# Valuation of FX vs. Commodities Barrier Options

With reference to my previous question about the computation of a barrier option delta, @LocalVolatility referenced a nice closed form solution to value barrier options on a stock paying a dividend yield. Out of curiosity, I wonder wether this model could "be abused" to value a similar barrier option on a commodities future? To value an FX option I would simply exchange the dividend yield with the foreign exchange rate.

Let's say I want to value an option on some commodites future, I understand that for a Vanilla option I could use the Black model (which is related to the Garman Kohlhagen model). The thought was, could I possibly "abuse" the model above to value a respective barrier option on a commodities future? My "idea of abusal" here would be, given the future, to extract the spot from the future via the interest rate parity relation and use this as input to the model. I know this is really vague, especially since it ignores things like storage costs in the cost of carry, just wondering?

EDIT

As mentioned in a comment below, I wonder wether I could also just exchange the dividend yield for the risk free rate and then price on the future. Intuitively, I got the idea from the derivation of the Black model, but I'm not sure at all.

If I understand your question correctly, then you have a barrier option pricer for spot model dynamics of the form

\begin{equation} \mathrm{d}S_t = (r - \delta) S_t \mathrm{d}t + \sigma S_t \mathrm{d}W_t. \end{equation}

Now you are wondering whether you can abuse the input parameters in a way such as to use the same model to price options on a forward contract. Normally, the forward dynamics under the Black-Scholes model are given by

\begin{equation} \mathrm{d}F_t = \sigma F_t \mathrm{d}W_t. \end{equation}

You could thus set $S_0 = F_0$ and $\delta = r$ in your model and get the correct price.

• Thx again, that was my intuition (see below the edit of my qestion).
– Tim
Feb 27 '17 at 8:08

I assume you mean a barrier option on a futures contract? It's not a straight forward thing to do analytically because of the Samuelson effect (futures tend to ramp up in volatility as they approach maturity). Most people build a dedicated vol surface for the futures contract and use a local vol based solution.

• Yes it's a barrier option on a futures contract, I'm not familiar with the Samuelson effect and will take a look at it. Does it hold in a "simple" GBM setting? I was also wondering wether I could exchange the dividend yield for the risk free yield as can be seen from the derivation of the Black model, but not sure here.
– Tim
Feb 26 '17 at 21:40
• No, the key problem is that most commodity futures will have a well defined term structure of volatility through time - i.e. realized volatility of a futures contract will be higher a week before expiration then a few month before expiration. Because of that, you can't use a flat vol to price it - the probability of touching the barrier would not be correct and your risk will be wrong too. Feb 27 '17 at 7:34
• I think you are talking about real life. This doesn't just apply to commidities? However, my question was only about the analytical solution derived in a Black Scholes like setting, where we assume a flat vol. Maybe I'm missing something here?
– Tim
Feb 27 '17 at 8:12