Problem Set III: Binomial Asset Pricing Model

ECE478 Financial Signal Processing
Problem Set III: Binomial Asset Pricing Model

Write code to simulate the BAPM and perform analysis as suggested. Notation and so forth
follows as given in Shreve, Stochastic Calculus for Finance vol. 1. Your simulation should
take r; d; u as given, and constant at all time steps. Building in error detection to ensure
these satisfy the no-arbitrage constraint is optional. Might as well take S0 = 1. We will
assume the coin toss distribution is constant over time, but the particular (p; q) values can
be variable. Two special cases are the ìactualî probabilities (p0; q0), and the risk-neutral
measure (~p; q~). Let Fn be the -algebra generated by the Örst n tosses, which covers the
time span f0  k  ng. Let ~!n = (!1


!n) 2 fH; Tg
n
denote a particular ëpathíthrough
the Örst n tosses. Assume every stochastic process here, say Xn is adapted to this Öltration.
Your model should cover the time steps f0  n  Ng. When you perform a Monte Carlo
simulation, M is the number of samples used (i.e., for purposes of Monte Carlo, you repeat
the experiment M times and average over those results).
There are two types of derivatives: one type has a payout VN = VN (SN ) that depends only
on the Önal value of the security; this is called ëpath independentí. Another type can depend
on the values of the security over the whole time span, i.e., VN = VN (S0; S1;


; SN ). Here
we will only build out simulation for the case of path-independent payouts, so feel free to
write your code assuming that condition holds. [A more complete BAPM simulation would
have to be able to handle the more general case.]
Finally, in what follows, for any stochastic process Xn, the discounted process is X~
n =
1
(1+r)
n Xn.
1. Distributions: Before you start coding: Assuming the coin-toss distribution (p; q),
Önd the distribution of Sn. That is, list all the possible values (there are n+1 of them),
and their probabilities. Also, if Rn = log (Sn=S0), show that Rn = cnYn + dn where
Yn binomial (n; p) (Önd the constants cn; dn). Note: In the continuous-time case,
something similar will occur: we will get Rt = log (St=S0)  Gaussian (what else?),
and hence the distribution of St will be what is called lognormal.
2. Exact simulation: Assume we have code to compute the payout function V (SN ) for
a path-independent derivative.
(a) Write code to compute V0 = Ep~

V~N

using the known distribution for SN .
(b) Write code to generate one step of the replicating portfolio. If at time n there
are
n shares of the stock, and wealth Xn, then the amount Mn = Xn  
nSn is
held in the money market. From n ! n + 1, the stock price changes from Sn to
Sn+1, and the amount in the money market grows to (1 + r) Mn. Thus, at time
n + 1, the wealth equation states:
Xn+1 =
nSn+1 + (1 + r) (Xn  
nSn)