Mellin Transform Method for the Valuation of the American Power Put Option with Non-Dividend and Dividend Yields

In this paper we present the Mellin transform method for the valuation of the American power put option with non-dividend and dividend yields, respectively. We use the Mellin transform method to derive the integral representations for the price and the free boundary of the American power put option. We also extend our results to derive the free boundary and the fundamental analytic valuation formula for perpetual American power put option which has no expiry date. Numerical experiments have shown that the Mellin transform method is a better alternative technique compared to the binomial model (BSM), recursive method (RM) and finite difference method (FDM) for the valuation of the American power put option. In general, the Mellin transform method is accurate, flexible and produces accurate prices for the optimal exercise boundary of the American power put option for a wide range of parameters. Hence the Mellin transform method is mutually consistent and agrees with the values of the analytic option valuation formula called the “BlackScholes model”.


Introduction
Option valuation has been studied extensively in the last three decades.Many problems in financial mathematics entail the computation of a particular integral.In many cases these integrals can be valued analytically and in some cases they can be computed using a partial differential equation, or valued using numerical integration.
Power option is defined as a contingent claim on the product of powers of several underlying assets.The holder has either the right, but not the obligation to buy, as in the case of the power call option, or the possibility to sell, as in the case of the power put option, an asset for a certain price at a prescribed date in the future.The difference between the American and the European power options is that the European power option can only be exercised at the maturity or expiry date while the American power option can be exercised by its holder at any time before the expiry date.This early exercise feature makes the valuation of the American power option mathematically challenging and therefore, creats a great field of research.
A perpetual American power option is an option that has no expiry date.In other words, this type of power option never expires.In a special case of a plain vanilla perpetual option, a closed form solution for the free boundary and price of the American put was derived by [1].
Mellin transforms in option theory were introduced by [2], [3] extended the results obtained in [2] and showed how the Mellin transform approach could be used to derive the valuation formula for the perpetual American put options on dividend-paying stocks.[4] considered the Mellin transform method for the valuation of some vanilla power options with non-dividend yield.They derived the fundamental valuation formula known as the Black-Scholes model using the convolution property of the Mellin transform method.The analytical valuation of the American options was considered by [5].An alternative approach to the valuation of American options and applications was considered by [6].
For the mathematical background of the Mellin transform method in derivatives valuation see [7]- [15], just to mention few.In this paper, we focus on the Mellin transform method for the valuation of the American power put option with non-dividend and dividend yields, respectively, and its extension to power option which has no expiry date, i.e. "perpetual American power put option".The rest of the paper is structured as follows: in Section 2, we present American power options and the payoffs for power call and put options.Section 3 presents the Mellin transform method for the valuation of the American power put option.Section 4 considers the extension of the Mellin transform method to the valuation of the perpetual American power put option.In Section 5, we present some numerical experiments.Section 6 concludes the paper.

American Power Options
The power options can be seen as a class of options in which the payoff at expiry is related to the th n power of the underlying price of the asset.American power options are options that can be exercised before or at the expiry date with non-linear payoff.The American power option comes in two forms, namely, the American power call option and the American power put option.The American power call option is an option with nonlinear payoff given by the difference between the price of the underlying asset at maturity raised to a strictly positive power and the exercise price.The American power put option is an option with non-linear payoff given by the difference between the exercise price and price of the underlying asset at maturity raised to a strictly positive power.For an American power option on the underlying price of the asset n T S with exercise price K and time to expiry T, we have the payoffs for the American power call and put options as , max , 0 respectively.Remark 1 • For 1 n = , the payoffs for American power call and put options in (1) and (2) become the payoffs for plain American call and put options, i.e.

The Mellin Transform Method for the Valuation of the American Power Put Option
There are many methods for the valuation of the American power option leading to different but equivalent mathematical formulations.We consider the derivation of the integral representation for the price of the American power put option and the integral equation to determine the free boundary of the American power put option via the Mellin transform method for the case of both non-dividend and dividend yields.

American Power Put Option with Non-Dividend Yield
Consider the non-homogeneous Black-Scholes partial differential equation for the American power put option with non-dividend yield given by ( ) where the early exercise function f defined on ( ) ( ) The final time condition given by ( ) The free boundary ˆn t S is determined by the smooth pasting conditions given by ( ) and ( ) Applying the Mellin transform to (5), we have that The Mellin transform of the early exercise function in (12) is obtained as Solving further, we have the particular solution of ( 12) as The complementary solution to the left hand side of ( 12) is obtained as , e where ( ) c ω is the integration constant obtained as


is the Mellin transform of the final time condition and is given by Using ( 16) and ( 17) in (15) we have that Hence the general solution to ( 12) is given by The Mellin inversion of (19) is obtained as where • The first term in (20) is the integral representation for the price of the European power put option (stems from the minimum guaranteed payoff of the American power put) which pays no dividend yield (see [4]).
The second term in (20) is called the early exercise premium (the value attributable to the right of exercising the option early) for the American power put option with non dividend yield denoted by ( ) in (21) and using the smooth pasting conditions given by ( 9) and (10), we have the integral representation for the free boundary of the American power put option with non-dividend yield as

American Power Put Option with Dividend Yield
The derivation of the integral representation for the price of the American power put option which pays dividend yield using the Mellin transform method is given in the following result.Theorem 3 Let n t S be the price of the underlying asset, K be the strike price, r be the risk interest rate, q be the dividend yield and T be the time to maturity.Assume n t S yields dividend, then the integral representation for the price of the American power put option ( ) Proof.Consider the non-homogeneous Black-Scholes partial differential equation for the American power put option with dividend yield given by ( ) and n t S the free boundary of the American power put option with dividend yield.The high contact condition is given by 0, ∞ .The other conditions are given by ( 7) and (8).With the smooth pasting conditions given by ( ) , The Mellin transform of (24) gives Following the same procedures for the case of non-dividend yield, the general solution to (30) is obtained as The Mellin inversion of (31) is given by Equation ( 32) is the integral representation for the price of American power put option with dividend yield, where ( ) • The first term in (32) is the integral representation for the price of the European power put option (stems from the minimum guaranteed payoff of the American power put) with dividend yield and the last two terms denote the early exercise premium (the value attributable to the right of exercising the option early) for the American power put option with dividend yield denoted by ( ) in (33) and using the smooth pasting conditions given by ( 26) and ( 27), we have the integral representation for the free boundary of the American power put option with dividend yield as Using the smooth pasting conditions given by ( ) the free boundary formulation of the American power put option with dividend yield is given by ( ) The following results present some special cases of (20) and (32).Theorem 6 If T t τ → − and 1 n = , then (i) The integral representation for the American power put option which pays no dividend yield (20) reduces to the integral equation derived by Kim [6] for the price of the plain American put option given by where (ii) The free boundary for the American power put option which pays no dividend yield (22) reduces to the integral equation derived by Kim [6] for the price of the plain American put option given by ( ) where The early exercise function is given by ( ) into (45), we have that Using the transformation given by ( ) (47) becomes → , then the optimal exercise boundary t S of the American power put option with 1 n = with dividend yield is given by lim min , where ( ) where ( ) and Notice first that critical stock price is bounded from above i.e. , 0 S K τ τ ≤ ∀ > .Taking the limits of ( 56) and (57) as 0 τ + → , we have that ( ) Using (63), the limit of ( 55) is obtained as The first integral I τ can also be written as Applying the residue theorem of complex number given by ( ) ( ) Then the inner integral in (67) becomes ( ) Substituting ( 70) and ( 71) into (66) for q r ≤ , we have that Using the l'Hospital rule, for q r > , (64) becomes Combining ( 72) and ( 73) The above results confirm the formula of Kim and Yu [6] Theorem 8 If the underlying asset price follows a lognormal diffusion process and the interest rate is a positive constant, then the optimal exercise boundary of the American power put option with 1 n = at maturity is given by , for lim , for In order to investigate the behaviour of the optimal exercise boundary S τ of the American power put option with 1 n = near maturity, we consider (55) which is of the form If q r > , the limit of the right hand side of (55) as 0 τ → can be evaluated using the l'Hospital's rule we have that If q r ≤ , the limit of the right hand side of (55) as 0 τ → is obtained directly as Combining ( 75) and (76), we have the optimal exercise boundary of the American power put option with 1 n = at maturity given by , for lim , for From (75), we notice that when q r > the American put can have a positive value at expiration given that it has not been exercised earlier.This indicates that large dividend payouts reduce the incentives of early exercise.
From (76), we deduce that when q r ≤ the American put will have a zero payoff at expiration even if it has not been exercised earlier.This is because it is not possible for the underlying asset price at expiration to fall below K without crossing the exercise boundary at an earlier time.
Theorem 9 The integral representation for the price of the American power put option which pays dividend yield given by (32) can be reduced to integral representation derived by Kim [6].where where ( ) Using the procedures of [3], (78) can be written as with the Mellin transforms of ( ) The price of the American power put option which pays dividend yield can be expressed as The integral ( ) 2r n α σ = .Using the following variables transformation given by ( ) For the first and second integrals in (83) respectively, we have that By changing y η = , ( Hence by setting 1 n = , this proves (77).

Application of the Results to Perpetual American Power Put Option Valuation
Now, we apply the results generated for the integral equations in (20) and (32) to power options which have no expiry date.The following results shows the derivation of the expression for the free boundary of perpetual the American power put option and its closed form solution for both non-dividend and dividend yields, using the Mellin transform method.
Theorem 10 (Non-Dividend Yield) If T → ∞ and , then the free boundary of the perpetual American power put option is given by ( ) ( ) and the price of the perpetual American power option becomes Proof.The integral representation for the price of the American power put option which pays no dividend yield given by (20) can be expressed as respectively.For (90) to hold as T → ∞ , it is necessary that ( ) ( ) 2 0 Re ω α < < , where 2 α is given by (89).The second smooth pasting condition (10) for a perpetual power put can be written as Differentiating (91) at n we have that ( ) ( ) ( ) ( ) ( ) where Taking the limit of (97) as T → ∞ , we have Therefore, ( ) ( ) where ( )( ) ˆ, 2 1 d ˆ2π 1 Since ( ) 2 0 Re ω α < < , application of the residue theorem given by (68) leads to ( ) Substituting ( 96) and ( 101) into (93) yields ( ) Equation ( 102) is the expression for the free boundary of a perpetual American power put option.Next, we use (102) to derive an expression for the price of perpetual American power put option ( ) Note that the price of a perpetual European power put option is zero, since it can never be exercised.Therefore, taking the limit as T → ∞ in (90), the price of perpetual American put option for n where ( ) Once again we apply the residue theorem (68) to get  α are given by (109).The second smooth pasting condition (10) for a per- petual power put which pays dividend can be written as , , , 1 as Therefore, by setting n n t S S ∞ = we have that ( ) ( ) In the same manner, setting T → ∞ and differentiating (113) w.r.
and solving (121) further we have that Once again by the application of residue theorem (68), then (120) and ( 122) yield and respectively.Substituting (117), ( 123) and ( 124) into (114), we obtain ( ) Equation ( 125) is called the free boundary of the perpetual American power put option which pays dividend yield.
The price for the perpetual American power put option is given by ( )

Numerical Experiments
In this section we present some numerical experiments and discussion of results.

Experiment 1
We consider the valuation of the American power put option for { } The result generated is shown in Table 1 below.

Experiment 2
We consider the valuation of the American power put option for 1 n = which pays non-dividend yield with the following parameters: The comparative results analysis of the Mellin transform method (MTM) in the context of Black-Scholes model (BSM), binomial model (BM), recursive method (RM) and Finite difference method (FDM) are shown in Table 8.American power put option values using Table 1.
Figure 2. The comparative results analysis using Table 2.
Figure 3.The comparative results analysis using Table 3.
Figure 4.The comparative results analysis using Table 4.
Figure 5.The free boundaries of American power put option with n = 1.

Figure 6.
The comparative results analysis using Table 8.
τ denote the price of the European put option with no dividend yield and free boundary for the American put option with no dividend yield respectively.Let Substituting the value of the early exercise function ( ) , f v y from (43) and ) into (41) we have the early exercise premium for the American put option with non-dividend yield as into (40) we get the integral equation (35) obtained byKim [6]  as the free boundary Ŝτ of the American put option which pay no dividend yield (37) derived byKim [ rearranging, ( the convolution theorem of the Mellin transform we have that the inner integral, (102) becomes < ≤ .For (110) to hold as T → ∞ , it is necessary that

3
The results generated for the price of the American power put option via Black-Scholes model (BSM), binomial model (BM) and the Mellin transform method (MTM) are shown in Tables2-4below.Also the results generated for the free boundary of the American power put option are shown in Tables5-7below.Experiment We consider the valuation of the American Power put option with the following parameters:

Figure 1 .
Figure 1.American power put option values using Table1. d

Table 1 .
American power put values.

Table 2 .
The price of American power put option using

Table 3 .
The price of American power put option using

Table 4 .
The price of American power put option using