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.
The sensitivity of the option price to volatility, expressed as Vega, in the Black-Scholes framework is given by:
Or equivalently:
Where:
- \( S_0 \): Current stock price
- \( T-t \): Time to expiration
- \( N'(d_1) \): Probability density function of the standard normal distribution evaluated at \( d_1 \)
- \( d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)(T-t)}{\sigma \sqrt{T-t}} \)
- \( K \): Strike price
- \( r \): Risk-free interest rate
- \( \sigma \): Volatility
- \( \exp \): Exponential function
- \( \pi \): Approximately 3.14159
To evaluate the change in Vega as volatility varies, we use the second partial derivative of the option price concerning volatility, known as Volga:
Where:
- \( d_2 = d_1 - \sigma \sqrt{T-t} \)
For most scenarios, \( d_1 \cdot d_2 > 0 \), ensuring that the option price behaves as a convex function of volatility. This convexity explains why stochastic volatility generally increases the option's price.
However, there are exceptions. For instance, when \( d_1 > 0 \) and \( d_2 < 0 \), Volga can turn negative, breaking the convexity assumption. This often occurs when the option is at-the-money or slightly in-the-money.
Thus, while the general belief is that stochastic volatility increases option prices, the relationship is not universally true, particularly under specific conditions where convexity is disrupted.
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