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PROFESSOR: Vasily.
Vasily Strela who works now for Morgan Stanley,
did his PhD here in the math department, and kindly said he
would tell us about financial mathematics.
So, it's all yours.
GUEST SPEAKER: Let me thank Professor Strang
for giving this opportunity to talk here,
and it feels very good to be back, be back to 18.086.
So, a few more words about myself.
I've been Professor Strang's student in mathematics
about ten years ago.
So after receiving my PhD, I taught mathematics
for a few years.
Then ended up working for a financial institution,
for investment bank, Morgan Stanley in particular.
I'm part of an analytic modeling group in fixed income division.
What we are doing, we are doing math applications
in finance and modeling derivatives, fixed income
derivatives.
That's actually what I'm going to talk about today.
I want to show how 18.086, it's a wonderful class
which I admire a lot, which applications
it has in the real world, and in particular in finance
and derivatives pricing.
Let's start with a simple example,
which actually comes not from finance, but rather
from gambling.
Well, let's look at horse racing or cockroach
racing, if you prefer.
Suppose there are two horses, and sure enough,
people bet on them and bookie is a clever one, very
scientific-minded guy and he made
a very good research of previous history of these two horses.
He found out that the first horse has 20% chance to win.
The second horse has 80% chance to win.
He is actually right about his knowledge about chances to win.
On the other hand, general public, people who bet,
they don't have access to all information,
and the bets are split slightly differently.
So 10,000 is placed on the first horse,
and 50,000 is placed on the second horse.
Bookie, sticking to his scientific knowledge,
splits the odds 4 to 1, meaning that if the first horse wins,
then whoever put on the horse gets his money back
and four times his money back on top of it.
Or if the second horse wins then whoever put money on this horse
will get the money back and 1/4 on top of it.
So, let's see.
What are chances for bookie to win or lose in this situation?
Well, if the first horse wins then he
has to give back 10,000 plus 40,000, 50,000,
and he got 60,000, so he gains 10,000.
Well, good, good for him.
On the other hand, if the second horse wins,
then he has to give back 50,000 plus 1/4 of 50,
which is 12,500, so 62.50 altogether,
and he loses $2,500.
After many runs, the expected win or loss of the bookmaker
is the probability of the first horse
to win times the expected win, plus the probability
of second horse to win times the expected loss, which turns out
to be exactly zero.
So in each particular run, bookie may lose or win,
but in the long run he expects to break even.
On the other hand, if he would put the chances,
he would set the odds according to the money bet, 5 to 1.
what would be the outcome?
Well, if the first horse wins he gives back 10,000 plus 50,000,
60,000, exactly the amount he collected.
Or if the second horse wins, again, he
gives back 50 plus 1/5 of that, 60, he breaks even.
So no matter which horse wins in this scenario,
the bookie breaks even.
How bookie operates, well he actually
charges a fee for each bet, right.
So the second situation is much more preferable for him.
When he doesn't care which horse wins, he just collects the fee.
Well here, he may lose or gain money.
This is quite beautiful observation,
which we will see how it works in derivatives.
So now back to finance, back to derivatives.
So we are actually interested in pricing
a few financial derivatives, and what is a financial derivative?
Well, a financial derivative is a contract,
payoff of which at maturity, at some time T,
depends on underlying security -- in our case,
we always we will be talking about a stock as underlying
security, and probably interest rates.
What are the examples of financial derivatives?
Well, the most simple example is probably a forward contract.
Forward contracts is a contract when you agree to purchase
the security for a price which is set today --
you've agreed to purchase the security in the future
for the price agreed today.
Well, for example, if you needed 1,000 barrel of oil to heat
your house, but not today, but rather for the next winter,
on the other hand, you don't want
to take the risks of waiting until the next winter
and buying oil then, you would rather agree on the price
now and pay it in the future and get the oil.
What the price should be?
What is the fair price for this contract?
Well, we will see how to price it.
Well, the few observation here is that this line represents
the payout -- it's always useful to represent the payout
graphically.
This is just a straight line because the payout
of our contract is S minus K at time T.
This actually gives the current price of the contract for all
different values of the underlying.
Usually, the price of the forward contract
is set such that for the current value of the underlying,
the price of the contract is zero.
It costs nothing to enter a forward contract,
so that's why it intersects zero here.
What are other common derivatives?
Another common derivative are calls and puts.
And I put European call and put here.
Don't be confused by European or American.
It has nothing to do with Europe or America,
it has to do with the structure of the contract.
European basically means that the contract
expires at certain time T. American means
that's you can exercise this contract at any time
between now and future.
We'll be talking only about European contracts.
So European call option is a contract
which gives you the right, but not obligation,
to purchase the underlying security at set price K, which
is called strike price, at a future time
T, which is expiration time.
So if your security at time T ends up below K,
below the strike, then sure enough
there is no point of buying the security
for a more expensive price.
So the contract expires worthless.
On the other hand, if your stock ends up being greater than K
at expiration time C, then you would
make money but my purchasing this stock for K
dollars and your payout will be S minus K,
and this is a graph of your payout.
This line here, as we will see, is the current price
of the contract, and we'll see how to obtain
this line in a few minutes.
Another common contract is a put.
While call was basically a bet that your stock will grow,
right, the put is the bet that your stock will not grow.
So, in this case, the put is the right, but not the obligation
to sell the stock for a certain price K.
Here is the payout, which is similar to the put,
but just flipped.
This is the current price of a put option.
Calls and puts, being very common contracts,
are traded on exchanges -- Chicago exchange is probably
the most common place for the calls and puts on stocks
to trade.
I just printed out a Bloomberg screen,
which gives the information about a few calls
and puts on IBM stock.
So I did it on March 8, and the IBM stock
was trading at this time at $81.14,
and here are descriptions of the contract,
they expire on 22nd of April, so it's
pretty short-dated contract.
They can go as far as two years from now, usually.
Here is a set of strikes, and here are a set of prices.
As you can see, there is no single price,
there is always a bid and ask, and that's how dealers
and brokers make their money -- like a bookie,
they basically charge you a fee for selling or buying
the contract.
That's how the money are made -- they are made on this spread,
but not on the price of the contract itself,
because as we will see in a second,
we actually can price the contract exactly,
and there is no uncertainty once the price of the stock is set.
There are plenty of other options.
Slightly more exotics contracts, either digital
which pays either zero or one depending
on where your stock ends up.
It probably is not exchange-traded,
also I'm not sure.
There are hundreds, if not thousands,
of exotic options where you can say that, well,
how much would be the right to purchase
a stock for the maximum price between today and two years
from now.
So it will be past-dependent.
Depending on how the stock will go,
the payout will be defined by this path.
There are American options where you
can exercise your option any time between now and maturity,
and so on and so forth.
So, just before we go into mathematics
of pricing, just a few observations and statements.
First of all, it turns out that thanks
to developed mathematics, mathematical theory,
if you make certain assumptions on the dynamics of the stock,
then there is no uncertainty in the price of the option.
You can say exactly how much the option costs now,
and that's what provides, and this
is a big driver for the market.
So dealers quote these contracts and there is a great agreement
on the prices.
The price of the derivative contract
is defined completely by the stock price
and not by risk preferences of the market participant.
So it doesn't matter what are your views on the growth
prospects of the stock.
It will not affect the price of the derivative contract.
As I said, so the mathematical part of it
comes into giving the exact price without any uncertainty.
So let's consider a simple example now.
Let's assume that we are in a very simple world.
Well, first of all, in our world there are only three objects --
the stock itself, the riskless money market account,
meaning that it is an account where we can either borrow
money or invest money at the riskless rate r,
and finally our derivative contract.
Here we are not making any assumptions of what kind
of derivative contract it is -- it could be forward,
it could be call, it could be put, it can be anything.
Moreover, our world is so simple, that first of all,
it's discrete, and second of all,
there is only one time step to the expiration of power
of contract, dt.
Not only there is only one step left,
we actually know exactly what our transition probabilities.
There are only two states at the end,
and we know the transition probability.
So with probability p, we move from the state zero
to the state one, and with probability of one minus p,
we move to the state two.
And just notice, because this is riskless money market account,
it's the same in both cases.
You just invest money and it grows with risk-free interest
rate.
So, what can we say about the price of our derivative f?
Well a simple-minded -- well, let's start with the forward
contract.
We know what the payout in delta t of our forward contract
will be, it will be just the difference between the stock
price and our strike.
Well, a simple-minded approach would be -- well,
we know the transition probabilities,
let's just compute the expected value of our contract,
and that's what we would expect to get if there were many such
experiments.
Well, you take the probability of going to state one,
you multiply by the payoff at stage one.
Take, minus p for probability of going to state two,
multiply by the payout in that state two.
Sum them up and you get the expression.
As I said, the common thing to choose
the strike such that the contract
has zero value now, so you get your strike.
Well, in particular you could say
that if you research the market well
and you know that the stock has equal probability of going
up and down, then actually you expect your strike
to be an average of end values of the stock.
But as we can imagine, following our bookie example,
this is not the right price.
There is actually a definite price
which doesn't depend on transition probability.
Here is the reason why there is a definite price.
Well let's just consider a very simple strategy.
Let's borrow just enough to purchase a stock.
So let's borrow S_0 dollars right now and buy
the stock for this money.
And let's enter the forward contract.
Well, by definition forward contract has price zero now,
so we enter the forward contract.
Now, at the time dt when our contract expires, what happens?
Well, we deliver our stock, which we already
have in our hand in exchange of K dollars.
That's our forward contract.
On the other hand, we have to repay our loan,
and because it was a loan, it grew.
It grew to S_0 times e to the r*dt.
Now, let's see.
What would happen if K was greater than S times e
to the r*dt?
Then we know for sure, we know now for sure,
that we would make money.
There is no uncertainty about it now.
Similarly, if K is less than this value,
then we know that we will lose money.
That's not how the rational market works.
If everybody knew that by setting this price
you would make money, people would do it all day long
and make infinite money.
So there will be no other side of the market.
So the price has to go down.
So the only choice for K, the only market-implied choice,
is that K has to be equal to S times e to the r*dt.
As you can see, it doesn't depend on transition
probabilities at all.
That's what market implies us.
That's the price of forward contract,
and that actually explains why, when
I was plotting the forward contract,
current price was just the straight line,
it's just discounted payoff.
The payout is linear, so just the parallel to the payoff.
That's the idea, basically.
The idea is to try to find such a portfolio of stock
and the money market account with such a payout ,
which will exactly replicate the payoff of our derivative.
If we found such of a portfolio, than we know for sure
that the value of this portfolio,
the replicating portfolio today is
equal to the value of the derivative, because otherwise,
you would make or lose money risklessly.
That's no-arbitrage condition.
So, can we apply it to our general one-step world?
Well, if we have a general payout f, what we want to do,
we want to form a replicating portfolio such
that at expiration time, it will replicate our payouts.
So we want to choose such constants a and b that such
that the combination of stock and money market
account in both states will replicate
the payout of our option.
Then, if we are able to find such constants a and b,
then we just look at the current price of the contract
and it has to be equal to the current price
of our derivative.
Well, but in our particular case, this is easy.
It's just two linear equations with two unknowns, easily
solved, and here is current price of our derivative.
No matter what payout is -- I mean you just substitute
the payout here, and if you know S_1 and S_2, that's it.
A useful way to look at this, just to re-write this equation,
is in this form, and then notice that actually the current price
of our derivative can be viewed as a discounted expected
payout of the derivative, but with very certain probability.
This probability, it doesn't come
from statistical properties of the stock or from any research,
it actually is defined by the market.
So it's called a risk-neutral probability.
So this probability doesn't depend
on the views on the market by the market participants.
An interesting observation is that actually,
the value of this stock, the discounted value of the stock
is actually also is expected value of our outcomes
under this risk-neutral probability.
That's basically general idea.
Now let's move one notch up and try to apply
these idea to continuous case.
Well, if you live in continuous world now,
we need to make some assumptions on the behavior of the stock.
The very common assumption is that the dynamics of the stock
is log-normal.
Log-normal meaning that the logarithm of the stock
is actually normally distributed.
So, here mu is some drift, sigma is the volatility of our stock,
and dW is a Wiener process, W is a Wiener process
such that dW is normally distributed
with mean zero and variance square root dt.
Our approach would be to find the replicating portfolio.
And what does it mean?
It means that we want to find such constants, over time dt --
so we assume that a and b are constant over the next step,
dt -- such that the change in our derivative is a linear
combination with this constant of the change of our underlying
security and the change of money market account.
Now we just need to look more closely at this equation.
First of all, let's concentrate on df.
So, f, our derivative, is a function
of stock value and time.
But unfortunately, our stock value is stochastic,
so df is not that simple, and to write df out we have to use
a famous -- Ito's formula from stochastic calculus,
which actually is analogous of Taylor's formula for stochastic
variables.
Let's see.
If our S will not be stochastic, if it
would be completely deterministic
and depend only on dt, then there
would be no term and differential f is just
the standard expression.
On the other hand, if we have dependence
on stochastic variables, then we have
to have more terms, and why this happens?
Well, in very rough words is that because the order
of magnitude of dW is higher than dt's --
it's square root of dt.
So we have to make into account more terms,
and in particular, we have to take into account next order
of dS squared.
Formally, dS square can be written this way,
and again, very rough explanation is as follows.
If we would square this equation there
will be three terms there.
One would come from the square of this term,
and this would be of the order of dt squared,
next order of magnitude -- much smaller than dt.
The second term will be cross-product of dW*dt.
What order of magnitude we are talking about,
it is dt to the power 3/2, again, much smaller than dt.
On the other hand, the third term will be the square of dW,
this is of order of magnitude of dt,
so that's what we have to keep.
And that's what Ito's formula is about.
Now, we are basically, we know all terms here,
and let me stress out that this term, dB, it is not stochastic,
it's completely deterministic because we
know that B grows with the rate r, that's what it is.
So we substitute all those terms into our replicating equation.
We collect the terms.
We get this equation.
And again, there is the deterministic part,
there is stochastic part.
So the only way for this equation to hold
is this term to be equal to this term, and this term
to be equal to this term, and that's what's written out here.
So again, two equations, these two unknowns, and here
is answer.
Finally, let's take a*S to another part.
Notice that this part of our equation
is completely deterministic.
So we know how it will grow.
So basically, d of f minus a*S, which is b times dB,
is r times b times dt.
And we know all other terms, we substitute them
here, take something to the left-hand side
and get this equation.
So this is partial differential equation for our derivative f,
as a function of S and t, of second order,
and this equation is the famous Black-Scholes equation.
It was derived by Fischer Black and Myron Scholes
in their famous paper published in 1973.
Myron Scholes and Robert Merton actually
received Nobel Prize for deriving and solving
this equation in '97.
Black was already dead by the time.
This is really the cornerstone of math finance.
The cornerstone is because using the replicating portfolio,
using this reasoning, we were able to find an exact equation
for our derivative.
So a few remarks on Black-Scholes.
So first of all, we made some assumptions
on the dynamic of the stock, but we never made any assumptions
on our derivative.
Which means that any derivative has to satisfy this equation,
and that's very strong result. So if you
assume that our stock is lognormal,
which is not a bad assumption and agrees quite well
with the market, then we basically, in principle,
can price any derivative.
We know the equation for any derivative.
The other thing is that our Black-Scholes equation
doesn't depend on the actual drift mu
in the dynamics of our stock.
So again, it is the manifest of risk-neutral dynamic.
Not only we wrote down the equation for our derivative,
we also found a replicating portfolio.
So in other words, we found a hedging strategy,
meaning that at any given time we
can form this portfolio with rates a and b.
If we hold both the derivative and both replicating portfolio,
altogether, this is zero sum gain.
We know that no matter where stock moves,
we will not lose money or gain money.
So if we just charge bid-offer on the derivative,
if we charge a fee on the contract,
we can hedge ourself perfectly, buy the contract
or sell the contract, hedge perfectly ourself
and just make money on the fee, that's it.
Finally, more mathematical remark
is that actually after a few manipulations, a few change
of variables, the Black-Scholes equation comes out
to be just a heat question, which you already
saw in this class.
This is very good news.
Why is this good news?
Well, because heat equation is very well studied.
So the solutions are well-known, and numerical methods, the ways
to solve it, in particular the numerical ways
to solve it are well-known.
So we are in business.
But as any partial differential equation,
the equation itself doesn't make much sense because to find
a particular solution we need boundary and initial condition.
And although any derivative satisfies Black-Scholes
equation, the final and boundary conditions
will vary from contract to contract.
Here are a few examples of the final and boundary conditions.
Here, an interesting remark that if, usually,
we would talk about initial condition, here
we are talking about final condition.
The time goes in reverse; we know
the state of the world at the end, at expiration, not today.
So, here are final and boundary conditions for call and put,
and let's look a little bit at the pictures for our call
and put to see where they come from.
So for example, for calls, well, this is our final condition,
right, this is defined by the payout.
On the other hand, the boundary condition, well, what happens,
we put them at zero at an infinity, we [INAUDIBLE PHRASE]
to put them at zero and infinity.
And why?
Well, because if stock hits zero then it stays at zero.
That's what our dynamics show.
So the value of our contract at maturity will become zero.
On the other hand, if the stock grows, grows to infinity,
a good assumption to make is that actually it
becomes similar to stock itself, so it would just
become parallel to the stock, and that's the conditions
which we impose here.
Similarly for the put, you can derive these conditions.
And again, just because it is a heat equation,
it turns out that for a simple derivative such as the calls
and puts, it is possible to find an exact analytic solution.
Here are exact analytic set of solutions for a call, put
and the digital contracts.
Well, not surprising again, I mean
they're all connected to the error function,
so to the normal distribution, basically,
as the solutions of heat equation ought to be.
Why do they look exactly the same?
If we have five minutes at the end,
we'll probably shed some light on the specific form
of equations.
But let me just stress that we can see that it's discounted,
and what I'm claiming, it's expected value of our payout
under risk-neutral measure.
Here is an example -- it's of a particular call option
on the same IBM stock.
So I chose the short-dated contract,
just to avoid the dividend payment.
So it's a contract expiring on March 18.
So there is 10 days to expiration.
The stock, as we saw, the expirations of stock,
as we saw, is what's trading at 81.14.
The volatility is somewhere around 14%,
estimated either from other options or historically.
Here is the price of our contract.
I also have a simple Black-Scholes calculator here,
and let's see if we can match this price.
So let's see, I believe the volatility was 13%, right,
13.47.
The interest rate, it's already here.
As we all know, Fed just bumped the interest rate,
so they are at 4.75% right now.
The strike of our option was 80.
Time to expiration was actually 10 days,
and this should be measured at a fraction of year.
So we divide 10 by 365.
This stock was trading at 81.14, if I'm not mistaken.
Here is the price of our call options contract, which is 150.
Well, it's within the offer.
So maybe our volatility's slightly off
and if we increase [UNINTELLIGIBLE] to say,
increase it to 14%, it will go slightly up.
152.
Well, in general, let's play a little bit with it.
Well, it is very short-dated option,
so the value of our option is very close to the payout.
So if we increase the time to maturity,
let's make it two years just to see where --
so now value of our option is -- well, that's what it is.
If increase volatility, sure enough, let's make it 30%.
So what do we expect?
We expect if volatility is higher, the uncertainty higher,
so the value of our contract should go up, and it sure does.
So basically that's how Black-Scholes works.
And plenty of those contracts trade
on the market, but unfortunately not all of these contracts
are so simple as calls and puts.
First of all, there are many more complicated products
with more difficult payout, which will constitute different
and probably discontinuous final conditions on our Black-Scholes
equation.
Moreover, we made an assumption that the volatility
is constant with time, and interest rate
is constant with time.
It is certainly not true for the real world.
Volatility probably should be time-dependent,
and this would make the coefficients
in our Black-Scholes equation time-dependent.
Unfortunately, these cannot be solve analytically.
So in most of the cases in practice,
we will have to use some kind of numerical solution.
Finite difference methods is the typical approach
for the heat equation.
As you know, both explicit and implicit schemes,
and you will discuss some of those in 18.086.
Tree methods.
Tree methods meaning that we go back to our one-step tree,
and basically assume that our time to expiration
is many time steps away and we'll grow the tree further,
so from this node we have two more nodes,
and so and so forth.
That would imply the final condition at the end,
and discount back using our risk-neutral probabilities,
and get the price now.
So those are called tree methods.
One can show that actually those tree methods are equivalent
to finite difference -- explicit finite difference schemes.
Those are very popular.
But again, in tree methods, what is very important
is to set the probabilities from your tree, the transition
probabilities, to the right ones,
and the right ones are risk-neutral probabilities.
Probabilities implied by the market, actually.
Another important numerical method is Monte Carlo
simulation where you would simulate many different
scenarios of the development of your stock up to the maturity,
and then, basically find -- using this path,
you will find the expected value of your payout.
But again, in order for this expected value
to be the same as the risk-neutral value,
as the arbitrage-free value, you have
to develop your Monte Carlo simulations
with risk-neutral probabilities.
So, risk-neutral valuation is extremely important.
Here is actually the general risk-neutral statement, which
one can prove, is that actually, the value of any derivative
is just discounted expected value of the payout
of this derivative at maturity, but you
have to take this expectation at the right measure.
Using the right measure, meaning that you have to set correctly
the transition probability -- you have to make them
market-neutral.
Under this measure, actually the dynamics of our stocks
looks slightly different, and as you can see,
our drift becomes the interest rate.
So under risk-neutral measure, everything
grows with our risk-free interest rate.
Just to shed a little bit of light on how we go
at the solutions for calls and puts,
Black-Scholes solutions for call and put,
well this is the distribution of our stock,
log-normal distribution of our stock at time T,
and if we take this distribution and integrate our payout
of our call option against this distribution -- in other words,
find the expected value of payout of our call option under
risk-neutral measure, then, sure enough,
you will get [UNINTELLIGIBLE PHRASE].
This illustrates the best -- because what is digital?
Digital is just the probability to end up
at above the strike at time T, right?
So if you integrate this log-normal pdf
from the strike K to infinity, that will be your answer.
This is a good exercise in integration,
to make sure that it's correct.
So let's see.
To conclude, what we've seen.
So, we have seen that modern derivatives business makes
use of quite advanced mathematics,
and what kinds of mathematics is used there?
Well, partial differential equations are used heavily.
Numerical methods for the solution
of this partial differential equations are naturally used.
In order to get these equations, we actually
need to operate in terms of stochastic calculus,
meaning that we need to know how to deal with Ito calculus,
Ito formula, Girsanov theorem, and so on and so forth.
The other thing is to be able to build simulations
to solve the heat equation and all other equations
that you might encounter.
The topic which we didn't touch upon
is statistics because, of course, very advanced
statistics is used for many, many things,
for analyzing historical data, which
can be quite beautiful for trading
strategies and many others.
Besides these five topics, there is much, much more
to mathematical finance, which makes it a very, very
exciting field to work in.
That's what I wanted to talk about.
Thank you very much for your attention.
PROFESSOR: Maybe I'll ask a firm question
about boundary conditions, because you
had said that those are different
for different contracts.
And how do you deal with them in the finite differences
or the tree model or whatever?
What would be a typical one?
GUEST SPEAKER: Well, typical one -- two very typical ones.
So those, you basically make a grid of your problem,
in particular, you build a tree, which
is actually a grid of all possible outcomes.
You set them up at the end, so your tree grows --
so you set your boundary here at the end, and well, you set,
probably, some initial -- this is final condition,
so you set some boundary conditions here.
So this is your time T. This is t = 0, time t -- this is 0,
this is 1, this is 2, this is T. So you set your payout here,
so it will be maximum of S minus K and 0.
PROFESSOR: How many time steps might you take in this?
GUEST SPEAKER: Well, you would do like daily for three months
-- if you three-month options.
PROFESSOR: Maybe 100 steps.
GUEST SPEAKER: Yeah, something like that.
Well, if it's two-year option, that you probably would do it
weekly or something like that.
PROFESSOR: You don't get into large,
what would be scientifically, large-scale calculating.
PROFESSOR: No, in finance we usually
don't keep this problem--.
PROFESSOR: In finite differences,
do you use like higher-order -- suppose, well,
you had second derivatives, would you always use second
differences or second-order accuracy?
GUEST SPEAKER: In general, yes.
In general, second-order accuracy.
In general you don't go higher.
I mean the precision -- well, it's within cents, right.
So you cannot do better than that.
So it depends -- well, it depends what kind of amount you
are dealing with.
If you're actually selling and buying units of stock,
you might consider something more precise.
But it's very problem-defined.
So that's how we deal with it.
PROFESSOR: Any questions?
You can put the mic on if you have a question.
STUDENT: [UNINTELLIGIBLE PHRASE].
GUEST SPEAKER: Well, it is Markov process.
Yes.
I mean, this is just a numerical solution.
So yeah, it is Markov process.
and basically all stochastic calculus
is about Markov process, continuous Markov process.
PROFESSOR: Is the mathematics that you
get involved with pretty well set now or is there
a need for more mathematics, if I
can ask the question that way?
PROFESSOR: Yeah, well, in this field
it is probably quite well set.
But if you get into more complicated fields,
especially into credit modeling, the model for the credits
of certain companies, then mathematics is not quite set,
because there, you start talking about jump processes
and not Wiener processes, not just log-normal processes.
This stochastic differential equation become very hard,
but maybe still analytically tractable.
So from this point of view there is need --
but it's not a fundamental mathematics,
it's not that you are opening a new field,
but definitely trying to solve a stochastic differential
equation -- which usually boils down to solving a partial
differential equation analytically --
can be pretty hard a mathematical problem,
viewed as a mathematical problem.
PROFESSOR: So you showed the example
of Black-Scholes solver.
Everybody has that available all the time?
GUEST SPEAKER: Oh yeah.
On Chicago trading floor, the traders
have calculators where they just press a button
and it's just hard-wired there.
PROFESSOR: And they're printing out error functions,
basically -- a combination of error function, yeah.
GUEST SPEAKER: Well, sure enough, nobody uses just --
I mean this was very approximate example and that's why I chose
such short-dated stock, that before it pays any dividends,
and where we can assume the volatility is constant,
and so on and so forth, to match the prices.
Otherwise, the prices wouldn't match.
PROFESSOR: Thank you.