# 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?

• I am a little confused as to your goal. A statistic is any function of data. The pricing models are Frequentist point estimators. So, to me, your question is a bit like asking is there a way to simulate data whose sample mean is 5. The answer is yes, infinitely many. What is the goal you are mentally working to. – Dave Harris Dec 31 '16 at 2:11
• My thinking is as follows: I know that under Black's model the forward prices are log normally distributed. Therefore I can simulate forward prices using this distribution. – User16473932 Jan 2 '17 at 6:58
• My thinking is as follows: I know that under Black's model the forward prices are log normally distributed. Therefore I can simulate forward prices using this distribution. I want to re price an option though. So my question is, can I make use of the log-normally simulated forward prices to price the option (and if so, how would I do this?) Or would I need to simulate spot prices by perhaps using gbm, where the drift is no longer the risk free but rather the growth implied by the forward prices I would like a simulation that is calibrated to the forward price as well as option price. – User16473932 Jan 2 '17 at 7:50
• If all you want to do is simulate log-normal prices choose a package, such as R, and generate random numbers. The code in R would just be: y<-exp(rnorm(1000,mu,sigma)) – Dave Harris Jan 2 '17 at 22:51
• I am well aware on how to simulate a log normal distribution. I am however not sure how to use this to reprice an option as what I have simulated if future prices. Does this mean under each simulation I again use Blacks model (though this seems like I would double count volatility) or do I need to convert the future prices to spot prices and then get a terminal value for the option and discount? – User16473932 Jan 3 '17 at 7:35

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