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>> In this module we're going to go through an example very quickly where we

Â price a European put on a futures contract.

Â Up until now we've seen how to price European and American options on the

Â underlying security. Very often in fact though we want to price

Â options on futures contracts, which are written on the underlying security.

Â So in fact many of the most liquid options are options on futures contracts.

Â They include for example the S & P 500 index in the US, the Eurostoxx 50 index in

Â the Euro zone, the FTSE 100 from the UK and the Nikei 225 from Japan.

Â In these cases, the underlying security is not actually traded, that is because if

Â you wanted to actually trade the S & P 500 we would need to trade 500 different

Â stocks. Now it would be very expensive and time

Â consuming. So generally we don't trade these indices,

Â we trade futures on these indices. So what we're going to do is we're going

Â to price a European put on a futures contract.

Â So consider the following parameters, we're going to assume the initial stock

Â price or in this case the initial index price is 100.

Â We'll assume n equals 10 periods. We will assume a risk free rate of 2%, a

Â dividend yield of 1% and a sigma of 20%. We're going to assume that the futures

Â expiration and the option expiration coincide and are equal to 0.5 years.

Â In practice this is what typically happens.

Â But of course, for more theory, we could if we like, still price options where the

Â futures expiration was greater than the option expiration.

Â We'll be able to obtain the futures price lattice on our spreadsheet, by using the

Â fact that s n equals f n, and then using the fact for t less than n, f t is equal

Â to the expected value of f t plus 1, using the risk mutual probabilities.

Â When we do that, we'll see that we get a put option value of 5.21.

Â So let's go to the spreadsheet and see what we get.

Â So here are our initial parameters. We've an initial price of 100, t equals

Â 0.5 years or 6 months. Sigma equals 20%.

Â We're assuming 10 periods and or, we have our r and dividend yield as well.

Â So u, d and q are calculated as we showed in an earlier module where we used the

Â binomial model to approximate geometric boundary in motion and the Black Shoals

Â Model. So our futures parameters, we're assuming

Â an expiration of t equals 10. So 10 time periods corresponding with 6

Â months, or half a year. Over here we can choose what sort of

Â option we want to price. 1 for a call, minus 1 for a put.

Â We're assuming a strike of 100, and the expiration, again, is going to be 10

Â periods. So the option expiration coincides with

Â the futures expiration. It's going to be a European option.

Â So here we have our lattice, or stock-price lattice.

Â In each time period, the stock price grows by a factor of u, or it falls by a factor

Â of d. If we scroll down, we have the futures

Â lattice. And in this case, we can see, well, by

Â definition, f n equals s n at time n. So we see that the futures prices in here

Â are time t equals 10, agree with the stock prices up here, at time t equals 10.

Â We then work backwards in this lattice, calculating the futures prices using that

Â formula we showed on this last slide. Okay.

Â So now, we have our futures price lattice. We can now price our options, by simply

Â calculating the payoff of the option at time t equals 10, the expiration.

Â So in this case it's a put option with strike 100.

Â So when the stock price is greater than 100, the payoff is 0.

Â When it is less than 100, we get a payoff of 100 minus the stock price.

Â So that gives us these values here, and then we can calculate the option by

Â working backwards in the binomial lattice as we've seen before.

Â In this case, we find the put option value is equal to $5.212.

Â It's worth making a point about how these models are used in practice.

Â In practice, we don't need a model to price liquid options.

Â Market forces, i.e., supply and demand, actually determines the price of options.

Â In this case, this amounts to determining , or the implied volatility.

Â And that is because, if you recall, so the price of a call options c 0 is equal to

Â the expected value at time 0, using our risk-neutral probabilities of e to the

Â minus r t times s t minus k, the positive part, or if you like this is, equal to,

Â the maximum, of s t minus k and 0. Where under the BlackScholes Model s t

Â equals s 0 e to the r minus sigma squared over 2 times t.

Â Plus sigma, w t. W t here is your standard Brownian motion.

Â So w T is a random variable that's got a normal distribution between 0 and variance

Â t. Anyway, given all of this we recognize

Â that c 0 is equal to some function of the initial stock price, the risk free rate,

Â the dividend yield, and I should have included the dividend yield up there.

Â Sigma, the time to maturity, and k. And in fact we actually know all of these

Â quantities. We can see them all in the market place.

Â We know what s 0 is, we know what r is we know what the dividend yield is, we know

Â what the maturity of the option is and we know what the strike of the option is.

Â The only thing we don't know is sigma. However in the case of liquid options, we

Â can actually observe c 0 the price of the option in the market place.

Â As I said supply and demand is what really sets the price of the liquid options in

Â the market place. So we can see c 0 and then we can use this

Â equation to back out the one unknown, which is sigma.

Â And then this is often called the implied volatility.

Â And we'll be returning to this later in the course as well.

Â So at this point you might ask if that's the case, if supply and demand sets the

Â price of options, why do we need a model? Well we need models for two reasons.

Â One is to help us price what are called exotic or more illiquid derivative

Â securities whose prices are not available in the marketplace.

Â We can also use models to hedge options. And I'll return to hedging later in the

Â course as well.

Â