Description
Notation here follows Shreve, Stochastic Calculus for Finance, vols. I & II, Springer, 2004.
Theoretical Problems
1. Let Ft be the Öltration generated by a Wiener process W (t). Let R (t) be the interest
rate process used to deÖne the discount process D (t). Assume there exists a unique
risk-neutral measure, leading to the Wiener process W~ (t) with respect to P~. If V (T)
is a random variable that is FT -measurable, and V (t) is deÖned via:
V (t) = 1
D (t)
E~ (D (T) V (T)jFt)
then D (t) V (t) is a martingale.
(a) Suppose V (T) > 0 a.s. Show that V (t) > 0 a.s. (from its deÖnition above).
(b) Show that there exists an adapted process ( ~ t) such that:
dV (t) = R (t) V (t) dt +
( ~ t)
D (t)
dW~ (t)
Hint: Start with a formula for d (D (t) V (t)) as per the martingale representation
theorem, then as you expand this out recognize that dV dt = 0.
(c) Show that there exists an adapted process (t) such that we can write:
dV (t) = R (t) V (t) dt + (t) V (t) dW~ (t)
By the way, (t) can be random and in particular it is Öne if the formula for
(t) you derived involves V (t). The point is there is SOME process you can
put there that works! How did we use strict positivity? (Think of D (t), V (t)
as continuous processes; D (t) is intrinsically positive, but what happens if V (t)
can take on negative as well as positive values?) This shows that V (t) is a
generalized geometric Brownian motion process. The point of this problem is
that every strictly positive asset is a generalized geometric Brownian motion.
2. Let X (t); Y (t) be ItÙ processes given by:
dX (t) = a (t) dt + b (t) dW (t)
dY (t) = c (t) dt + d (t) dW (t)
where a; b; c; d are adapted processes. Assume Y (t) > 0 a.s., and V (t) = X (t) =Y (t).
Obtain an SDE satisÖed by V (t), simpliÖed so it has the above form (i.e., in the form
of an ItÙ process). Note that X (t); Y (t), but not dX (t) or dY (t), can appear in your
Önal expression for dV (t).
Simulating Stochastic Di§erential Equations
We are going to be viewing continuous-time deterministic functions and stochastic processes
in discretized time. For simplicity, we will take equally spaced time intervals, say t = n,
0 n N, with Önal time T = N. (Careful, including t = 0, this is N + 1 points). We
will use the notation:
x [n] = x (n) = xn
Consider a stochastic di§erential equation of the form:
dX (t) = (t; X (t)) dt + (t; X (t)) dW (t)
where (); () are deterministic functions, and as usual W (t) is a Wiener process. The
parameter controls the variance of the increment dW; speciÖcally, in the discretization, let
us use the notation:
dW [n] = W [n + 1] W [n]
where fdW [n]g are iid N (0; ). The SDE actually means:
X (t) = X (0) + Z t
0
(u; X (u)) du +
Z t
0
(u; X (u)) dW (u)
In discretized form, with t = n this becomes:
X [n] = X [0] + Xn1
m=0
(m; X [m]) +
Xn1
m=0
(m; X [m]) dW [m]
Letís default with = 0:01 and N = 250 (which is approximately the number of days per
year for a typical Önancial instrument).
1. Start with basic geometric Brownian motion:
dSt = Stdt + StdWt
with S (0) = 1, = 0:1, = 0:2. Also assume a constant underlying interest rate
r = 0:05. Under the risk-neutral measure, dSt satisÖes a modiÖed SDE involving dt
and dW~
t
. See Problem 3 below: If you ever detect St 0 trap that condition,
note that it occurred, and discard that path (donít use it).
(a) Write the modiÖed SDE involving dW~
t
. SpeciÖcally, calculate the coe¢ cients that
appear in this SDE from the speciÖc values of ; ; r provided. Unless speciÖed
otherwise, however, grow your paths using the ORIGINAL formulation (i.e., the
ACTUAL probabilities).
(b) Generate 1000 paths of S [n].
(c) Use a Monte Carlo approach to estimate E (S [N=2]) and E (S [N]) directly.
(d) We now want to connect this to the Black-Scholes model. Let V (T) = (S (T) K)
+
,
the payout of a European call option with strike price K. In our discrete notation,
V [N] = (S [N] K)
+
. Use your estimate for E (S [N]) as your value for K. Use
the Black-Scholes model to obtain a formula for V [N=2] in terms of S [N=2], and
graph it (for the speciÖed parameters ; ; r
(e) Now take the Örst 10 paths you generated, S
(i)
[n], 1 i 10. For each S
(i)
[N=2],
you can compute V
(i)
[N=2] from the Black-Scholes formula. On the other hand,
you can use a Monte Carlo approach to compute V
(i)
[N=2] using the martingale
property of the discounted stock price. So, for each i, grow 1000 paths from N=2
to N and average to estimate V
(i)
[N=2]. Compare these estimated values with the
exact values in these cases. Report the results how you see Öt: a table; you could
superimpose two scatter plots (S
(i)
[N=2] versus actual V
(i)
[N=2], and S
(i)
[N=2]
versus actual V
(i)
[N=2]). Comments: Careful. First, you need to grow NEW
paths, emanating from time t = T=2, towards t = T, taking S
(i)
[N=2] as an initial
condition. Also, you are computing V
(i)
[N=2] as an expectation by virtue of the
martingale property, BUT the question is WHICH SDE FOR St DO YOU
USE?
2. Cox-Ingersoll-Ross Interest Rate Model
dR (t) = (a R (t)) dt +
p
R (t)dW (t)
with ; ; positive, and let R (0) = r > 0. This model for interest rate R (t) guarantees R (t) > 0. This model is typically used for short term interest rates, which tend to
exhibit volatility. Although no closed form solution exists, it is possible to determine
certain properties for it. In particular the mean and variance of R (t) are given by:
E (R (t)) = e
tr +
1 e
t
!
var (R (t)) =
2
r
e
t e
2 t
+
2
2
2
1 2e
t e
2 t
!
2
2
2
Discretizing this SDE for simulation purposes is very tricky because the property that
R (t) > 0 does not carry over if you use the simple discretization as in the previous problem! This actually brings up a larger questionñ does the discretized system
accurately reáect properties of the original SDE?
(a) Select: = 1, = 0:10 , r = 0:05, = 0:5. Generate 1000 paths for R (t) over
a span 0 t 10, using = 0:01. In your code, trap the condition that R 0
(if this occurs, your code should display an exception, should halt that one path,
but should continue computing the other paths). So the goal is to generate 1000
valid paths (with R (t) never reaching 0). How many ìbadî paths do you get
in order to reach 1000?
(b) Graph the Örst 10 paths for R (t), just to see what it looks like.
(c) Use a Monte Carlo approach to estimate the mean and variance of R (t) at t = 1
and t = 10, and compare with the exact formulas given above. Here you should
use the valid paths only.
3. Go back to Problem 1 again. We know the actual St (in continuous-time) is geometric
Brownian motion so we necessarily have St > 0 a.s., and yet is that property guaranteed in your simulation? On the other hand, with all your simulations, did you ever
encounter simulated St 0? I expect not. Why not? [Even if you got a negative
answer, why did I guess that you wouldnít? The hint is that it is a guess
