Forward volatility is a measure of the expected future volatility of an underlying asset over a specific period in the future. It is particularly important in the pricing of financial derivatives, as the expected future volatility affects the option’s value.
To more explicitly understand the concept, recall that volatility is the standard deviation of the underlying asset’s returns, and it is a critical input in the pricing of options using models like the Black-Scholes model. Expected future volatility is different from historical or realized volatility, as it represents an expectation or forecast of future price fluctuations.
With this understanding, we can define the forward volatility as follows. Define a time period starting at T1 and ending at T2, where 0 ≤ T1 < T2, then the forward volatility, σF, over the time period [T1, T2] is given by:
$$\sigma_{F}(T_1,T_2) = \sqrt{\frac{1}{T_2 - T_1} \cdot \left( Var(S_{T_2}) - Var(S_{T_1}) \right)}$$
Where Var(St) represents the expected variance of the underlying asset price at time t.
Now, let’s see how forward volatility is useful in pricing financial derivatives, particularly options.
A key ingredient in any option pricing model, such as the Black-Scholes Model or Local Volatility Model, is the volatility of the underlying asset. In most models, a constant volatility is assumed for simplicity, and the success of the model often relies on how well this constant represents the actual volatility over the life of the option.
However, in practice, markets are dynamic, which means future volatility may not be constant over time. Hence, it is better to use forward volatility because it incorporates the market participants’ expectations of future market conditions.
One major application of forward volatility in the pricing of financial derivatives is to price exotic options, such as barrier options, Asian options, and forward start options. These options often have complicated payoffs and depend on volatility over multiple time periods, making the use of forward volatility crucial to accurately price them.
For instance, consider a forward-start call option with a strike price K, where the option becomes exercisable at time T1 and expires at time T2. The price of this option will be a function of the risk-neutral expectation of the value of the option at time T1, considering the forward volatility from T1 to T2. As forward volatility incorporates the anticipated future volatility into the option pricing model, it allows for better pricing of such options.
In conclusion, the concept of forward volatility is essential in pricing financial derivatives, particularly exotic options, as it helps to account for the anticipated level of market volatility in the future. Using forward volatility improves pricing accuracy and reduces the risk of mispricing these complex financial instruments.