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Time dependent Heston model. (English) Zbl 1198.91203

Using volatility of volatility expansion, the authors derive a fairly accurate analytical approximation formula for the price of vanilla options for the Heston model with time dependent parameters, and establish tight error estimates. This work may be considered as an extension of some results of A. L. Lewis [Option valuation under stochastic volatility. With Mathematica code. Newport Beach, CA: Finance Press (2000; Zbl 0937.91060)] to time dependent parameters.

For every $\epsilon \in \left[0,1\right]$, consider the stochastic differential equation

$d{X}_{t}^{\epsilon }=\sqrt{{v}_{t}^{\epsilon }}d{W}_{t}-\frac{{v}_{t}^{\epsilon }}{2}dt,\phantom{\rule{3.33333pt}{0ex}}{X}_{0}^{\epsilon }={x}_{0},$
$d{v}_{t}^{\epsilon }=\kappa \left({\vartheta }_{t}-{v}_{t}^{\epsilon }\right)dt+\epsilon {\xi }_{t}\sqrt{{v}_{t}^{\epsilon }}d{B}_{t},\phantom{\rule{3.33333pt}{0ex}}{v}_{0}^{\epsilon }={v}_{0},$
$d{〈W,B〉}_{t}={\rho }_{t}dt,$

where ${\left({B}_{t},{W}_{t}\right)}_{0\le t\le T}$ is a two-dimensional correlated Brownian motion, ${\left({X}_{t}^{\epsilon }\right)}_{0\le t\le T}$ stands for the logarithm of the forward price process and ${\left({v}_{t}^{\epsilon }\right)}_{0\le t\le T}$ is the square of the volatility process. The expected payoff of a put option with strike $K$ is

$g\left(\epsilon \right)={e}^{-{\int }_{0}^{T}{r}_{t}dt}𝔼\left[{\left(K-{e}^{{\int }_{0}^{T}{r}_{t}-{q}_{t}dt+{X}_{T}^{\epsilon }}\right)}_{+}\right],$

where $T$ is the time of maturity, ${\left({r}_{t}\right)}_{0\le t\le T}$ and ${\left({q}_{t}\right)}_{0\le t\le T}$ are the risk-free rate and the dividend yield, respectively.

The main result of the paper is an approximation of $g\left(1\right)$ by an analytical formula. To this end, it is observed that

$g\left(\epsilon \right)=𝔼\left[{P}_{BS}\left({x}_{0}+{\int }_{0}^{T}{\rho }_{t}\sqrt{{v}_{t}^{\epsilon }}d{B}_{t}-{\int }_{0}^{T}\frac{{\rho }_{t}^{2}}{2}{v}_{t}^{\epsilon }dt,{\int }_{0}^{T}\left(1-{\rho }_{t}^{2}\right){v}_{t}^{\epsilon }dt\right)\right],\phantom{\rule{2.em}{0ex}}\left(☆\right)$

where $\left(x,y\right)↦{P}_{BS}\left(x,y\right)$ is the Black-Scholes price of a put option with spot price ${e}^{x}$, strike price $K$, total variance $y$, risk-free rate ${r}_{eq}={\int }_{0}^{T}{r}_{t}dt/T$, dividend yield ${q}_{eq}={\int }_{0}^{T}{q}_{t}dt/T$, and maturity $T$. The approximation of $g\left(1\right)$ is obtained from ($☆$) by taking second order Taylor series expansions of the function $\epsilon ↦{v}_{t}^{\epsilon }$ and of ${P}_{BS}$ with respect to its first and second variables, and by evaluating the expectation of the resulting approximative expression. It is shown that the error of approximation is

$O\left({T}^{2}{sup}_{t\in \left[0,T\right]}{\xi }_{t}^{3}\right)$

provided ${sup}_{t\in \left[0,T\right]}|{\rho }_{t}|<1$ and ${inf}_{t\in \left[0,T\right]}{\xi }_{t}>0,\phantom{\rule{3.33333pt}{0ex}}{inf}_{t\in \left[0,T\right]}2\kappa {\vartheta }_{t}/{\xi }_{t}^{2}\ge 1·$

By calibrating the model to market data from the $S&P$ 500 Index, the authors show that their approach, using piecewise constant parameters, allows a significant reduction of the calibration error, e.g. as compared to the above mentioned constant parameter approach of Lewis. However, the obtained parameters vary greatly in time, which makes the use of the Heston model with piecewise constant parameters questionable. Moreover, the assumption ${inf}_{t\in \left[0,T\right]}2\kappa {\vartheta }_{t}/{\xi }_{t}^{2}\ge 1$, which is of crucial importance for the error estimate, may not hold in practice.

##### MSC:
 91G20 Derivative securities 60H07 Stochastic calculus of variations and the Malliavin calculus 91G80 Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)