# Option value based on a vwap

I need to calculate the value of an European option on a listed share. The payout is a cash payout of the 5 day volume weighted average price (VWAP) above the strike price at expiry date. The 5 day vwap is calculated by taking the total value divided by the total volume for the 5 days before expiry (including the expiry date)

I want to calculate the value using the Black Scholes formula. I have the risk free rate and dividend yield. I'm unsure what to use for the spot price and the volatility.

For the spot price, I believe I should use the current 5 day VWAP since that is what will be used to calculate the payout (as opposed to using the closing share price on the valuation date)

There is no actively traded options for this listed entity, so I'll be using historical prices to calculate the historical volatility. Here I'll use daily closing share prices to calculate volatility.

Does this approach make sense?

If you're close to the expiry you may want to refine Black & Scholes by replacing $\sigma \sqrt{T}$ with $\sigma \sqrt{T_1 + (T-T_1)/3}$ where $T$ is the expiry date and $T_1 = T - 5 \text{ days}$. This will give you a good enough approximation.
The rationale for using the share price on the valuation date is that even though the payoff is on the final 5 days VWAP, you would still delta-hedge the option with the share, hence the spot price is your underlying. As for the $\sqrt{T_1 + (T-T_1)/3}$ term it comes from the fact that conditional on the spot price on $T_1$ the VWAP computed on period $T_1$ to $T$ is approximately log normal with log standard deviation $\sigma \sqrt{(T-T_1)/3}$.
• The formula I wrote applies when you are before the first day of fixing : $t \leq T_1$. Once you are within the VWAP observation period, that is $t > T_1$, you need to move the known part of the 5 days VWAP to the strike and apply BS to the unknown part. Say for the sake of the argument that daily volumes are the same every day and you are 1 day before expiry, then 5 days VWAP = 4/5 x 4 days VWAP (which is known) + 1/5 1 day VWAP on the last day (which is unknown). This is in essence an option on average when you are within the averaging period, for which you will find a number of references. May 17, 2017 at 18:14