In the last module we introduced the 1-Period binomial model. And we discussed no arbitrage conditions in that model. In this module, we're actually going to continue working with the 1-Period binomial model. But now we're going to price options in that model. We're going to assume the same 1-period model that we've been assuming up till now. Stock price begins at a 100. It goes up to 107, which is u times S 0, or it falls to d times S 0, which in this case is 93.46. We're going to assume that the gross risk free rate is 1.01. So if I invest $100 in the cash account at time zero, it will be worth $101 at time t equals 1. So now what I want to do is, I want to figure out how much is a call option that pays this quantity here worth at t equals 1. So, the strike is $102. Okay? In this case the $102 falls somewhere in between the 93.46 and the 107. And we need to figure out how much the security is worth. We'll also be interested in answering the question how does the price of the option vary as p varies? Remember p is the true probability of the stock price going up and 1 minus p is the true probability of the stock price going down. So to answer these questions, we're going to construct what's called a replicating portfolio. The idea of a replicating portfolio is extremely important in finance and derivatives pricing. So we'll see what that means in the next slide when we actually price the option. Okay. So what we're going to do is, we're going to consider the following strategy. We're going to consider buying x shares or x units of the stock. And investing y dollars in cash or in the cash account, at time t equals 0. We don't know what x and y are yet. We're going to figure out what they will be, soon. At t equals 1, this portfolio will be worth 107x plus 1.01y, when S equals 107. And it will be worth 93.46x plus 1.01y when S equals 93.46. So, this is where we had an up move in the stock price at time t equals 1, and this where we had a down move. Okay. So we buy x units of stock. So in a up move, our stock position's worth 107x. In a down move, the stock position will be worth 93.46x. And our position in the cash account will be worth 1.01y regardless of whether you had an up move or down move because the cash account is riskless. Its value at time t equals 1 does not depend on whether the stock went up or down. Okay, so what we're going to try and do is choose x and y so that this portfolio is equal to the option payoff at t equals 1. What is the option payoff at t equals 1? Well, remember it's a strike of 102, so therefore the option payoff at t equals 1 will be 0 down here, okay? The stock prices were at 93.46, the strike is 102, so you would not exercise the option and the maximum is 0. However, if the stock prices were at 107 at t equals 1 then you're in the money. You would exercise the option and receive 107 minus 102, which is $5. So if you like, that's the value of the option c1 at time t equals 1. So what we're going to try and do is choose x and y so that this portfolio we've built here in the stock and cash account at time t equals 0. We're going to choose x and y so that they equal the option value at t equals 1. The option value at t equals 1 is 5 in an up move and 0 in a down move. So 5 in the up move. We've already seen what the portfolio's worth in an move. It's 107x plus 1.01y, which is exactly what we have here. And in a down move, the portfolio is worth 93.46x plus 1.01y. Now, if you look at this, this, of course, is just two equations, two linear equations, and two unknowns. It's easy to solve it. If you do, you'll find out that x equals 0.3693, and y equals minus 34.1708. So, what we've actually done, at this point, very simple. We've constructed a replicating portfolio. We've constructed a portfolio which invests at time zero, in the stock and in the cash account. And we've chosen the quantities x and y in such a way that the value of this portfolio at time t equals 1 is exactly equal to the payoff of the option at t equals 1. So, we have succeeded in replicating the portfolio. Okay. What does a negative value of y means? Well a negative value of y just means that we're borrowing. We saw on the previous slide that y equals minus 34.1708. So y equals minus 34.1708. This is our position in the cash account. If y was positive, we would have invested in the cash account. In other words we would have lent money at an interest rate of r, but because it's negative what we're actually doing is we're borrowing money. And we're going to pay interest on that borrowings at an interest rate of r. We found the value of x of 0.3693. And this means that at time t equals 0, we purchased 0.3693 shares, or units of the stock. If x was negative, we would have, we would have actually been short-selling the stock. We would have been borrowing it and selling it in the marketplace. How much does this replicating portfolio cost at t equals 0? Well, the stock costs a $100. You can think of the cash account as costing $1. Okay? When we said we borrow $34.1708, that's like short selling the cash account. Okay. So the position is worth .3693 times 100 minus 34.1708 times 1 and that's equal to 2.76 modulo some rounding here. I didn't want to show any more than two decimal places. Okay. So the value of this replicating portfolio is $2.76. This is how much it costs to actually put on the position. Okay. So I borrow $31.1708, and I purchase 0.3693 units of share, of of the shares. And that comes to a total of 2.76. So that, I argue, is the arbitrage free value of the option. Now let's pause for a second here and think about this. I'm claiming 2.76 is the fair value of the option in this model. In fact, I'm saying it's the arbitrage free value of the option in this model. To see this suppose for example, that I tell you, suppose the option price in the marketplace is $2 not 2.76, then. There is an arbitrage. How do I know this? Well, here's what I can do. I can buy the option, so I can do two things. I can buy. The option. For $2. And the second thing I can do is I can sell the replicating portfolio. So I can do the exact opposite of this replicating portfolio. I could short sell 0.3693 units of the stock. And I could invest $34.1708 in the cash account. That will bring me in $2.76. This will cost me minus $2, so I will get a net profit of 76 cents at t equals 0. What happens if t equals 1? Well, if t equals 1, the two positions offset. Remember, this portfolio replicates the value of the option. Okay, so I've replicated the option but I also own the option, which I've purchased for $2, so the two will offset each other and I will have 0 at t equals 1 and so this is an example of a type A arbitrage. Similarly if the option price was selling for a quantity greater than 2.76, maybe the option price was selling for $4. Then what I would do is I would do the opposite. I would sell the option for $4, and I would buy the replicating portfolio for $2.76. So that will give me a net of 4 minus 2.76 which would be $1.24. I would take that in a t equal 0 and at t equals 1, the two positions would offset each other because the portfolio replicates the payoff of the option. So again, I would have an overcharge. So in fact, this value of 2.76 is the arbitrage free value. Of the option. Okay, so we've just succeeded in pricing a call option and that example with a strike of 102. We can do the same thing more generally. Okay, so we can actually price any derivative security in our one period binomial model. So how do we go about doing this? Well, again. We're going to do this now without any actual dollar numbers here. We start off with a stock price of S0. Probability of p of an up move, 1 minus p of a down move. If it goes up, the stock price ends at u times S0. And if it goes down, it ends up at d times S0 at time t equals 1. Assume we want to compute the fair value, or the arbitrage-free price of a derivative security whose payoff will cost C1 of S1, and this is the following payoffs. It pays off C subscript u up here, and it pays off C subscript d down here, if the stock price fell. So this is the payoff of the derivative security at t equals 1. Well, we want to compute the fair value of that security. And we can do it in the exact same way as before. What we do is we construct at time t equals 0 a replicating portfolio. How do we do that? Well, we purchase x units of the stock. And we invest y dollars in the cash account. Or if you like, we purchase y units of the cash account, assuming the cash account is worth $1 at t equals 0. So what we do is we set up our two equations and two unknowns. This is the payoff of the derivative security in the event of an up move, here on the right hand side. And this is the payoff of the derivative security in the case of a down move. And these are at t equaqls 1. And on the left hand side, we have the value of our portfolio at time one. If we purchased x shares at time 0. Then those x shares would be worth u times s 0 x at time 1, if the stock price went up. And then it would be worth d times S 0 times x at time 1 if the stock price fell. Of course, our y dollars that we invest in the cash account will be worth 4 times y regardless of whether the stock price went up. Or down. So, now you can solve these two linear equations and two unknowns. It's very straightforward, it only takes a few minutes to do so, so I'll leave that to you. When you solve those two equations, you will have a particular value for X and a particular value for Y. We can actually combine them together to get the fair value of the derivative security at times 0. Remember the replicating portfolio purchased x units of the stock at time 0. So x units of the stock cost this, this much. And it invested y dollars in the cash account. Or if you like, it purchased y units of the cash account with a unit costing $1. And that cost y. So the total replicating portfolio costs x times S0 plus y, and that must be the fair value of the derivative security at time 0, for the exact same argument we gave a moment ago. If it was, if the price of the derivative security was less than this quantity, we could purchase, or we could construct an arbitrage. If it was greater than the value of the replicating portfolio, we could again construct an arbitrage. Okay. So, if you actually go ahead and you solve these two equations and two unknowns, what you will find is the following. It's just two or three lines of algebra, but you can check that you will find that C0, the fair value. Of the derivative security can be written as 1 over R times R minus d, u, over u minus d times cu, plus u minus R divided by u minus d times cd. Okay. What we do is we call this guy q. This then, you can check is actually 1 minus q. Okay. If I take 1 minus this quantity here, you'll see I'm left with this here. So this is 1 minus Q. And because of the no arbitrage assumption, we have d is less than R is less than u. So in fact, R is greater than d. And u is greater than d, but because R is less than u, we see that 0 is less than q is less than 1, which of course also means that 0 is less than 1 minus q is less than 1. So in particular, I can write the derivative price, C0 as being 1 over R times q Cu. Plus 1 minus qCd. And I can actually, in fact, this is what I do, these are what are called the risk-neutral probabilities. They're risk-neutral probabilities, they are probabilities because they both sum to 1. q plus 1 minus q. Equals 1, and each probability is strictly greater than 0. These are called risk mutual probabilities. And so I can write my derivative security price as 1 over R, which is the discount factor, times the expected value of the payoff of the derivative which is C1. Under the probability mass function q, okay so what I've done is I've computed the price of the derivative payoff and its price can be expressed as the expected value of the payoff at time 1, discounted under the risk neutral probabilities. And this then is called risk-neutral pricing. So this is a really important concept in finance. We've developed the [INAUDIBLE] context of a 1-period binomial model. We'll develop it soon in the context of multi period binomial models. But we'll also see later in the course that it generalizes very easily to more complicated models. And it is, it is how derivative securities. Are priced and practiced. Okay. We can also answer our earlier question. A question we asked earlier was, how does the option price depend on p? Remember, p was the probability. The true probability of an up move. And 1 minus p was the true probability of a down move. And if we look at what we've done here, we're seeing that the fair value of the derivative security is equal to this quantity here. Well, all that's appearing here is R, capital R, and the payoff of the derivative security. These risk-neutral probabilities are expressed in terms of R, d, and u. I don't see p appearing anywhere here. So, it looks like the value of the option does not depend on p. And this seems to be a crazy situation. So just to see how crazy it is, consider the following situation. Consider two different securities. We've got stock ABC. And we've got stock XYZ. And we're going to assume that each stock follows a one period binomial model. Both stocks start off with the same initial price, $100. And at time t equals 1, the first stock can either climb to $110 or fall to $90. And it's the exact same for stock XYZ. It climbs to $110 or it falls to $90. The only difference between the two is that the probability of stock ABC going up, the true probability, is 0.99. Whereas for stock XYZ the true probability of the stock price going up is 0.01. So these are clearly two very, very different securities. They have the same values. Okay, at time t equals 0 and possible values of time t equals 1, but the probabilities are clearly very, very different. I'm sure most of you would think well you'd far prefer to own this stock than this stock. Likewise if I ask you how much is a call option Which strike $100 worth. Many of you would assume that the call option price, in this case, should be much higher than the call option price in this case. After all, the probability of going up is much higher. So remember the payoff of a call option will be $10.00 and $0 in each case. But with probability 0.99, I'm going to get the $10.00 here, but with probability 0.01, I am going to get $10.00 here. So it looks like owning a call option here should be a lot more valuable than owning a call option on stock XYZ, okay? But, that is not the case. If you believe the assumptions of the model, that there is no transactions cost and that you can borrow and lend at the risk-free rate of R, and that you can buy or short sell the stock, then our previous analysis shows us That the true value of the derivative security depends only on R and u and d. It does not depend on the true probability's p. And in fact, if my calculations are correct, in both cases the option is worth approximately $4.80. So C0 equals $4.80 here, and C0 equals $4.80 here. Now, at this point, a lot of people get upset. They go, there's no way this could be the case. There's something wrong here. The theory that we've developed is wrong, you're, you're doing something incorrect. But in fact, nothing I've done is incorrect. What I've said here is correct. The fair price of the option in both cases is $4.80. So what I want you to do now is to think about this for a while. And ask yourself what's going on. Why is the fair option price $4.80 in both cases. And to think about this. I'll return to this in a couple of modules' time, and we'll discuss this again, and hopefully we'll, we'll clarify what's going on and explain why this apparent contradiction actually is not a contradiction at all.
