A Note on the Pricing of American Capped Power Put Option

( ) ( ) + = − i V x K x = x b (6) Abstract We give an explicit solution to the perpetual American capped power put option pricing problem in the Black- Scholes-Merton Model. The approach is mainly based on free-boundary formulation and verification. For completeness we also


Yoshitaka Sakagami
Faculty of Management, Otemon Gakuin University, Osaka Prefecture 567-0008, Japan The steps to solve this free-boundary problem is same as in the case of the standard American put option (i=1) [5] so we only outline the steps. Since ( ) ≤ V x K , the equation 5 implies Di r (11) 1 1 / Thus ( ) V x is written as Now we have the following theorem.
, and the optimal stopping time is given by Tb The steps to prove this theorem is same as in the case of the standard American put option (i = 1) [5], so we omit.
It should be noted that for i < 1,

Introduction
A standard American power put option is a financial contract that allows the holder to sell an asset for a prescribed amount at any time. The price of this asset is raised to some power. The case of power being one corresponds to the usual American put option [1]. For the European power put option, the value of this option is given [2][3][4]. For the perpetual American power (≥ 1) put option, for completeness, we give the value of this option and the optimal stopping time. A capped power option is a power option whose maximum payoff is set to a prescribed level. For the European capped power put option, the value of this option is given [2][3][4][5]. For the perpetual American capped power put option, we give the value of this option and the optimal stopping time. Throughout this note, the approach is mainly based by freeboundary formulation and verification. Only one exception is Theorem 2.2. This note is organized as follows. In Section 2, we explicitly solve the perpetual American power put option pricing problem. In Section 3, we explicitly solve the perpetual American capped power put option pricing problem [6][7][8].

The Perpetual American Power Put Option
The arbitrage-free price of the perpetual American power put option is given by where K is the strike price, T is a stopping time, i is a positive constant greater than or equal 1, and x>0 is the initial value of the stock price process X=(Xt)t ≥ 0. In equation 1, the supremum is taken over all stopping times T of the process X started at x. The stock price process X=(Xt)t ≥ 0 is assumed to be a geometric Brownian motion. That is, Where r , 0 σ ≥ . The infinitesimal generator of X is given by As in the case of standard perpetual American put option, we suppose that there exists a point is optimal in equation 1 Then we solve the following free-boundary problem for unknown V and b.

Abstract
We give an explicit solution to the perpetual American capped power put option pricing problem in the Black-

The Perpetual American Capped Power Put Option
The arbitrage-free price of the perpetual American capped power put option is given by( Where ( ) < C K and i are positive constants. First we suppose there exists a point is optimal in (14). Here h satisfies Then we solve the following free-boundary problem for unknown V and b.
It is clear that for 0 < ≤ x h the arbitrage-free price from equation14 is given by equation 21. Thus, 0 < ≤ x h is the stopping region.
Since we suppose that 1 ( , ) ∈ i b h K , b is same as in the case of the American power put option. Thus the free-boundary b is given by x h it is easily seen to hold. Thus Thus it is easily verified by standard means (using the localization of M and Fatou's lemma) that we get that s T X b h . Hence the fourth term in the right-hand side of this equation is zero. Moreover using equation16, we find the second term in the right-hand side is also zero. Finally using the optional sampling theorem we get Letting n go to infinity and using the dominated convergence and, we get that . The proof is completed.

Theorem 2.2: Suppose that
The optimal stopping time is given by h T .