Financial Engineering is a multidisciplinary field drawing from finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part I will be on the use of simple stochastic models to price derivative securities in various asset classes including equities, fixed income, credit and mortgage-backed securities. We will also consider the role that some of these asset classes played during the financial crisis. A notable feature of this course will be an interview module with Emanuel Derman, the renowned ``quant'' and best-selling author of "My Life as a Quant". We hope that students who complete the course will begin to understand the "rocket science" behind financial engineering but perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism. The follow-on course FE & RM Part II will continue to develop derivatives pricing models but it will also focus on asset allocation and portfolio optimization as well as other applications of financial engineering such as real options, commodity and energy derivatives and algorithmic trading.
The 1-Period Binomial Model
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From the course by Columbia University
Financial Engineering and Risk Management Part I
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From the lesson
Introduction to Derivative Securities
The mechanics of forwards, futures, swaps and options. Option pricing in the 1-period binomial model.
Meet the Instructors
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
We're not going to discuss option pricing in the 1-period binomial model.
We'll see that option pricing in this model amounts to no more then of solving a
series of two linear equations and two unknowns.
The great advantage of working with the 1-period binomial model is that we will
see it easily extends to pricing options in the multiperiod binomial model.
So this is the one period binomial model, we are assuming that the stock price
starts off at S0 equals $100. The stock price at time, t equals 1, will
either have grown to uS 0, which is $107 in this case, so here I am assuming u
equals 1.07. R will have fallen to $93.46, which is d
times S0 and d is equal to 1 over u. Okay.
So this is the 1-period binomial model. Stock price goes up by a factor of u or it
falls by a factor of d. We have a probability p, which is the
probability of an up move and we have a probability 1 minus p, which is the
probability of a down move. Okay.
So we also assume that we have a cash account available, which enables us to
borrow or lend at a gross risk free rate of R.
So, if we invest $1 in the cash account at time t equals 0.
It would be worth R dollars at time t equals 1.
Similarly, if we were to borrow $1 at time t equals 0, we would have to pay back R
dollars at t equals 1. We've also seen that short sales are
allowed. What that means is, if I want to short
sell a stock, what I can do is I can borrow the stock.
I assume I can borrow the stock at no cost and sell it in the marketplace.
Then later on I can buy the stock back and return to the person who lent it to me.
So that's how short sales work. If I want to short sell a stock.
I borrow it, I sell it in the marketplace, later on, I buy it back, and then return
it to the person who lent it to me in the first place.
We're also implicitly going to be assuming that there are no transactions costs.
Okay, so basically, I can buy and sell and borrow and lend with no transactions
costs. Okay.
So two questions, two simple questions to begin with.
The first question is, how much is a call option that pays the maximum of S1 minus
107 and 0 at t equals 1, 1 worth? So in this case, 107 is the strike, and
this is a call option. And the second question is, how much is a
call option that pays the maximum of S1 minus 92 and 0 at t equals 1 worth?
So in this case, 92 is the strike. Well, in fact, here, we'll see that we can
answer these questions very easily. In the case of question one, well, if we
go back to our 1-period model, we see the maximum security price at time one is 107.
So therefore, in fact, the maximum of S1 minus 107, 0 is always going to actually
be zero at t equals 1, because in our binomial model, the stock price is never
greater than 107. So the call price at time 0, we'll call it
C0, must be zero. Okay, for question two, it's actually also
quite straight-forward. Again, with the, the strike here is $92,
if I go back to my binomial model here, I see that the smallest possible at time t
equals 1 is 93.46. So no matter what, in fact, I can write
the maximum of S1 minus 92 and 0. Well, that is going to be S1 minus 92.
And that is because the stock price at time 1 is always greater than or equal to
92. So the max will always occur at S2 minus
92. So if I'm going to get S1 minus, minus 92
at time t equals 1, how much is this payoff worth today at time 0?
Well, linear pricing tells me it must be worth S0 minus 92 over R.
Okay. That's because what I'm getting is, I'm
getting S1 dollars at t equals 1. Well, S1 dollars at t equals 1 must be
worth S0 dollars today. I'm also getting minus $92 at t equal 1.
Well, 92 is just a fixed determinant to cash growth, so I just discount its value
back to t equals 0 to see if this is a total value of the option price at time 0.
Okay. So that was two simple questions that we
could answer. But what happens if the strike lies in
between 107 and 93.46? Okay, in order to answer that, I actually
need to introduce, first of all, some ideas of arbitrage, Type A and Type B
arbitrage. And we're going to need these more general
definitions, these are more general than the earlier definitions of weak and strong
arbitrage that we used in the deterministic world.
We're going to need them because we're introducing randomness Into our models.
So, we have the two following definitions. A Type A arbitrage is a security or
portfolio that produces immediate positive reward at t equals 0 and it has a
nonnegative value at t equals 1. So, that is a security with initial cost
V0 less than 0. So if its cost is negative, that means, if
we buy, it we actually receive money. Okay, it's a little bit confusing.
But when you have a negative initial cost, you actually receive money when you buy
something. So a Type A arbitrage is a security with
initial cost V0 less than 0 and time t equals 1 value V1 greater than or equal to
0. So an example of this type of arbitrage
would be maybe finding $10 in the street. So if you find $10 in the street right
now, well, you're going to receive a positive amount, it's like having a
negative cost of $10 and at time 1 you'd have V1 equals 0.
So you find the money right now, you get $10, and you have no liability at time t
equals 1. Okay.
A type B arbitrage, is a security or portfolio that has a non-positive initial
cost, has positive probability of yielding a positive payoff at t equals 1 and zero
probability of producing a negative payoff then.
So let's translate them. The type B Arbitrage is a security with
initial cost less than or equal to 0. So in other words, if you enter or you
purchase this security, you're not going to pay anything for it and indeed you
might receive something. You receive something if this is strictly
negative and its terminal value times t equals 1 is nonnegative and it's not equal
to 0. So , an example of such a security would
be maybe someone coming up to you in the street and handing you a free lottery
ticket. A free lottery ticket means you pay
nothing at times 0. Okay, so we pay nothing for the free
lottery ticket. And a time one, maybe you'll win
something. Okay.
But you definitely won't lose anything. So V1 is greater than or equal to 0,
because only the lottery ticket means you're never going to have to actually pay
out. And there'll be a chance, maybe a small
chance, but there will be a chance that you'll actually receive something, so it's
not equal to zero. So, over here, you can think of this as
being the different states of the lottery. Maybe there's one state where you win.
Maybe this is a positive payoff for you. And all the other states are zero.
So this is an example of a security which is greater than or equal to 0, but it's
not strictly equal to 0. Okay, so that's a Type B arbitrage.
Now, let's return to our 1-period binomial model and discuss what conditions must
hold in order to have no arbitrage in the model.
Recall that we can borrow or lend at gross risk free rate, R per period.
We're also assuming that short sales are allowed.
So we have the following theorem. The theorem states that there is no
arbitrage if and only if d, which is this quantity here, is less than the gross risk
free rate, which in turn is less than u. So let's see how we might prove this.
So consider the first situation here. Suppose R is less than d is less than u,
then we can construct the following portfolio.
Let's borrow S0 dollars and invest in the stock.
This will actually gives us a Type B arbitrage.
How do I know that? Well, let's see.
Let's look at the cash flows. So, our cash flows R, so this is under
condition one, so we have R less than d less than u, and we're going to follow
this portfolio here. So, at t equals 0, we borrow S0 dollars,
so that gives us plus S0 dollars, but we invested in the stock.
So we're going to buy one unit of stock, so that means we're going to spend the S0
dollars we got from borrowing it and we're going to spend it on the stock, so the net
cash flow times 0 is 0. What happens at t equals 1?
At t equals 1, so this is from our borrow, and this is our position in the stock, and
at t equals 1, we have to pay back our borrowing.
So we have to pay back S0, but we have to pay it with interest.
So we're going to be paying back S0 times R, okay?
But we own one unit of stock, okay? Remember, we invested SS0 dollars into
stock at time t equals zero. That's going to be worth us 0 or dS0 at
time t equals 1. And the fact we're going to use these
proceeds to pay back the borrowings at t equals 1.
The net cash flow here then is u minus RS0 if the stock price went up or it's d minus
RS0 if the stock price went down. But by assumption, d is greater than R and
u is greater than R so this component is positive and this component is positive.
So regardless of whether the stock went up or not, we're going to have made money at
time t equals 1 and that is an example of a Type B arbitrage.
Okay, so that's case one. How about case two?
Well, we could do something very similar. In this case, if R is too large, then it
suggests that it might be a good idea to short sell the stock, take S0 dollar,
dollars in from short selling the stock, invested in the cash account and earn R.
And then the time t equals 1, you can buy back the stock for either d times S0 or u
times S0, but in either case, you're going to have less than S0 times R.
So in fact, you can do the same sort of argument that I did up here to show that
you would also get a type B arbitrage in case two.
Okay, so we're always going to assume, later on, that d is less than R is less
than u. And, that's because we don't want
arbitrage to exist in our models. It's a standard economic assumption that
we assume there's no arbitrage and the reason is if there was an arbitrage then
market forces would act very quickly to dispel that arbitrage.
Supply and demand would drive the arbitrage away.
So we're always going to be assuming there's no arbirtrage.
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