Lecture 7. Stochastic Volatility Models.md

In practice, in the the industry jump processes are often not used this is because the jumps are hard to hedge. The interpretation of the impact on implied volatility shapes is not straightforward

Today we talk about Stochastic volatility models as an alternative and obtaining

When we talk about Stochastic Volatility we talk about events with higher dimensions than 1, hence we need more advanced techniques to obtain IV in an efficient way

Contents

• Towards Stochastic Volatility
• The Stochastic Volatility Model of Heston
• Correlated Stochastic Differential Equations
• Itô's Lemma for Vector Processes Pricing PDE for the Heston Model
• Impact of SV Model Parameters on Implied Volatility
• Black-Scholes vs. Heston Model
• Characteristic Function for the Heston Model

Towards Stochastic Volatility

Deficiencies of the Black-Scholes Model

• The idea of implied volatility does not fit to the Black-Scholes model
• Look for market consistent asset price models
• Use a local volatility, model stochastic volatility model, or a model with jumps, to better fit market data, and incorporate smile effects

A look at what parts of volatility surface we can calibrate using the models we have learnt so far x-axis = moneyness

• once you see market data and it's associated with moneyness, it is typically represented by the strike $K$ which is divided by $s_{0}$ giving $\frac{K}{s_{0}}$
• In this way if the market moves a lot, upwards or downwards, then the smile doesn't depend on a level but on the percentage the price is away from the strike (how far away from in the money)

Towards stochastic volatility

We have already seen the market:

$$ \left{\begin{array}{c c l}{{\mathrm{d}M(t)}}&{{=}}&{{r M(t)\mathrm{d}t,}}\ {{\mathrm{d}S(t)}}&{{=}}&{{\mu S(t)\mathrm{d}t+\sigma S(t)\mathrm{d}W^{\mathbb{p}}(t),}}\end{array}\right. $$ where under $\mathbb{Q}$ measure $\mu = r$, i.e.: $$ \mathrm{d}S(t)=r S(t)\mathrm{d}t+\sigma S(t)\mathrm{d}W^{\mathbb{Q}}(t). $$ in the alternative process we aim to generalise the assumptions about constant parameters $r$ and $\sigma$ We can choose:

1. Constant: $r,\sigma$
2. Deterministic-Piecewise constant: $r_{i}, \sigma_{i}, on [T_{i-1}, T_{i}]$
3. Stochastic-time dependent: $r(t) = f(t,W_{r}(t)), \sigma(t) = g(t,W_{\sigma}(t))$

• Modelling volatility as a random variable is confirmed by practical data that indicate the variable and unpredictable nature of volatility. (Hull and White, Stein and Stein, Heston, Schöbel and Zhu).
• The resulting SD for the variance process can be recognised as a mean-reverting square-root process, a process originally proposed by Cox, Ingersoll & Ross (1985) to model the spot interest rate. If the variance exceeds its mean, it is driven back to the mean with the speed of mean reversion.
• Return distributions under stochastic volatility models also typically exhibit fatter tails than their log-normal counterparts, but the most significant argument to consider the volatility to be random is the implied volatility smile/skew, which can be accurately recovered by stochastic volatility models, especially for medium to long time to maturity options.

Stochastic Volatility: Model of Heston

1. The variance process is a so-called CIR (Cox-Ingersoll-Ross) stochastic process: $$ \begin{array}{l l l}{{\mathrm{d}v(t)}}&{{=}}&{{\kappa({\bar{\nu}}-,\nu(t))\mathrm{d}t\ +\gamma\sqrt{\nu(t)}\mathrm{d}W_{v}(t).}}\end{array} $$

2. For a given time $t > 0$, variance $v(t)$ is distributed as $\bar{c}(t)$ times a non-central chi-squared random variable, $\mathcal{X}^{2}(\bar{d}, \bar{\lambda}(t))$, with $\bar{d}$ the "degrees of freedom" parameter and non-centrality parameter $\bar{\lambda}(t)$, i.e. $$ \nu(t)\sim\bar{c}(t)\chi^{2}\left(\bar{d},\bar{\lambda}(t)\right),\quad t>0, $$ with $$ \bar{c}(t)=\frac{1}{4\kappa}\gamma^{2}(\mathrm{\bar{i}}-\mathrm{\bf{e}^{-\kappa t}}),\quad\bar{d}=\frac{4\kappa\bar{\nu}}{\gamma^{2}},\quad\bar{\lambda}(t)=\frac{4\kappa\nu_{0}\mathrm{e}^{-\kappa t}}{\gamma^{2}(1-\mathrm{e}^{-\kappa t})}. $$

3. The square-root process for the variance precludes negative values for $v(t)$, and if $v(t)$ reaches zero it can subsequently become positive. It is the Feller condition, $2\kappa \bar{v} > \gamma^{2}$, which guarantees that $v(t)$ stays positive; otherwise, if the Feller condition is not satisfied, the variance process may reach zero.

Here we talk about a distribution, we have an initial point $v_{0}$. The distribution represented is a transition from $v_{0}$ to $v(t)$. Hence we could sample $v_{0}$ with any given point in the future, then we would have $v_{t} - v_{s}$. All we would have to substitute in the model would be $v(t)|v(s)$ which means "vt given vs". Then we would only have to substitute occurrences of $t$ with $t - s$

Noncentral $\chi^{2}$ distribution

Let $X_{1}, X_{2},,,,,X_{i},,,,,,X_{\bar{d}}$) be $\bar{d}$ independent, normally distributed random variables with means $\mu_{i}$ and variances $\sigma^{2}{i}$. Then the random variable $$ \sum{i=1}^{\bar{d}}\left( \frac{X_{i}}{\sigma i} \right)^{2} $$ is distributed according to the noncentral chi-squared distribution. It has two parameters: $\bar{d}$ which specifies the number of degrees of freedom (i.e. the number of $X_{i}$), and noncentrality parameter $\bar{\lambda}(t)$ which is related to the mean of the random variables $X_{i}$ by: $$ \bar{\lambda}(t) = \sum_{i=1}^{\bar{d}}\left( \frac{\mu_{i}}{\sigma_{i}} \right)^{2} $$ For this distribution we know the pdf, the characteristic function, the moment-generating function, etc.

The corresponding cumulative distribution function (CDF): $$ F_{v(t)}(x)=P[v(t)\leq x]=P\left[\chi^{2}\left(\bar{d},\bar{\lambda}(t)\right)\leq\frac{x}{\bar{c}(t)}\right]=F_{\chi^{2}(\bar{d},\bar{\lambda}(t))}\left(\frac{x}{\bar{c}(t)}\right), $$ where: $$ F_{\chi^{2}(\bar{d},\bar{\lambda}(t))}(y)=\sum_{k=0}^{\infty}\exp\left(-\frac{\bar{\lambda}(t)}{2}\right) $$

$$ \Gamma(a,z)=\int_{0}^{z},t^{a-1}\mathrm{e}^{-t}\mathrm{d}t,\Gamma(z)=\int_{0}^{\infty}t^{z-1}\mathrm{e}^{-t}\mathrm{d}t. $$ The corresponding density function reads: with $$ \mathcal{B}_{s}(z)=\left(\frac{z}{2}\right)^{a}\sum_{k=0}^{\infty}\frac{\left(\frac{1}{4}z^{2}\right)^{k}}{k!\Gamma(a+k+1)}, $$