Simulating Black 76 Option Prices from Futures Prices

Summary

The document explains how a Monte Carlo simulation can align with Black 76 without first converting futures prices into spot prices. It starts from a lognormal spot-price process with drift and volatility, then substitutes the spot-forward relationship so the simulation is expressed using the futures or forward price. The resulting terminal price distribution has the volatility adjustment and random shock applied to that futures-based starting value.

This reframes Black 76 as the Black Scholes framework written in forward or futures units. The answer is a concise conceptual recipe for simulating terminal prices before applying an option payoff and discounting consistently. It does not provide a numerical example, address contract-specific futures dynamics, or detail the pricing and discounting assumptions needed in a full implementation, so users should confirm that those assumptions match their instrument and market setup.

Key ideas

• A Black 76 Monte Carlo can be formulated directly from the futures or forward price without converting it to spot.
• The simulation applies a volatility adjustment and random shock to the futures-based starting value.
• The relationship between spot drift and the forward price makes the spot-process form reusable in futures units.
• Black 76 can be viewed as Black Scholes expressed with forward or futures prices, subject to appropriate model assumptions.

Tags

# Black's model and Monte Carlo


# Black's model and Monte Carlo












It is well know that one uses the Black 76 model to price commodity derivatives. I would however like to perform a Monte Carlo simulation that ties back to this number.

How would one go about this process? Is there a way to make use of the known future prices to simulate suitable spot prices that will result in the Monte Carlo tying back to the formula approach?

## Answer by ZRH (score 1)

https://quant.stackexchange.com/a/43868

No need to convert futures prices to spot prices. Your simulation should look like:

$S_{T}=S_{0}*exp(\mu T-0.5\sigma^{2}T+\sigma \sqrt{T}z)$

where $\mu$ is the drift of spot prices. If you use the spot-forward relationship $F=S_{0}*exp(\mu T)$, you can rewrite the equation in your simulation to be:

$S_{T}=F*exp(-0.5\sigma^{2}T+\sigma \sqrt{T}z)$

Put simply, Black76 is just standard BS rewritten to use forward/futures prices instead of spot prices

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.