A NEW STOPPING PROBLEM AND THE CRITICAL EXERCISE PRICE FOR AMERICAN FRACTIONAL LOOKBACK OPTION IN A SPECIAL MIXED JUMP-DIFFUSION MODEL

2018 ◽  
Vol 34 (1) ◽  
pp. 27-52
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Market Pricing ◽  
Early Exercise ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary ◽  
Critical Boundary ◽  
Jump Diffusion Model

A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.

scholarly journals Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model

2017 ◽  
Vol 04 (02n03) ◽  
pp. 1750033
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Market Pricing ◽  
Early Exercise ◽  
Lookback Options ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary ◽  
Jump Diffusion Model

This study presents an efficient method for pricing the American fractional lookback option in the case where the stock price follows a mixed jump diffusion fraction Brownian motion. By using It ô formula and Wick–It ô–Skorohod integral, a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

scholarly journals Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Early Exercise ◽  
Optimal Exercise Boundary ◽  
Lookback Options ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary

A new framework for pricing the American fractional lookback option is developed in the case where the stock price follows a mixed jump-diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

scholarly journals The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Chao Wang ◽  
Shengwu Zhou ◽  
Jingyuan Yang
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Fractional Brownian Motion ◽  
Stock Price ◽  
Jump Diffusion ◽  
Default Intensity ◽  
Stochastic Interest ◽  
Jump Diffusion Model ◽  
Vulnerable Option ◽  
Vulnerable Options

Under the assumption of the stock price, interest rate, and default intensity obeying the stochastic differential equation driven by fractional Brownian motion, the jump-diffusion model is established for the financial market in fractional Brownian motion setting. With the changes of measures, the traditional pricing method is simplified and the general pricing formula is obtained for the European vulnerable option with stochastic interest rate. At the same time, the explicit expression for it comes into being.

scholarly journals Is the Jump-Diffusion Model a Good Solution for Credit Risk Modeling? The Case of Convertible Bonds

2019 ◽  
Author(s):  
Tim Xiao
Keyword(s):  
Credit Risk ◽  
Diffusion Model ◽  
Stock Price ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Reduced Form ◽  
Risk Modeling ◽  
Jump Diffusion Model ◽  
Credit Risk Modeling ◽  
Arbitrage Strategy

This paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

scholarly journals LONG MEMORY VERSION OF STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL WITH STOCHASTIC INTENSITY

2020 ◽  
Vol 38 (2) ◽  
Author(s):  
Somayeh Fallah ◽  
Farshid Mehrdoust
Keyword(s):  
Stochastic Volatility ◽  
Long Range ◽  
Diffusion Model ◽  
Long Memory ◽  
Stock Price ◽  
Real Data ◽  
Jump Diffusion ◽  
Heston Model ◽  
Long Range Dependence ◽  
Jump Diffusion Model

It is widely accepted that certain financial data exhibit long range dependence. We consider a fractional stochastic volatility jump diffusion model in which the stock price follows a double exponential jump diffusion process with volatility described by a long memory stochastic process and intensity rate expressed by an ordinary Cox, Ingersoll, and Ross (CIR) process. By calibrating the model with real data, we examine the performance of the model and also, we illustrate the role of long range dependence property by comparing our presented model with the Heston model.

THE BRITISH KNOCK-OUT PUT OPTION

2015 ◽  
Vol 18 (02) ◽  
pp. 1550008 ◽  
Author(s):  
LULUWAH AL-FAGIH
Keyword(s):  
Stock Price ◽  
Financial Analysis ◽  
Closed Form Expression ◽  
Form Expression ◽  
Knock Out ◽  
Early Exercise ◽  
New Class ◽  
Put Option ◽  
Exercise Boundary ◽  
Symmetry Relations

Following the economic rationale introduced by Peskir & Samee (2011, 2013) we present a new class of barrier options within the British payoff mechanism where the holder enjoys the early exercise feature of American type options whereupon his payoff (deliverable immediately) is the best prediction of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Should the option holder believe the true drift of the stock price to be unfavorable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. In this paper, we focus on the knock-out put option with an up barrier. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results, we perform a financial analysis of the British knock-out put option. We spot some of the trends previously seen in Peskir & Samee (2011) but observe some behavior unique to the knock-out case. Finally, we derive the British put-call and up-down symmetry relations which express the arbitrage-free price and the rational exercise boundary of the British down-and-out call option in terms of the arbitrage-free price and the rational exercise boundary of the British up-and-out put option.

Pricing Option on Jump Diffusion and Stochastic Interest Rates Model

2011 ◽  
Vol 50-51 ◽  
pp. 723-727
Author(s):  
Bo Peng ◽  
Zhi Hui Wu
Keyword(s):  
Interest Rates ◽  
Stock Price ◽  
Special Kind ◽  
Jump Process ◽  
Jump Diffusion ◽  
Time Interval ◽  
Martingale Method ◽  
Stochastic Interest Rates ◽  
Stochastic Interest ◽  
Jump Diffusion Model

This paper assumed that the stock price jump process for a special kind of renewal jump process, that is incident time interval for independent and subordinate to Gamma distribution random variable sequence. We obtain the European bi-direction option pricing formulas on jump diffusion model under the stochastic interest rates by simply mathematical induce by means of martingale method.

scholarly journals Valuation of Game Options in Jump-Diffusion Model and with Applications to Convertible Bonds

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Lei Wang ◽  
Zhiming Jin
Keyword(s):  
Free Boundary ◽  
Diffusion Process ◽  
Stock Price ◽  
Free Boundary Problems ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Simple Application ◽  
Boundary Problems ◽  
Game Options ◽  
Jump Diffusion Model

Game option is an American-type option with added feature that the writer can exercise the option at any time before maturity. In this paper, we consider some type of game options and obtain explicit expressions through solving Stefan(free boundary) problems under condition that the stock price is driven by some jump-diffusion process. Finally, we give a simple application about convertible bonds.

A TWO-FACTOR JUMP-DIFFUSION MODEL FOR PRICING CONVERTIBLE BONDS WITH DEFAULT RISK

2016 ◽  
Vol 19 (06) ◽  
pp. 1650046 ◽  
Author(s):  
RADHA KRISHN COONJOBEHARRY ◽  
DÉSIRÉ YANNICK TANGMAN ◽  
MUDDUN BHURUTH
Keyword(s):  
Interest Rate ◽  
Default Risk ◽  
Stock Price ◽  
Numerical Experiments ◽  
Diffusion Processes ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Minor Effect ◽  
Jump Diffusion Model ◽  
The Interest Rate

The current literature on convertible bonds (CBs) comprises only models where the stock price and the interest rate are governed by pure-diffusion processes. This paper fills a gap by developing and implementing a two-factor model where both underlying factors follow jump-diffusion processes, and which also incorporates default risk. We derive the partial integro-differential equation satisfied by the CB price in our model, and solve it by a spectral method based on Chebyshev discretizations and Clenshaw–Curtis quadratures. The conversion, call, and put constraints give rise to a linear complementarity problem, which is solved by an operator-splitting (OS) method. Through numerical experiments, we investigate the effects that the various parameters have on the CB price. In particular, our numerical experiments show that jumps in the stock price have a significant impact on the CB price, while jumps in the interest rate tend to have a minor effect on the price. In general, the dynamics of the stock price have more impact in pricing the CB than the dynamics of the interest rate.

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