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.

SHOULD AN AMERICAN OPTION BE EXERCISED EARLIER OR LATER IF VOLATILITY IS NOT ASSUMED TO BE A CONSTANT?

2011 ◽  
Vol 14 (08) ◽  
pp. 1279-1297 ◽  
Author(s):  
SONG-PING ZHU ◽  
WEN-TING CHEN
Keyword(s):  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Put Options ◽  
American Put Option ◽  
Put Option ◽  
Exercise Price ◽  
Volatility Model ◽  
American Put

In this paper, we present a correction to Merton (1973)'s well-known classical case of pricing perpetual American put options by considering the same pricing problem under a stochastic volatility model with the assumption that the volatility is slowly varying. Two analytic formulae for the option price and the optimal exercise price of a perpetual American put option are derived, respectively. Upon comparing the results obtained from our analytic approximations with those calculated by a spectral collocation method, it is shown that our current approximation formulae provide fast and reasonably accurate numerical values of both option price and the optimal exercise price of a perpetual American put option, within the validity of the assumption we have made for the asymptotic expansion. We shall also show that the range of applicability of our formulae is remarkably wider than it was initially aimed for, after the original assumption on the order of the "volatility of volatility" being somewhat relaxed. Based on the newly-derived formulae, the quantitative effect of the stochastic volatility on the optimal exercise strategy of a perpetual American put option has also been discussed. A most noticeable and interesting result is that there is a special cut-off value for the spot variance, below which a perpetual American put option priced under the Heston model should be held longer than the case of the same option priced under the traditional Black-Scholes model, when the price of the underlying is falling.

scholarly journals The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yanli Zhou ◽  
Shican Liu ◽  
Shuang Li ◽  
Xiangyu Ge
Keyword(s):  
Finite Difference ◽  
Stochastic Volatility ◽  
Asymptotic Approximation ◽  
Implied Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
American Put Option ◽  
Put Option ◽  
Volatility Model ◽  
American Put

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

On Pricing American Put Option on a Fixed Term: A Monte Carlo Approach

2021 ◽  
pp. 2050010
Author(s):  
Perpetual Andam Boiquaye
Keyword(s):  
Monte Carlo ◽  
Optimal Stopping ◽  
Geometric Mean ◽  
Option Price ◽  
Stopping Time ◽  
Optimal Stopping Time ◽  
American Put Option ◽  
Put Option ◽  
One Year ◽  
American Put

This paper focuses primarily on pricing an American put option with a fixed term where the price process is geometric mean-reverting. The change of measure is assumed to be incorporated. Monte Carlo simulation was used to calculate the price of the option and the results obtained were analyzed. The option price was found to be $94.42 and the optimal stopping time was approximately one year after the option was sold which means that exercising early is the best for an American put option on a fixed term. Also, the seller of the put option should have sold $0.01 assets and bought $ 95.51 bonds to get the same payoff as the buyer at the end of one year for it to be a zero-sum game. In the simulation study, the parameters were varied to see the influence it had on the option price and the stopping time and it showed that it either increases or decreases the value of the option price and the optimal stopping time or it remained unchanged.

THE EARLY EXERCISE PREMIUM IN AMERICAN OPTIONS BY USING NONPARAMETRIC REGRESSIONS

2018 ◽  
Vol 21 (07) ◽  
pp. 1850039
Author(s):  
WEIPING LI ◽  
SU CHEN
Keyword(s):  
Mean Square Error ◽  
Nonparametric Regression ◽  
American Options ◽  
Mean Square ◽  
Early Exercise ◽  
American Put Option ◽  
Put Option ◽  
Out Of Sample ◽  
American Put ◽  
Early Exercise Premium

The early exercise premium and the price of an American put option are evaluated by using nonparametric regression on the time to expiration, the moneyness and the volatility of underlying assets. In terms of mean square error (MSE), our nonparametric methods of American put option pricings outperform the existing classical methods for both in-the-sample (1 September 2011–31 January 2012) and out-of-sample (1 September 2012–28 February 2013) testings on the S&P 100 Index (OEX). Our methods have better predictions and more accurate approximations. The Greek letters for both the early exercise premium and the American put option are computed numerically.

Error Estimates for Approximations of American Put Option Price

2012 ◽  
Vol 12 (1) ◽  
pp. 108-120
Author(s):  
David Šiška
Keyword(s):  
Approximation Error ◽  
Option Price ◽  
Discrete Problem ◽  
American Put Option ◽  
Finite Difference Approximations ◽  
Time Discretisation ◽  
Put Option ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
American Put

Abstract Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.

scholarly journals Examining the Efficiency of American Put Option Pricing by Monte Carlo Methods with Variance Reduction

2018 ◽  
Vol 10 (2) ◽  
pp. 10
Author(s):  
George Chang
Keyword(s):  
Monte Carlo ◽  
Option Pricing ◽  
Variance Reduction ◽  
American Options ◽  
Balance Sheet ◽  
Put Options ◽  
Control Variate ◽  
American Put Option ◽  
Put Option ◽  
American Put

We apply the Monte Carlo simulation algorithm developed by Broadie and Glasserman (1997) and the control variate technique first introduced to asset pricing via simulation by Boyle (1977) to examine the efficiency of American put option pricing via this combined method. The importance and effectiveness of variance reduction is clearly demonstrated in our simulation results. We also found that the control variates technique does not work as well for deep-in-the-money American put options. This is because deep-in-the-money American options are more likely to be exercised early, thus the value of the American options are less in line (or less correlated) with those of their European counterparts. the same FPESS can also be observed when investigators partition large datasets into smaller datasets to address a variety of auditing questions. In this study, we fill the empirical gap in the literature by investigating the sensitivity of the FPESS to partitioned datasets. We randomly selected 16 balance-sheet datasets from: China Stock Market Financial Statements Database™, that tested to be Benford Conforming noted as RBCD. We then explore how partitioning these datasets affects the FPESS by repeated randomly sampling: first 10% of the RBCD and then selecting 250 observations from the RBCD. This created two partitioned groups of 160 datasets each. The Statistical profile observed was: For the RBCD there were no indications of Non-Conformity; for the 10%-Sample there were no overall indications that Extended Procedures would be warranted; and for the 250-Sample there were a number of indications that the dataset was Non-Conforming. This demonstrated clearly that small datasets are indeed likely to create the FPESS. We offer a discussion of these results with implications for audits in the Big-Data context where the audit In-charge would find it necessary to partition the datasets of the client. 

scholarly journals Option Pricing under the Jump Diffusion and Multifactor Stochastic Processes

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Shican Liu ◽  
Yanli Zhou ◽  
Yonghong Wu ◽  
Xiangyu Ge
Keyword(s):  
Diffusion Process ◽  
Option Pricing ◽  
Stochastic Volatility ◽  
Term Structure ◽  
Implied Volatility ◽  
Jump Diffusion ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Volatility Model ◽  
Jump Diffusion Process

In financial markets, there exists long-observed feature of the implied volatility surface such as volatility smile and skew. Stochastic volatility models are commonly used to model this financial phenomenon more accurately compared with the conventional Black-Scholes pricing models. However, one factor stochastic volatility model is not good enough to capture the term structure phenomenon of volatility smirk. In our paper, we extend the Heston model to be a hybrid option pricing model driven by multiscale stochastic volatility and jump diffusion process. In our model the correlation effects have been taken into consideration. For the reason that the combination of multiscale volatility processes and jump diffusion process results in a high dimensional differential equation (PIDE), an efficient finite element method is proposed and the integral term arising from the jump term is absorbed to simplify the problem. The numerical results show an efficient explanation for volatility smirks when we incorporate jumps into both the stock process and the volatility process.

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