分享去年宾州大学投资理论考题

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Exercise 1.

An investment project gives access to one and only one of two dividends P
and Q where P and Q follow two Geometric Brownian Motions
dP = αPPdt + σPPdzP
dQ = αQQdt + σQQdzQ
and γdt is the expected value of dzP dzQ. For an active project it is possible
to switch between the two dividends: The cost of switching from P to Q is
cQ where c > 0 and the cost of switching from Q to P is cP. Let ρ > 0 be
the discount rate and suppose that δP = ρ − αP , δQ = ρ − αQ > 0 and that
δP − δQ, δQ − δP < ρ.
Let V (P,Q) denote the value of an active project when receiving P and
let W(P,Q) denote the value of an active project when receiving Q.

1.1 Give an example of a project which fits with the above project.

1.2 Discuss the optimal strategy for an active investment project. Use your
discussion to state how V (P,Q) and W(P,Q) are related.

1.3 Find the Bellman equations for V (P,Q) and W(P,Q).

1.4 Find dV (P,Q) and dW(P,Q) and find (partial) differential equations
that determine V (P,Q) and W(P,Q).

1.5 Suppose that V (P,Q) = Pv(Q/P) and that W(P,Q) = Qw(P/Q),
rewrite dV (P,Q) and dW(P,Q) and rewrite the differential equations
that determine V (P,Q) and W(P,Q).

1.6 Find V (P,Q) and W(P,Q).

1.7 Give a short interpretation of your expressions for V (P,Q) andW(P,Q).

1.8 Find undetermined constants in you expressions for V (P,Q) andW(P,Q)
and find equations to determine the constants.

1.9 Determine the undetermined constants in your expressions for V (P,Q)
and W(P,Q) for some set of parameters where σP , σQ > 0.

Exercise 2.

Consider a market P(t) = 2/􀀰Y (t), where P(t) is the price and Y (t) is
the supply. There is a monopoly on the market. The monopoly has an old
technology with the cost function (1/C)Y (t).
The monopoly may switch to a new technology (1/D(t))Y (t) where D(t)
follows a Geometric Brownian Motion
dD = αDdt + σDdz
with σ > 0. The cost of changing technology is I > 0. The monopoly may
use one and only one technology and it is not possible to switch back from
the new technology to the old technology.
The interest rate is r. There is an asset in the economy. The price of the
asset is X(t) where X(t) follows a Geometric Brownian Motion
dX = (α + δ)Xdt + σXdz
with δ > 0.
Let V (D) denote the value of the new technology and let F(D) denote
the value of the option to invest in the new technology.

2.1 Give an example of investment projects which fits with the above
project.

2.2 Find the value of the old technology.

2.3 Discuss the optimal strategy for an active investment project.

2.4 Find the value of the new technology and give a short interpretation of
your expression.

2.5 Find the value of the option to invest and give a short interpretation
of your expression.

2.6 Find undetermined constants in your expressions for V (D) and F(D),
find equations to determine the constants and determine.

2.7 Find the optimal strategy.

2.8 Find the total surplus (total surplus = profit and consumer surplus) of
the old technology. Find the total surplus of the new technology (let
W(D) denote the total surplus). Find the total surplus of the option
to invest in the new technology (let G(D) denote the total surplus of
the option).

2.9 Find the strategy that maximizes the total surplus and compare it with
the optimal strategy for the monopoly.

[此贴子已经被作者于2006-12-19 1:36:31编辑过]

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