STOCHASTIC CALCULUS 627 HW 6 PROBLEMS (PRELIMINARY) –
FALL 2024
JOHN C. MILLER
1. Exercise 5.5 in Shreve II
Please do Exercise 5.5 in Shreve II (i.e. prove Corollary 5.3.2).
2. Find α and β for the transformation of the B-S-M PDE to the heat
equation
Show why α was chosen to be −12(k − 1) and β was chosen to be −14(k + 1)2.
3. Example of risk neutral and PDE valuation
Suppose r is the risk free rate and suppose S(t) follows geometric Brownian motion with
constant volatility σ ̸= 0. Let
V (T ) =
{
S(T ) if S(T ) ≥ K,
0 if S(T ) < K.
(i) Find V (0) using risk-neutral valuation. V (0) should be a function of S(0), K, T , r, σ,
and the standard normal cumulative distribution function N(·).
(ii) Also find V (0) using PDE valuation (following example given in class).
4. American digital call
Suppose r is the risk free rate and suppose S(t) follows geometric Brownian motion with
constant volatility σ ̸= 0. Let
V (T ) =
{
1 if S(t) ≥ B for some 0 ≤ t ≤ T,
0 otherwise.
Find V (0) using the reflection principles for solutions of the Black-Scholes-Merton PDE. V (0)
should be a function of S(0), B, T , r, σ, and the standard normal cumulative distribution
function N(·).
5. Expectation of exit time
For constants a, b > 0, let τ be the first time that Brownian motion W (t) exits the interval
[−a, b] (see example from class). What is E[τ ]?
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2 JOHN C. MILLER
6. American exercise put
In the notes, we did an example of a three-period model with u = 2, d = 1/2, r = 1/4 and
S0 = 4, with a payout of a put struck at $5. Using the same parameters, calculate the value
of a put with strike $5, using a four period model.
7. Arbitrage
Consider the multidimensional market model with one Brownian motion (d = 1) and two
assets (m = 2):
dS1 = α1S1 dt+ σ1S1 dW,
dS2 = α2S2 dt+ σ2S2 dW
with constant risk free rate r.
(i) What are the market price of risk equations for this market model?
(ii) What are the conditions (expressed in terms of α1, α2, σ1, σ2, r) for a risk neutral
measure to exist?
(iii) If those conditions do not hold, please describe an arbitrage strategy.
8. Induction proof
Please complete the induction proof (see Tuesday’s lecture) to show that
vT−k(x) = log
(x
k
)k
(1 + ρ)k(k−1)/2.
Email address: john.miller@jhu.edu