By- Xiaohong Chen, May 2015 Graduate

**Introduction:**

With the Black-Scholes model, the volatility of stock price is assumed to be constant, but we have observed that the implied volatilities of tradable options vary from day to day. This can be caused by changing risk preference of market participants like in the jump-diffusion model. An alternative explanation is that the instantaneous volatility of a stock itself follows a stochastic process. One of the first research papers of stochastic volatility model was published by Hull and White [1]. In this blog post, we focus on discussing the content of that paper below:

**Stochastic model and risk-neutral pricing****:**

Generally, volatility is chosen to follow a diffusive process. Let us consider a derivative *f* with the underlying *S*. Assume the instantaneous variance *V*=σ^{2}. Then, in Hull and White’s paper [1], we have the following stochastic processes:

The variable φ is a parameter that may depend on *S*, σ, and *t*. The variable *μ*and *ξ* may depend on σ and *t*, but it is assumed, for the present, that they do not depend on *S*. The Brownian motions *z* and *w* have correlation *ρ*. Also, we assume the risk-free rate, which will be denoted by *r*, is constant or at least deterministic.

A quick application of Ito’s lemma shows that the drift of stock price must be the risk-free rate *r* in risk-neutral world. Invoking Girsanov’s theorem, we conclude that all risk-neutral measures are associated to the process of the form

Where *w* tilt and *z* tilt are Brownian motions with correlation coefficient *ρ*under risk-neutral measure. Notice that the drift of instantaneous variance *a*is arbitrary and could be any reasonable function of σ and *t*. This reflects the fact that volatility is not a tradable quantity. Hence our market has two sources of uncertainty but only one underlying and so is incomplete.

**A closed form formula in the uncorrelated case:**

Hull and White [1] deduce a closed-form formula for pricing European options when the correlation coefficient *ρ* is zero. By using risk-neutral pricing formula, the price of an option can be expressed as

where

*T *: time to maturity;

*S*_{t} : security price at time *t*;

*σ*_{t} : instantaneous standard deviation at time *t*;

*p*~(*S*_{T}|*S*_{t}, *σ*^{2}_{t}): conditional distribution of *S*_{T} given the security price and variance at time* t* under the risk-neutral world;

*f*(S_{T},* σ*_{t}, *T*): max[0, *S*-*K*] and *K* is strike price.

Define *V* bar as the mean variance over the life of the derivative security defined by the stochastic integral

Using this, the distribution of *S*_{T} may be written as

where g~(S_{T}|V-bar) and h~(V-bar|*σ*^{2}_{t}) are conditional distributions of *S*_{T} and*V*-bar under risk-neutral world respectively. Thus one can get

which can then be written as

Under the assumptions that *ρ*=0, *μ* and *ξ* are independent of *S*, the inner term is the Black-Scholes price for a call option on a security with a mean variance*V-*bar, which will be denoted as C(*V*-bar) and expressed as

where

and N(x) is the cumulative density function of standard normal distribution. Thus, the option value is given by

**Monte Carlo simulation procedure**

In this part, we relax some assumptions made before. We now allow the correlation coefficient *ρ *to be nonzero and let *V* follow a mean-reverting process. One example is

where *κ* and *θ *are constants. Here, the instantaneous variance follows a CIR process [2]. It reverts to level *θ *at rate *κ*.

Now we introduce a Monte Carlo simulation procedure described in Hull & White [1], we divide the time interval *T *- *t* into *n* equal subintervals. Two independent normal variates *u*_{i} and *v*_{i} are sampled and used to generate the stock price *S*_{i} and variance *V*_{i} at time *i *in a risk-neutral world using the formula:

where Δ*t *= (*T *– *t*)/*n*.

In order to speed up the simulation, we need to apply some variance reduction techniques. Hull & White proposed a procedure as following. The value of

is calculated to give one “sample value”, *P*_{1}, of the option price. A second price, *P*_{2}, is calculated by replacing *u*_{i} with -*u*_{i} (1 ≤ *i *≤ *n*) and repeating the calculations; *P*_{3} is calculated by replacing *v*_{i} with -*v*_{i} (1 ≤ *i *≤ *n*) and repeating the calculation; *P*_{4 }is calculated by replacing *u*_{i} with -*u*_{i} and *v*_{i} with -*v*_{i} (1 ≤ *i*≤ *n*) and repeating the calculations. Finally, two sample values of the B-S price *q*_{1} and *q*_{2} are calculated by simulating *S* using {* u*_{i} } and { -*u*_{i} }, respectively, with *V* kept constant at *V*_{0}. This provides the following two estimates of the pricing bias:

These estimates of bias are averaged over a large number of simulations, and the final estimator of the option price is

where* C*(*V*_{0}) is the B-S price with *V* kept constant at *V*_{0}.

This procedure uses the antithetic variables technique twice and the control variate technique. More details about these techniques are described in Glasserman [3].

**Stochastic volatility smile**

Since the possibility of the stochastic volatility getting large increases the possibility of the large movement of the underlyings, the model, therefore, gives rise to a fatter tails distribution for the terminal log stock price. This leads to implied-volatility smiles which pick up out-of-money. See Figure 1.

One can introduce skewness by letting the underlying and the volatility correlated. Roughly speaking, the smile is downwards sloping when the correlation is negative while it becomes upwards sloping for large moneyness when the correlation is positive. See Figure 2 and 3.

The major difference between stochastic volatility model and jump-diffusion model is in their time decay. The amount of stochasticity in the volatility increases over time and this leads to long-maturity smiles not decaying. However, the time behavior could be controlled by the mean-reversion parameter to some degree. The faster the mean-reversion, the flatter long-time smiles will be. See Figure 1 and 4.

**Figure ****1** Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is uncorrelated with spot. The reversion speed is 1 and the volatility of variance is 0.5. Initial volatility is 10%.

**Figure ****2** Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is negatively correlated (-0.6) with spot. The reversion speed is 1 and the volatility of variance is 0.5. Initial volatility is 10%.

**Figure ****3** Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is positively correlated (0.6) with spot. The reversion speed is 1 and the volatility of variance is 0.5. Initial volatility is 10%.

**Figure ****4** Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is uncorrelated with spot. The reversion speed is 2 and the volatility of variance is 0.5. Initial volatility is 10%.

**Reference**

[1] |
J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” *Advances in Futures and Options Research, *vol. 3, pp. 27-61, 1988. |

[2] |
J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” *Econometrica, *vol. 53, pp. 385-407, 1985. |

[3] |
P. Glasserman, Monte Carlo Methods in Fiancial Engineering, New York: Springer Science+Business Media, 2003. |

[4] |
M. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge University Press, 2003. |

[5] |
J. C. Hull, Options, Futures, And Other Derivatives, Pearson Education Limited, 2012. |

[6] |
S. E. Shreve, Stochastic Calculus for Finance II, Springer Science+Business Media, LLC, 2004. |