Abstract
Stochastic volatility models are used in mathematical finance to describe the dynamics of asset prices. In these models, the asset price is modeled as a stochastic process depending on time implicitly defined by a stochastic differential Equation. The volatility of the asset price itself is modeled as a stochastic process depending on time whose dynamics is described by a stochastic differential Equation. The stochastic differential Equations for the asset price and for the volatility are coupled and together with the necessary initial conditions and correlation assumptions constitute the model. Note that the stochastic volatility is not observable in the financial markets. In order to use these models, for example, to evaluate prices of derivatives on the asset or to forecast asset prices, it is necessary to calibrate them. That is, it is necessary to estimate starting from a set of data the values of the initial volatility and of the unknown parameters that appear in the asset price/volatility dynamic Equations. These data usually are observations of the asset prices and/or of the prices of derivatives on the asset at some known times. We analyze some stochastic volatility models summarizing merits and weaknesses of each of them. We point out that these models are examples of stochastic state space models and present the main techniques used to calibrate them. A calibration problem for the Heston model is solved using the maximum likelihood method. Some numerical experiments about the calibration of the Heston model involving synthetic and real data are presented.
Highlights
Stochastic processes, stochastic differential Equations and partial differential Equations are used to describe the dynamics of random phenomena
Note that when option prices are used as data of the calibration problem it is necessary to compute their theoretical value in the stochastic volatility model considered as a function of the values assigned to the parameters
In the formulation of the calibration problem that follows we use the joint probability density function of the variables St, vt,t > 0, of the Heston model conditioned to the observations S i,C i made at the times ti such that ti
Summary
Stochastic processes, stochastic differential Equations and partial differential Equations are used to describe the dynamics of random phenomena. In 1973, Black and Scholes [3] introduced stochastic differential Equations in mathematical finance to model the random behavior of asset prices. The assumption made by Black and Scholes that the asset price volatility is a constant has been criticized To overcome this difficulty, around 1990, several stochastic volatility models of the asset price dynamics had been introduced in the theory and in the practice of mathematical finance. Around 1990, several stochastic volatility models of the asset price dynamics had been introduced in the theory and in the practice of mathematical finance Models of this type are: the Hull and White model, the Stein and Stein model, the Heston model, the SABR model and many others that are not possible to mention individually. This review paper introduces the stochastic volatility models, explains some of the calibration problems associated to them, illustrates the main calibration methods, and considers in detail as an example the problem of calibrating the Heston model with the maximum likelihood method using synthetic and real data
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