Stochastic models for option pricing- stochastic volatility model

By- Xiaohong Chen, May 2015 Graduate


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 V2. 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 μand 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 ρunder risk-neutral measure. Notice that the drift of instantaneous variance ais 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



T :  time to maturity;

St : security price at time t;

σt : instantaneous standard deviation at time t;

p~(ST|St, σ2t): conditional distribution of ST given the security price and variance at time t under the risk-neutral world;

f(ST, σt, 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 ST may be written as


where g~(ST|V-bar) and h~(V-bar|σ2t) are conditional distributions of ST andV-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 varianceV-bar, which will be denoted as C(V-bar) and expressed as




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 ui and vi are sampled and used to generate the stock price Si and variance Vi at time i in a risk-neutral world using the formula:


where Δt = (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”, P1, of the option price. A second price, P2, is calculated by replacing ui with -ui (1 ≤ i n) and repeating the calculations; P3 is calculated by replacing vi with -vi (1 ≤ i n) and repeating the calculation; P4 is calculated by replacing ui with -ui and  vi with -vi (1 ≤ in) and repeating the calculations. Finally, two sample values of the B-S price q1 and q2 are calculated by simulating S using { ui } and { -ui }, respectively, with V kept constant at V0. 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(V0) is the B-S price with V kept constant at V0.

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%.


[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.


Test drive the Quantitative Finance BETA site from Stack Exchange

Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. This website covers questions on real-life problems you face such as:

  • securities valuation
  • risk modeling
  • market microstructure
  • portfolio management
  • financial engineering
  • econometrics

And this is not a discussion forum...Quantitative Finance Stack Exchange is all about answers. Pretty straight forward- Ask a question, get an answer! Great resource to check out while it is in BETA.

Other resources for Quantitative Finance:

1. is a quantitative financial portal created by Paul Wilmott. One can find job postings, technical articles, up-to-date news, and other useful resources.

2. QuantStart

QuantStart is a personal website discussing Algorithmic Trading strategy research, development, backtesting and implementation. The author behind this website once worked in a hedge fund as a quantitative trading developer in London. Therefore, his articles are very practical. There are several articles on career development, which are very useful for those unfamiliar with the industry.

3. QuantLib

The QuantLib project is aimed at providing a comprehensive software framework for quantitative finance. QuantLib is a free/open-source library for modeling, trading, and risk management in real-life. It is written in C++ with a clean object model, and is then exported to different languages such as C#, Objective Caml, Java, Perl, Python, GNU R, Ruby, and Scheme.

4. Yahoo Finance

You can get free stock quotes, option prices, up to date news, portfolio management resources, international market data, message boards, and mortgage rates from here.

5. Data and Charts of U.S. Department of the Treasury

Great resource to find all kinds of interest rates related with the U.S. treasury.

6. QuantNet

It is a leading resource on Financial Engineering education and news, but the main focus is education. Quantnet provides detail information about Financial Engineering programs in North America.

By- Xiaohong Chen, Financial Math Intern, May 2015 Graduate