Risk-Neutral and Forward Measures for Options on Futures

# Why is the forward measure needed for options on futures when the risk-neutral measure suffices?


# Why is the forward measure needed for options on futures when the risk-neutral measure suffices?












I have been running around in circles with this attempting to make sense of this.

Universal pricing theorem Given a numéraire asset $(N(t))_{t \geq 0}$ such that for all tradeable assets $(S(t))_{t \geq 0}$, the following process $\left(\frac{S(t)}{N(t)}\right)_{t \geq 0}$ is a $\mathbb{Q}^{N}$-martingale, then for any $\mathcal{F}_{T}$-contingent claim $X$, we have that its price at $t$ satisfies

$$V(t) = N(t)\mathbb E[N(T)^{-1}X\lvert \mathcal{F}_{t}]$$

Context Consider a commodity that has its futures quoted/traded in the market throughout the day across monthly maturities as $F(t,T)$ where $t$ is current date and $T$ is the maturity. Now this commodity also has a spot price $S(t)$. By definition, $$F(t,T) = \mathbb E^{\tilde{\mathbb Q}}[S(T)\lvert S(t)], \tag{*}$$

where to be clear $\tilde{\mathbb Q}$ is the risk-neutral probability measure, i.e., it is associated with the numérarie asset $M(t) = e^{rt}$, where $r > 0$.

Now, by $(*)$ $F(t,T)$ is a $\tilde{\mathbb Q}$-martingale, and we write the dynamics as follows:

$$ dF(t,T)=\sigma(T-t)dW^{\tilde{\mathbb Q}}(t), \tag{**}$$ where $W^{\tilde{\mathbb Q}}$ is a Brownian motion under $\tilde{\mathbb Q}$, and $\sigma(t)$ is a deterministic function.

My problem: Now I am having trouble writing down the clean justification of pricing a call under certain measures as per the Universal Pricing Theorem above. Clearly, $(**)$ is normal under $\tilde{\mathbb Q}$.Since $F(t,T)$ likes its spot $S(t)$, is tradeable on the market and the market is free of arbitrage so by the first fundamental theorem of asset pricing, the risk-neutral measure must exist and by the universal pricing theorem, the price of the call option with strike K, which expires at expiry of the future, (i.e., $T$) (also note $F(T,T)=S(T)$) has the price at time $t$ of

$$C(t,F(t,T),K) = M(t)\mathbb E^{\tilde{\mathbb Q}}[M(T)^{-1} (F(T,T)-K)^{+}\lvert \mathcal{F}_{t}] = e^{-r(T-t)}\mathbb E^{\tilde{\mathbb Q}}[(F(T,T)-K)^{+}\lvert \mathcal{F}_{t}]$$

and since we know the distribution of $F(T,T) = F(t,T)+\int_{t}^{T}\sigma(T-s)dW^{\tilde{\mathbb Q}}(s)$ we are able to calculate the price.

BUT assuming the above is correct why do we always take the forward measure for options on futures when the risk-neutral measure also suffices, as shown below? In addition, where does the assumption that the forward measure exists actually come from?

## Answer by Rylan (score 2, accepted)

https://quant.stackexchange.com/a/83692

A few comments:

The definition you give above is for a future, not a forward. You're correct that futures are martingales under the risk-neutral measure; however, forward prices are not necessarily. This could come up if your hedging strategy needs to be based on forwards, rather than futures. For the rest of the answer, I'll ignore this part.

As river_rat notes, the ability to pull the discount factor out is a special case. It requires the assumption that rates are either constant or at least independent from the thing it's multiplying in the expectation, which is $(F(T, T) - K)^+$, in this case.

If $r$ is stochastic (and not independent from $(F(T, T) - K)^+$), you could still compute the expectation under the risk neutral measure, but it might be really painful, so you'd probably prefer to use the forward measure.

As a side note, I mostly hear about the forward measure in terms of bond options. For a commodity you could maybe make the assumption that the commodity price is independent from rates if you want to, but you definitely can't for a bond-based payoff.

As for the question "where does the assumption that the forward measure exists actually come from", I would say that in most cases, using a different measure is basically a trick to make computations easier. In Black-Scholes, for example, we start with a dynamic in the physical (real world) measure, which is ultimately the measure we care about. We do some tricks to "invent" another measure where the discounted asset is a martingale, which gives us (after a few arguments which I'm omitting) that the asset price has to be the expectation of its payoff under that measure we invented.

Similarly, in the example you give, we can basically do a little algebra to re-express our "expectation under the risk neutral measure" as "something times an expectation under the forward measure", which is another "invented" measure where things are sometimes easier to calculate.