Calibrating the Italian Smile with Time-Varying Volatility and Heavy-Tailed Models

Abstract

In this paper, we consider several time-varying volatility and/or heavy-tailed models to explain the dynamics of return time series and to fit the volatility smile for exchange-traded options where the underlying is the main Italian stock index. Given observed prices for the time period we investigate, we calibrate both continuous-time and discrete-time models. First, we estimate the models from a time-series perspective (i.e. under the historical probability measure) by investigating more than 10 years of daily index price log-returns. Then, we explore the risk-neutral measure by fitting the values of the implied volatility for numerous strikes and maturities during the highly volatile period from April 1, 2007 (prior to the subprime mortgage crisis in the US) to March 30, 2012. We assess the extent to which time-varying volatility and heavy-tailed distributions are needed to explain the behavior of the most important stock index of the Italian market.

Appendix

Change of Measure for Lévy process

Before explaining how to find a proper change of measure between the market measure $\mathrm {P}$ and the risk-neutral measure $\mathrm {Q}$, we review, some useful definitions.

Theorem 1

(Lévy-Khintchine formula) A probability law $\mu$ of a real-valued random variable X on $\mathbb {R}$ is infinitely divisible with characteristic exponent $\psi$,

$$\begin{aligned} \int _{\mathbb {R}} e^{i\theta x}\mu (dx) = e^{\psi (\theta )}\quad \text {for}\;\theta \in \mathbb {R}\end{aligned}$$

if and only if there exists a triple $(a_h,\sigma ,\nu )$ where $a_h\in \mathbb {R}$, $\sigma \ge 0$, $\nu$ is a measure on $\mathbb {R}\backslash \{0\}$ satisfying

$$\begin{aligned} \int _{\mathbb {R}\backslash \{0\}} (1\wedge x^2)\nu (dx)<\infty \end{aligned}$$

and h is a given truncation function such that

$$\begin{aligned} \psi (\theta ) = ia\theta - \frac{1}{2}\sigma ^2\theta ^2 + \int _{\mathbb {R}\backslash \{0\}} ( e^{i\theta x}- 1 - i\theta h(x))\nu (dx) \end{aligned}$$

(37)

for every $\theta \in \mathbb {R}$.

We say that our infinitely divisible distribution $\mu$ has Lévy triplet $(a_h, \sigma , \nu )$. The measure $\nu$ is called the Lévy measure of $\mu$, $\sigma$ represents the Gaussian component, and a is a constant depending on the truncation function h. If the Lévy measure is of the form $\nu (dx) = u(x)dx$, we call u(x) the Lévy density. If $\mu$ is an infinitely divisible distribution, there exists a Lévy process $(X_t)_{t\ge 0}$ such that the distribution of $X_1$ is $\mu$.

Now, we want to find conditions under which the Lévy process $X_t$ under the measure $\mathrm {P}$ is still a Lévy process under a new measure $\mathrm {Q}$. In order to find an equivalent measure, we will consider the general result of density transformation between Lévy processes proven in Sato (1999). Even if we restrict our attention to structure-preserving measures, the class of probabilities equivalent to a given one is surprisingly large. Nonetheless, as stated in the following theorem (statement 3.), we cannot freely change the drift $a_h$ if a diffusion component is not present, that is if $\sigma = 0$.

Theorem 2

Let $(X_t, \mathrm {P})$ and $(X_t^*, \mathrm {Q})$ be Lévy processes on $\mathbb {R}$ with generating triplets $(a_h,\sigma ,\nu )$ and $(a_h^*,\sigma ^*,\nu ^*)$, respectively. Then, $\mathrm {P}$ and $\mathrm {Q}$ are equivalent for each t if and only if

• 1. $\sigma = \sigma ^*$;

• 2. The following integral is finite

  $$\begin{aligned} \int _{\mathbb {R}}(e^{\varphi (x)/2}-1)^2\nu (dx)<\infty , \end{aligned}$$

  with the function $\varphi (x)$ defined by $\frac{d\tilde{\nu }}{d\nu } = e^{\varphi (x)}$;

• 3. The constant b is such that

  $$\begin{aligned} a_h^* - a_h - \int _{\mathbb {R}}h(x)(\nu ^* -\nu )(dx) = b \sigma ^2 \end{aligned}$$

  if $\sigma > 0$ and zero if $\sigma =0$.

Proof

See Theorem 33.1 in Sato (1999) and Cont and Tankov (2004). $\square$

Particle Filter

In continuous-time stochastic volatility models, the unobserved volatility can be inferred by the observed stock returns. Since in most of the cases one deals with daily returns, the model has to be discretized. In all the cases we are interested in, the model can be written as

$$\begin{aligned} \begin{aligned} v_t&= f(v_{t-1},\varTheta ,\xi _{t-1})\\ y_t&= h(v_{t},\varTheta ) + \sqrt{v_t}\varepsilon _{t}\\ \end{aligned} \end{aligned}$$

where $v_t = \sigma ^2_t$, t is the day counter, $v_t$ is the square of the stochastic volatility modeled as a Markov process with initial distribution $p(v_0)$, and transition law $p(v_t|v_{t-1})$. Both p and the transition function f depends on the dynamics described by model chosen. The variable $y_t$ represents the set of given observations (in our case the observed index log-returns). It is assumed to be conditionally independent given the state $v_t$ and with distribution $p(y_t|v_t)$. Then, $\xi _{t-1}$ depends on the volatility dynamics and the noise $\varepsilon _t$ is normally distributed noise with mean zero and unit variance, at least in the stochastic volatility models we are interested in studying. The function h is the so-called measurement function, that in our case is given by the stock price returns model and $\varTheta$ is a set of static parameters.

Particle filter is a sequential Monte Carlo method for recursively approximating the posterior density $p(v_t|y_{1:t})$ by assuming a known measurement density $h(y_t|v_t)$ and the ability to simulate from the Markov transition density $f(v_{t+1}|v_t)$. The algorithm estimates the posterior density by considering a set of random samples with associated weights $\{v_t^i, w_t^i\}_{i=1}^N$ where N is the number of samples at each given point in time t. The algorithm, also known as the bootstrap filter, includes three main steps: (a) sampling, (b) weights computation, and (c) resampling. In our empirical test we proceed as follows:

• 1. we sample $v_t^i$ from the distribution $p(v_t|v_{t-1})$ with i ranges from 1 to N, where N is equal to 5000;

• 2. we compute the weight as follows

  $$\begin{aligned} w_t^i = -\frac{1}{\sqrt{2\pi v_t^i}} \exp \left( -\frac{(y_t^{observed} - h(v_t^i,\varTheta ))^2}{2v_t^i}\right) , \end{aligned}$$

  evaluate the likelihood estimate

  $$\begin{aligned} \widehat{L_t} = \frac{1}{N}\sum _{i=1}^N w_t^i \end{aligned}$$

  and then normalize the weights in order to have $\sum _{i=1}^N w_t^i = 1$.

• 3. we resample by taking into consideration the smooth resampling algorithm (see Douc and Cappé 2005 and Malik and Pitt (2011)) and we obtain a new set of samples $\tilde{v}_t^i$ approximately distributed according to $p(v_t|y_{1:t})$ and evaluate the state

  $$\begin{aligned} \widehat{v}_t = \frac{1}{N}\sum _{i=1}^N v_t^i; \end{aligned}$$

To estimate the parameters $\varTheta$, we build the joint log-likelihood over the entire observation period, that is

$$\begin{aligned} LL(\varTheta , y_{1:T}) = \sum _{t=1}^T \log \left( \widehat{L_t}\right) , \end{aligned}$$

and, finally, we insert this function into an optimization procedure. We use the Matlab r2011a function fmincon as the optimization routine. The classical Heston and the SV-GammaOU models have a better convergence property compared to other competitor stochastic volatility continuous-time models. This is mainly due to the number of parameters to be estimated.