The Problem with measuring risk sensitivities in Black-Scholes Option pricing model
May 23, 2009
In a recent interview for a middle level risk management position the Head of Market Risk of a large European bank in Asia asked the interviewee: "what is the real issue with measuring vega in a closed form Black-Scholes model?" The candidate did not understand the question well and started talking about vega being variable and the second mathematical derivative of the option price with respect to the volatility, i.e. the sensitivity of vega with the volatility of the option. The interviewer interrupted him immediately and re-phrased the question: "I think you misunderstood. What I am asking is what is the fundamental difference between the risk measures (“greek”) delta and vega as they are estimated in a closed form Black-Scholes model? I am not talking about what they are – obviously vega is the volatility sensitivity, delta is the sensitivity with respect to underlying stock price and so on. I am talking about the way they are estimated in a closed form model…by taking the mathematical derivatives. Is there any fundamental inconsistency between the way delta is measured and the way vega is measured in Black-Scholes?" The candidate remained uncertain about the meaning of this question and how to answer it and quickly they moved on to other more technical topics of discussion.
However, the question asked above was as fundamental as it gets in Black-Scholes Option pricing model.
The short answer to the above question would have been that estimation of vega, the sensitivity of an option price to asset volatility was totally superfluous and inconsistent with the Black-Scholes model. Why? This is because in Black-Scholes model asset volatility (volatility of the asset return) is assumed to be constant. And hence one cannot measure the sensitivity of an option price with the asset volatility. The first mathematical derivative of a function with respect to a constant (parameter) is zero. Vega is the first mathematical derivative of an option price with respect to the asset volatility.
If is the volatility of asset returns and if it is constant and is the call option price, then .
On the other hand, delta is the sensitivity of the option with respect to the underlying stock price (a variable). The stock price is random and changes all the time and it enters the Black-Scholes model (formula) as a variable. Therefore, delta is not only an important and valid sensitivity (risk) measure but also totally consistent with the Black-Scholes model. The first mathematical derivative of the option price with respect to the underlying asset price will also be non-zero and changing. If is the stock price and is the call option price then . The delta is always non-zero.
Thus, theoretically speaking, in a strict Black-Scholes model (a world where the volatility is constant) a risk measure (“greek”) such as vega is totally meaningless and more importantly inconsistent with the model itself. On the other hand delta is valid risk measure and consistent with the Black-Scholes model.
Now let’s go for the long answer. The Black-Scholes Option pricing model, perhaps the most famous formula in the world of finance, expresses the value (loosely speaking, the price) of a call option as a function of five variables, the stock (asset) price, the strike price, the risk free rate, the volatility of the stock return and the time to maturity. Of course, if the stock pays dividend then the sixth variable becomes the dividend yield.
The Black-Scholes formula is given by:
If you look at the above formula, then it indeed appears that the call option price is a function of all the five variables, the underlying stock (asset) price , the strike price , the risk free interest rate , the asset (return) volatility , and the time to maturity . And in reality, a call option price is dependent all these five variables because in financial markets all these five variables change.
However, strictly speaking in the above formula, and this is the moot point, there are only two “variables”, the stock price, and the time, . Mathematically speaking, the stock price and time are independent variables and the call option price is a dependent variable. Even though the time shown is the time to maturity, in a Black-Scholes model the “time” factor enters as a variable . The rest, the volatility, , the strike and the interest rate, are all “parameters”, with the strike price being pre-defined. And a parameter is estimated a priori before being input in any model, because these are mathematical constants. And a risk measure, which measures sensitivity of an option price, with respect to a parameter (a constant) does not make sense.
The original, and the first, derivation of the Black-Scholes formula was via the partial differential equation (PDE) method by Fischer Black and Myron Scholes. The stock price was assumed to follow a geometric Brownian motion (GBM) and GBM is a kind of diffusion equation. Diffusion equations are special kind of partial differential equations in physics and many natural systems follow these diffusion equation. Therefore, the call option on the stock, which was a function of only two variables, the underlying stock price and the time followed a diffusion equation. If then the partial diffusion equation for the call option price is given by:
As we can see from the above PDE the Call option price, is dependent only on two variables and , the stock price and the time respectively. The volatility, and the short term interest rate, both appear as coefficients in the above PDE. In fact, in a stochastic differential equation for the asset price, the starting point of Black-Scholes’ derivation, volatility of asset returns (asset volatility) is the diffusion coefficient and constant. Coefficients of any equations, by definition, are constants.
Therefore, in a Black-Scholes world, where we have closed form solutions of a call option price, the risk measures of vega and rho are superfluous and inconsistent with the theoretical model, whereas delta is a valid and consistent risk measure.
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