of the double Mellin transform, for readers who are unfamiliar. Most of the properties of the double Mellin transform are similar to those of the single Mellin transform. Readers who are interested in the Mellin transform can refer to Bertrand et al [1] or Sneddon [6] for further details.
If this integral converges for a < Re(x * ) < b, then the inverse of the Mellin transform is Definition 2.2 (Definition of the double Mellin transform and inverse double Mellin transform). Let g(x, y) be a locally integrable function on R + × R + . Then, the double Mellin transform M xy (g(x, y))(x * , y * ) of g(x, y) is defined by If this integral converges for a 1 < Re(x * ) < b 1 , a 2 < Re(y * ) < b 2 , then for a 1 < c 1 < b 1 , a 2 < c 2 < b 2 the inverse double Mellin transform is given by Proposition 1 (Convolution property of the double Mellin transform). Let f (x, y) and g(x, y) be locally integrable functions on R + × R + . For a 1 < Re(x * ) < b 1 , a 2 < Re(y * ) < b 2 , suppose that the double Mellin transformsf (x * , y * ) andĝ(x * , y * ) exist. Then the double Mellin convolution is given by the inverse double Mellin transform off (x * , y * ) ·ĝ(x * , y * ) as follows: Proof. See Eltayeb and Kilicman [4].
For a positive integer n, the followings hold: We provide some conditions to guarantee the existence of double Mellin transform in the following proposition. Since the proof is straightforward, we omit it here.
Proposition 2. Suppose that the locally integrable function f (x, y) on R + × R + satisfies the following conditions:

Notation.
In the rest of the paper we shall use the following notations: 3. Main results. We extend PDE problem (1) to the domain R and derive an analytic solution using the double Mellin transform. The following lemma is required to prove the main theorem.
Lemma 3.1. Consider the following PDE with Dirichlet boundary condition: Then, a solution V D (t, x, y) of the PDE (7) can be extended to a solution of the following PDE: where k y = 2r y /σ 2 y .
Proof. To solve (7), we consider the following PDE on the unrestricted domain R: LetV D (t, x * , y * ) denote the double Mellin transform of V D (t, x, y). Then, for the PDE in (8), we have The solution of (9) is given bŷ By applying the inverse double Mellin transform, we obtain that To compute (10), let us consider the following: From Yoon and Kim [7], we have where k x = 2r x /σ 2 x , k y = 2r y /σ 2 y . Let us call G L (t, x, y) the Green function of the Black-Scholes PDE operator L. By the double Mellin convolution property, we get Moreover, since A(x * , y * ) is a quadratic form of y * , it holds that Then we obtain from the properties of the Mellin transform that which implies that where we have used 1 Indeed, a direct calculation shows that x, y). Then we have x, y) is the solution of the PDE (7), and is given by Now we can get the following main theorem.
Theorem 3.2. The solution V (t, x, y) of the PDE (1) with mixed boundary condition can be extended to following PDE: Proof. We define the differential operator H[·] as follows: By Lemma 7, the solution P (t, x, y) of the PDE (13) can be expressed by whereP D (t, x, y) satisfies the PDE Let us define the right inverse operator H −1 of the differential operator H as follows: x, y). By applying the double Mellin transform, we obtain P (t, x * , y * ) = −(1 + y * )V (t, x * , y * ), whereV ,P ,V D ,P D ,V * D , andP * D represent the double Mellin transforms of V, P,V D , P D ,V * D , andP * D , respectively. Especially, we see that Applying double Mellin transform to (14) and using the relations (16), we get We observe that Furthermore, we note that By the Mellin convolution property of a single variable, we have We obtain from the properties of the Mellin transform that By the definition of H and its linearity, we see that if H[g 1 (y)] = H[g 2 (y)], then g 1 (y) = g 2 (y) + cy for some constant c. It follows from (18), (21) and (22) that for some function c(t, x) being independent of y.
This and (23) yield where c(x) := c(T, x) is a function depending only on x. An integration by parts gives

JUNKEE JEON AND JEHAN OH
Similarly, we have (k y + 1) Therefore, we finally obtain Since V (T, x, y) = h(x, y) in R x,y , we deduce that c(x) ≡ h(x, 1). This gives which is the desired result.
The following lemma is useful for deriving the analytic representation of the solution.
From the double Mellin transform approach, we get whereV D is the double Mellin transform of V D . Thus, we can deduce from (21) in Theorem 3.2 that Theorem 3.4. The solution V (t, x, y) of the PDE (1) with mixed boundary condition has the following analytic representation: Proof. By Theorem 3.2, the PDE (1) with mixed boundary condition is extended to following PDE: where By the Mellin transform approaches in Lemma 7, the value function V (t, x, y) is given by Thus, we have just proved the desired result.
4. Application to option pricing. In this section, we consider the European maximum-exchange quato lookback option defined in Dai et al. [2]. Let F t denote the exchange rate at time t, which means that F t represents the domestic price at time t of one unit of foreign currency. Let S t be the foreign currency price at time t, and let r d and r f be the constant domestic and foreign riskless interest rates, respectively. Then, the stochastic dynamics of S t and F t are described by where σ s and σ f are the volatility of S t and F t , respectively. Here, W d s and W d f represent standard Brownian motion under the domestic risk neutral measure Q d , and dW d s dW d f = ρdt. By using the quanto prewashing technique introduced by Dravid et al. [3], we obtain that δ d s = r f − q − ρσ f σ s , where q is the dividend yield of the foreign asset S t in a foreign country.
For the exchange rate process, define the maximum process of F t as M t = max 0≤γ≤t F γ , t ≥ 0.
Consider the European maximum exchange-rate quanto lookback call option, whose terminal payoff function in the domestic currency world is given by on the domain {(t, s, f, m) | 0 ≤ t < T, 0 < s < ∞, 0 < f < m, 0 < m < ∞}. Here the operator L is defined by We where h(s) ≡ (s − K) + and k f = 2(r d − r f )/σ 2 f . Theorem 4.1. The price of the European maximum exchange rate quanto lookback option, C(t, s, f, m), is given by