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Pricing a European Option on a Final-Period VWAP

Article Quant Q&A · Author: Kritz
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Summary

The document explains how to adapt a Black–Scholes approximation for a European option whose cash payoff depends on the volume-weighted average share price over the five trading days ending at expiry. The underlying spot input should be the share price on the valuation date, rather than the current five-day VWAP, because the share is the instrument used to hedge the option. Volatility can be estimated from historical share prices or inferred from comparable securities when suitable implied volatility data is available.

For options close to expiry, the response proposes adjusting the volatility-time term to reflect that only the final part of the option’s life uses an average price. Conditional on the share price at the start of that averaging window, the VWAP is approximated as lognormal, with a reduced variance contribution. This is presented as a practical approximation, especially near expiry; the answer does not supply a full exact pricing model or discuss details such as discrete trading volumes and sampling conventions.

Key ideas

  • Use the share price on the valuation date as the spot input, even when the payoff is based on a later VWAP window.
  • The share remains the relevant hedge instrument for the option.
  • Historical volatility or comparable securities’ implied volatility can inform the volatility estimate.
  • Near expiry, adjust the volatility-time term to account for the shorter averaging period.
  • The approximation assumes the VWAP over the final window is approximately lognormal conditional on its starting share price.

Tags

OptionsEquitiesDerivatives pricingVolatility
Full text
# Option value based on a vwap


# 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?

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/34272

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.