Solutions 297
9.8With 95% probability the logarithmic return on the exchange rate satisfies
k>m+xσ∼=− 23 .68%,wherex∼=− 1 .645, so thatN(x)∼=5%. The 1, 000
dollars converted into euros, invested without risk at the raterEUR, and con-
verted back into dollars after one year, will give 1,000erEURekdollars. With
probability 95% this amount will satisfy1 ,000erEURek> 1 ,000erEURem+xσ∼= 821 .40 dollars.
On the other hand, 1,000 dollars invested at the raterUSDwould have grown
to 1,000erUSD∼= 1 , 051 .27 dollars. As a result,
VaR = 1,000erUSD− 1 ,000erEURem+xσ∼= 229 .88 dollars.
9.9A single call costs $21.634. We purchase approximately 46.22 options. With
probability 5% the stock price will be less than $49.74. We shall still be able
to exercise the options, cashing $450.18 in the borderline case. The alternative
risk-free investment of $1,000 at 8% would grow to $1, 040 .81. Hence VaR∼=
590 .63 dollars. If the stock grows at the expected rate, reaching $63.71, then
we shall obtain $1, 095 .88 when the options are exercised. With 5% probability
the stock price will be above $81.6 and then our options will be worth at least
$1, 922 .75.
9.10The cost of a single bull spread is $0.8585, with expected return 29.6523%,
standard deviation 99.169%, and VaR equal to $15,000 (at 74.03% confidence
level). If 92.95% of the capital is invested without risk and the remainder in
the bull spread, then the expected return will the same as on stock, with risk
of 6.9957% and VaR equal to $650.
9.11A put with strike price $56 costs $0.426. A put with strike price $58 costs
$0.9282. The expected return on the bear spread is 111.4635%, the risk reach-
ing 177.2334%. The worst case scenario (among those admitted by the an-
alyst) is when the stock price drops to $58.59. In this scenario, which will
happen with conditional probability 0.3901, the investor will lose everything,
so VaR = 15,000 dollars at 60.99% confidence level.Chapter 10
10.1The yields arey(0)∼= 14 .08% andy(3)∼= 13 .63%. ThusB(0,3) = e−^3 τy(0)∼=
0 .9654 dollars. Arbitrage can be achieved as follows:- At time 0 buy a 6-month bond forB(0,6) = 0.9320 dollars, raising the
money by issuing 0.9654 of a 3-month bond, which sells atB(0,3)∼= 0. 9654
dollars. - At time 3 (after 3 months) issue 0.9989 of a 3-month bond, which sells
atB(3,6) = 0.9665 dollars, and use the proceeds of $0.9654 to settle the
fraction of a 3-month bond issued at time 0. - At time 6 (after half a year) the 6-month bond bought at time 0 will pay $1,
out of which $0.9989 will settle the fraction of a 3-month bond issued at
time 3.
The balance of $0.0011 will be the arbitrage profit.
10.2The implied rates arey(0)∼= 12 .38% andy(6)∼= 13 .06%.Investing $100,
we can buy 106.38 bonds now and 113.56 after six months. The logarithmic
return over one year is ln(113. 56 /100)∼= 12 .72%,the arithmetic mean of the
semi-annual returns.