0:02

In this module we're going to discuss replicating strategies

Â in the Binomial Model.

Â We've already seen replicating strategies in the one period binomial model, but

Â we're actually going to see as well that you can construct replicating strategies

Â that replicate the pay off of an option in the multi-period binomial model.

Â And we'll see in fact that we can price options by constructing

Â a replicating portfolio.

Â In fact, that's what we were doing all along when we were computing option

Â prices by working with the one period risk mutual probabilities, and

Â working backwards in the binomial lattice.

Â They're actually the same thing, but we're gonna make that clear in this module.

Â 0:37

We're going to as usual, let St denote the stock price at time t.

Â We're now going to let Bt denote the value of the cash account at time t.

Â We haven't spoken about Bt before, but

Â it's always been in there background where we spoke about the gross risk fee rate or.

Â So what we're going to do now is formally recognize that as a security.

Â We're gonna start off at time 0 with B0 equals $1.

Â So this is the value of the cash account.

Â At time 0, and

Â then the values of the cash account at time t is simply R to the power of t.

Â So it's just a deterministic security that grows by a factor of R in each period.

Â Okay, so now we're explicitly viewing the cash account as a security.

Â We're going to let xt denote the number of shares held between times t minus 1 and

Â t for each t equals 1 up to n.

Â We're also going to let yt denote the number of units of the cash account

Â that we hold between times t minus 1 and t, for t equals 1 up to n.

Â And then we're going to define theta t to be xt, yt, and

Â this is the portfolio that is held at time t.

Â So it is the portfolio that's going to be held immediately after trading

Â at time t minus 1, and immediately before trading at time t.

Â So, theta t is the portfolio that's held between times t minus 1 and t.

Â 2:02

So, here's an example of a three period binomial model.

Â And in this we have theta

Â 0 applying from this time to this time.

Â Of course, we can have another theta 0 from this node down to this node.

Â And of course theta 0's identical in both cases because it has been chosen at this

Â time, here in account look into the future when it's making its trading decisions.

Â For another example theta 2 will be chosen at this node,

Â theta 2 will be chosen at this node, theta 2 will be chosen in this node, and

Â in each case the theta 2 will be actually different.

Â We'll have a different theta 2 here.

Â A different theta 2 here, and a different theta 2 here, and of course,

Â theta 2 then applies at time t equals 3.

Â Okay, so these are examples.

Â So, this theta here is an example of a trading strategy.

Â And of course, you'd also have a theta 1 chosen there, and a theta 1 chosen there,

Â and the predictor's amounts of securities of the security and the cash account that

Â you purchase at time 1 will depend on whether your down here or your up here.

Â 3:17

Now I want to discuss the concept of self financing, and

Â in particular self financing trading strategies.

Â Before we do that, we just need to define what we mean by the value process,

Â Vt of theta, that is associated with the trading strategy theta t equals xt, yt.

Â We define Vt to be equal to xtSt plus ytBt for t greater than or

Â equal to 1, and of course, this is just the value of the portfolio time t.

Â Xt, xt's the number of units of the stock that we hold, so

Â xtSt is our stock position, and ytBt is our position in the cash account,

Â our value of our cash account position, so this is the total value of the portfolio.

Â We need a slightly different def, definition of t equal to 0 and

Â that's because we don't have an x0, y0.

Â If you recall,

Â xt is the number of units of shares that we purchased at time t minus 1.

Â So, if we had an x0, we would be referring to time minus 1, but

Â we don't have a time minus 1.

Â So we have a slightly different definition for t equal to 0, but

Â we still end up with the value of the strategy times 0.

Â So when t equal 0, we get Vt equals x1 times s0 plus y1 times b0.

Â But of course x1 has chosen at times 0, and y1 has chosen at times 0.

Â So if you like,

Â this is the value of the portfolio immediately after trading at times 0.

Â Okay, so that's the value process.

Â We now want to define what we mean by a self-financing trading strategy.

Â A self-financing trading strategy is a trading

Â strategy where changes in Vt are due entirely to trading gains or losses,

Â rather than the addition or withdrawal of cash funds.

Â In particular, a self-financing trading strategy satisfies

Â Vt equals xt plus 1 st plus yt plus 1 Bt.

Â And if you notice, Vt is therefore the value of the portfolio immediately

Â after trading at time t, because xt plus 1 is the number of units of shared

Â help between times t and t plus 1 and yt plus 1 is the number of the units of

Â the cash account held between times t and t plus 1.

Â So this is the value of the portfolio immediately after trading at time t.

Â Up here, this is the value of the portfolio immediately before trading

Â at time t.

Â We now have the following proposition.

Â If a trading strategy, theta t is self-financing

Â then the corresponding value process Vt satisfies this relationship here.

Â So what's going on on this relationship here?

Â Well the left hand side is just the change in value of the portfolio,

Â and on the right hand side we have xt plus 1 which is the number of units

Â of the share held between times t and t plus 1 times the gain on the stock.

Â 6:13

So, all this is saying is that if the strategy theta t is self-financing,

Â then the gain in the portfolio is equal to the number of stocks held times the gain

Â on the stock,

Â plus number of units of the cash account times the gain on the cash account.

Â So in particular, changes in the value of the portfolio can only be due to

Â capital gains, so a gain in the stock or a gain in the cash account, or

Â capital losses, and not the injection or withdrawal of funds.

Â In other words, nobody is injecting funds into the portfolio at time t or t plus 1,

Â and nobody is withdrawing funds from the port, portfolio times t at plus 1.

Â So all the changes in the value of the portfolio are coming from capital gains,

Â or losses on the stock on cash account.

Â And it's actually easy to check this result.

Â So, we just go through it for the case t is greater than or equal to 1.

Â In that case, Vt plus 1 minus Vt is as follows.

Â Well, Vt plus 1 is actually equal to, we can see it

Â here it is equal to, we're gonna use this definition at t plus 1.

Â So it's xt plus 1 st plus 1, plus yt plus 1 Bt plus 1.

Â So that's what this term is here, and when we subtract Vt we're

Â going to use the self-financing condition and use this over here, and

Â thus this term here, you put them together and you get the desired quantity.

Â We get a similar simple proof for the case where t equals 0.

Â So that's what a self-financing con, trading strategy is,

Â it's a very intuitive idea.

Â It's simply what you would get from a trading strategy where all the trades

Â are funded by the gains or losses in the portfolio through time.

Â They're not funded by injecting fresh capital into the portfolio or

Â taking capital out of the portfolio.

Â 8:12

So we have seen how to price derivative securities in the binomial model.

Â And the key we saw to this was the use of one period risk mutual probabilities.

Â But actually we first pressed options in the one period models using a replicating

Â portfolio argument.

Â And we did this without ever needing to define risk-neutral probabilities.

Â If you go back to that earlier module you'll see, we actually

Â priced the 1-period option construct by constructing a replicating portfolio.

Â We only constructed risk-neutral probabilities afterwards and

Â we used these risk-neutral probabilities that defines all securities.

Â But the first thing we did was we constructed a replicating

Â portfolio in the 1-period model.

Â Well, in the multi-period model, we can do the same.

Â We can construct a self-financing trading strategy

Â that replicates the payoff of the option.

Â This is called dynamic replication.

Â And the initial cost of this replicating strategy

Â must equal the value of the option.

Â Otherwise, there's an arbitrage opportunity.

Â And the way to see this is, is as follows.

Â So, as before let theta t be

Â our self-financing trading strategy.

Â 9:49

Of the strategy or Vn.

Â We've been using n for number of periods so we can stick with n.

Â Now because this is self-financing strategy,

Â there are no cash flows between time 0, which I'll highlight here.

Â There are no cash flows between time 0 and tie n, and time n.

Â There's no cash flow coming out.

Â There's no cash flow going in.

Â The strategy self-finances itself.

Â So, the only time there's a cash flow is at time 0,

Â which is when the portfolio's constructed, and the time n has some value.

Â So Vn, if Vn is equal to the value of an option say at time n,

Â Cn then the fair value of the optional time 0 must be V0,

Â otherwise there would be an albatross strategy.

Â For example, a V0 was less than C0 you could buy V0,

Â buy the self-financing trading strategy at times 0,

Â sell the option at times 0, so you would make money for nothing.

Â You would get C0 minus V0 which is greater than 0, and

Â then at maturity at time n, Vn would be equal to Cn, so

Â the cash flows would cancel each other out at time n, and that would be an arbitrage.

Â Okay.

Â So, the initial cost of this replicating strategy must equal the value of

Â the option.

Â This dynamic replication price is, of course equal to the price obtained from

Â using the risk-neutral probabilities and working backwards in the binomial lattice.

Â And in any node, the value of the option is equal to the value of

Â the replicating portfolio at that node.

Â 11:24

So, in black here we have the stock price, in blue

Â we have the option price, and in red xt, yt, we have the replicating strategy.

Â So the way we price this option initially whilst we determine the payoff

Â at time t equals 3, remember the strike here was $100.

Â So it was a call option, with strike equals $100.

Â So these are the payoffs of the options 22.5, 7, 0, 0.

Â And the way we priced this option originally was we computed its price at

Â time t equal to 2 by looking at each of the one period models at time t equals 2,

Â so this is one, one period model.

Â This is another one period model, and this up here is another one period model.

Â I'm gonna use their one period knowledge, in particular the one period risk-neutral

Â probabilities, to compute the prices of the option at these points.

Â 12:18

Okay.

Â Then given these prices of time, t equals 2, we work backwards to t equals 1 to

Â compute the price of the option at t equals 1, again using our knowledge of

Â one period binomial models in the 1 period risk future probabilities.

Â So for example, here we found that the price was 10.23, and in fact,

Â 10.23 was calculated as the price of a derivative security

Â that paid 15.48 this node, and 3.86 at this node, i.e.,

Â the value of the option at this node and the value of the option at this node.

Â 12:48

But in fact, when we're doing that, all we're really doing implicitly is

Â constructing the strategy at this point, which replicates 15.48 and 3.86.

Â So, when we were working backwards in the lattice, what we were really doing is

Â constructing a replicating self-financing trading strategy.

Â Okay.

Â So, we can see here in the next slide, we can see the replicating strategy.

Â Modular rounding errors is not enough decimal,

Â if I put in additional decimal points the slide would look to cluttered.

Â So modular rounding errors these, these numbers are correct so xt, yt, xt, yt.

Â And what you can actually see for example, down here we have it,

Â so 0.8, so now I'm referring to this node here.

Â Okay, so at this node here, what we are saying is that 0.802 x 107,

Â 107 is the value of the stock at that time minus 74.84 times 1.01.

Â Well this is the value of the cash account.

Â Time 1, and at this node.

Â Well that's equal to 10.23, which is the value of the option at that node.

Â And you can check that this strategy actually at time 2

Â will replicate 15.48 if the stock price goes up.

Â If we go up this way or the strategy will be worth 3.86 if the stock price falls.

Â 15:01

Times the cash account which is worth 1.01 at that node.

Â So this is the value of the portfolio just before trading at this node,

Â just after trading at that node we are holding 0.802 units of the stock.

Â And minus 74.84 units of the cash account, and as I said,

Â modular rounding errors these two numbers must be the same.

Â That is the self-financing condition.

Â So, when you look at option pricing in this binomial model,

Â you can think of it as using the risk-neutral probabilities,

Â working backwards one period at a time to compute the price.

Â But what you're all implicitly doing when you do that is you're constructing

Â the self-financing trading strategy that replicates the payoff of the option,

Â and the value of the self-financing trading strategy times 0 is 6.57, and

Â this is called dynamic replication.

Â What you're doing is you're using a trading strategy which adjusts the holding

Â in the stock and the cash account at each time, so that at time t equals 3,

Â at maturity, we replicate the payoff of the option.

Â