0:00

We're not going to discuss option pricing in the 1-period binomial model.

Â We'll see that option pricing in this model amounts to no more then of solving a

Â series of two linear equations and two unknowns.

Â The great advantage of working with the 1-period binomial model is that we will

Â see it easily extends to pricing options in the multiperiod binomial model.

Â So this is the one period binomial model, we are assuming that the stock price

Â starts off at S0 equals $100. The stock price at time, t equals 1, will

Â either have grown to uS 0, which is $107 in this case, so here I am assuming u

Â equals 1.07. R will have fallen to $93.46, which is d

Â times S0 and d is equal to 1 over u. Okay.

Â So this is the 1-period binomial model. Stock price goes up by a factor of u or it

Â falls by a factor of d. We have a probability p, which is the

Â probability of an up move and we have a probability 1 minus p, which is the

Â probability of a down move. Okay.

Â So we also assume that we have a cash account available, which enables us to

Â borrow or lend at a gross risk free rate of R.

Â So, if we invest $1 in the cash account at time t equals 0.

Â It would be worth R dollars at time t equals 1.

Â Similarly, if we were to borrow $1 at time t equals 0, we would have to pay back R

Â dollars at t equals 1. We've also seen that short sales are

Â allowed. What that means is, if I want to short

Â sell a stock, what I can do is I can borrow the stock.

Â I assume I can borrow the stock at no cost and sell it in the marketplace.

Â Then later on I can buy the stock back and return to the person who lent it to me.

Â So that's how short sales work. If I want to short sell a stock.

Â I borrow it, I sell it in the marketplace, later on, I buy it back, and then return

Â it to the person who lent it to me in the first place.

Â We're also implicitly going to be assuming that there are no transactions costs.

Â Okay, so basically, I can buy and sell and borrow and lend with no transactions

Â costs. Okay.

Â So two questions, two simple questions to begin with.

Â The first question is, how much is a call option that pays the maximum of S1 minus

Â 107 and 0 at t equals 1, 1 worth? So in this case, 107 is the strike, and

Â this is a call option. And the second question is, how much is a

Â call option that pays the maximum of S1 minus 92 and 0 at t equals 1 worth?

Â So in this case, 92 is the strike. Well, in fact, here, we'll see that we can

Â answer these questions very easily. In the case of question one, well, if we

Â go back to our 1-period model, we see the maximum security price at time one is 107.

Â So therefore, in fact, the maximum of S1 minus 107, 0 is always going to actually

Â be zero at t equals 1, because in our binomial model, the stock price is never

Â greater than 107. So the call price at time 0, we'll call it

Â C0, must be zero. Okay, for question two, it's actually also

Â quite straight-forward. Again, with the, the strike here is $92,

Â if I go back to my binomial model here, I see that the smallest possible at time t

Â equals 1 is 93.46. So no matter what, in fact, I can write

Â the maximum of S1 minus 92 and 0. Well, that is going to be S1 minus 92.

Â And that is because the stock price at time 1 is always greater than or equal to

Â 92. So the max will always occur at S2 minus

Â 92. So if I'm going to get S1 minus, minus 92

Â at time t equals 1, how much is this payoff worth today at time 0?

Â Well, linear pricing tells me it must be worth S0 minus 92 over R.

Â Okay. That's because what I'm getting is, I'm

Â getting S1 dollars at t equals 1. Well, S1 dollars at t equals 1 must be

Â worth S0 dollars today. I'm also getting minus $92 at t equal 1.

Â Well, 92 is just a fixed determinant to cash growth, so I just discount its value

Â back to t equals 0 to see if this is a total value of the option price at time 0.

Â Okay. So that was two simple questions that we

Â could answer. But what happens if the strike lies in

Â between 107 and 93.46? Okay, in order to answer that, I actually

Â need to introduce, first of all, some ideas of arbitrage, Type A and Type B

Â arbitrage. And we're going to need these more general

Â definitions, these are more general than the earlier definitions of weak and strong

Â arbitrage that we used in the deterministic world.

Â We're going to need them because we're introducing randomness Into our models.

Â So, we have the two following definitions. A Type A arbitrage is a security or

Â portfolio that produces immediate positive reward at t equals 0 and it has a

Â nonnegative value at t equals 1. So, that is a security with initial cost

Â V0 less than 0. So if its cost is negative, that means, if

Â we buy, it we actually receive money. Okay, it's a little bit confusing.

Â But when you have a negative initial cost, you actually receive money when you buy

Â something. So a Type A arbitrage is a security with

Â initial cost V0 less than 0 and time t equals 1 value V1 greater than or equal to

Â 0. So an example of this type of arbitrage

Â would be maybe finding $10 in the street. So if you find $10 in the street right

Â now, well, you're going to receive a positive amount, it's like having a

Â negative cost of $10 and at time 1 you'd have V1 equals 0.

Â So you find the money right now, you get $10, and you have no liability at time t

Â equals 1. Okay.

Â A type B arbitrage, is a security or portfolio that has a non-positive initial

Â cost, has positive probability of yielding a positive payoff at t equals 1 and zero

Â probability of producing a negative payoff then.

Â So let's translate them. The type B Arbitrage is a security with

Â initial cost less than or equal to 0. So in other words, if you enter or you

Â purchase this security, you're not going to pay anything for it and indeed you

Â might receive something. You receive something if this is strictly

Â negative and its terminal value times t equals 1 is nonnegative and it's not equal

Â to 0. So , an example of such a security would

Â be maybe someone coming up to you in the street and handing you a free lottery

Â ticket. A free lottery ticket means you pay

Â nothing at times 0. Okay, so we pay nothing for the free

Â lottery ticket. And a time one, maybe you'll win

Â something. Okay.

Â But you definitely won't lose anything. So V1 is greater than or equal to 0,

Â because only the lottery ticket means you're never going to have to actually pay

Â out. And there'll be a chance, maybe a small

Â chance, but there will be a chance that you'll actually receive something, so it's

Â not equal to zero. So, over here, you can think of this as

Â being the different states of the lottery. Maybe there's one state where you win.

Â Maybe this is a positive payoff for you. And all the other states are zero.

Â So this is an example of a security which is greater than or equal to 0, but it's

Â not strictly equal to 0. Okay, so that's a Type B arbitrage.

Â Now, let's return to our 1-period binomial model and discuss what conditions must

Â hold in order to have no arbitrage in the model.

Â Recall that we can borrow or lend at gross risk free rate, R per period.

Â We're also assuming that short sales are allowed.

Â So we have the following theorem. The theorem states that there is no

Â arbitrage if and only if d, which is this quantity here, is less than the gross risk

Â free rate, which in turn is less than u. So let's see how we might prove this.

Â So consider the first situation here. Suppose R is less than d is less than u,

Â then we can construct the following portfolio.

Â Let's borrow S0 dollars and invest in the stock.

Â This will actually gives us a Type B arbitrage.

Â How do I know that? Well, let's see.

Â Let's look at the cash flows. So, our cash flows R, so this is under

Â condition one, so we have R less than d less than u, and we're going to follow

Â this portfolio here. So, at t equals 0, we borrow S0 dollars,

Â so that gives us plus S0 dollars, but we invested in the stock.

Â So we're going to buy one unit of stock, so that means we're going to spend the S0

Â dollars we got from borrowing it and we're going to spend it on the stock, so the net

Â cash flow times 0 is 0. What happens at t equals 1?

Â At t equals 1, so this is from our borrow, and this is our position in the stock, and

Â at t equals 1, we have to pay back our borrowing.

Â So we have to pay back S0, but we have to pay it with interest.

Â So we're going to be paying back S0 times R, okay?

Â But we own one unit of stock, okay? Remember, we invested SS0 dollars into

Â stock at time t equals zero. That's going to be worth us 0 or dS0 at

Â time t equals 1. And the fact we're going to use these

Â proceeds to pay back the borrowings at t equals 1.

Â The net cash flow here then is u minus RS0 if the stock price went up or it's d minus

Â RS0 if the stock price went down. But by assumption, d is greater than R and

Â u is greater than R so this component is positive and this component is positive.

Â So regardless of whether the stock went up or not, we're going to have made money at

Â time t equals 1 and that is an example of a Type B arbitrage.

Â Okay, so that's case one. How about case two?

Â Well, we could do something very similar. In this case, if R is too large, then it

Â suggests that it might be a good idea to short sell the stock, take S0 dollar,

Â dollars in from short selling the stock, invested in the cash account and earn R.

Â And then the time t equals 1, you can buy back the stock for either d times S0 or u

Â times S0, but in either case, you're going to have less than S0 times R.

Â So in fact, you can do the same sort of argument that I did up here to show that

Â you would also get a type B arbitrage in case two.

Â Okay, so we're always going to assume, later on, that d is less than R is less

Â than u. And, that's because we don't want

Â arbitrage to exist in our models. It's a standard economic assumption that

Â we assume there's no arbitrage and the reason is if there was an arbitrage then

Â market forces would act very quickly to dispel that arbitrage.

Â Supply and demand would drive the arbitrage away.

Â So we're always going to be assuming there's no arbirtrage.

Â