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?

1 Answer

For the spot price you should use the share price on the valuation date, not the 5 day VWAP. Once you've estimated the volatility (historical or by comparison to similar stock's implied volatilities if available) you may use Black & Scholes if the expiry is far enough.

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 $\sigma \sqrt{(T-T_1)/3}$.

• That makes sense, but what I'm unsure of is if you're very close to maturity. Let's say you're one day from expiry, the vwap is near to the spot (at the money), but the closing price is for some reason 5% lower. If you then use the closing price, won't it understate the FV? – Johan May 17 '17 at 17:58
• 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. – Antoine Conze May 17 '17 at 18:14
• Excellent thanks. Do you know of any references off hand? – Johan May 17 '17 at 18:23
• I do not have a specific one off hand but if you google "asian options closed form formulas" you should find a number of them. – Antoine Conze May 17 '17 at 18:30
• You cannot buy the share at the current VWAP (it is an average of prices in the past) so you cannot use the current VWAP as your "spot" in the valuation formula. The fact hat the final payoff is written on the VWAP rather than on a single spot just makes the option asian-like (option on average), but does not change the valuation principles. Also regarding price manipulation it is often the reason why an average is used rather than a single spot in the payoff (it's harder to manipulate an average) especially for illiquid stocks. – Antoine Conze May 19 '17 at 7:07