MATH 459
Test 2
Apr. 13, 2018
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 in a calculator such as the 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
5 10
6 10
TOTAL 60
1
1. (10 points) An asset has value S0 = $60 at t = 0, and will have unknown value S(T )
at time T . A call option with strike K = $50 is currently (t = 0) selling for C0 = $11, 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 obtained by selling (short) one unit of the asset at t = 0, buying
one call option, and investing the present value of the strike K in a bank at the risk-free
rate. Show there is an arbitrage opportunity. Give all actions at t = T .
b) If the call option price is instead C0 = $10, does the arbitrage opportunity still exist?
c) If the call option price is instead C0 = $12, 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 +S0 (income from shorting the asset), −C0 (buy call option),
and −Ke−rT (deposit present value of strike in bank). The total cash flow at t = 0 is
S0 − C0 −Ke−rT = 60− 11− 50(98/100) = 49− 49 = 0.
aT ) At t = T , withdraw (Ke
−rT )/e−rT = +K = 50 from bank, reacquire and return asset.
If S(T ) ≥ K, reacquire the asset by exercising the call option, spending K:
at time T the cash flow is +K −K = +50− 50 = 0.
If S(T ) < K, reacquire the asset by purchasing on the spot market, spending S(T ):
at time T the cash flow is +K − S(T ) = +50− S(T ) > 0.
The cash flow at t = T is ≥ 0, and there is a positive probability of a positive payoff.
Since there is no outlay of cash at t = 0 and a positive probability of income at t = T , there
is an arbitrage opportunity (“type B”)
b) If the call option price is C0 = 10, then the cash flow at t = 0 becomes 60− 10− 49 = 1.
There is an immediate payoff at t = 0 and a positive probability of income at t = T , so there
is an arbitrage opportunity (“type A”)
c) If the call option price is C0 = 12, then the cash flow at t = 0 becomes 60− 12− 49 = −1.
There is an expenditure at t = 0 and no guaranteed income at t = T , so there is not an
arbitrage opportunity.
1

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