Optimal stopping problem of American put options for the diffusion risk model

Introduction

Option is a kind of financial derivative that first appeared in the United States in the mid-1970s and has been developing rapidly. Options can be divided into European options and American options according to the transaction time. However, American options can be traded at any time during the period of validity, they are more widely used in international finance. In recent years, the pricing problem and trading time of American options has become one of the most popular optimal-stopping problems in the field of financial mathematics. And domestic and foreign scholars have widely used the optimal stopping time methods to study American options. For example, Bensoussan (1984) [1] used the optimal stopping time problem to characterize the execution time of American-style undetermined equity pricing under complete market conditions. Merton and Pliska (1995) [2] used the optimal stopping time theory to study the problem of securities investment with fixed transaction costs. Ekstrom (2004) [3] showed that the convexity of the optimal stopping boundary for the American put option in the standard Black-Scholes model. Chiarolla and Angelis (2015) [4] studied the optimal stopping problem of pricing American put option on a zero coupon bond in the Heath-Jarrow-Morton model for forward interest rates, and showed the first price of the American bond option equals the optimal payoff. Armerin (2019) [5] considered an optimal stopping problem of a killed exponentially growing process and obtained the valuation of American call option. Woo and Choe (2020) [6] solved the optimal stopping problem related to maximum and minimum and found the closed form pricing formula of the perpetual American look back spread option. Based on these, the optimal stopping time problem of American put options for the diffusion risk model is considered.

, which is a gain function equal to the American put option. 0 r  Represents a constant rate and K is a constant that represents strike price. The esssup denotes the essential supremum (see Peskir and Shiryaev (2006) [7] for the definition). In order to get the optimal value function of the American put option and the optimal trading time, the problem is transformed into solving an optimal stopping problem on { , 0} t Ut  . That is, finding an optimal stopping time *

Where () Vu is the optimal value function of America put option and *  is the optimal trade time.

The remainder of the paper is organized as follows. In section 2, the formulation of the optimal stopping times problem for the diffusion risk model are given. In section 3, the optimal stopping times are obtained. Furthermore, numerical examples are presented to illustrate the applications of the optimal stopping problem in section 4.

Formulation of the optimal stopping times

The model (1.1) is a time-homogeneous Markov process taking values in with generator A (see Klebaner (1998) [8] ) that

Where f belongs to the domain () DA of the generator A of { , 0} t Ut  and f is twice continuously differentiable with respect to x . For a fixed 0

Obviously, for (0, ] ua  , (0, ) 0

The Laplace-Stieltjes transforms of the hitting time for the diffusion risk model can be obtained (see He and Yang (2017) [9] ) as follows. aK  , it is meaningless and * a does not depend on .

, ).

and the value function * ()

The optimal stopping problem Theorem 3.1: Let { , 0}

and ()

Vu be the value function of the optimal stopping problem defined in ( )

For a family of functions { ( ), (

is an optimal stopping time.

Proof. It is known that there exists a * (0, ) aK  such that * ( ) ( )

, for all (0, ) a  and all (0, )

The operator of the model ( )

= -

References

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2. Merton A, Pliska S. Optimal portfolio management with fixed transaction costs. Math Finance 1995;5(4):337-356.
3. Ekstrom E. Convexity of the optimal stopping boundary for the American put option. Journal of Mathematical Analysis and Applications 2004;299(1):147-156.
4. Chiarolla MB, Angelis TD. Analytical pricing of American Put options on a Zero Coupon bond in the Heath-Jarrow-Morton model. Stochastic Processes and their Applications 2015;125(2):678-707.
5. Armerin F. Optimal stopping of a killed exponentially growing process. Applied Mathematics and Computation 2019;353:208-214.
6. Woo MH, Choe GH. Pricing of American look back spread options. Stochastic Processes and their Applications 2020. https://doi.org/10.1016/j.spa.2020.05.012.
7. Peskir G, Shiryaev A. Optimal stopping and free boundary problems. Zurich: Birkhauser Verleg 2006.
8. Klebaner FC. Introduction to stichastic calculus with applications. Imperial Coleege Press 1998.
9. He JM, Yang YT. The exit times for the diffusion risk model with drift coefficient. Dynamical Systems and Differential Equations 2017;7(2):136-141.