0:16

We're going to assume the same 1-period model that

Â we've been assuming up till now.

Â Stock price begins at a 100.

Â It goes up to 107, which is u times S 0, or

Â it falls to d times S 0, which in this case is 93.46.

Â We're going to assume that the gross risk free rate is 1.01.

Â So if I invest $100 in the cash account at time zero,

Â it will be worth $101 at time t equals 1.

Â So now what I want to do is, I want to figure out how much is a call option

Â that pays this quantity here worth at t equals 1.

Â So, the strike is $102.

Â Okay?

Â In this case the $102 falls somewhere in between the 93.46 and the 107.

Â And we need to figure out how much the security is worth.

Â 1:08

We'll also be interested in answering the question

Â how does the price of the option vary as p varies?

Â Remember p is the true probability of the stock price going up and

Â 1 minus p is the true probability of the stock price going down.

Â So to answer these questions,

Â we're going to construct what's called a replicating portfolio.

Â The idea of a replicating portfolio is extremely important in finance and

Â derivatives pricing.

Â So we'll see what that means in the next slide when we actually price the option.

Â 1:37

Okay.

Â So what we're going to do is, we're going to consider the following strategy.

Â We're going to consider buying x shares or x units of the stock.

Â And investing y dollars in cash or in the cash account, at time t equals 0.

Â We don't know what x and y are yet.

Â We're going to figure out what they will be, soon.

Â At t equals 1, this portfolio will be worth 107x plus 1.01y, when S equals 107.

Â And it will be worth 93.46x plus 1.01y when S equals 93.46.

Â So, this is where we had an up move

Â 2:23

Okay. So we buy x units of stock.

Â So in a up move, our stock position's worth 107x.

Â In a down move, the stock position will be worth 93.46x.

Â And our position in the cash account will be worth 1.01y regardless of whether you

Â had an up move or down move because the cash account is riskless.

Â Its value at time t equals 1 does not depend on whether the stock went

Â up or down.

Â 2:48

Okay, so what we're going to try and do is choose x and y so

Â that this portfolio is equal to the option payoff at t equals 1.

Â What is the option payoff at t equals 1?

Â Well, remember it's a strike of 102, so

Â therefore the option payoff at t equals 1 will be 0 down here, okay?

Â The stock prices were at 93.46, the strike is 102,

Â so you would not exercise the option and the maximum is 0.

Â However, if the stock prices were at 107 at t equals 1 then you're in the money.

Â You would exercise the option and receive 107 minus 102, which is $5.

Â So if you like, that's the value of the option c1 at time t equals 1.

Â 4:01

So 5 in the up move.

Â We've already seen what the portfolio's worth in an move.

Â It's 107x plus 1.01y, which is exactly what we have here.

Â And in a down move, the portfolio is worth 93.46x plus 1.01y.

Â Now, if you look at this, this, of course, is just two equations,

Â two linear equations, and two unknowns.

Â It's easy to solve it.

Â If you do, you'll find out that x equals 0.3693, and

Â y equals minus 34.1708.

Â So, what we've actually done, at this point, very simple.

Â We've constructed a replicating portfolio.

Â We've constructed a portfolio which invests at time zero, in the stock and

Â in the cash account.

Â And we've chosen the quantities x and

Â y in such a way that the value of this portfolio at time t equals

Â 1 is exactly equal to the payoff of the option at t equals 1.

Â So, we have succeeded in replicating the portfolio.

Â 5:05

Okay.

Â What does a negative value of y means?

Â Well a negative value of y just means that we're borrowing.

Â We saw on the previous slide that y equals minus 34.1708.

Â So y equals minus 34.1708.

Â This is our position in the cash account.

Â If y was positive, we would have invested in the cash account.

Â In other words we would have lent money at an interest rate of r, but

Â because it's negative what we're actually doing is we're borrowing money.

Â And we're going to pay interest on that borrowings at an interest rate of r.

Â 5:48

And this means that at time t equals 0,

Â we purchased 0.3693 shares, or units of the stock.

Â If x was negative, we would have,

Â we would have actually been short-selling the stock.

Â We would have been borrowing it and selling it in the marketplace.

Â 6:03

How much does this replicating portfolio cost at t equals 0?

Â Well, the stock costs a $100.

Â You can think of the cash account as costing $1.

Â Okay?

Â When we said we borrow $34.1708, that's like short selling the cash account.

Â Okay.

Â So the position is worth .3693 times 100 minus 34.1708 times 1 and

Â that's equal to 2.76 modulo some rounding here.

Â I didn't want to show any more than two decimal places.

Â Okay.

Â So the value of this replicating portfolio is $2.76.

Â This is how much it costs to actually put on the position.

Â Okay.

Â So I borrow $31.1708, and

Â I purchase 0.3693 units of share, of of the shares.

Â And that comes to a total of 2.76.

Â So that, I argue, is the arbitrage free value of the option.

Â Now let's pause for a second here and think about this.

Â I'm claiming 2.76 is the fair value of the option in this model.

Â In fact, I'm saying it's the arbitrage free value of the option in this model.

Â To see this suppose for example,

Â that I tell you, suppose the option

Â price in the marketplace is $2 not 2.76, then.

Â 8:23

So I can do the exact opposite of this replicating portfolio.

Â I could short sell 0.3693 units of the stock.

Â And I could invest $34.1708 in the cash account.

Â That will bring me in $2.76.

Â This will cost me minus $2, so

Â I will get a net profit of 76 cents at t equals 0.

Â What happens if t equals 1?

Â Well, if t equals 1, the two positions offset.

Â Remember, this portfolio replicates the value of the option.

Â Okay, so I've replicated the option but I also own the option,

Â which I've purchased for $2, so the two will offset each other and

Â I will have 0 at t equals 1 and

Â so this is an example of a type A arbitrage.

Â 9:20

Similarly if the option price was selling for

Â a quantity greater than 2.76, maybe the option price was selling for $4.

Â Then what I would do is I would do the opposite.

Â I would sell the option for $4, and

Â I would buy the replicating portfolio for $2.76.

Â So that will give me a net of 4 minus 2.76 which would be $1.24.

Â I would take that in a t equal 0 and at t equals 1, the two positions would offset

Â each other because the portfolio replicates the payoff of the option.

Â So again, I would have an overcharge.

Â So in fact, this value of 2.76 is the arbitrage free value.

Â Of the option.

Â 10:03

Okay, so we've just succeeded in pricing a call option and

Â that example with a strike of 102.

Â We can do the same thing more generally.

Â Okay, so we can actually price any derivative security

Â in our one period binomial model.

Â So how do we go about doing this?

Â Well, again.

Â We're going to do this now without any actual dollar numbers here.

Â We start off with a stock price of S0.

Â Probability of p of an up move, 1 minus p of a down move.

Â If it goes up, the stock price ends at u times S0.

Â And if it goes down, it ends up at d times S0 at time t equals 1.

Â 10:40

Assume we want to compute the fair value, or

Â the arbitrage-free price of a derivative security whose payoff will cost C1 of S1,

Â and this is the following payoffs.

Â It pays off C subscript u up here, and

Â it pays off C subscript d down here, if the stock price fell.

Â 10:59

So this is the payoff of the derivative security at t equals 1.

Â Well, we want to compute the fair value of that security.

Â And we can do it in the exact same way as before.

Â What we do is we construct at time t equals 0 a replicating portfolio.

Â How do we do that?

Â Well, we purchase x units of the stock.

Â And we invest y dollars in the cash account.

Â Or if you like, we purchase y units of the cash account,

Â assuming the cash account is worth $1 at t equals 0.

Â So what we do is we set up our two equations and two unknowns.

Â 11:37

This is the payoff of the derivative security in the event of an up move,

Â here on the right hand side.

Â And this is the payoff of the derivative security in the case of a down move.

Â And these are at t equaqls 1.

Â And on the left hand side, we have the value of our portfolio at time one.

Â If we purchased x shares at time 0.

Â Then those x shares would be worth u times s 0 x at time 1,

Â if the stock price went up.

Â And then it would be worth d times S 0 times x at time 1 if the stock price fell.

Â 12:32

When you solve those two equations, you will have a particular value for X and

Â a particular value for Y.

Â We can actually combine them together to get the fair value of

Â the derivative security at times 0.

Â Remember the replicating portfolio purchased x units of the stock at time 0.

Â So x units of the stock cost this, this much.

Â And it invested y dollars in the cash account.

Â Or if you like, it purchased y units of the cash account with a unit costing $1.

Â And that cost y.

Â So the total replicating portfolio costs x times S0 plus y, and

Â that must be the fair value of the derivative security at time 0, for

Â the exact same argument we gave a moment ago.

Â If it was, if the price of the derivative security was less than this quantity,

Â we could purchase, or we could construct an arbitrage.

Â If it was greater than the value of the replicating portfolio,

Â we could again construct an arbitrage.

Â Okay. So, if you actually go ahead and you solve

Â these two equations and two unknowns, what you will find is the following.

Â It's just two or three lines of algebra,

Â but you can check that you will find that C0, the fair value.

Â Of the derivative security can be written as 1 over

Â R times R minus d, u, over u minus d times cu,

Â 14:04

This then, you can check is actually 1 minus q.

Â Okay.

Â If I take 1 minus this quantity here, you'll see I'm left with this here.

Â So this is 1 minus Q.

Â And because of the no arbitrage assumption,

Â we have d is less than R is less than u.

Â So in fact, R is greater than d.

Â And u is greater than d, but because R is less than u,

Â we see that 0 is less than q is less than 1,

Â which of course also means that 0 is less than 1 minus q is less than 1.

Â So in particular, I can write the derivative price,

Â C0 as being 1 over R times q Cu.

Â Plus 1 minus qCd.

Â And I can actually, in fact, this is what I do,

Â these are what are called the risk-neutral probabilities.

Â They're risk-neutral probabilities,

Â they are probabilities because they both sum to 1.

Â q plus 1 minus q.

Â 15:05

Equals 1, and each probability is strictly greater than 0.

Â These are called risk mutual probabilities.

Â And so I can write my derivative security price as 1 over R, which is the discount

Â factor, times the expected value of the payoff of the derivative which is C1.

Â 15:23

Under the probability mass function q, okay so

Â what I've done is I've computed the price of the derivative payoff and

Â its price can be expressed as the expected value of the payoff at time 1,

Â discounted under the risk neutral probabilities.

Â 15:43

And this then is called risk-neutral pricing.

Â So this is a really important concept in finance.

Â We've developed the [INAUDIBLE] context of a 1-period binomial model.

Â We'll develop it soon in the context of multi period binomial models.

Â But we'll also see later in the course that it generalizes very easily to

Â more complicated models.

Â And it is, it is how derivative securities.

Â Are priced and practiced.

Â Okay.

Â 16:11

We can also answer our earlier question.

Â A question we asked earlier was, how does the option price depend on p?

Â Remember, p was the probability.

Â The true probability of an up move.

Â And 1 minus p was the true probability of a down move.

Â 16:34

And if we look at what we've done here, we're seeing that the fair

Â value of the derivative security is equal to this quantity here.

Â Well, all that's appearing here is R, capital R, and

Â the payoff of the derivative security.

Â These risk-neutral probabilities are expressed in terms of R, d, and u.

Â 17:18

And we've got stock XYZ.

Â And we're going to assume that each stock follows a one period binomial model.

Â Both stocks start off with the same initial price, $100.

Â And at time t equals 1, the first stock can either climb to $110 or fall to $90.

Â And it's the exact same for stock XYZ.

Â It climbs to $110 or it falls to $90.

Â The only difference between the two is that the probability

Â of stock ABC going up, the true probability, is 0.99.

Â Whereas for stock XYZ

Â the true probability of the stock price going up is 0.01.

Â So these are clearly two very, very different securities.

Â They have the same values.

Â Okay, at time t equals 0 and possible values of time t equals 1, but

Â the probabilities are clearly very, very different.

Â I'm sure most of you would think well you'd far prefer to own this stock

Â than this stock.

Â Likewise if I ask you how much is a call option

Â 18:32

Which strike $100 worth.

Â Many of you would assume that the call option price,

Â in this case, should be much higher than the call option price in this case.

Â After all, the probability of going up is much higher.

Â So remember the payoff of a call option will be $10.00 and $0 in each case.

Â But with probability 0.99, I'm going to get the $10.00 here, but

Â with probability 0.01, I am going to get $10.00 here.

Â So it looks like owning a call option here should be a lot more

Â valuable than owning a call option on stock XYZ, okay?

Â But, that is not the case.

Â If you believe the assumptions of the model, that there is no transactions cost

Â and that you can borrow and lend at the risk-free rate of R, and

Â that you can buy or short sell the stock, then our previous analysis shows us

Â That the true value of the derivative security depends only on R and u and d.

Â It does not depend on the true probability's p.

Â And in fact, if my calculations are correct,

Â in both cases the option is worth approximately $4.80.

Â So C0 equals $4.80 here, and

Â C0 equals $4.80 here.

Â Now, at this point, a lot of people get upset.

Â They go, there's no way this could be the case.

Â There's something wrong here.

Â The theory that we've developed is wrong, you're, you're doing something incorrect.

Â But in fact, nothing I've done is incorrect.

Â What I've said here is correct.

Â The fair price of the option in both cases is $4.80.

Â So what I want you to do now is to think about this for a while.

Â And ask yourself what's going on.

Â Why is the fair option price $4.80 in both cases.

Â And to think about this.

Â I'll return to this in a couple of modules' time, and

Â we'll discuss this again, and hopefully we'll, we'll clarify what's going on and

Â explain why this apparent contradiction actually is not a contradiction at all.

Â