MATH 459
Test 2
Apr. 12, 2019
NAME (please print legibly):
Your University ID Number:
• No books, collaboration or access to outside material is allowed, with two
exceptions. Each student may bring one sheet of notes. Each student may
bring one or more TI calculators BA-35, BA II Plus, BA II Plus Profes-
sional, TI-30Xa, TI-30X II (IIS solar or IIB battery), TI-30XS MultiView
(or XB battery), allowed for use on the SOA FM exam, or TI-83 Plus and
the TI-84 Plus family, allowed for use on the PSAT, the SAT Subject Tests,
Math Level 1 and 2 Tests, AP Calculus exam and ACT Test.
• Please show all your work. You may use back pages if necessary. You may
not receive full credit for a correct answer if there is no work shown.
The Black-Scholes formulas are
Ceuro(S, t) = SN(d1)−Ke−r(T−t)N(d2), P euro(S, t) = Ke−r(T−t)N(−d2)− SN(−d1),where
d1 =
lnS + (r + 1
2
σ2)(T − t)− lnK
σ
√
T − t , d2 =
lnS + (r − 1
2
σ2)(T − t)− lnK
σ
√
T − t
N(z) =
1√
2pi
∫ z
−∞
e−
x2
2 dx
QUESTION VALUE SCORE
1 10
2 10
3 10
4 10
TOTAL 40
1
1. Ms. S owns a portfolio containing some special securities that she believes will perform
well compared to the market. However, Ms. S realizes that her portfolio has a high beta
relative to the market and so is exposed to a significant degree of market risk.
Let P represent the value of Ms. S’s portfolio. Then P (T ) = P0(1 + rP )
where P0 is present value and rP fractional return for the period from now to time T .
Let M be the market as represented by the S&P index. Then M(T ) = M0(1 + rM) where
M0 is present value and rM fractional return for the period from now to time T .
Ms. S constructs a hedged portfolio by writing h futures contracts based on the S&P index.
At time T the new portfolio has value y(h) = P (T )− h(M(T )−M0).
a) Find var(y(h)) as a quadratic in h with coefficients involving var(P (T )), var(M(T )), and
cov(P (T ),M(T )). Simplify, removing any terms that can be shown to be zero.
b) Find var(y(h)) as a quadratic in h with coefficients involving var(rP ), var(rM), cov(rP , rM).
c) Find the value h = hmin that minimizes the expression for var(y(h)) given in b).
Rewrite the formula for hmin in terms of the beta of the original portfolio βP =
cov(rP ,rM )
var(rM )
.
d) Write out a formula for the beta of the hedged portfolio after writing h = hmin futures.
a) var(y(h)) = var(P (T )− hM(T ) + hM0) = var(P (T )− hM(T ))
= cov(P (T )− hM(T ), P (T )− hM(T )) = var(P (T ))− 2hcov(P (T ),M(T )) + h2var(M(T ))
b) var(y(h)) = var(P0(1 + rP ))− 2hcov(P0(1 + rP ),M0(1 + rM)) + h2var(M0(1 + rM)))
= P 20 var((1 + rP ))− 2hP0M0cov(1 + rP , 1 + rM) + h2M20var(1 + rM))
= P 20 var(rP )− 2hP0M0cov(rP , rM) + h2M20var(rM))
c) ∂
∂h
var(y(h)) = −2P0M0cov(rP , rM) + 2hM20var(rM))
hmin =
P0M0cov(rP ,rM )
M20 var(rM ))
= P0cov(rP ,rM )
M0var(rM ))
= P0
M0
βP
d) A formula for the beta of y(hmin) is βy =
cov(ry ,rM )
var(rM )
where ry, the return rate for the hedged
portfolio, is such that P0(1 + ry) = y(hmin), that is ry =
y(hmin)
P0
− 1.
(It turns out that cov(y(hmin), rM) = 0 so cov(ry, rM) = 0 and βy = 0)
1
2. An asset has value S0 = $40 at t = 0, and will have unknown value S(T ) at time T . A
put option with strike K = $50 is currently (t = 0) selling for P0 = $9, and the discount
factor for the period 0 to T is e−rT = 0.98 = 98/100. The risk-free rate is r.
a) Consider the portfolio constructed at t = 0 by borrowing Ke−rT from an ideal bank,
buying one unit of the asset, and buying one put option.
Show there is an arbitrage opportunity. Give all actions at t = T .
b) If the put option price is instead P0 = $8, does the arbitrage opportunity still exist?
c) If the put option price is instead P0 = $10, does the arbitrage opportunity still exist?
Solution:
a) The strike K is cash that may need to be spent at t = T , so “the present value of the
strike” is Ke−rT = 50(98/100) = 49, the strike discounted to present.
a0) At t = 0, the cashflow is +Ke
−rT (borrow present value of strike from bank), −S0 (spent
to buy the asset), and −P0 (spent to buy put option).
+Ke−rT − S0 − P0 = 49− 40− 9 = 0, so at t = 0, net cash flow is 0.
aT ) At t = T , if S(T ) ≥ K, sell the asset on the spot market for S(T ) and return amount
(Ke−rT )/e−rT = +K owed to the bank: Net cash flow is S(T )−K ≥ 0.
At t = T , if S(T ) < K, exercise the put option and sell the asset for K. Return the amount
K to the bank. Net cash flow K −K = 0.
Since there is no outlay of cash at t = 0 and at t = T the cash flow is ≥ 0, and there is a
positive probability of a positive payoff, this is an arbitrage opportunity (“type B”)
b) If the option price is P0 = 8, then the cash flow at t = 0 becomes 49− 40− 8 = 1. Since
there is an immediate payoff at t = 0 and at t = T the cash flow is ≥ 0, this is an arbitrage
opportunity (“type A”)
c) If the option price is P0 = 10, then the cash flow at t = 0 becomes 49 − 40 − 10 = −1.
There is an expenditure at t = 0 and no guaranteed income at t = T , so this is not an
arbitrage opportunity.
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F
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