# Finding optimal trading of option on a foward

Assume you have a option on a forward $$F$$ with a payoff: $$\max(F_T - K, 0)$$.

Assume also, that you have a bullish view on the forward in such a way that $$E_{0}[F_T] > F_0 = E_{0}^{*}[F_T]$$ (where the star indexes risk neutral expectation). You can buy (or short) the forward $$F$$ up to the volume $$M$$. What is your optimal investment strategy?

I am not sure how to proceed with this, does anyone have any suggestions?

My guess is that given that there is no assumption of risk aversion or transaction costs we would buy $$M$$ of $$F$$ from time $$0 < t < T$$. I do not really see what else we could do here.

• What does the option do here? Can I also buy or short the option? Do I have a position on the option? May 20 '19 at 10:13

The optimal investment strategy depends on the investment goals, or equivalently your utility function (which the investment strategy is supposed to maximize).

The forward will trade at $$\mathbf{E}^*_0(F_T)$$ in the market when you invest at $$t=0$$.

If you buy your maximum volume $$M$$, then gain/loss at $$T$$ is given by $$M(F_T-\mathbf{E}^*_0(F_T))$$ (which is unknown at $$t=0$$).

If it is a forward contract, you do not have any upfront investment. If the contract is under CSA (Credit Support Annex) i.e. collateralized, you are exposed to margin calls. If you were to buy options (long call option), you won't have any margin calls but an upfront investment in the form of option premium is required.

If it is an uncollateralized forward contract, you are only exposed to the gain or loss at $$T$$.

It's all a question of whether your risk appetite requires you to buy the downside protection (which a call option offers), inspite of your bullish view.

Or, whether you are open to a large downside risk in exchange for a larger return offered by a naked forward position (when compared to a call option).

Btw.. I think the question becomes more interesting if the constraint of volume $$M$$ does not exist; rather you have a limited capital. In this case you would want to maximize the volume (given your bullish view) - by factoring in the margin / collateral calls you may need to provide in case of naked forwards or the maximum option premium you can afford given the capital constraint. Options provide leverage, so that you deal with a larger volume - and can enhance returns on a limited capital.