Heston model
Heston’s Stochastic Volatility
Model Implementation,
Calibration and Some Extensions
Sergei Mikhailov, Ulrich Nögel
Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany,
Mikhailov@itwm.fhg.de; Noegel@itwm.fhg.de
1 Introduction
The paper discusses theoretical properties, shows the performance and presents some extensions of Heston’s (1993) stochastic volatility model.
The model proposed by Heston extends the Black and Scholes (1993) model and includes it as a special case. Heston’s setting take into account non-lognormal distribution of the assets returns, leverage effect, important mean-reverting property of volatility and it remains analytically tractable. The Black-Scholes volatility surfaces generated by Heston’s model look like empirical implied volatility surfaces. The complication is related to the risk-neutral valuation concept. It is not possible to build a riskless portfolio if we formulate the statement that the volatility of the asset varies stochastically. This is principally because the volatility is not a tradable security.
2 Heston’s Stochastic Volatility Model
In this section we specify Heston’s stochastic volatility model and provide some details how to compute options prices.
We use the following notations:
S(t)
V(t)
C
74
Equity spot price, financial index. . ..
Variance.
European call option price.
K
W1,2
r q κ θ V0 σ ρ t0 T
Strike price.
Standard Brownian movements.
Interest rate.
Dividend yield.
Mean reversion rate.
Long run variance.
Initial variance.
Volatility of variance.
Correlation parameter.
Current date.
Maturity date.
Heston’s stochastic volatility model (1993) is specified as followed dS(t) = µdt +
S(t)
V(t)dW 1 ,
(1.1)
dV(t) = κ(θ − V(t))dt + σ V(t)dW 2 .
(1.2)
To take into account leverage effect, Wiener stochastic processes W1 , W2 should be correlated dW 1 · dW 2 = ρdt . The stochastic model (1.2) for the variance is related to
Model Implementation,
Calibration and Some Extensions
Sergei Mikhailov, Ulrich Nögel
Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany,
Mikhailov@itwm.fhg.de; Noegel@itwm.fhg.de
1 Introduction
The paper discusses theoretical properties, shows the performance and presents some extensions of Heston’s (1993) stochastic volatility model.
The model proposed by Heston extends the Black and Scholes (1993) model and includes it as a special case. Heston’s setting take into account non-lognormal distribution of the assets returns, leverage effect, important mean-reverting property of volatility and it remains analytically tractable. The Black-Scholes volatility surfaces generated by Heston’s model look like empirical implied volatility surfaces. The complication is related to the risk-neutral valuation concept. It is not possible to build a riskless portfolio if we formulate the statement that the volatility of the asset varies stochastically. This is principally because the volatility is not a tradable security.
2 Heston’s Stochastic Volatility Model
In this section we specify Heston’s stochastic volatility model and provide some details how to compute options prices.
We use the following notations:
S(t)
V(t)
C
74
Equity spot price, financial index. . ..
Variance.
European call option price.
K
W1,2
r q κ θ V0 σ ρ t0 T
Strike price.
Standard Brownian movements.
Interest rate.
Dividend yield.
Mean reversion rate.
Long run variance.
Initial variance.
Volatility of variance.
Correlation parameter.
Current date.
Maturity date.
Heston’s stochastic volatility model (1993) is specified as followed dS(t) = µdt +
S(t)
V(t)dW 1 ,
(1.1)
dV(t) = κ(θ − V(t))dt + σ V(t)dW 2 .
(1.2)
To take into account leverage effect, Wiener stochastic processes W1 , W2 should be correlated dW 1 · dW 2 = ρdt . The stochastic model (1.2) for the variance is related to
References: I Cox, J., J. Ingersoll and S. Ross (1985): A theory of the term structure of interest rates. I Feller, W. (1951): Two singular diffusion problems, Annals of Mathematics 54, 173–182. I Heston, S. (1993): A closed-form solutions for options with stochastic volatility, Review of Financial Studies, 6, 327–343. I Heston, S. and S. Nandi (1997): A Closed Form GARCH Option Pricing Model, Federal Reserve Bank of Atlanta Working Paper 97-9, 1–34.