In the next sequence of modules we're going to discuss equity derivatives in practice, but before we get on to discussing equity derivatives in practice, we're going to spend some time in this first module discussing and reviewing the binomial model. So we'll call again our pricing of a European call option in the binomial model We're going to assume an exploration of t equals to 3, a strike of $100 and a gross risk free rate of r equals 1.01. So the pay off of the option is given to us by the maximum of 0 and st minus k which is 100. So we see here the pay off of the option. Its 22.5, 7, 0, 0 and recall how we priced this option. We computed our risk neutral probabilities, which are given to us down here, q u and q d and then we work backwards in the binomial ladder. So for example the value 15.48 is the value of the core option at this node and is given to us by one over the risk free interest rate, 1.01 Times the expected value of the payoff of the option one period ahead. So we can work backwards in the lattice and price the option that way. Note also that when we price using this mechanism here, we're guaranteed to have no arbitrage by construction. And that's as long as D is less than or less than U. Remember that this is our new Arbitras condition in the binomial model. And it ensures that QU and QD are both strictly great in zero as we have down here. And of course what that means therefore is that in a price like this It's impossible to have an arbitrage because it would be impossible to get a payoff here and here, which is strictly positive, and have a value that's strictly negative here, for example. And that's because qu and qd are strictly positive. So this is how we price securities or derivative securities in the binomial model. We ensure that this condition is satisfied to ensure no arbitrage, and then we work backwards in the lattice as usual. And so we can continue on in this fashion. Compute the price of the option at every node, working backwards until we find an initial option price of 6.57 dollars. Now, we can also write the option price, or compute it in one shot. When one calculation, using this expression here. So, this just reflects the fact that the option price is the expected value under Q of the discounted payoff of the option. The discount factors won over are acute. And these are the probabilities, so for example, 3Q squared times 1 minus Q, while this is equal to 3, reduced to. Times Q squared times one minus Q to the power of three minus one. So this is a binomial probability, it counts the number of ways in which the stock price can go up into periods and fall in the third period. And in fact, this is equal to three here, and we know that there's three ways to get up to this point, so down one and up two, up one, down one, and up one, or up two and down one. So we can also basically combine all the one period probabilities into three period probabilities given to us by here, and compute the value of the option in this manner. We also discussed, trading strategies in the binomial model. So let's quickly review again what we did there. St is going to denote the stock price at time t. Vt denotes the value of the cash account at time t, without any loss of generality we assume that b0 equals one dollar, so that bt equals r to the power of t. So now we're explicitly viewing the cash account as security. We let xt denote the number of shares. Held between times t minus one and t. We also let yt denote the number of units that the casher account have between times t minus one and t. Then theta t equals x t y t as the portfolio held immediately after trading at time t minus one and therefore is known at time t minus one. And immediately before trading at time t. So, basically, if this is t minus 1, and this is time t, then we know theta t at this point, and this represents the portfolio that's held immediately after trading at time t minus 1, until trading at time t. So theta t is a trading strategy. We also discussed the value process associated with a trading strategy. It is defined to be vt equals xtst plus ytbt for t greater or equal to one. So this, if you like, is the value just before, trading, at time t. And a t equal to zero, well we can't talk about trading just before time t equal to zero, because time t equal to zero is the beginning of our horizon. So this is equal to the value of the portfolio just after trading at time zero. We also have a definition of self financing strategy. Self financing trading strategy is a trading strategy where changes in VT, are do entirely to trading games or loses, rather than the addition of withdrawal of cash funds. In particular self-financing trading strategy satisfies this condition here. And of course we know that this, is equal to the value. Of the portfolio, just after, trading at time t. So basically, what we're saying here, is that a portfolio, or a trading strategy, is self-financing. If it's value just before trading is equal to its value just after trading. And what that means is that no funds have been deposited or withdrawn at time t. In other words, it is self-financing - it finances itself. No new cash injections at time t, no cash withdrawals at time t. We also have the following proposition. We said if a trading strategy theta t is self-financing, so s.f. is self-financing, then the corresponding value process vt satisfies the following. So this is the profit and loss from the trading strategy between times t and t plus one, so if we like, this is our p and l At time t plus 1. It's equal to x t plus 1 times the change in the stock price between t and t plus 1 plus y t plus 1 times the change in the cash account between times t and t plus 1. So a valid question at this point is why do we care about self financing trading strategies. Well, we care for multiple reasons. In the multi-period binomial model, we can actually construct a self-financing trading strategy that replicates the payoff of the option, or, indeed, any derivative security. This is called dynamic replication. And the initial cost of this replicating strategy must equal the value of the options, otherwise there is an arbitrage opportunity. The dynamic replication price is of course equal to the price obtained from using the risk-neutral probabilities and working backwards in the lattice. Indeed this is exactly how we computed the risk neutral probabilities in the first place, we initially considered a one period model, and actually with the way we computed QU and QD was by replicating the payoff of the option at this node, and this node. So if we recall, we solve two linear equations and two unknowns. What we were doing at that point, is replicating the payoff of the security, in this one period model. So the risk neutral probabilities came from a replication, in this one period model, and you can actually splice all the one period models together, to construct a multi-period model. And use the replicating strategies in each single one period model to construct a dynamic replicating strategy for the option. And so over here we find the replicating strategy for a European option. This is the option we began the module with, so for example, we see up here Up here we see 0.802 times 107. 107 is the value of the stock. 0.802 is the number of shares that we hold in the stock immediately after trading at this point. So 0.802 times 107 plus minus 74.84 times 1.01, which is the value of the cash account at that node. Is equal to 10.23 here, the value of the option at that node. We could also check that in fact this is also equal to 0.598 times 107 plus Minus 53.25 times 1.01. Now where does this come from? Well, 0.598 is the number of units of the stock held between times 0 and time 1. So 0.598 is the value here, and the minus 53.25 is the value here. And what we're seeing is that up at this node, the value of the option is 10.23. So this must be equal to the value of the replicating strategy at this node, which is this value here, but this must also equal the value of 0.598 times 107 minus 53.25 times 1.01 And that's because the replicating strategy is self financing. This here is the value of the self financing strategy immediately before trading at this node and this value here is the value of the self financing strategy immediately after trading at this node and those two value must be the same and they must equal the value of the option which is 10.23. So that's the end of our review of pricing in the binomial model and replicating strategies in the binomial model.
