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A Note on the Pricing of American Capped Power Put Option | OMICS International
ISSN: 2167-0234
Journal of Business & Financial Affairs
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A Note on the Pricing of American Capped Power Put Option

Yoshitaka Sakagami*

Faculty of Management, Otemon Gakuin University, Osaka Prefecture 567-0008, Japan

*Corresponding Author:
Yoshitaka Sakagami
Faculty of Management
Otemon Gakuin University
Osaka Prefecture 567-0008
Japan
Tel:+81-72-641-9520
E-mail: [email protected]

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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Abstract

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.

Keywords

The perpetual American capped power put option; Geo-etric Brownian motion; Free-boundary

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 [2-4]. 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 Perpetual American Power Put Option

The arbitrage-free price of the perpetual American power put option is given by

Equation            (1)

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,

Equation                             (2)

Where ,Equation.The infinitesimal generator of X is given by

Equation                    (3)

As in the case of standard perpetual American put option, we suppose that there exists a point Equation such that

Equation                   (4)

is optimal in equation 1 Then we solve the following free-boundary problem for unknown V and b. Here 0 < bi < K.

Equation                  (5)

Equation                  (6)

Equation                  (7)

Equation                  (8)

Equation                  (9)

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 Equation, the equation 5 implies

Equation                                            (10)

Where Equation and c is an undetermined constant. Using equation 10, we solve two equations 6 and 7 to give

Equation                                  (11)

Equation                                  (12)

Thus V (x) is written as

Equation                              (13)

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) [5], so we omit.

It should be noted that for Equation,so we assume that i ≥ 1 in this section.

The Perpetual American Capped Power Put Option

The arbitrage-free price of the perpetual American capped power put option is given by^

Equation                     (14)

Where Equation and i are positive constants. First we suppose there exists a point Equation such that

Equation                                                    (15)

is optimal in (14). Here h satisfies Equation. Then we solve the following free-boundary problem for unknown V and b.

Equation                                          (16)

Equation                                          (17)

Equation                         (18)

Equation                         (19)

Equation                         (20)

Equation                         (21)

It is clear that for Equationthe arbitrage-free price from equation14 is given by equation 21. Thus, Equation is the stopping region.

Since we suppose that Equation, b is same as in the case of the American power put option. Thus the free-boundary b is given by

Equation                      (22)

Equation is written as

Equationif Equation                                (23)

Note that Equation is equivalent to Equation

Theorem 2.1: Suppose that Equationthen Equation coincides with Equationand the optimal stopping time is given by Equation

Proof: From our earlier consideration we can suppose that EquationSince Equationand Equation, the change-ofvariable formula (see Remark 2.3 in [5]) with the smooth-fit condition (18) gives

Equation                       (24)

Equation

Equation

S When Equation, we see that Equationfor Equation. For Equation,it clearly holds. For Equation it is easily seen to hold. Thus

Equation

Where Equation is defined by

Equation

is a continuous martingale (because Equation 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

Equation

For all Equation

Next we set Equation in equation 24. Here Equation is a localization sequence of bounded stopping times for M. For Equation.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

Equation

Equation                     (25)

Letting n go to infinity and using the dominated convergence and, we get that

Equation

for all Equation. The proof is completed.

Theorem 2.2: Suppose that Equationthen V (x) is written as

Equationif Equation                (26)

x∈(h,∞)

The optimal stopping time is given by Th .

Proof: We set Equation. Then

Equation                            (27)

Equation

Equation

Equation

Equation                         (28)

Where the final equality follows by the formula for the expected first hitting time for a geometric Brownian motion

Now we show for Equation

Equation                     (29)

We set f (x) to be equal to the left-hand side minus the right-hand side in equation 29. ClearlyEquation To show that Equation, it

Equation                   (30)

Because x > h . Since Equation (equation30) holds. Since Equation 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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