HOMEWORK 12 MATH 7349
1. Let St be a stock satisfying
dSt = R(t, St)Stdt + σ(t, St)StdB˜t
for some interest rate process R(t, St) and voltaility σ(t, St). Consider a situation where an investor
must pay a (random) amount at a rate C(t, St) to an outside source. Specifically, if the investor
holds a single-stock portfolio Xt with ∆t units of a stock St, then the portfolio satisfies
dXt = ∆tdSt +Rt(Xt −∆tSt)dt −C(t, St)dt.● Let Dt be the discount process given by d(Dt) = −R(t, St)dt, and fix T > 0. Write
D(T )X(T ) as the sum of a Reimann and an Ito integral.● Prove that there exists a choice of ∆t and a nonrandom value of X0 such that X(T ) = 0
with probability 1.
Hint: Let M˜t = E˜[∫ T0 DsC(s,Bs)ds ∣Ft]. Use the Martingale Representation to write∫ T0 DsC(s,Bs)ds as an expectation of a Riemann integral plus an Ito integral with respect
to B˜t. Make a judicious choice for ∆t.
2. Let
v∗L(x) = ⎧⎪⎪⎨⎪⎪⎩(K − x) 0 ≤ x ≤ L(K −L)(x/L)−2r/σ2 x > L.
Show that this function is differentiable for all x > 0 if and only if L = L∗ = 2r2r+σ2K.
3. Let v1(x) and v2(x) be the prices of a perpetual American puts with strike prices K1 an K2,
respectively, with K1 v2(x) ≥ (K1 − x)+ and − rv2(x) + rxv′2(x) + σ2x22 v′′2 (x) ≤ 0.
Show that there exist values of x such that neither inequality is tight.
4. We will show that the only continuous, differentiable solution to the system of inequalities
v(x) ≥ (K1 − x)+ and − rv(x) + rxv′(x) + σ2x2
2
v′′(x) ≤ 0
such that at least one of the two inequalities is tight is v∗L∗(x).● Show that any solution f(x) on an interval [x1, x2] to
− rf(x) + rxf ′(x) + σ2x2
2
f ′′(x) = 0 (1)
must be of the form f(x) = Ax +Bx−2r/σ2 for some pair of constants A and B.● Assume that f(x) is continuous and has a continuous derivative. Let x1 < x2, and assume
that f(x) satisfies (1) on [x1, x2]. Furthermore, assume that, for x ∈ (x1 − ε, x1) and
x ∈ (x2, x2 + ε), f(x) = (K − x)+ for some K. Then f(x) = 0 on [x1, x2].● Assume that f(x) is continuous and has a continuous derivative. Show that, if f(0) = K,
then f(x) cannot satisfy (1) on [0, x2] for any x2.
1
2 HOMEWORK 12 MATH 7349
● Now assume that v is continuous, differentiable, with v(0) =K and
v(x) ≥ (K1 − x)+ and − rv(x) + rxv′(x) + σ2x2
2
v′′(x) ≤ 0
such that at least one of the two inequalities is tight. Show that v(x) = (K − x)+ on [0, x1]
for some x1, and then solve (1) for [x2,∞). Prove that v(x) = v∗L∗(x).

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