6CCM388A Mathematical Finance in Discrete time:
Homework 6
Week starting 9th November, 2015
2. Introduction: This problem is on the Capital Asset Pricing method. We are in the
N period binomial asset pricing model, containing with one risky asset and an interest
rate r per period on cash. In this method, the investor gauges his satisfaction with his
investments depending on the result of a so-called “Utility Function” U : R → R∪{−∞}:
supposing that the investor’s portfolio is worth XN after N periods, the satisfaction of
the investor is given by U (XN ).
In order to give more economic meaning to the utility function, one assumes that the
utility increases with the wealth XN , but that the increments of the utility are diminishing as the wealth increases. One also assumes that U (x) = −∞ for all x < c for
some c ∈ R, to represent an unacceptable situation to the investor. To be precise, either
U (x) = −∞, or the first derivative U 0 (x) ≥ 0 and the second derivative U 00 (x) < 0.
The investor has an initial wealth X0 , and naturally wishes to find the optimal self(0)
(1)
(N −1)
? of
financing investment strategy (∆? , ∆? , . . . , ∆?
) such that the final wealth XN
the resulting self-financing portfolio maximizes the expected utility, that is
?
E[U (XN
)] ≥ E[U (XN )]
where XN is the final wealth of any other self financing portfolio (X0 , ∆(0) , . . . , ∆(N −1) )
with initial wealth X0 .
NB: The expectation is with respect to the true probability measure, not
the risk-neutral one.
In this question, we show how to use the Radon-Nickodym derivative of the real-world
probability measure P and the risk-neutral probability measure P̃ in order to construct
the optimal portfolio in the binomial asset pricing model.
(a) Let XN be any random variable depending on the first N coin tosses only and
satisfying
XN
Ẽ
= X0 .
(1 + r)N
1
NB: The expectation this time is with respect to the risk-neutral probability measure P̃.
Explain how to construct a self financing portfolio (X0 , ∆(0) , . . . , ∆(N −1) ) with
initial wealth X0 whose final wealth is exactly equal to XN , no matter what the
scenario is.
Hint: Consider the XN to be the pay off of a derivative, and replicate it using the
algorithm learnt in lectures.
(b) Thanks to question (a), the findingh the optimal
portfolio is equivalent to finding the
i
?
XN
?
? )] ≥ E[U (X )]
random variable XN satisfying Ẽ (1+r)N = X0 such that E[U (XN
N
i
h
XN
= X0 .
for any other random variable which also satisfies Ẽ (1+r)
N
Let Ω = {ω1 , . . . , ωM } be the possible scenarios in the sample space, and let XN be
a random variable. Then, denote pi = P(ωi ), xi = XN (ωi ), and zi = Z(ωi ) :=
P̃(ωi )
P(ωi ) .
Observe that the random variable Z is the Radon-Nickodym derivative of P̃ with
respect to P. For convenience, we denote also ζi := zi /(1 + r)N . Explain why
finding the optimal random variable is equivalent to the problem:
PM
P
maximize M
i=1 ζi pi xi =
i=1 pi U (xi ) with respect to (x1 , . . . , xM ) and subject to
X0 .
(c) For λ ∈ R and x = (x1 , . . . , xM ), the Lagrangian is defined by
!
M
M
X
X
L(x, λ) =
pi U (xi ) − λ
ζi pi xi − X0 .
i=1
i=1
Let I : R → R be the inverse function of the first derivative U 0 (x), i.e. I(U 0 (x)) =
x. For fixed λ, prove that the vector y = (y1 , . . . , yM ) with
yi = I(λζi )
is such that partial derivatives ∂xi L(x, λ)|xi =yi = 0 for every i. Prove that, for every
λ, this y is unique. Explain why L(y, λ) ≥ L(x, λ) for all possible x = (x1 , . . . , xM ).
Hint: Use the condition on the second derivative U 00 (x).
P
?
(d) Suppose we can compute λ? ∈ R such that M
0 . Explain why
i=1 ζi pi I(λ ζi ) = X
?
XN
?
?
the random variable is given by XN (ωi ) = I(λ ζi ) satisfies Ẽ[ (1+r)N ] and
?
E[U (XN
)] ≥ E[U (XN )]
XN
for any other random variable XN satisfying Ẽ[ (1+r)
N ].
(e) Let U (x) = log(x) for x > 0 and U (x) = −∞ for x ≤ 0. Compute λ? for this
?.
utility function, and use it to compute XN
2
(f) Suppose now we are in a two period model, with constant up and down factors
u = 2 = 1/d and interest rate r = 0.25. Let also S0 = 4. For any given coin
toss, the probability of H is p = 32 independently of any other toss. Using your
(0)
(N −1)
algorithm from Part (a), compute the optimal trading strategy (∆? , . . . , ∆?
for the utility function in question (e). Leave the answer in terms of X0 .
3
)