###### STOCHASTIC VOLATILITY WITH LEVY´ PROCESSES: CALIBRATION AND PRICING

###### Postgraduate

ABSTRACT

In this thesis, stochastic volatility models with L´evy processes are treated in parameter calibration by the Carr-Madan fast Fourier transform (FFT) method and pricing through the partial integro-differential equation (PIDE) approach.

First, different models where the underlying log stock price or volatility driven by either a Brownian motion or a L´evy process are examined on Standard & Poor’s (S&P) 500 data. The absolute percentage errors show that the calibration errors are different between the models. Furthermore, a new method to estimate the standard errors, which can be seen as a generalization of the traditional error estimation method, is proposed and the results show that in all the parameters of a stochastic volatility model, some parameters are well-identified while the others are not.

Next, the previous approach to parameter calibration is modified by making the volatility constrained under the volatilty process of the model and by making the other model parameters fixed. Parameters are calibrated over five consecutive days on S&P 500 or foreign exchange (FX) options data. The results show that the absolute percentage errors do not get much larger and are still in an acceptable threshold. Moreover, the parameter calibrating procedure is stabilized due to the constraint made on the volatility process. In other words, it is more likely that the same calibrated parameters are obtained from different initial guesses.

Last, for the PIDEs with two or three space dimensions, which arise in stochastic volatility models or in stochastic skew models, it is in general inefficient or infeasible to apply the same numerical technique to different parts of the system.

First, different models where the underlying log stock price or volatility driven by either a Brownian motion or a L´evy process are examined on Standard & Poor’s (S&P) 500 data. The absolute percentage errors show that the calibration errors are different between the models. Furthermore, a new method to estimate the standard errors, which can be seen as a generalization of the traditional error estimation method, is proposed and the results show that in all the parameters of a stochastic volatility model, some parameters are well-identified while the others are not.

Next, the previous approach to parameter calibration is modified by making the volatility constrained under the volatilty process of the model and by making the other model parameters fixed. Parameters are calibrated over five consecutive days on S&P 500 or foreign exchange (FX) options data. The results show that the absolute percentage errors do not get much larger and are still in an acceptable threshold. Moreover, the parameter calibrating procedure is stabilized due to the constraint made on the volatility process. In other words, it is more likely that the same calibrated parameters are obtained from different initial guesses.

Last, for the PIDEs with two or three space dimensions, which arise in stochastic volatility models or in stochastic skew models, it is in general inefficient or infeasible to apply the same numerical technique to different parts of the system.

An operator splitting method is proposed to break down the complicated problem into a diffusion part and a jump part. The two parts are treated with a finite difference and a finite element method, respectively. For the PIDEs in 1-D, 2-D and 3-D cases, the numerical approach by the operator splitting is carried out in a reasonable time. The results show that the operator splitting method is numerically stable and has the monotonicity perserving property with fairly good accuracy, when the boundary conditions at volatility are estimated by Neumann conditions.

**Introduction**

The main purpose of the thesis is to deal with stochastic volatility models using L´evy processes. Throughout the thesis, L´evy processes play a central role. For detailed treatments on L´evy processes see [Sat99, Ber96, JS03] and recently [App04]. Fundamental concepts on continuous martingales and stochastic calculus can be found in [KS97, RY99].

Stochastic Volatility Models

In the original Black-Scholes-Merton option pricing model ([BS73], [Mer73]), the price of an underlying asset St follows

dSt = rStdt + σStdBt

In the original Black-Scholes-Merton option pricing model ([BS73], [Mer73]), the price of an underlying asset St follows

dSt = rStdt + σStdBt

under the risk neutral measure, where r is the interest rate, Bt is the standard Brownian motion and σ denotes the volatility which is constant across time t.

However, the constant volatility assumption contradicts the options data from the market. The daily set of the options data, which is composed by options with different strikes and maturities, results in a smile surface rather than flat, when converted to the volatility by the Black Scholes formula.

However, the constant volatility assumption contradicts the options data from the market. The daily set of the options data, which is composed by options with different strikes and maturities, results in a smile surface rather than flat, when converted to the volatility by the Black Scholes formula.

This contradiction indicates that the constant volatility asumption is not appropriate and there are different approches to circumvent this contradiction. One is to model asset returns as processes with jumps, for example, [Mer76], [MCC98], [BN97], [Kou02]. Another approach is to assume that the volatility is not constant over time t. It includes the local volatility model and stochastic volatility models. The local volatility model proposed by [Dup94] assumes that the volatility is a deterministic function of the time t. Under this assumption, the market is still complete as in the Black Scholes case, and the derivative’s risk can be perfectly hedged by the underlying asset without need to estimate the volatility risk premium.

Stochastic volatility models assume that the volatility is stochastic across time t and this is the assumption we take in the dissertation. The origin of the stochastic volatilty models goes back to as early as 1982 in Engle’s ARCH model [Eng82] where the conditional variance is modeled in a discrete setting by

vt|It−1 ∼ N(0,ht),

ht = α0 + α1vt2−1,

where It−1 denotes information set up to time t−1 and N(0,ht) denotes the normal distribution with mean 0 and variance ht. In continuous time setting the volatility can be modeled as a mean reverting process, for example, Cox-Ingersoll-Ross (CIR) process [CIR85]

,

where κ denotes the rate of mean reversion, θ denotes the long term mean and β denotes the volatility of the process.

Stochastic volatility models assume that the volatility is stochastic across time t and this is the assumption we take in the dissertation. The origin of the stochastic volatilty models goes back to as early as 1982 in Engle’s ARCH model [Eng82] where the conditional variance is modeled in a discrete setting by

vt|It−1 ∼ N(0,ht),

ht = α0 + α1vt2−1,

where It−1 denotes information set up to time t−1 and N(0,ht) denotes the normal distribution with mean 0 and variance ht. In continuous time setting the volatility can be modeled as a mean reverting process, for example, Cox-Ingersoll-Ross (CIR) process [CIR85]

,

where κ denotes the rate of mean reversion, θ denotes the long term mean and β denotes the volatility of the process.

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