scholarly journals Complications with stochastic volatility models

1998 ◽  
Vol 30 (1) ◽  
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.

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.

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.

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.

ROLE OF INFORMATION IN PRICING DEFAULT-SENSITIVE CONTINGENT CLAIMS

2015 ◽  
Vol 18 (01) ◽  
pp. 1550007 ◽  
Author(s):  
MONIQUE JEANBLANC ◽  
MARTA LENIEC
Keyword(s):  
Financial Market ◽  
Stock Price ◽  
Stopping Time ◽  
Contingent Claims ◽  
Contingent Claim ◽  
Martingale Measures ◽  
Equivalent Martingale Measures ◽  
Equivalent Martingale ◽  
Role Of Information

We consider a financial market with a savings account and a stock S that follows a general diffusion. The default of the company, which issues the stock S, is modeled as a stopping time with respect to the filtration generated by the value of the firm that is not observable by regular investors. We assume that the stock price and the value of the firm are correlated. We study three investors with different information levels trading in the market who aim to price a general default-sensitive contingent claim. We use the density approach and Yor's method to solve the pricing problem. Specifically, we find the sets of equivalent martingale measures in three cases and, when needed, we choose one of them using f-divergence approach.

scholarly journals Variance-Optimal Hedging in General Affine Stochastic Volatility Models

2010 ◽  
Vol 42 (1) ◽  
pp. 83-105 ◽  
Author(s):  
Jan Kallsen ◽  
Arnd Pauwels
Keyword(s):  
Stochastic Volatility ◽  
Continuous Time ◽  
Stock Price ◽  
Forthcoming Paper ◽  
Parametric Models ◽  
Stochastic Volatility Models ◽  
Optimal Hedging ◽  
Volatility Models ◽  
Optimal Hedge ◽  
General Affine

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

scholarly journals Variance-Optimal Hedging in General Affine Stochastic Volatility Models

2010 ◽  
Vol 42 (01) ◽  
pp. 83-105 ◽  
Author(s):  
Jan Kallsen ◽  
Arnd Pauwels
Keyword(s):  
Stochastic Volatility ◽  
Continuous Time ◽  
Stock Price ◽  
Forthcoming Paper ◽  
Parametric Models ◽  
Stochastic Volatility Models ◽  
Optimal Hedging ◽  
Volatility Models ◽  
Optimal Hedge ◽  
General Affine

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

scholarly journals STRICT LOCAL MARTINGALES VIA FILTRATION ENLARGEMENT

2019 ◽  
Vol 23 (01) ◽  
pp. 2050001
Author(s):  
ADITI DANDAPANI ◽  
PHILIP PROTTER
Keyword(s):  
Stochastic Volatility ◽  
Sufficient Conditions ◽  
Local Martingale ◽  
Change Of Measure ◽  
Stochastic Volatility Models ◽  
Strict Local Martingales ◽  
Volatility Models ◽  
Strict Local Martingale

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with and without jumps and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale. We provide examples of initial enlargement that effect this change.

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

1999 ◽  
Vol 36 (2) ◽  
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.

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