Faculty of Management, Otemon Gakuin University, Osaka Prefecture 567-0008, Japan
Received March 17, 2015; Accepted April 28, 2015; Published June 15, 2015
Citation: Sakagami Y (2015) DA Note on the Pricing of American Capped Power Put Option. J Bus Fin Aff 4:139. doi:10.4172/2167-0234.1000139
Copyright: © 2015 Sakagami Y. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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We give an explicit solution to the perpetual American capped power put option pricing problem in the BlackScholes-Merton Model. The approach is mainly based on free-boundary formulation and verification. For completeness we also give an explicit solution to the perpetual American standard power (_ 1) option pricing problem.
The perpetual American capped power put option; Geo-etric Brownian motion; Free-boundary
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 . For the European power put option, the value of this option is given [2-4]. 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 [2-5]. 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 [6-8].
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 ,.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 such that
is optimal in equation 1 Then we solve the following free-boundary problem for unknown V and b. Here 0 < bi < K.
The steps to solve this free-boundary problem is same as in the case of the standard American put option (i=1)  so we only outline the steps. Since , the equation 5 implies
Where and c is an undetermined constant. Using equation 10, we solve two equations 6 and 7 to give
Thus V (x) is written as
Now we have the following theorem.
Theorem 1.1: V (x) coincides with V*(x), 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) , so we omit.
It should be noted that for ,so we assume that i ≥ 1 in this section.
The arbitrage-free price of the perpetual American capped power put option is given by^
Where and i are positive constants. First we suppose there exists a point such that
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 the arbitrage-free price from equation14 is given by equation 21. Thus, is the stopping region.
Since we suppose that , b is same as in the case of the American power put option. Thus the free-boundary b is given by
is written as
Note that is equivalent to
Theorem 2.1: Suppose that then coincides with and the optimal stopping time is given by
Proof: From our earlier consideration we can suppose that Since and , the change-ofvariable formula (see Remark 2.3 in ) with the smooth-fit condition (18) gives
S When , we see that for . For ,it clearly holds. For it is easily seen to hold. Thus
Where is defined by
is a continuous martingale (because is bounded for all x > 0. Thus it is easily verified by standard means (using the localization of M and Fatou’s lemma) that we get that
Next we set in equation 24. Here is a localization sequence of bounded stopping times for M. For .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
for all . The proof is completed.
Theorem 2.2: Suppose that then V (x) is written as
The optimal stopping time is given by Th .
Proof: We set . Then
Where the final equality follows by the formula for the expected first hitting time for a geometric Brownian motion
Now we show for
We set f (x) to be equal to the left-hand side minus the right-hand side in equation 29. Clearly To show that , it
Because x > h . Since (equation30) holds. Since equation 29 and 30 imply that (h,∞) is a continuous region. On the other hand, (0,h] is a stopping region. Thus h T is the optimal stopping time and V (x) is given by equation 26. The proof is completed.
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