MATI{3 OT 5 /39 7 5 Financial Derivatives
Tutorial 2: Solutions
Exercise 1 It suffices to sketch the graphs of functions representing the portfolio payoff as a
firnction of the terminal price
^97,
which rnay take any noruregative value.
(u) & : f{Sr) : Sr * (If – Sr)+ (the initial value should be strictly positive)
67
(b) & : fz(Sr) – -Sr + (Sr – K)+ (the initial value should be strictly negative)
/
//
crtK) //
/
/
/
//
_sr
-l
(“) xr :
“fe(Sr)
: (Kr – .9r)+ * (Sr – Xr)+ (the sign of the initiat value is undetermined)
(‘l
\ ?r(Kr)
sr
\\
-CT(KIN
(K, – Sy)* + (Sr – X)+ (the initiat value should be strictly positive)
){
-l
\
(d) xa: fd(sr)
Kz
I
/
I
\\
5r
11 | V2
(u) xs :
“fs(,sr)
: (,sr – xi+ – (sr – xz)+ (the initiar varue shourd be strictly positive)
Xs
o
cr(Kr)
,/
– -l-
\
\
\
V1
cr(Kzl
sr
where 0 I Kt I Kz.Hence the payoff X at time ? is strictly positive for every value of ,Sr > 0.
Therefore, the price zro : ,S0 – Co(Ki + Co(K2) paid for the portfolio at time 0 should be strictly
positive (but less than ,96 since X (
^97
for ,Sr > 0) and the profits/losses at time ? satisfy
PkL – Sr – (Sr – K1)* + (Sr – K2)* – ro.
Exercise 2 (u) We note that the portfolio’s payoff at tim e T equals
X – Sy – Cr(Kt) + Cr(Kz) – Sy – (Sr – K1)* + (Sr – Nz)+
Sr
4 P&L
(1)
(2)
//
I
I
,” c1 Gz)
\
Kr \r
\
Ef
q’ ?t(4)
\
K2- Tio
(b) the maximum loss occurs when ,97 :0 and equals -216. The maximum profit equals *oo when
sr -+ *oo, that is, the profit is unlimited. The unique break even point is sT : ro.
(c) We assume that
^9a may take any nonnegative value. Then profits and losses occur at time
7 whenever the following inequality holds z16 – ,s0 – co(K) + co(K2) > 0, that is, if ,s6 )
co(K) – co(Kz).since 1(1 I K2, we also expect that cs(K1) > co(Kz) since c7(.I(1) > c”(x”)
and C7(K1) > Cr(Kz) for ,97 ) K1. Hence, assuming that Ss is known, we obtain the following
conditions 0 < Co(,I(l) - Co(K2) < So.
(d) The view of the investor is bullish.
(e) One can perform an analysis similar to (a)-(d), by noting that we now have
_ xz)+X - 57 - 2Cr(Kr) + C7(K") - Sr - 2(Sr - K1)* + (Sr (3)
Exercise 3 We consider the portfolio with the payoff
x : -cr(Ki - 2cr(Kz) : -(sr - xi+ - 2(sr - xz)+ (4)
with the strictly positive price z16 : Co(K) + 2Co(K2) ) 0 received at time 0, which becomes
(1 + r)zre at time ?. Hence the profits/losses at time ? are given by
PkL: (1+r)ns -Cr(K)-2C'r(Kz): (1+ r)no- (Sr-l(i)+ -2(Sr- Xz\+. (b)
\
K:_\
buy shares and a < 0 means that we sell short shares). Then the initial and terminal values of the
combined portfolio become -a,So*Co(Kt)+2Co(Kz) and cS7 -Cr(Ki-2C,r,(K2), respectively.
If an investor wishes to make profits when ,97 is in the interval (Kr, Kz) then he/she should take
a : \, that is, buy one share of the stock ,S.
sr
( trr) (f;o-s,)
\
\
'Cr(t ,)tt
(b) If we buy one share of the stock ^9 at time
Kr * (1 + r)ro - (1 + r)So when ,S7 € (Kr, Kz).
Break even points are Sr: (1 + r)(So - ?i-0) and
(.) Since the maximum profit equals Kr + (1 +
^96 lno+(1 +r)-tKr.
0, then the maximum profit at time T \^rilt be
The loss in unlimited when ,S7 tends to infinity.
Sr :0.5(/(1 + 2Kz + (1 + ,)(no - ^gs))
r)no - (1 + r),Ss, it is strictly positive whenever
Exercise 4 Recall that 0
the paycffs X : rnirr (i,5'r - Ki, L) and Y - max (lSt - Kl,L) as
It is easy to sketch the graph of
a function af the stock price Sr,
/
X
ST
plt L 11+ L
(a) The decomposition of X in terms of long/short positions in standard call options combined
with a constant payoff.L is
X : L - Cr(K - L) +2Cr6) - C7(K + L).
Notice that other decompositions of the payoff X are possible (for instance, if we include in our
portfolio long/short positions in the put option Pr(K) and the stock S7).
(b) From the law of one price, we deduce that the price of X at time t e [0, ?] satisfies
nr(X) : LB(t,T) - C^K - I,) + 2C^K) - C{K + L).
(c) The method is analogous to the case of the payoff X. The decomposition of Y in terms of
iong/short positions in standard call and put options reads
Y : Pr(K) + Cr(K - L) - Cr6) + C7(K + L)
so that the price of Y at time t e [0,
"l
satisfies
"r(Y)
: Pt(K) + C^K * L) - Ct(K) + C{K + L).
Exercise 5 (a) It suflflces to generalise the approach used in Exercise 4 by first mimicking the
payoff function on the interval [Ko, Kr]: [0,1(r]. Then the 'initial portfoiio' should be adjusted by
a suitable number of long/short positions of call options with strikes K1,K2,...,Kn-r to match
the slope of the payoffon each interval [Kt,,,K+l for i: I,2,...)TL- 1. As soon as the portfolio
in the first step is chosen, the solution to the second step (that is, when we consider the interval
[l(r, *)) is unique.
(b) It suffices to apply the law of one price to your solution to part (a).
(c) Since the payoff X is already known at time ?, its price at time t e lT,Ul equals a(X) :
B(t,U)g(97). Hence before time 7, the price satisfies, for every t e [0,7],
nt(x) : nr(B (7, u) g (Sr)) : B (7, U)rv(s (Sy))
where the second equality holds oniy under the assumption that B(T,U) is deterministic (for
instance, B(f ,U) - .-r(u-T) if the continuously compounded short-term rate is constant). Notice
that our approach hinges on the 'backward induction' argument.
MATH3075_3975_sol2.pdf
P&L_Options.pdf