A STOCHASTIC VOLATILITY MODEL FOR RISK-REVERSALS IN FOREIGN EXCHANGE

2009 ◽  
Vol 12 (06) ◽  
pp. 877-899 ◽  
Author(s):  
CLAUDIO ALBANESE ◽  
ALEKSANDAR MIJATOVIĆ
Keyword(s):  
Stochastic Volatility ◽  
Foreign Exchange ◽  
Continuous Time ◽  
Efficient Algorithms ◽  
Stochastic Volatility Model ◽  
Foreign Exchange Markets ◽  
Local Volatility ◽  
Volatility Model ◽  
Exchange Markets

It is a widely recognized fact that risk-reversals play a central role in the pricing of derivatives in foreign exchange markets. It is also known that the values of risk-reversals vary stochastically with time. In this paper we introduce a stochastic volatility model with jumps and local volatility, defined on a continuous time lattice, which provides a way of modeling this kind of risk using numerically stable and relatively efficient algorithms.

scholarly journals Local volatility in the Heston model: a Malliavin calculus approach

2005 ◽  
Vol 2005 (3) ◽  
pp. 307-322 ◽  
Author(s):  
Christian-Oliver Ewald
Keyword(s):  
Stochastic Volatility ◽  
Malliavin Calculus ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Explicit Formulas ◽  
Local Volatility ◽  
Girsanov Transformation ◽  
Volatility Model ◽  
Log Price ◽  
Derivatives Of

We implement the Heston stochastic volatility model by using multidimensional Ornstein-Uhlenbeck processes and a special Girsanov transformation, and consider the Malliavin calculus of this model. We derive explicit formulas for the Malliavin derivatives of the Heston volatility and the log-price, and give a formula for the local volatility which is approachable by Monte-Carlo methods.

scholarly journals MULTIVARIATE ECOGARCH PROCESSES

2010 ◽  
Vol 27 (2) ◽  
pp. 344-371 ◽  
Author(s):  
Stephan Haug ◽  
Robert Stelzer
Keyword(s):  
Stochastic Volatility ◽  
Continuous Time ◽  
Stochastic Volatility Model ◽  
Leverage Effect ◽  
Mixing Properties ◽  
Multivariate Extension ◽  
Volatility Model

A multivariate extension of the exponential continuous time GARCH (p, q) model (ECOGARCH) is introduced and studied. Stationarity and mixing properties of the new stochastic volatility model are investigated, and ways to model a component-wise leverage effect are presented.

IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY

2001 ◽  
Vol 04 (01) ◽  
pp. 45-89 ◽  
Author(s):  
ROGER W. LEE
Keyword(s):  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Asymptotic Methods ◽  
Stochastic Volatility Model ◽  
Slow Variation ◽  
Option Prices ◽  
Local Volatility ◽  
Volatility Model ◽  
Zero Correlation

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we call small-variation asymptotics, and against Fouque, Papanicolaou, and Sircar's rapid-variation asymptotics. We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

scholarly journals THE HESTON STOCHASTIC-LOCAL VOLATILITY MODEL: EFFICIENT MONTE CARLO SIMULATION

2014 ◽  
Vol 17 (07) ◽  
pp. 1450045 ◽  
Author(s):  
ANTHONIE W. VAN DER STOEP ◽  
LECH A. GRZELAK ◽  
CORNELIS W. OOSTERLEE
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Error Analysis ◽  
Stochastic Volatility ◽  
Numerical Experiments ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Local Volatility ◽  
Local Volatility Model ◽  
Volatility Model

In this paper we propose an efficient Monte Carlo scheme for simulating the stochastic volatility model of Heston (1993) enhanced by a nonparametric local volatility component. This hybrid model combines the main advantages of the Heston model and the local volatility model introduced by Dupire (1994) and Derman & Kani (1998). In particular, the additional local volatility component acts as a "compensator" that bridges the mismatch between the nonperfectly calibrated Heston model and the market quotes for European-type options. By means of numerical experiments we show that our scheme enables a consistent and fast pricing of products that are sensitive to the forward volatility skew. Detailed error analysis is also provided.

FOREIGN EXCHANGE OPTIONS UNDER STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES

2008 ◽  
Vol 11 (03) ◽  
pp. 277-294 ◽  
Author(s):  
REHEZ AHLIP
Keyword(s):  
Exchange Rate ◽  
Interest Rates ◽  
Stochastic Volatility ◽  
Foreign Exchange ◽  
Stochastic Volatility Model ◽  
Stochastic Interest Rates ◽  
Volatility Model ◽  
Stochastic Interest ◽  
The Exchange Rate ◽  
Inversion Techniques

In this paper, we present a stochastic volatility model with stochastic interest rates in a Foreign Exchange (FX) setting. The instantaneous volatility follows a mean-reverting Ornstein–Uhlenbeck process and is correlated with the exchange rate. The domestic and foreign interest rates are modeled by mean-reverting Ornstein–Uhlenbeck processes. The main result is an analytic formula for the price of a European call on the exchange rate. It is derived using martingale methods in arbitrage pricing of contingent claims and Fourier inversion techniques.

Sign in / Sign up

Export Citation Format

Share Document