Download Introduction to Financial Theory

Introduction to Financial Theory.
Instructor : Rama CONT
Solutions
Consider a one-period market with two securities, one risky asset whose price is denoted by S(t) and a bond
whose value at time t is denoted by B(t).. The date at the beginning of the period is denoted t=0 and the end is
denoted T. R is the return on the bond, assumed to be non-random and B(0)=1. The stock price can assume M
possible values at date T :
P( S(T)= Sk )= pk for k=1,2,..M
A straddle with maturity T is a contingent claim with the following payoff h:
h( S(T) ) = S(T) – K
h( S(T) ) = K - S(T)
1.
What does the absence of arbitrage opportunities imply on R, S k k=1,2,..M?
Answer:
2.
if S(T) > K
if S(T) < K
SM < S(0) (1+R ) < S1
Write the conditions a portfolio of stocks and bonds has to verify in order to replicate the payoff of a
straddle. Is it always possible to find such a portfolio ?
If the portfolio,contains x stocks and y bonds then for each k=1..M, one must have: x S k + y B(T) = h(Sk)
which gives M equations for 2 unknowns x and y. There is a unique solution in general if M =2.
3.
Explain what is meant by «risk neutral probability associated to the state k». This quantity is denoted by
q(k). What are the properties of q(k), k=1..M ?
q(k) = (1+R) (k) where (k) is the value of the Arrow Debreu security associated to state k.
We have: q(k)>0, q(1) +..+ q(M) = 1 and
q(1) S1 +q(2) S2 +…+q(M) SM = S(0) ( 1+R) (martingale property).
4.
Express the price of a general contingent claim in terms of q(k), k=1..M and the interest rate R. What does
this relation become in the case where the contingent claim is the stock itself?
If C(h) is the price at t=0 of a contingent payoff h() then C(h) = (q(1) h 1 +q(2) h2 +…+q(M) hM ) /(1+R)
In the case pf the stock one recovers the martingale property
q(1) S1 +q(2) S2 +…+q(M) SM = S(0) ( 1+R) (martingale property).
5.
Show that a straddle can be replicated using a call option and a put option.
Remark that: h(S(T)) = max(S(T)-K,0) + max (K-S(T), 0).
6.
Using put-call parity, show that a straddle can be replicated using a call option, stocks and bonds.
Put call parity : C(T,K) – P(T,K) = S(T) – K / (1+R) implies P(T,K)=C(T,K) –S(T) + K/(1+R)
where C(T,K) and P(T,K) are respectively the prices of a call and a put option on S. Now since
Straddle = Call + Put
Therefore : Straddle = Long 2 Calls + short one stock + long K/(1+R) in bonds
In the following we assume M=2. S(0)= 100$, S1 =105 $, S2= $ 95, R = 3 %. P(S(T)=105 )= 50 %.
The strike price for the straddle is given by K=99 $
7.
Are the no arbitrage conditions of Question 1) verified?
Yes : 95 <100 ( 1.03) < 105
8.
In the case where M=2, find a replication portfolio for the straddle
The straddle gives a payoff of $6 if the price goes up and $4 if the stock goes down.
Solving the replication conditions in 2) with h(1)=$6, h(2)=$4 gives : x= 0.4 , y = - 34.4
9.
Using the no-arbitrage principle, give the value at t=0 of a straddle.
V= x S(0) + y B(0) = 100 x + y = 40 – 34.4 = 5.6 $
10. An investor estimates the value of a straddle by its expected payoff. Show how you can construct an
arbitrage strategy by entering into a deal with this investor.
E(h)= P(S(T)=105 )= 50 %.) h(105) + P(S(T)=105 )= 50 %.) h(95) = 0.5 ( 6 + 4) = 5 $
which is cheaper than the arbitrage value V of the straddle. So we can buy a straddle from the investor and then
short x stocks and long (–y ) bonds. The resulting strategy is an arbitragr (check) with profit 0.60 $.
11. Calculate q(1), q(2). Compare with p1 and p2.
Using the martingale condition q(1) S1 +q(2) S2 = S(0) ( 1+R) plus q(1) + q(2) = 1 one obtains
q(1) = [S0 (1+R) – S2] / [S1 – S2] = 8/10 = 0.8
12. Express the value at t=0 of a straddle in terms of q(1) and q(2). Compare this value with the one obtained in
Question 9).
C(h) = 0.8 (6) + 0.2 (4) = 5.6 $ which is the same as in 9).
13. Use q(1) and q(2) to calculate the price of a call and a put on S(T) with strike K. Verify the relation implied
by question 5).
You do this last one, come on!