Overview and Major Findings:
This paper examines the problem of pricing a European call on an asset (Stock) that has a stochastic or variable volatility. Addressing this problem was done by investigating two cases: the first case is to determine the option price when the stochastic volatility is independent of stock price. The second case is to determine the option price when the stochastic volatility is correlated with the stock price.
This paper provides a solution in series form for the stochastic volatility option, in addition to a discussion about the numerical methods that are used to examine pricing biases, and an investigation about the occurrence of the biases in the case of stochastic volatility.
As for the results obtained, this paper presents interesting results for each of the two cases.
When the stochastic volatility is independent of stock price, the results show that the price calculated using Black-Scholes equation is overestimated for at-the-money options and underestimated for deep in-and out-of-the-money options. This overpricing takes place for stock prices within about ten percent of the exercise price. Moreover, it is shown that the degree of the pricing bias can be up to five percent of the Black-Scholes price.
For the second case when the stock price is positively correlated with the volatility, the results show that the Black-Scholes formula overprices in-the-money options and underprices out-of-the-money options. On the other hand, when the stock price is negatively correlated with the volatility, the Black-Scholes formula overprices out-of-the-money options and underprices in-the-money options.
Although these results are obtained for European call option, they can be directly used for European put options by applying the put-call parity equation. Furthermore, these results can be also applied for American call options on non-divided-paying stocks.
Research Methodology: First Approach: Series Form Solution
To derive a solution for the price of a call option on a security with a stochastic volatility that is uncorrelated with the security prices, the authors of this paper follows the below steps:
1-Define a derivative asset that depends on a security price S and its instantaneous variance V which are governed by the following stochastic processes: dS=φS dt+σS dW dV=μV+εV dZ
2- Use the general differential equation of Garman and adjust it to represent the two state variable S and V: df/dt+1/2 [σ^2 S^2+(d^2 f )/(dS^2 )+2ρσ^3 ε S (d^2 f)/(dS dV)+ε^2 V^2 (d^2 f)/(dV^2 )]-rf=-rS df/dS-[μ- β(μ-r)]σ^2 df/dV Where ρ is the instantaneous correlation between S and V.
3-Assume ρ is zero and use risk-neutral valuation procedure: The price of the option will be governed by the equation below: f(S_t ,〖σ^2〗_t,t)=e^(-r(T-t)) ∫▒〖f(S_T ,〖σ^2〗_T,T) p(S_T⁄S_t 〗 ,〖σ^2〗_t) dS_T ; Where p(S_T⁄S_t ,〖σ^2〗_t ) is the conditional distribution of S_T given the security price and variance at time t. Second Approach: Numerical Solutions
To calculate the option price, the authors use Monte Carlo Simulation for both correlated and uncorrelated case. Specifically to run the Monte Carlo Simulation, the time interval T-t is divided into n equal subintervals in addition to sampling independent standard normal variances.
The results obtained using Monte Carlo Simulation are compared with the results of the first approach as well as with the Black-Scholes formula outcomes. The percent bias of the option price is deduced from the comparison made.

Critique:
In the article “The Pricing of Options on Assets with Stochastic Volatilities” by JOHN HULL and ALAN WHITE from the journal of Finance, the authors provide a solution to the problem of pricing European call options on an asset with stochastic volatility, which has not previously been solved due to the fact that there are no assets which are clearly perfectly correlated with the variance. This paper is considered a major improvement in the financial field since it provides various solution forms for an important financial problem (Pricing European call with stochastic volatility): one solution in series form in addition to two numerical solutions with and without correlation between stock price and volatility.
Finally, the authors mentioned some observations of Rubinstein about the effect of the time-to-maturity (strong for out-of the money options and weak for in-the-money options). These observations are inconsistent with the model presented in the paper unless the correlation between stock prices and the associated volatility reverse its sign from one year to the next. Moreover, the occurrence of this event is hard to explain.