Probability distribution of returns in the Heston model with stochastic volatility

By A. Drăgulescu, Victor M. Yakovenko. 2002

312 citations recorded by the source corpus.

Abstract We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker‐Planck equation exactly and, after integrating out the variance, find an analytic formula for the time‐dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow‐Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log‐returns with a time‐dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow‐Jones data for 1982–2001 follow the scaling function for seven orders of magnitude.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/7b8bf929ec39f7ec3ee8c408a2ddac88230bec03

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Citations in the corpus

Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion (2018, background): “As already remarked in [67] in this setting the resulting stochastic equation for D(t) is nothing else than the Heston model, that is widely used in financial mathematics and specifies the time evolution of the stochastic volatility of a given asset [90, 97, 98].”
A CLOSED‐FORM EXACT SOLUTION FOR PRICING VARIANCE SWAPS WITH STOCHASTIC VOLATILITY (2010, methodology): “This set of parameters for the square root process was also adopted by Dragulescu and Yakovenko (2002). As for the MC simulations, we took asset price S0 = 1 and the number of the paths N = 200, 000 for all the simulation results presented here.”
Moment explosions in stochastic volatility models (2006, unknown): “Proof Follows from the calculations of the density in the Heston model from [ 10 ].”
The characteristic function of rough Heston models (2016, background): “The popularity of this model is probably due to three main reasons: • It reproduces well several important stylized facts of low frequency price data, namely leverage effect, time-varying volatility and fat tails, see [7, 9, 13, 35].”