E-mail to the class: 02/05/00
Hi all,
It was pointed out the following important typos in the solution to the mock midterm. My apologies.
Part (C), point 3: we should have
equation (0.11) and (0.12)
de = (r$ - rE) e dt + sigmaE dX1
dS = r$ S dt +sigmaS dX2
and equations (0.13) (0.14)
e(t+dt) = e(t) + (r$-rE) e(t) dt + sigmaE sqrt(dt) epsilon1
S(t+dt) = S(t) + r$ S(t) + sigma sqrt(dt) epsilon2
Q:
on page 84 of TN#3, we want to find the PDE for Options on
Futures. some
questions: 1) why are futures non-traded factors? don't we have an
established (and huge) market for futures? i guess i'm misunderstanding
the definition of non-traded factors;
A:
A traded factor is something you can buy for the price S. Even though the
futures price is F, you can't buy anything at that price. You can enter
into a contract to deliver at F. It is subtle but different.
Q:
2) assuming that futures are
non-traded, why is that we construct the portfolio as if it were a
traded one? shouldn't we have 2 derivatives and no futures at all?;
A:
You are right: but this is one of those cases where it turns out that you
can actually use a standard model. The reason is that the future price
changes and generates profits itself. Instead, any other factor (say
volatility) would not generate profits/losses by its own movements.
(Indeed, you need to buy an option that generates profits/losses with
the movements of sigma)
Q:
3)
i understand that the cost of entering into a futures is zero, but the
value of a portfolio comprised of long 1 derivative and short delta
futures should change with the change in futures price, shouldn't it?
ie, I don't get why PI = V(F,t) and not PI = V(F,t) - delta.F thx for
the help,
A:
This is related to point (1) above. The value of a future contract at the
current future price is always zero by its own definition. Instead,
between now and tomorrow it is going to generate profits/losses and hence
you need to put it into the Total returns. An any instant, the value of
your porfoltio is just equal to the value of the option (notice that the
option is valued at the CURRENT F, which implies that your current
position in the future has zero value)
Q:
(1) HW2: Part A # (2c)(iv): You made a statement that "Since the right
hand side
of (0.15) does not depend on r, we must
have 1 + B'(t) = 0" How can you draw that conclusion?
A:
if 1+B'(t) is not equal to zero, a change in r would change the left hand
side but not the right hand side. Hence, the equality would not be met
anymore, violating the equation. It must be the case that 1+B'(t) = 0.
Q:
Part B: You state that we cannot estimate dr = adt + (sigma)dx using a
regression technique because r(t) is not stationary. Could you please
explain
this? How can we tell if this condition is met?
A:
Well: a random walk
r(t+1) = r(t) + a + sigma epsilon
is a non stationary process in the sense that it does NOT have finite unconditional moments (that is, unconditionally E(r(t)) = infinite, E(r(t)^2) = infinite etc). Since the regression coefficient would be
Cov(r(t) r(t+1))/Var(r(t))
you see that if all these quantity are infinite, you cannot compute.
When you have a process like
r(t+1) = a + phi r(t) + sigma epsilon
you must have |phi|<1 to have stationarity.
As explained, you can estimate directly
dr = a + sigma epsilon
because instead dr *is* stationary.
Q: Feynman-Kac Formula: V(S,t)=E( exp(-r*(t'-t)) * g(S')| S(t)= S) assume r is constant & g(S') is final payoff.
For a call option, g(S') is the {integral from K to infinity} for: (S'-K)*(p*)*dS'
For a forward contract, g(S') is just (S'-K).
Question: Why doesn't the forward contract have the p* term or the integral sign, as in the call option (is it because p*=1 or because r=D and there is no drift?)
A: NO! g(S') is the payoff at maturity (how much money you get from your option) that is,
g(S')=max(S'-K,0)
The integral is the expectation! For any random variable X,
E(X) = integral [X f(X) dX]
where f(X) is the probability density of X.
Hence, V(S,t) = exp(-r (T-t)) * integral [ g(S') p*(S') dS' ]
= exp(-r (T-t)) * integral [ max(S'-K,0) p*(S') dS' ]
See TN3 page 79. Hence, there is no difference between the forward contract and the option contract.
Q: on H4,
Since many of the options at various strike prices weren't traded or were
traded when the index was at a different price, the "last sale" column can
be way off. Should we use the ask price or a bid/ask average for the
market price of the option?
A:
YES: This (Ask) is what I used
Q: On HW3
(1) For 1.(a) "...rewrite the PDE on page 62 using the
coefficients..." Is that supposed to be the PDE on page 63?
A:
No, it is the one on page 62 (the general one).
Q:
(2) On question 2 (a), the drift term in both the stock and the
volatility seems really nasty. Is the V supposed to be the derivative
itself that we're pricing? So there's some feedback? If not, what does
it represent?
A:
It's a typo: the V is supposed to be sigma. Thank you
Q:
For (B), it looks like all we're supposed to do is take the excess
return of the S&P over the (annualized == r/365) interest rate and use a
simple method of moments to get the mean and standard deviation. So, a
few minutes in Excel, essentially. Is that right?
A:
YES
Q:
On TN#4, p. 101, we have Si(t+1) = Si(t) + rSi(t) +
sigma*Si(t)(dt)^1/2*e(t+1).
Should the second term on the right hand side be r*Si(t)*dt? So, are
we missing a dt?
A:
YES, thanks
Q:
Old question: what does it really mean for a security to be a solution
to the B-S PDE? For example, a bond. Does it simply mean that our
closed form B-S formula can price a bond just like it can an option?
A:
A bond also satisfies the PDE. Remember HW1? You showed that exp(-rT)
satisfies the PDE. Now, exp(-rT) is the price of a bond in BS world.
THe only difference between a bond an option is its payoff at maturity.
Q: I have a question on part (C) of HW2
In Figure 2.2 of Wilmott I see options on the S&P100 but not on the S&P500. Am I missing them? Also, shall we use different options with different strike price or just one?
A: Do use the SP100, thanks. Also, use all the options as stated, with different strike prices. Maybe you can plot them to see how the implied volatility curve looks like.
E-mail to the class: 01/17/00
I noticed that Exercise 2 (b) in Homework 2 is quite difficult to solve. So, SKIP IT! You can still do Exercise 2 (c) from point (i)-(iii) in the "Hint"
In summary, you must do exercise 2 (a) and (c) (i)-(iii)
*If* you have extra-time: you can try to do the following. (It is *not* required because we will do this later on in the course). Solve the PDE explicitly:
Assume that the solution has the following form
V(t,r) = exp( A(t) - B(t) r ) (*)
for two functions of time A(t) and B(t) to be determined.
So, you can follow the following steps:
(1) compute all the partial derivatives of the function (*)
(2) plug them in into the PDE obtained in part (c) (iii)
(3) simplify and collect all the terms involving "r"
(4) Consider the coefficient of "r": it should involve only the function B(t) (or its derivative with respect to time). Set this coefficient equal to 0
(5) Using this condition, you can then solve for B(t)
(6) The PDE now does not have a term in r anymore and you know B(t): Hence, solve for A(t)
Q:
I have a question from the computer exercise. The code on page 110 says
at
some point Vega=Asset*sqr(Expiry/3.1415926/2)....
Is that true or should it be inside the (), Expiry/3.14*2
A: It is correct as in the book. With computer codes, the statement
(Expiry/3.1415926/2) = ((Expiry/3.1415)/2) = Expiry/(3.14*2)
That is, the "A/B" divide A by B independetly on whether there A is a fraction or whatever
Q2: Also, how do I execute the code, how do I input the data into the main in order to run it?
A: You are defining a function: Hence, it going to work as any type of function in Excel. Suppose you have
A1=Call Price
A2=Strike
A3=Time
A4=Stock Price
A5=Int. Rate
A6=Error (say, .0001)
Then, to execute the code you are writing you must simply write the name of the function and the inputs (make sure you put the inputs in the RIGHT order. If you get something wrong, the computer could stall in the attempt to solve an impossible program!) In the specific, you should write
ImpVolCall(A1,A2,A3,A4,A5,A6)
E-mail to the class: 01/13/00
Hi all,
There is an important typo in TN3 (it is important because you might find confusing solving HW2, question 1). On page 62 of the teaching notes, the PDE does NOT have F1^2, F2^2 and F1 x F2 that multiply the second partial derivatives with respect to F1, F2 and F1, F2 respecively. The notes on the web have already been updated accordingly.
Another tip to solve Question1 in HW2: Notice that the stock S is a traded security. Hence, it must also satisfy 1.19. What is the "sigma_i2" in the case of stocks? Does dX2 affect stocks at all? Can we then express lambda1 as a function of the parameters of the stock?
Q: 1. For the Euler scheme, I understand that X(i)-X(i-1)~N(0,T/n). So, we can just generate a draw from normals and multiply it with square root of T/n and we will have it. We also need to generate S(t)=exp(X(t)). We only know that X(i)-X(i-1)~N(0,T/n), but know nothing about X(i)'s. Are the X(i)'s ~N(X(i-1),T/n), or do we use what we were using previously, as X(i)=sum over all j of Z(j), where j=1,...,i, and Z(j) is defined as before.
A: Please, read my Electronic Board. The Answer is there already. To say it again:
(1) Simulate a PATH of X(i)'s. Start at X(0)=0 and then use X(i)-X(i-1)~N(0,T/n) to simulate the others. That is, You can simulate a bunch of normal draws (n exactly)
Z(i)~N(0,T/n)
and then X(i)=sum(up to i) Z(j)
(2) S(i)=exp(X(i))
Q: 2. The Millstein scheme requires that we know the partial derivative of sigma(S-hat) with respect to S. Is there a closed form function of sigma(S)? How should I interpret this?
A: Well, this you should figure out. You are using
dS=1/2 S dt + S dX
What is sigma(S)? Hence, what is d sigma(S)/dS?
(Of course, in the teaching notes I meand d sigma(Shat)/d Shat. It is a typo)
Q1: To calculate Sthat for Figure 3, we need to know St, where St = exp(Xt) (TN1 (2.3)). If Xt ~ N(0,t) and n=5 t=10, does this imply that Xt~N(0,2)? Furthermore, if Xt~N(0,2) then is Xt=Zt x sqrt(2) where Zt ~ N(0,1). So, if my realization of Z is -.6, Xt = sqrt(2) x -.6 and St = exp(sqrt(2) x -.6). Or does Xt ~ N(0,t) imply that Xt = +sqrt(2) with prob =.5 and - sqrt(2) with prob = .5, so that if my realization from a random normal distribution is -.6, Xt = - sqrt(2) and St = exp(- sqrt(2)). If next period my realization of z = -.7 then is Xt = exp(- sqrt(2))* exp(- sqrt(2)). Hence X2 = exp(- sqrt(2))* exp(- sqrt(2)).
A: I am a little bit lost in this question. First, to compute Shat you DO NOT need St. The point of the exercise is exactly to see what is the difference between Shat and St.
So: you must do the following.
(1) Simulate one path for X(t). Suppose that T=10 and n=5, then
t=2,4,6,8,10. If T=10 and n=100, then t=.1,.2,.3....,9.9,10
(2) Each X(t)-X(t-1)~N(0,T/n)
(3) This last can be approximated Either with the Binomial (+/-
SqRoot(T/n) with Prob. 1/2) or with a nomal draw.
Now, GIVEN X(t)'s
(4) Compute S(t)=exp(X(t))
(5) Compute Shat(t) using the scheme 2.4.
Q2: TN1, page 20, does Euler assume u =1/2 or 0 and s = 1
A:
No: Euler assumes that X(i)-X(i-1) is Normally distributed: hence, the u
and d are not there anymore (those were for the binomial approximation)
Q3: How does TN1 2.2 compare to TN1 2.10. In particular is the
term
X(i) - X(i-1) the same for both equations, i.e., +sqrt(t/n) with prob =
1/2 and
-sqrt(t/n)
with prob = 1/2?
A:
The X(i)-X(i-1) are either discrete (2.2) or normally distributed (2.10)
Q: In TN1, expression 2.7: shouldn't it be p=(exp(mu delta) - d)/(u-d)? Also, the expression for u shouldn't it be exp(sigma sqrt(delta)), and not exp(sigma^2 sqrt(delta))?
A: YES: Thank you for catching the typos.
Q: In TN#1, I have a question about the Milstein Approx: doesn't the term {[X(i) - X(i-1)]^2-delta} = 0?
since X(i) - X(i-1) = +/- sqrt(t/n) and therefore [X(i) - X(i-1)]^2 = t/n BUT delta is also defined as t/n.
If so, the 2nd part of the Milstein scheme disappears, and it's the same as Euler.
A: You are right if you approximate the B< using a discrete scheme. If you use a normal distribution to approximate draws from a BM then the two schemes are different and the Milstein works better.
Q :
Is minitab applicable?
A :
I went to check minitab and it does not have a maximization routine. I.e.,
you cannot do Max. Likelihood. For the exercise in the first homework, it
is not a problem because you can use OLS. To estimate a GARCH model (that
is, a stochastic volatility model) you won't be able to use regressions.
You need to start looking at other things, such as S-plus, Matlab,
Mathematica etc. Unfortunately, none of these are in the computer lab.
Q : Professor Veronesi,
I am familiar with Gauss and, to a lesser extent S-Plus, but I am
considering buying Matlab since it can do everything in one package. I'm
checking out the add-on products; you mentioned the Optimization Toolbox
in
class this evening, and there is also a Financial Toolbox, a Partial
Differential Equation Toolbox, and a Statistics Toolbox, anf course many
others but these seem to be likely candidates for being useful in this
class. Which of these do you recommend?
A :
I can tell you what I use:
I use frequently the Maximization Toolbox and the Statistics Toolbox. I do
not have the PDE solver and do not have any idea whether it is good or
bad. It is not necessary to buy it for course for sure, given that some
techniques are in Wilmott book. On the other hand, if your work involve
obtaining frequently the solution to PDE, then it could be useful. The
Financial Toobox improved a lot: I never use it (but I have it) because I
like to program things, but if you are lazy, it could be very useful.
Basically, it contains a lot of standard stuff (like the financial
functions in Excel) plus binomial trees, BS option pricing and a little
more than that. If you instead do a lot of fixed income, the Spline
toolbox can become handy to interpolate the yield curve etc.
A couple of points.
(1) In Homework 1, solve the econometric exercise using only one method, of your choice. Remember though that even if for this exercise you will be able to use both OLS and MLE, for future exercises OLS may not be a possibility. You may want to start to look on ways to get MLE estimates.
(2) Some asked me about Matlab. There is a student edition which is very powerful (MATLAB is not restricted in any way) and costs $99.00. It cannot be bought through bookstores but it can be ordered from their OnLine Store. Although you won't need anything else for this course, you could also get other toolboxes for $59 each. These are quite usefull, as well. You can check the information on the website: http://www.mathworks.com/products/studentversion/index.shtml If you need to learn something, this is what I would learn. Of course, if you already know S-plus or VBA or anything else, you are welcome to use whatever you like. The coding using matlab is way easier than using VBA, so it would also be easier to perform the solutions of the various numerical procedures needed to solve our problems. (Examples of these codes are in Wilmott's CDrom as well). Hope you are going to have fun! Best, Pietro