>> In this module we're going to go through an example very quickly where we price a European put on a futures contract. Up until now we've seen how to price European and American options on the underlying security. Very often in fact though we want to price options on futures contracts, which are written on the underlying security. So in fact many of the most liquid options are options on futures contracts. They include for example the S & P 500 index in the US, the Eurostoxx 50 index in the Euro zone, the FTSE 100 from the UK and the Nikei 225 from Japan. In these cases, the underlying security is not actually traded, that is because if you wanted to actually trade the S & P 500 we would need to trade 500 different stocks. Now it would be very expensive and time consuming. So generally we don't trade these indices, we trade futures on these indices. So what we're going to do is we're going to price a European put on a futures contract. So consider the following parameters, we're going to assume the initial stock price or in this case the initial index price is 100. We'll assume n equals 10 periods. We will assume a risk free rate of 2%, a dividend yield of 1% and a sigma of 20%. We're going to assume that the futures expiration and the option expiration coincide and are equal to 0.5 years. In practice this is what typically happens. But of course, for more theory, we could if we like, still price options where the futures expiration was greater than the option expiration. We'll be able to obtain the futures price lattice on our spreadsheet, by using the fact that s n equals f n, and then using the fact for t less than n, f t is equal to the expected value of f t plus 1, using the risk mutual probabilities. When we do that, we'll see that we get a put option value of 5.21. So let's go to the spreadsheet and see what we get. So here are our initial parameters. We've an initial price of 100, t equals 0.5 years or 6 months. Sigma equals 20%. We're assuming 10 periods and or, we have our r and dividend yield as well. So u, d and q are calculated as we showed in an earlier module where we used the binomial model to approximate geometric boundary in motion and the Black Shoals Model. So our futures parameters, we're assuming an expiration of t equals 10. So 10 time periods corresponding with 6 months, or half a year. Over here we can choose what sort of option we want to price. 1 for a call, minus 1 for a put. We're assuming a strike of 100, and the expiration, again, is going to be 10 periods. So the option expiration coincides with the futures expiration. It's going to be a European option. So here we have our lattice, or stock-price lattice. In each time period, the stock price grows by a factor of u, or it falls by a factor of d. If we scroll down, we have the futures lattice. And in this case, we can see, well, by definition, f n equals s n at time n. So we see that the futures prices in here are time t equals 10, agree with the stock prices up here, at time t equals 10. We then work backwards in this lattice, calculating the futures prices using that formula we showed on this last slide. Okay. So now, we have our futures price lattice. We can now price our options, by simply calculating the payoff of the option at time t equals 10, the expiration. So in this case it's a put option with strike 100. So when the stock price is greater than 100, the payoff is 0. When it is less than 100, we get a payoff of 100 minus the stock price. So that gives us these values here, and then we can calculate the option by working backwards in the binomial lattice as we've seen before. In this case, we find the put option value is equal to $5.212. It's worth making a point about how these models are used in practice. In practice, we don't need a model to price liquid options. Market forces, i.e., supply and demand, actually determines the price of options. In this case, this amounts to determining , or the implied volatility. And that is because, if you recall, so the price of a call options c 0 is equal to the expected value at time 0, using our risk-neutral probabilities of e to the minus r t times s t minus k, the positive part, or if you like this is, equal to, the maximum, of s t minus k and 0. Where under the BlackScholes Model s t equals s 0 e to the r minus sigma squared over 2 times t. Plus sigma, w t. W t here is your standard Brownian motion. So w T is a random variable that's got a normal distribution between 0 and variance t. Anyway, given all of this we recognize that c 0 is equal to some function of the initial stock price, the risk free rate, the dividend yield, and I should have included the dividend yield up there. Sigma, the time to maturity, and k. And in fact we actually know all of these quantities. We can see them all in the market place. We know what s 0 is, we know what r is we know what the dividend yield is, we know what the maturity of the option is and we know what the strike of the option is. The only thing we don't know is sigma. However in the case of liquid options, we can actually observe c 0 the price of the option in the market place. As I said supply and demand is what really sets the price of the liquid options in the market place. So we can see c 0 and then we can use this equation to back out the one unknown, which is sigma. And then this is often called the implied volatility. And we'll be returning to this later in the course as well. So at this point you might ask if that's the case, if supply and demand sets the price of options, why do we need a model? Well we need models for two reasons. One is to help us price what are called exotic or more illiquid derivative securities whose prices are not available in the marketplace. We can also use models to hedge options. And I'll return to hedging later in the course as well.