In this module we're going to discuss replicating strategies in the Binomial Model. We've already seen replicating strategies in the one period binomial model, but we're actually going to see as well that you can construct replicating strategies that replicate the pay off of an option in the multi-period binomial model. And we'll see in fact that we can price options by constructing a replicating portfolio. In fact, that's what we were doing all along when we were computing option prices by working with the one period risk mutual probabilities, and working backwards in the binomial lattice. They're actually the same thing, but we're gonna make that clear in this module. We're going to as usual, let St denote the stock price at time t. We're now going to let Bt denote the value of the cash account at time t. We haven't spoken about Bt before, but it's always been in there background where we spoke about the gross risk fee rate or. So what we're going to do now is formally recognize that as a security. We're gonna start off at time 0 with B0 equals $1. So this is the value of the cash account. At time 0, and then the values of the cash account at time t is simply R to the power of t. So it's just a deterministic security that grows by a factor of R in each period. Okay, so now we're explicitly viewing the cash account as a security. We're going to let xt denote the number of shares held between times t minus 1 and t for each t equals 1 up to n. We're also going to let yt denote the number of units of the cash account that we hold between times t minus 1 and t, for t equals 1 up to n. And then we're going to define theta t to be xt, yt, and this is the portfolio that is held at time t. So it is the portfolio that's going to be held immediately after trading at time t minus 1, and immediately before trading at time t. So, theta t is the portfolio that's held between times t minus 1 and t. Theta t is also a random process, and in particular, we call this a trading strategy. So, here's an example of a three period binomial model. And in this we have theta 0 applying from this time to this time. Of course, we can have another theta 0 from this node down to this node. And of course theta 0's identical in both cases because it has been chosen at this time, here in account look into the future when it's making its trading decisions. For another example theta 2 will be chosen at this node, theta 2 will be chosen at this node, theta 2 will be chosen in this node, and in each case the theta 2 will be actually different. We'll have a different theta 2 here. A different theta 2 here, and a different theta 2 here, and of course, theta 2 then applies at time t equals 3. Okay, so these are examples. So, this theta here is an example of a trading strategy. And of course, you'd also have a theta 1 chosen there, and a theta 1 chosen there, and the predictor's amounts of securities of the security and the cash account that you purchase at time 1 will depend on whether your down here or your up here. Now I want to discuss the concept of self financing, and in particular self financing trading strategies. Before we do that, we just need to define what we mean by the value process, Vt of theta, that is associated with the trading strategy theta t equals xt, yt. We define Vt to be equal to xtSt plus ytBt for t greater than or equal to 1, and of course, this is just the value of the portfolio time t. Xt, xt's the number of units of the stock that we hold, so xtSt is our stock position, and ytBt is our position in the cash account, our value of our cash account position, so this is the total value of the portfolio. We need a slightly different def, definition of t equal to 0 and that's because we don't have an x0, y0. If you recall, xt is the number of units of shares that we purchased at time t minus 1. So, if we had an x0, we would be referring to time minus 1, but we don't have a time minus 1. So we have a slightly different definition for t equal to 0, but we still end up with the value of the strategy times 0. So when t equal 0, we get Vt equals x1 times s0 plus y1 times b0. But of course x1 has chosen at times 0, and y1 has chosen at times 0. So if you like, this is the value of the portfolio immediately after trading at times 0. Okay, so that's the value process. We now want to define what we mean by a self-financing trading strategy. A self-financing trading strategy is a trading strategy where changes in Vt are due entirely to trading gains or losses, rather than the addition or withdrawal of cash funds. In particular, a self-financing trading strategy satisfies Vt equals xt plus 1 st plus yt plus 1 Bt. And if you notice, Vt is therefore the value of the portfolio immediately after trading at time t, because xt plus 1 is the number of units of shared help between times t and t plus 1 and yt plus 1 is the number of the units of the cash account held between times t and t plus 1. So this is the value of the portfolio immediately after trading at time t. Up here, this is the value of the portfolio immediately before trading at time t. We now have the following proposition. If a trading strategy, theta t is self-financing then the corresponding value process Vt satisfies this relationship here. So what's going on on this relationship here? Well the left hand side is just the change in value of the portfolio, and on the right hand side we have xt plus 1 which is the number of units of the share held between times t and t plus 1 times the gain on the stock. Plus yt plus 1 times the gain on the cash account. So, all this is saying is that if the strategy theta t is self-financing, then the gain in the portfolio is equal to the number of stocks held times the gain on the stock, plus number of units of the cash account times the gain on the cash account. So in particular, changes in the value of the portfolio can only be due to capital gains, so a gain in the stock or a gain in the cash account, or capital losses, and not the injection or withdrawal of funds. In other words, nobody is injecting funds into the portfolio at time t or t plus 1, and nobody is withdrawing funds from the port, portfolio times t at plus 1. So all the changes in the value of the portfolio are coming from capital gains, or losses on the stock on cash account. And it's actually easy to check this result. So, we just go through it for the case t is greater than or equal to 1. In that case, Vt plus 1 minus Vt is as follows. Well, Vt plus 1 is actually equal to, we can see it here it is equal to, we're gonna use this definition at t plus 1. So it's xt plus 1 st plus 1, plus yt plus 1 Bt plus 1. So that's what this term is here, and when we subtract Vt we're going to use the self-financing condition and use this over here, and thus this term here, you put them together and you get the desired quantity. We get a similar simple proof for the case where t equals 0. So that's what a self-financing con, trading strategy is, it's a very intuitive idea. It's simply what you would get from a trading strategy where all the trades are funded by the gains or losses in the portfolio through time. They're not funded by injecting fresh capital into the portfolio or taking capital out of the portfolio. Okay, in this slide, I want to talk about Risk-Neutral Prices of securities and emphasize them in fact, they're the same thing. Those prices are the same prices you get from constructing replicating strategies. Okay. So we have seen how to price derivative securities in the binomial model. And the key we saw to this was the use of one period risk mutual probabilities. But actually we first pressed options in the one period models using a replicating portfolio argument. And we did this without ever needing to define risk-neutral probabilities. If you go back to that earlier module you'll see, we actually priced the 1-period option construct by constructing a replicating portfolio. We only constructed risk-neutral probabilities afterwards and we used these risk-neutral probabilities that defines all securities. But the first thing we did was we constructed a replicating portfolio in the 1-period model. Well, in the multi-period model, we can do the same. We can construct a self-financing trading strategy that replicates the payoff of the option. This is called dynamic replication. And the initial cost of this replicating strategy must equal the value of the option. Otherwise, there's an arbitrage opportunity. And the way to see this is, is as follows. So, as before let theta t be our self-financing trading strategy. Let V0 be the initial value of the strategy. And let V capital T be the terminal value. Of the strategy or Vn. We've been using n for number of periods so we can stick with n. Now because this is self-financing strategy, there are no cash flows between time 0, which I'll highlight here. There are no cash flows between time 0 and tie n, and time n. There's no cash flow coming out. There's no cash flow going in. The strategy self-finances itself. So, the only time there's a cash flow is at time 0, which is when the portfolio's constructed, and the time n has some value. So Vn, if Vn is equal to the value of an option say at time n, Cn then the fair value of the optional time 0 must be V0, otherwise there would be an albatross strategy. For example, a V0 was less than C0 you could buy V0, buy the self-financing trading strategy at times 0, sell the option at times 0, so you would make money for nothing. You would get C0 minus V0 which is greater than 0, and then at maturity at time n, Vn would be equal to Cn, so the cash flows would cancel each other out at time n, and that would be an arbitrage. Okay. So, the initial cost of this replicating strategy must equal the value of the option. This dynamic replication price is, of course equal to the price obtained from using the risk-neutral probabilities and working backwards in the binomial lattice. And in any node, the value of the option is equal to the value of the replicating portfolio at that node. So for example, this is our earlier European option. So, in black here we have the stock price, in blue we have the option price, and in red xt, yt, we have the replicating strategy. So the way we price this option initially whilst we determine the payoff at time t equals 3, remember the strike here was $100. So it was a call option, with strike equals $100. So these are the payoffs of the options 22.5, 7, 0, 0. And the way we priced this option originally was we computed its price at time t equal to 2 by looking at each of the one period models at time t equals 2, so this is one, one period model. This is another one period model, and this up here is another one period model. I'm gonna use their one period knowledge, in particular the one period risk-neutral probabilities, to compute the prices of the option at these points. Okay. Then given these prices of time, t equals 2, we work backwards to t equals 1 to compute the price of the option at t equals 1, again using our knowledge of one period binomial models in the 1 period risk future probabilities. So for example, here we found that the price was 10.23, and in fact, 10.23 was calculated as the price of a derivative security that paid 15.48 this node, and 3.86 at this node, i.e., the value of the option at this node and the value of the option at this node. But in fact, when we're doing that, all we're really doing implicitly is constructing the strategy at this point, which replicates 15.48 and 3.86. So, when we were working backwards in the lattice, what we were really doing is constructing a replicating self-financing trading strategy. Okay. So, we can see here in the next slide, we can see the replicating strategy. Modular rounding errors is not enough decimal, if I put in additional decimal points the slide would look to cluttered. So modular rounding errors these, these numbers are correct so xt, yt, xt, yt. And what you can actually see for example, down here we have it, so 0.8, so now I'm referring to this node here. Okay, so at this node here, what we are saying is that 0.802 x 107, 107 is the value of the stock at that time minus 74.84 times 1.01. Well this is the value of the cash account. Time 1, and at this node. Well that's equal to 10.23, which is the value of the option at that node. And you can check that this strategy actually at time 2 will replicate 15.48 if the stock price goes up. If we go up this way or the strategy will be worth 3.86 if the stock price falls. Another check you can do here is you can actually check the self-financing condition. So here's how we can do that. So, the self-financing condition at this node should say that the value of the portfolio just before trading at this node equals the value of the portfolio just after trading at this node. So, just before. What is the value of the portfolio? Well just before trading we're holding 0.598. Units of stock and the stock is worth 107, and we are short minus 53.25. Times the cash account which is worth 1.01 at that node. So this is the value of the portfolio just before trading at this node, just after trading at that node we are holding 0.802 units of the stock. And minus 74.84 units of the cash account, and as I said, modular rounding errors these two numbers must be the same. That is the self-financing condition. So, when you look at option pricing in this binomial model, you can think of it as using the risk-neutral probabilities, working backwards one period at a time to compute the price. But what you're all implicitly doing when you do that is you're constructing the self-financing trading strategy that replicates the payoff of the option, and the value of the self-financing trading strategy times 0 is 6.57, and this is called dynamic replication. What you're doing is you're using a trading strategy which adjusts the holding in the stock and the cash account at each time, so that at time t equals 3, at maturity, we replicate the payoff of the option.
