Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part II will be on the use of simple stochastic models to (i) solve portfolio optimization problems (ii) price derivative securities in various asset classes including equities and credit and (iii) consider some advanced applications of financial engineering including algorithmic trading and the pricing of real options. We will also consider the role that financial engineering played during the financial crisis. We hope that students who complete the course and the prerequisite course (FE & RM Part I) will have a good understanding of 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.
Pricing Derivatives Using the Volatility Surface
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Do curso por Columbia University
Financial Engineering and Risk Management Part II
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Equity Derivatives in Practice: Part II
More about Black-Scholes, the Greeks and delta-hedging; the volatility surface; pricing derivatives using the volatility surface; model calibration.
Conheça os instrutores
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
We're now going to see how we can use the volatility surface to see how we can
price different types of derivative securities.
Obviously, we can use the volatility surface to price European call and put
options. After all, we actually constructed the
volatility surface using European call and put option prices.
But we will see that there are other derivative securities that can also be
priced. These are derivative securities whose
value only depends On the marginal risk-neutral distributions of the stock
price. So, we're going to see some examples.
In particular we will see how to price a digital option.
And we will also see how to price a so-called range accrual using the
information in the implied volatility surface.
Suppose we wish to price a digital option, which pays $1 if the time t stock
price, ST, is greater than K, and 0 otherwise.
[INAUDIBLE] . We actually know that we can price the
security given the implied volatility surface.
Now why is that? Well, the reason for that, as we saw in
one of the more recent modules, that if you know the implied volatility surface,
then you know the marginal risk-neutral distribution of The security processor on
a fixed time 'T', so the pay off of this option is going to be equal to the
maximum of '0' and the indicative function '1' which pays off '1' is 'ST'
is greater than 'K'. So this is the pay off of the digital
option. So the risk neutral distribution of this
option only depends on the marginal risk neutral distribution of st.
And so we know from the volatility surface that we can calculate this
marginal risk neutral distribution. And therefore, we can evaluate.
This quantity here. Which is the initial value of this
digital option. So lets see how we can actually go ahead
and price this. It is easy to see.
That the digital price. Were going to call it dkt.
K is the strike. T is the maturity.
Is given by the following. So dkt is equal to, well we have this
limit as delta k goes to 0, of the s, the market price of a call option with strike
k maturity t, minus the market price for a call option with strike k plus delta k
and maturity t, all divided by delta k. Now why is that?
Well, it's easy to see this if we draw a picture and that's what we'll do.
So we will draw a picture. This is going to be our payoff of this
strategy, so buying 1 over delta K times and option of strike K maturity T and
selling 1 over delta K times the call option strike K times delat K and
maturity T T. So this is represents S T, the stock
price at maturity. We have this value here, whchi is K.
And we have the value here K plus delta K.
Now it is easy to check. That this option, this strategy here, of
going long option and shortest option gives a payoff of 0 up as far as K, and
then it grows linearly up to a value of 1.
At k plus delta k. And thereafter, gives a constant value
one. And you can see this easily.
So you see that if the terminal stock price is greater than k plus delta k.
Then you're going to make a profit of delta k from these two positions.
Divide that by the delta k here, and you get a profit of 1.
So, this is the payoff. And so it should be clear that, as we let
delta k go to zero. Then this line here is becoming more and
more vertical. And we're getting closer and closer to
the payoff of a digital option which pays 1 dollar, only if st is greater than k.
And so that's why we get this limiting argument here.
So dkt is equal to the limit as delta k goes to 0 with this.
We're just going to multiply through by minus 1, take the minus outside here and
we get it's the limit is delta k goes to 0 of this quantity here.
And we just recognize this as being the partial derivative of the market price of
a call option with strike k maturity t with respect to the strike k.
So it's very straightforward to see that the price of the digital option is given
to us by this. Now recall how we defined the implied
volatility of an option. What we did is, we equate the market
price of the option with the Black Scholes price.
And we figure out what is the implied volatility parameter, sigma of k and t.
Which makes the Black-Scholes price match the market price.
Just to simplify notation, I haven't bothered to include the other parameters
that I often include here, R C and S 0. So we know that sigma K, T is the
volatility parameter that must go into the Black-Scholes formula so that we get
a price on the right-hand side that is equal to the market price of the option.
As I said before, sometimes this is likened to Plugging the wrong number into
the wrong formula to get the right price. So now, if you recall from the previous
slide. What we need to do is, we need to compute
this partial derivative here with respect to k.
Well, we know that the partial derivative of this with respect to k is the partial
derivative of the right hand side with respect to k.
And there are actually two terms that come into this.
We see, the first argument. K appears here.
But also in the sigma argument. K also appears in there.
So were actually going to have two terms corresponding to, to, to k here.
So were going to get that the partial derivative, with respect of k.
Is equal to the partial derivative of the Black Scholes formula with respect to the
strike. Plus the Black-Scholes formula with
respect to Sigma. Well, that's our vega times delta sigma,
delta K. And that's actually what we will call the
skew. Of course, we have a minus, because we
have in both terms. Because we had a minus outside here as
well. So, we're going to get D cave T is equal
to minus delta CBS, delta K, minus the vega.
Times the skew. And so the skew we're just going to refer
to this as being delta sigma delta k. And remember that for a fixed time
maturity. Maturity in the Acuity market we'll see a
skew like this. So this will be s t or if you like k.
Either one. And this is sigma kt.
So, we can actually calculate delta cbs, delta k, and the vega from the Black
Scholes formula. These are straightforward to compute,
because we know the Black Scholes formula.
And there, we can compute these derivatives.
The skew can be estimated from the implied volatility surface.
So we will have calculated our, our volatility surface, and we'll be able to
estimate the sku. In other words, we'll be able to compute
what delta sigma, delta k is. For example, suppose this is the strike
here. Well, this, therefore, has.
A value of, let's call it Sigma K. Maybe you go up to K plus Delta K.
This has a value, Sigma K plus Delta K, and so we can estimate the partial
derivative, the skew, or Delta Sigma Delta K...
As being approximately equal to sigma of k plus delta k minus sigma k divided by k
plus delta k minus k, which is delta k. And of course as I let delta k go to
zero, this approximation becomes better and better.
So we can approximate. Delta sigma delta k, or if you like the
skew we're calling it, from the implied volatility surface that we will have
available to us. So this is an example of how the
Black-Scholes terminology, or technology, is used in practice.
Even though the Black Scholes model is known to be wrong.
We can still compute option prices with, with the, with the Black Scholes
terminology. In this case, we're u-, computing option
prices from the implied volatility surface.
The implied volatility surface, if you recall, has been set up, so that, by
construction. Call and put options will match the
prices of call and put options in the marketplace.
And we're going to be able to use this volatility surface to compute other types
of options as well. And in this case, we're going to compute
the price of a digital option. As I also mentioned before, we can use
the volatility surface to price any security Who's payoff only depends on the
stock price at a given fix time tee. That is because we know the marginal
retribution distribution once we know[UNKNOWN] and we know the marginal
retribution then we can compute the derivative who's payoff only depends on
the stock price and a fix time T So here's an example.
This example is taken from the book, the Volatility Surface, by Jim Gatheral.
It's an advanced text. So I wouldn't necessarily advise anyone
to go out and look at it. It's more of a doctoral text on financial
mathematics and financial engineering. But there's a nice example in that text
that we'll go through here. So what we're going to do, is, we're
going to, we're going to be pricing a digital option.
The digital option is, has a strike of k equals 100.
The current stock price is 100. So the digital is at the money.
Remember, just to be clear, the payoff of this digital time t is equal to.
The maximum of zero. And the indicator function of st being
greater than or equal to k. And the fact if you stop for a second,
you can see you don't really need the maximum here because this is simply the
indicator function of st being greater than or equal to k.
So, if you recall the vega from the Black-Scholes formula, well it is as
follows. We know that vega, I'll write it here.
We know that vega is equal to e. To the minus c T times S square root
capital T times five of D one. Where D one was equal to the, the log of
S zero divided by K. Plus R minus C plus sigma squared over
two times T divide, all divided by sigma square root T.
Now, in this situation, in this example, we're going to assume r equals c equals
zero. T equals 1 year.
And s zero equals k. So in that case, d is equal to, it turns
out to be simply sigma over 2. And it also implies that the vega, in
this case. Is equal to.
While c is zero. S is 100.
T is one. So it's five of sigma over two.
So what were actually going to do. Is were going to get this as equal to.
S0 times 5 of sigma over 2. So now we can go to the To the task at
hand, which is to compute the price of this digital option.
We know this price is given to us by this quantity here.
So we can actually calculate delta cbs, delta k from the Black Scholes formula.
You can check. But it actually turns out to be this
quantity here. The vega is given to us by s 0 phi of
sigma atm over 2. Sigma atm is the octamum implied
volatility. And we're told it's 25 percent.
Finally, how about the skew? Well, we are told that the skew is 2.5
percent per 10 percent change in strike. So a 10 percent change in strike is equal
to 10 percent of s zero. Because the strike is equal to s zero.
And it's 2.5% per 10% change. So it's going to be 0.025 divided by .1
s0. This is the skew.
However, we also have a minus sign here. Now the minus sign is, we're not
explicitly told there's a minus sign here.
But we know that there must be, because we know the equity markets we see askew
like this. So clearly as, so this is K.
Clearly as K increases, the implied volatility falls.
So this number here, which if you recall, is delta sigma delta K, that's going to
be negative. And so I implicitly understand that the
skew here presents the negative 2.5% per 10% change in strike, and so that's how I
get this quantity here. The S zero accounts with the S zero her.
I can evaluate this quantity using, I can do it simply in Excel, and I get a
digital price of 0.55... What's interesting is if we ignored the
skew component. In other words, if I just took the market
price, see market price of the call option.
To be equal to just the black[UNKNOWN] price, as a function of KT and sigma at
the money. And ignore the fact that sigma is also a
funcion of K as we see in this skew here. If I ignore that, then I would only get a
delta cbs delta k term appearing. Remember when we take partial derivatives
here with respect to k, we get a term from the k argument.
But we also get a term from the implied volatility argument.
Because the implied volatility's a function of k.
And that's why I get this second term here.
But if I ignored the second term, pretend that signal was a constant, that would be
the case of the Black-Scholes model held, then I would only get the 0.45 value
here. And, so in fact, by taking the skew into
account correctly, I see the price of the digital option as 0.55 and not 0.45, and
actually this is significant. This represents ten cents extra.
On .54 dollars. So consider now a 3 month range accrual
on the S & P 500 index with range of 1,500 to 1,550.
After 3 months the product pays X% of notional where x equals the percentage of
days over the 3 months that the index is inside the range.
So, for example, the notional is $10M and the index is inside the range 70% of the
time; then the payoff will be $7M. The question is, is it possible to
calculate the price of this range accrual using the volatility surface?
The answer is, yes. Consider a portfolio consisting of a pair
of digital's For each date between now and the expiration.
So, actually lets expand on this answer and see how indeed we can use the
volatility surface to price this range accrual.
So lets assume that there are N Trading days in the three month period in
question. Then the payoff at time t Let's write it
here. The payoff at time T, that's called a P
subscript T, will be equal to 10,000,000 times the summation from I equals 1 to N
of the indicator function. That the underlying security price, we'll
call it si, is inside the range in question and this range is 1,500 less
than or equal to si, less than or equal to 1,550.
We have to divide by n because its the percentage of days that were inside the
range. And so we must divide.
So this summation is the total number of days that were inside the range that the
security price, the underlined security price, in this case the S&P 500, is
inside the range. So the percentage of days that were
inside the range is the summation divided by M.
So this is the payoff at time, capital T. So how might we price this security?
Well, we know, let's write it here. From Risk-neutral pricing that the
initial value of the security is equal to the expected value under the Risk-
neutral probability distribution of E to the minus R capital T.
Capital T is assumed to be the maturities of To three months.
Of times pt. So let's expand on that.
This is equal to 10 divided by n. So I'm going to omit the m, for
millionths. So now my units are in millionths.
So it's 10 over n. Times the summation from i equals 1 to n
of e to the minus r times t minus ti. Times the expected value of e to the
minus, or ti, times this quantity here. Now, what can we do with this.
We'll notice by the way, that I have a minus minus TI which is a plus TI and
that counts as with a minus TI inside here.
So let's look at this expression here. So we have the expected value of e to the
minus rti, times the indicator function. That 1,500 is less than or equal to si.
Is less than or equal to 1,550. Where S I is the underlining security
price on day I. Well I can write this as the following,
this is equal to, E to the minus R T I TImes the indicator function of si being
greater than or equal to 1,500 minus the indicator function of si being greater
than or equal to 1,550. So it's quite straightforward to see that
this indicator function here. One on the event s size between 1,500 and
1,550 is equal to the difference of these 2 indicator functions here.
And if you think about it, what you will see is that this is equal to.
Well, using our earlier notation for the price of the digital option.
This is equal to D 1500 on Date TI minus D 1550 on Date TI.
And so actually what we've managed to do is we've managed to break down the range
accrual into a strip of digital pairs... A different pair for each[UNKNOWN] TI,
and so therefore the price of the rang accrual is equal to 10 million dollars
divided by N, the number of days, times the summation...
From i equals 1 to n of e to the minus r times capital T maturity minus t little
i. Times the digital option.
Would strike 1500 maturity ti minus the digital option would strike 1550 in
maturity ti. And indeed, we can compute these digital
option prices from the implied volatility surface as we saw a short while ago.
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