ANALYTICAL PRICING OF DOUBLE-BARRIER OPTIONS UNDER A DOUBLE-EXPONENTIAL JUMP DIFFUSION PROCESS: APPLICATIONS OF LAPLACE TRANSFORM

2004 ◽  
Vol 07 (02) ◽  
pp. 151-175 ◽  
Author(s):  
ARTUR SEPP
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
Asset Price ◽  
Option Price ◽  
Jump Diffusion ◽  
Time Dependent ◽  
Barrier Option ◽  
Double Exponential ◽  
Jump Diffusion Process ◽  
Risk Parameters

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility. Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black–Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.

First passage times of a jump diffusion process

2003 ◽  
Vol 35 (2) ◽  
pp. 504-531 ◽  
Author(s):  
S. G. Kou ◽  
Hui Wang
Keyword(s):  
Diffusion Process ◽  
Diffusion Processes ◽  
Jump Diffusion ◽  
First Passage Times ◽  
First Passage ◽  
Jump Diffusion Processes ◽  
Lookback Options ◽  
Double Exponential ◽  
Jump Diffusion Process ◽  
Passage Times

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

scholarly journals Pricing of American Put Option under a Jump Diffusion Process with Stochastic Volatility in an Incomplete Market

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Shuang Li ◽  
Yanli Zhou ◽  
Xinfeng Ruan ◽  
B. Wiwatanapataphee
Keyword(s):  
Diffusion Process ◽  
Stochastic Volatility ◽  
American Options ◽  
Option Price ◽  
Jump Diffusion ◽  
Incomplete Market ◽  
American Put Option ◽  
Put Option ◽  
Jump Diffusion Process ◽  
American Put

We study the pricing of American options in an incomplete market in which the dynamics of the underlying risky asset is driven by a jump diffusion process with stochastic volatility. By employing a risk-minimization criterion, we obtain the Radon-Nikodym derivative for the minimal martingale measure and consequently a linear complementarity problem (LCP) for American option price. An iterative method is then established to solve the LCP problem for American put option price. Our numerical results show that the model and numerical scheme are robust in capturing the feature of incomplete finance market, particularly the influence of market volatility on the price of American options.

scholarly journals The distribution of refracted Lévy processes with jumps having rational Laplace transforms

2017 ◽  
Vol 54 (4) ◽  
pp. 1167-1192
Author(s):  
Jiang Zhou ◽  
Lan Wu
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
Jump Diffusion ◽  
Laplace Transforms ◽  
Variable Annuities ◽  
State Dependent ◽  
Approximating Procedure ◽  
Jump Diffusion Process ◽  
The Laplace Transform ◽  
Interesting Approach

Abstract We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulae for the Laplace transform of its distribution. Our formulae are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. In addition, the results are used to price variable annuities with state-dependent fees.

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