Use the Garman-Kohlhagen model to calculate the premium of a European US$ put with a strike price of 90 Yen given
Spot exchange rate: 95 Yen per dolar
Volatility: 40 percent
Time to expire: 1 year
US interest rate: 3 percent per annum
Japan interest rate: 6 percent per annum
Show your computations of -d1, -d2, N(-d1) and N(-d2)
The price of a European Put option with strike price K and time of maturity T is given by the formula:
P = K*e^(-rd*t) * N(-d2) - S*e^(-rf*t) *N(-d1)
where d1= ( ln(S/K) + (rd-rf + s^2/2) *t ) / (s*t^0.5)
d2 = ( ln(S/K) + (rd-rf - s^2/2) *t ) / (s*t^0.5) = d1 -s*t^0.5
K is the strike exchange rate = 90 Yen/$
rd is the domestic interest rate =3% = 0.03
t is the time till maturity in years =1
S is the spot exchage rate = 95 Yen/$
rf is the foreign interest rate = 6% =0.06
s is the volatility of the exchange rate = 40% = 0.4
So, d1= (ln(95/90)+(0.03-0.06+0.4^2/2)*1/ (0.4*1^0.5) = 0.260168
So -d1 = -0.260168
d2 == (ln(95/90)+(0.03-0.06-0.4^2/2)*1/ (0.4*1^0.5) = -0.13983
d2 = d1-s*t^0.5 = 0.260168 - 0.4*1^0.5 = -0.13983
So, -d2= 0.13983
N(-d1) = N(-0.260168) = area under normal distribution curve below -0.260168 = 0.397367
N(-d2) = N(0.13983) = area under normal distribution curve below 0.13983 =0.555604
So,
P = K*e^(-rd*t) * N(-d2) - S*e^(-rf*t) *N(-d1)
=90*e^(-0.03*1)* 0.555604 - 95*e^(-0.06*1)*0.397367
=12.9750 yen/$ is the required premium of the European Put option
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