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This paper presents integral representations for the price of vanilla put options, namely, European and American put options on a basket of two-dividend paying stocks using integral method based on the double Mellin transform. We show that by the decomposition of the integral equation for the price of American basket put option, the integral equation for the price of European basket put option can be obtained directly.

An option is a contingent claim that presents its holder with the right, but not obligation, to purchase a given amount of underlying asset at some future date. In practice, the underlying asset is often the price of stock, commodity, foreign exchange rate, debit instrument, stock indices or future contract. Although the history of options extended back to several decades, it was not until 1973 that the trading of option was formalized by the establishment of the Chicago Board of Options of Exchange (CBOE). This same year was also a trading point for research in the valuation of financial derivatives.

Black and Scholes [

Basket option is defined as an option on a collection or basket of stocks. In other words, basket options are options whose payoff depends on the value of a basket, i.e., a portfolio of assets. Equity index options and currency basket options are classical examples of basket options. However, basket options are becoming increasingly widespread in commodity and particularly energy markets. The volatility of the basket is lower than the individual volatilities of the stocks and therefore these options are popular as hedging tools. The valuation of basket options is a challenging task because the underlying value is a weighted sum of individual asset prices. The common assumption of log-normality (and hence, the famous Black-Scholes formula) cannot be applied directly, because the sum of log-normal random variables is not log-normal. The valuation problem for American option on a basket of two-dividend paying stocks leads to the solution of a multi-dimensional free boundary problem.

Panini and Srivastav [

For mathematical background, applications of the Mellin transforms and various numerical methods for the valuation of basket options (see [

In this paper, we apply double Mellin transform to derive integral representations for the prices of vanilla basket put options, namely, European and American basket put options with dividend yields. The rest of the paper is structured as follows. Section 2 presents the overview of the Black-Scholes partial differential equation for vanilla basket options of multi-dividend paying stocks. In Section 3, we apply the double Mellin transform method to derive the integral equations for the representations of the price of both European and American put options on a basket of two-dividend paying stocks. Section 4 concludes the paper.

Consider the price of the underlying asset for the multi-stocks given by

Equation (1) is defined on the probability space

where

The non-homogeneous Black-Scholes partial differential equation for vanilla options on a basket of multi- stocks with dividend yield is given by

Equation (3) can also be written as

Suppose

Consider a vanilla put option; recall that when the option is granted exercise rights for any

Equation (4) is called the terminal condition for vanilla put option. For the put option, the continuation region

C exists for

be stated as

When

Similar to the single asset option, the payoff function is equivalent to (4). As usual the option must satisfy the condition

for any

The Mellin transform of

By definition, the inversion formula for the Mellin transform of (8) is given by

where

This section presents the Mellin transform method for the valuation of European and American put options on a basket of two stocks with dividend yields.

Setting

where

The boundary conditions for (10) are given by

Now, we find an integral representation for the price of European put option

where the wiener processes

Let

Conversely, the inversion formula of (13) is given by

Taking the Mellin transform of (10), yields

where

Substituting (16) into (15), we have

Simplifying further, (17) becomes

Setting

Thus, (18) becomes

The general solution of (20) is called the complementary solution since it is homogeneous first order differential equation. The general solution is given by

where

where

We introduce dimensionless variables

and using the localizing assumption that

where

Let

Solving for the new domain gives

Thus,

Substituting (25a) and (25c) into (25), we have that

From (24), for

Using (22) and (26), then (21) yields

Substituting (27) into (14), we have that

Equation (28) is the integral representation of the price of European put option on a basket of two stocks with dividend yield via the double Mellin transform method.

Now we consider the double Mellin transforms in order to derive the expression for the price of American put option on a basket of two stocks. American put option on a basket of two stocks gives the option’s holder the right to sell the basket stocks at any time from 0 to

The non-homogeneous Black-Scholes partial differential equation for American put option on a basket of two stocks with dividend yield is given by

where

The final time condition or terminal condition is given by

The other boundary conditions are given by

Let us assume that

Using the same procedures of the double Mellin transform as for the case of European put option on a basket of two stocks with dividend yield. Let us denote the double Mellin transform for the price of American basket put option

Conversely the inversion formula of the double Mellin transform is given by

Taking the Mellin transform of (29), we have that

where

where

where

Therefore,

Taking the double Mellin transform of (39) yields

Let us consider the nonhomogeneous part of (36) whose solution is given by

But

Setting

So,

Solving (42a), we have respectively

Substituting (44), (45) and (46) into (43)

Substituting (47) into (41) yields

Taking the double Mellin transform of (48), we have that

The inverse double Mellin transform of (37) is given by

Substituting (40) and (49) into (50)

where

Remarks

・ The integral equation for the price of European put option on a basket of two stocks with dividend yield can be obtained directly from the price of its counterpart, the American put option on a basket of two stocks. Hence, (51) can be written as

・ The second term in (51) is called early exercise premium.

・ For the case of non-dividend yield, (51) becomes

In this paper, we have considered vanilla basket put options, namely, European and American put options on a basket of two stocks with dividend yield. We used the integral method based on the double Mellin transform to derive the integral representations for the price of European and American put options on a basket of two-divi- dend paying stocks. We deduce from our results that by the decomposition of the price of American put option on a basket of two stocks, its counterpart “European put option” can be obtained directly as shown in (52).