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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 multiplications 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 uses 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.
When that becomes general public, people who bet, they
don't have access to all information, and the bets are
split slightly differently.
So the 10,000 is placed on the first horse, and 50,000 is
placed on the second horse.
Bookie, sticking to his scientific knowledge, places
the odds 4:1.
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, 5:1, 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, pay of which at
maturity at some time c, 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 where it's 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
keep your house, but not today but rather for the next
winter, on that account 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 f minus k at time c.
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 call, and I put European call
input 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 c.
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 the 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 c, which is expiration time.
So if your security at time c 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 f 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 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-rated, 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 decide 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 effect 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
options -- the stock itself, the riskless with money market
account, meaning that it is an account where we can either
borrow money or invest money at the request 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, but 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 I just noticed, 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 I say it about the price of our derivative f?
Well it's 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 strikes is 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
condition 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 f zero 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 for an 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 zero times e to the rdt.
Now let's see.
What would happen if k was greater than f
times e to the rdt?
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 had to go down.
So the only choice for k, the only market-implied choice, is
that k has to be equal to f times e to the rdt.
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 watching 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 riskless.
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 now a 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
s1 and s2, and s2 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 on 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 lognormal.
Lognormal meaning that the logrithm of the stock is
actually normally distributed.
So, here u is some drift, sigma is the volatility of our
stock, and dw is the [INAUDIBLE] process, w's the
[INAUDIBLE] of process such that dw is normally
distributed with mean zero and variance square root dt.
Our approach would be to find the replicating portfolio.
What that 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 you time.
But unfortunately, our stock value is stochastic, so the f
is not that simple, and to write dfo we have to use a
famous -- it's a formula from stochastic calculus, which
actually is analagous of Taylor's formula for
stochastic variables.
Let's see.
If our f will not be stochasticated, 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 --
its square root of dt.
So we have to make into account more terms, and
particularly we have to --
to take in part next order of df squared.
Formally, df 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 that we are talking about, this 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 the order of magnitude of dt, so that's what dw is.
And that's what either formula is about.
Now we are basically, now all terms here, and let me stress
out that this term, dv it is not stochastic, it's
completely deterministic because we know that d 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 af 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 as, which is d times dv, is r
times d times dc.
We know all other terms, we substitute them here, take
something to the left hand side and
we have this equation.
So this is partial differential equation for our
derivative f, as a function of f and t, of second order, and
this equation is the famous Black-Scholes equation.
It was derived by Fisher 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 would remarks on Black-Scholes.
So first of all, we make 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 was normal, 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 [UNINTELLIGIBLE]
[? 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 hedge and 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
all together, this is zero sum gain.
We know that no matter where stock moves, we will not move
money or gain money.
So if we just charge bid offer on the derivative, if we
charge a few 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, and 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 which defines Black-Scholes
equation, the final and boundary condition will vary
from contract to contract.
Here are a few examples of the final and boundary conditions.
Here, an interesting remark that usually we would talk
about initial condition, here we are talking about final
condition if 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.
Under the boundary condition, well, what happens, we put
them at zero at an infinity, which shows
[INAUDIBLE PHRASE], and why, well, because if stock hits
zero then it stays in zero.
That's what our dynamics show.
So the value of power of contract at maturity will
become zero.
[UNINTELLIGIBLE PHRASE]
the stock grows, grows to inifinity, 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 between both here.
Similar, for the put you can derive these conditions, and
again, that's because it is a heat equation.
It turns out that for a simple derivative such that the calls
and puts, it is possible to find an
exact analytic solution.
Here are exactimated 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 just trust that we can see that it's discounted, and what
I'm saying is expected value of our
payout on the [? risk ?]
[UNINTELLIGIBLE] for 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 avoid the
dividend payment.
So it's a contract expiring on March 18.
So there is some based 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 from 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 the Fed just bumped the interest rate, so
at 4.75% right now.
The striker of our option was 80 times 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 increase
at, let's say, 2.14% 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 three
years just to see where, so now what if 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 higher, then certain things
higher, so the value of our contract should go
up, and it sure does.
So basically that's how Black-Scholes works.
[? Plane ?]
[? Hills ?] does contract 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
[UNINTELLIGIBLE] will discontinue 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.
Finding different methods is the typical approach for the
heat equation.
As you know, both [UNINTELLIGIBLE PHRASE]
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 times expiration is many
[UNINTELLIGIBLE] time steps away and will grow the tree
further, so from this node we have some more nodes, and so
and so forth.
That would imply the final condition at the end, and
discount back using risk-neutral probabilities,
and get the price down.
So those are called tree methods.
One can show that actually those tree methods are
equivalent to find a difference -- expect to find a
difference scheme.
Those are very popular.
But again, in tree methods, what is very important is to
set the probability from your tree the transition
probability was the right one.
Since the right one risk-neutral probability.
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 after the maturity.
Then basically 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 is
risk-neutral probability.
So, risk-neutral volition 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 and divide 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 either [UNINTELLIGIBLE PHRASE]
you 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 Shoals solutions for
call and put, well this is the distribution of our stock, the
normal distribution of our stock at time t, and if we
take this distribution and integrate our payout of our
co-option against this distribution -- in other
words, find the expected value of payout of our full option
under the 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 both
a strike at time t, right?
So if you integrate this lognormal pdf from the strike
k to the infinite, 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 more than 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 the integrations, we actually need
to operate in some stochastic calculus, meaning that we need
to know how to deal with either calculus, either
formula, Gaussian 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 one 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 as the ends, so your tree grows.
So you set your boundaries 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 zero times t -- this is zero, 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 zero.
PROFESSOR: How many time steps might you take in this?
GUEST SPEAKER: Well, you would do like daily for three months
-- you do 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 you usually
don't keep this problem--.
PROFESSOR: In finite differences, do you use like
higher order as opposed, where you had second derivatives,
would you always use second differences or
second order accuracy?
GUEST SPEAKER: In general, yes.
In general, second order access.
In general you don't go higher.
I mean the precision -- well, it's within tenths, right.
So you can not 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?
We can put the mic on if you have a question.
STUDENT: [UNINTELLIGIBLE PHRASE].
GUEST SPEAKER: Well, it is market proc--
Yes.
I mean this is just a numerical solution.
So yeah, it is market process.
and basically all stochastic calculus is about market
process, continuous market 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 [UNINTELLIGIBLE]
Weiner processes, not just lognormal processes.
This stochastic differential equations become very hard,
but may be analytically practical.
So from this point of view there is --
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 in mathematical problem, view 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 their approximate example and that's
why I chose such short-dated stock that before it pays any
dividends, and where we cannot the volatility is constant,
and so on and so forth, match the prices.
Otherwise, the prices wouldn't match.
PROFESSOR: Thank you.