scholarly journals Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models

2009 ◽  
Vol 61 (3) ◽  
pp. 287-315 ◽  
Author(s):  
Archil Gulisashvili ◽  
Elias M. Stein
Keyword(s):  
Asymptotic Behavior ◽  
Stochastic Volatility ◽  
Stock Price ◽  
Implied Volatility ◽  
Distribution Density ◽  
Stochastic Volatility Models ◽  
Price Distribution ◽  
Volatility Models

PRICING TIMER OPTIONS: SECOND-ORDER MULTISCALE STOCHASTIC VOLATILITY ASYMPTOTICS

2021 ◽  
pp. 1-19
Author(s):  
XUHUI WANG ◽  
SHENG-JHIH WU ◽  
XINGYE YUE
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Option Price ◽  
Second Order ◽  
Approximation Formula ◽  
Stochastic Volatility Models ◽  
Regular Perturbation ◽  
Volatility Models ◽  
Order Expansion ◽  
High Level

Abstract We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

scholarly journals Super-replication in stochastic volatility models under portfolio constraints

1999 ◽  
Vol 36 (02) ◽  
pp. 523-545 ◽  
Author(s):  
Jakša Cvitanić ◽  
Huyên Pham ◽  
Nizar Touzi
Keyword(s):  
Stochastic Volatility ◽  
Stock Price ◽  
Convex Set ◽  
Short Selling ◽  
Contingent Claims ◽  
Stochastic Volatility Models ◽  
Portfolio Constraints ◽  
Volatility Models ◽  
Closed Convex Set

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a ‘hedger’ in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

scholarly journals Complications with stochastic volatility models

1998 ◽  
Vol 30 (01) ◽  
pp. 256-268 ◽  
Author(s):  
Carlos A. Sin
Keyword(s):  
Stochastic Volatility ◽  
Stock Price ◽  
Local Martingale ◽  
Stochastic Volatility Models ◽  
Martingale Measures ◽  
Equivalent Martingale Measures ◽  
Volatility Models ◽  
Equivalent Martingale

We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of equivalent martingale measures, and provide one explicit example.

ESTIMATION IN CONTINUOUS-TIME STOCHASTIC VOLATILITY MODELS USING NONLINEAR FILTERS

2000 ◽  
Vol 03 (02) ◽  
pp. 279-308 ◽  
Author(s):  
JAN NYGAARD NIELSEN ◽  
MARTIN VESTERGAARD
Keyword(s):  
Differential Equations ◽  
Stochastic Volatility ◽  
Stock Prices ◽  
Continuous Time ◽  
Stock Price ◽  
Estimation Method ◽  
Small Sample ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Filtering Problem

The stylized facts of stock prices, interest and exchange rates have led econometricians to propose stochastic volatility models in both discrete and continuous time. However, the volatility as a measure of economic uncertainty is not directly observable in the financial markets. The objective of the continuous-discrete filtering problem considered here is to obtain estimates of the stock price and, in particular, the volatility using discrete-time observations of the stock price. Furthermore, the nonlinear filter acts as an important part of a proposed method for maximum likelihood for estimating embedded parameters in stochastic differential equations. In general, only approximate solutions to the continuous-discrete filtering problem exist in the form of a set of ordinary differential equations for the mean and covariance of the state variables. In the present paper the small-sample properties of a second order filter is examined for some bivariate stochastic volatility models and the new combined parameter and state estimation method is applied to US stock market data.

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