108156.pdf

Solutions 287

You will be left with an arbitrage profit of

131 −120e

(^1012) ×12%
+1e
124 ×12%
+2e
121 ×12%∼
= 1. 44
dollars.
6.4No arbitrage profit can be realised in these circumstances. Though the theoret-
ical no-arbitrage forward price is about $87.83, the first strategy in the proof
of Proposition 6.2 brings a loss of 89−83e10%+2e^0.^5 ×7%∼=− 0 .66 dollars and
the second one results in a loss of−89 + 83e7%−2e^0.^5 ×10%∼=− 2 .08 dollars.
6.5The euro plays the role of the underlying asset with dividend yield 3%. Hence
the forward price (the exchange rate) is
F(0, 1 /2) = 0.9834e^0 .5(4%−3%)∼= 0. 9883
euros to a dollar.
6.6At timet

• borrow and pay (or receive and invest, if negative) the amountV(t)to
  acquire a short forward contract with forward priceF(0,T) and delivery
  dateT,


• initiate a new long forward contact with forward priceF(t, T)atnocost.
  Then at timeT


• close out both forward contracts receiving (or paying, if negative) the
  amountsS(T)−F(0,T)andS(T)−F(t, T), respectively;


• collectV(t)er(T−t)from the risk-free investment, with interest.
  The final balanceV(t)er(T−t)−[F(t, T)−F(0,T)]>0 will be your arbitrage
  profit.
  6.7By (6.3) the initial forward price isF(0,1)∼= 45 .72 dollars. This takes into
  account the dividend paid at time 1/2.
  a) IfS(9/12) = 49 dollars, thenF(9/ 12 ,1)∼= 49 .74 dollars by (6.2). It
  follows by (6.8) thatV(9/12)∼= 3 .96 dollars.
  b) IfS(9/12) = 51 dollars, thenF(9/12)∼= 51 .77 dollars andV(9/12)∼=
  5 .96 dollars.
  6.8Lett=1/ 365 ,T=1/ 4 .We apply the formula (6.11) to get

f(t, T)−f(0,T)=S(t)er(T−t)+S(0)erT=0

ifS(t)=S(0)ert,that is, if the stock grows at the risk-free rate.

6.9Sincef(t, T)=S(t)er(T−t),the random variablesS(t)andf(t, T) are perfectly

correlated withρS(t)f(t,T)=1andσf(t,T)=er(T−t)σS(t).It follows thatN=

e−r(T−t).

6.10Observe that Theorem 6.5 on the equality of futures and forward prices applies

also in the case of an asset with dividends paid continuously. We can, therefore,

use (6.6) to obtain

rdiv=8%− 0.^175 ln^1413 ,,^100500 ∼= 2 .20%.