MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 1, 2020
Tutorial sheet 2
Exercise 1 Consider the portfolios composed of shares and options:
(a) one share of stock and put option with strike K > 0,
(b) short position in one share of stock and call option with strike K > 0,
(c) put with strike K1 and short call with strike K2 where 0 < K1 < K2,
(d) put with strike K2 and call with strike K1 where 0 < K1 < K2.
(e) call with strike K1 and short call with strike K2 where 0 < K1 < K2.
All options are of European style and have the same maturity date T . Sketch
the payoff profile at time T of each portfolio. Can you formulate a conjecture
about a level of the initial value of each portfolio?
Exercise 2 Assume that you hold the following portfolio: one share, a short
call with strike K1 and a call with strike K2. Assume that K1 < K2 and
denote by π0 = S0 − C0(K1) + C0(K2) the price you paid for this portfolio’s
at time 0. Assume that the interest rate equals zero.
(a) Sketch the graph of your profit and loss at time T as a function of ST .
(b) Find the maximum profit, the maximum loss and the break even point(s).
(c) Determine any necessary condition(s) for this portfolio to have the proper-
ty that both profits and losses may occur when ST ranges from 0 to infinity.
(d) What is the market view of an investor holding this portfolio?
(e) Give answers to questions (a)-(d) when a short call with strike K1 is
replaced by two short calls with strike K1.
Exercise 3 Suppose that you hold the following portfolio: short call with
strike K1 and two short calls with strike K2 with K1 < K2. Denote by
π0 = C0(K1) + 2C0(K2) > 0 the price you received at time 0 and assume
that the simple interest rate between 0 and T equals r > 0.
(a) How many shares with initial price S0 would you need to buy/short at
time 0 to augment your portfolio if you expect that the market value at time
T of one share will be in the interval (K1, K2)?
(b) Compute the profit and loss at time T of your augmented portfolio as a
function of the stock price ST . Find the maximum profit, the maximum loss
and the break even point(s) of your portfolio.
(c) Assuming that C0(K1) and C0(K2) are given, derive a condition on the
stock price S0 for which you would make a profit with such a portfolio when
ST ∈ (K1, K2).
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Exercise 4 Let 0 < L < K be real numbers. Consider the contingent claim with the payoff X at maturity date T > 0 given as
X = min
(
|ST −K|, L
)
, ∀ST ∈ [0,+∞).
Notice that we interpret here ST as a real variable, as opposed to the random
variable ST (ω).
(a) Sketch the profile of the payoff X as a function of the stock price ST at
maturity date T and find the decomposition of the payoff X in terms of the
payoffs of standard call and put options with different strikes and expiration
date T , the payoff of long/short positions on the stock and a constant payoff
of maturity T .
(b) Using the law of one price, deduce from part (a) the decomposition of
the price of the payoff X at time t ∈ [0, T ] in terms of the prices at time t of
call and put options, the price at time t of the stock and the price at time t
of the zero-coupon bond with maturity T .
(c) Give answers to questions (a) and (b) for the payoff Y = max(|ST−K|, L).
Exercise 5 (MATH3975) Let g : R+ → R be an arbitrary continuous and
piecewise linear function such that g(0) = c ∈ R and g is linear on each
interval [Ki, Ki+1] for i = 0, 1, . . . , n − 1 where K0 = 0 < K1 < K2 < · · · <
Kn−1 < Kn = +∞ where n ≥ 2 is any (fixed) natural number. Denote by αi
the derivative of g on the interval (Ki, Ki+1) for i = 0, 1, . . . , n− 1.
(a) Show that the payoff X = g(ST ) can be represented by the terminal
payoff of a static portfolio composed of long/short positions in a finite family
of call and put options written on the stock S, with the expiration date T
and strikes K1, K2, . . . , Kn−1. You may also include long/short positions in
S (equivalently, call and put options with strike zero) and a constant payoff
at time T . Notice that several alternative solutions may exist.
(b) Using your solution from part (a), express the price at time t ∈ [0, T ] of
the payoff X = g(ST ) which settles at time T .
(c) What is the price of X = g(ST ) at time t ∈ [0, U ] if the payoff X is
determined at time T but it settled at time U (that is, the actual transfer of
the cash amount X between the counterparties occurs at time U).
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