Constant vs Stochastic Volatility

You're tasked with pricing a European call option under two different scenarios:

1. Using a constant volatility of 20%.

2. Drawing volatility from a random distribution with an average of 20%.

________

Which option do you anticipate would generally be more expensive, and why?

At a first glance, many would assume that the option with stochastic volatility would come with a heftier price tag. This is because when the stock price fluctuates more (i.e., becomes more "volatile"), there's a higher chance that it might exceed the strike price, thereby raising the call option's value.

 

The mathematical basis for this line of thought is rooted in the concept of convexity when it comes to option pricing vis-à-vis volatility.

 

Volatility's sensitivity which is expressed as Vega (σ) in the Black-Scholes framework, shows how the option's price changes as volatility changes. Specifically, for a European call option:

 

For a European call option, the Vega (often represented by ν or sometimes by σ in some contexts) in the Black-Scholes formula is:

 

ν = S₀ * √(T-t) * N'(d1)

 

=>

 

ν = S₀ * √(T-t) * (1 / √(2π)) * exp(-d1^2 / 2)

 

Where:

 

- S₀ is the current stock price.

- T is the expiration time of the option.

- t is the current time.

- d1 is defined as:

  d1 = [ln(S₀/K) + (r + (σ^2 / 2)) * (T-t)] / [σ * √(T-t)]

  

- ln() denotes the natural logarithm.

- K is the strike price of the option.

- r is the risk-free interest rate.

- σ is the volatility of the underlying asset.

- exp() denotes the exponential function.

- π is approximately 3.14159.

 

This formula provides the rate of change of the option price concerning a change in the underlying asset's volatility. The term involving exp(-d1^2 / 2) and the division by √(2π) is the standard normal probability density function.

 

Where:

- S₀ is the current stock price

- T-t is the time to expiration

- N'(d1) is the probability density function (PDF) of the standard normal distribution evaluated at d1.

 

When we look at the term representing the rate of change in Vega for a change in volatility, known as Volga, it gives us the second partial derivative of the option price in relation to volatility. For a European call option, you'd compute this as:

 

Volga = Volga = S₀ * S₀ * √(T-t) * (1 / √(2π)) * exp(-d1^2 / 2) * d1 * d2 / σ

 

To summarize:

 

Vega (ν) = S₀ * √(T-t) * N'(d1)

 

Volga = ν * (d1 * d2 )/ σ

 

In this equation, v is always a positive value.

 

Now, if you were to run the numbers for d1 and d2 using the Black-Scholes model, you'd generally find that the outcome of d1 multiplied by d2 will be positive for both out-of-the-money and in-the-money call options. This means, in most cases, the option price will behave as a convex function of the volatility when d1 * d2 is greater than 0.

 

But, and there's always a but, there are certain conditions where this might not stand true. For instance, there might be situations where d1 is positive, but d2 is in the negatives. This would cause the second derivative (Volga) to go negative, indicating that the option price isn't always convex for all volatility levels. This kind of situation is more likely to crop up when the option hovers around the money or is just slightly in the money.

 

So, to wrap it up, even though the prevalent notion is that options with stochastic volatility would burn a bigger hole in your pocket because of the convex nature of the option price in relation to volatility, there are scenarios where this might not hold water.