Goal: stochastic volatility model for the WTI crude oil futures curve

In Black-Scholes, S is modelled as a process driven by Brownian motion Wt with deterministic drift,

satisfying the SDE:

The familiar solution to the SDE is:

.

Empirically, volatility is not constant

volatility exhibits autocorrelation and the distribution is heavy-tailed

We need to allow volatility to vary stochastically over time

dS(t) = S(t)dt + (t)S(t)dW(t)

This relaxes the usual assumption of homoskedasticity

Can fit market option prices more accurately

Random volatility increases kurtosis of log returns

Correlation in volatility process induces correlation in square of log returns

The implied volatility surface exhibits extreme skew

Assumptions about skew dynamics have an important effect on delta-hedging. Given a change in the

underlying forward price, what inference can be made about changes in the implied volatility surface?

The floating-skew convention is that volatility surface shifts in tandem with the forward price with the

shape unchanged.

For WTI, crude oil volatility surfaces have historically exhibited both call and put skew regimes. As the

tenor of the contract decreases, the implied volatility typically increases (the Samuelson effect). On

longer time scales, fundamental drivers, particularly inventory, drive skew.

Put skew (corresponding to a negative slope) increases systematically as inventory levels increase.

Heuristically, at high inventory levels, negative fluctuations in demand (increases in net supply) are

harder to absorb into inventory than positive fluctuations are to alleviate. This results in skew to the

downside.

To examine relative skew, one can normalize the implied volatility surface by the prevailing ATM

volatility at each date.

Dynamics of stochastic volatility

VIX vs. S&P: historically around -0.6

Down markets, volatility would go up

Heston Stochastic Volatility Model

Mean-reverting behavior of the VIX

Any observable in the market is stochastic

We can apply a term structure of correlation but correlation is not generally modeled as stochastic

Stock price

CIR- evolution of volatity

where , the instantaneous variance, is a CIR process:

and are Wiener processes (i.e., random walks) with correlation , or equivalently, with

covariance dt.

The source of randomness is correlated (with correlation ) with the randomness of the underlying's

price processes

The parameters in the above equations represent the following:

is the rate of return of the asset.

is the long-term variance, or long run average price variance; as t tends to infinity, the

expected value of t tends to .

is the rate at which t reverts to .

is the vol of vol, or volatility of the volatility; as the name suggests, this determines the

variance of t.

If the parameters obey the following condition (known as the Feller condition) then the process is strictly positive

An extension iis to make time-dependent.

Here , the instantaneous variance, is a time-dependent CIR process:

and are Wiener processes (i.e., random walks) with correlation .

Heston- two correlated Brownian Motions

Both drawn from a multi-variate normal distribution Z ~ N(0, )

Cholesky decomposition of AAT

=

Cholesky decomposition of the covariance matrix

Source: quantedu.com

SV models have found comparably little use in applied work; prior to 2013, there were no standard

packages for estimating SV models, whereas for GARCH, most statistical packages have many options.

An MCMC algorithm provides draws from a posterior distribution with the desired RVs.

Process (Kastners stochvol notes):

(1) Prepare the data

(2) specify the prior distributions and parameters

(3) run the sampler

(4) assess the output and display the results

Example using EUR/USD (exrates data):

For the Bayesian normal linear model with homoskedastic errors, typically the Gibbs sampler is used for

drawing from the posterior distribution.

We need to compare the Bayesian normal linear model with homoskedastic errors to the Bayesian

normal linear model with SV errors.

To assess the predictive performance of a model, we can use the posterior predictive distribution.

Kastner Algorithm

1. Reduce the data set to a training set

2. Run the posterior sampler using data from the training set only to obtain M posterior draws

3. Simulate M values from the conditional distribution by drawing from a normal distribution

CME data from the equity market:

WTI implied volatility surface from Bloomberg (May 23, 2014):

CBOE historical data:

March 2011 December 2011 correlation between the VIX and OVX was 0.71

References:

Gander and Stephens, Inference for Stochastic Volatility Models Driven by Levy Processes, Working

Paper, 2005

Schneider, A Stochastic Volatility Model for Crude Oil Futures Curves and the Pricing of Calendar Spread

Options, Working Paper, 2014

Heston, A closed-form solution for options with stochastic volatility with applications to bond and

currency options, Review of Financial Studies, 1993

Kastner, Dealing with Stochastic Volatility in Time Series Using the R Package stochvol, 2013