A nine-month long forward contract on the currency that will allow you to buy one AUD for USD 0.9100. Calculate also the theoretical value of the forward contract. Compare and comment. 2.4678(1.5578)1.93651.5195(0.6095)1.51951.19231.19240.93560.93560.9356(0.0256)0.73410.73420.57610.5761(0)0.45210.3547(0) PriceCu4 (uuuF)1.067536Cuud (uudF)0.310042uuF0.677993udF0.151048uF0.407055Cd0.073575C0.235695Theoretical ValueCurrent PriceerTe2.7183r0.0265Value0.954381 The Price using Binomial Tree gives 0.23570.911.1457 while the theoretical value is 0.954381. The difference in the prices is because of the adjustments made to the binomial model reflecting both the up-ward moves and down-ward moves of the currency. A -month European call option to buy one AUD for USD 0.9100. Calculate also the value of the option by using the Black-Scholes formula. Compare and comment. 1.4481(0.5381)1.29831.1639(0.2539)1.1641.04361.04350.93560.93560.9355(0.0255)0.83880.83870.7520.7519(0)0.67410.6044(0) Where Vthe price e is the exponential function p is the probability adjusted for the risks Vu represent value of up-ward move and Vd represent down-ward moves. 9-month callStrike Price0.91T9/120.75hT/n0.1875rUSD r-AUD r-0.012u1.1154d0.896539p0.4697261-p0.5303Valuee-rh(pVu(1-p)Vu)Cuuu0.385466Cuud0.132122Cud0.061741Cuu0.249876Cu0.149375Call Value at to0.069779 Black-Scholes CCurrent Value0.9356Exercise0.91exponential constant, e2.7183risk-free rate0.0265time to expiration0.75N(d1)Cumulative Normal Probability Density(ln(S/E)(rfvolatility2/2)t)/(volatilityt0.5)S/E1.028132ln(S/E)0.027744d10.198541N(d1) N(0.20)0.5793d2d1-volatilityt0.5-0.04395N(d2)N(-0.04)0.484European CallValue of CallSN(d1)-Ee-rft(Nd2)0.110221 By use of a four time-step binomial tree, the calculated price for European call is 0.0698. When the Black-Scholes Equation is used, the price of this option is 0.1102. The difference comes because the binomial value is a time-discrete approximation as opposed to the Black-Scholes model. The calculation involved only four time-intervals, which permitted only five possible prices at the expiry time. The binomial model would converge to the Black-Scholes model as the number of time-steps approach infinity. A nine-month American call option to buy one AUD for USD 0.9100. 9-month callStrike Price0.91T9/120.75hT/n0.1875rUSD r-AUD r-0.012u1.1154d0.896539p0.4697261-p0.5303Valuee-rh(pVu(1-p)Vu)exercise valueCuuu0.3854660.3883Since 0.38830.3855, use 0.3883 in further calculationsCuud0.1321360.1335Since 0.13350.1321, use 0.1335 in further calculationsCud0.0623950.0252Since 0.06240.0256, use 0.0624 in further calculationsCuu0.2519240.254Since 0.2540.252, use 0.254 in further calculationsCu0.1516390.1336Since 0.15160.1336, use 0.1516 in further calculationsC0.070854 Black-Scholes CCurrent Value0.9356Exercise0.91exponential constant, e2.7183risk-free rate0.0265time to expiration0.75N(d1)Cumulative Normal Probability Density(ln(S/E)(rfvolatility2/2)t)/(volatilityt0.5)S/E1.028132ln(S/E)0.027744d10.198541N(d1) N(0.20)0.5793d2d1-volatilityt0.5-0.04395N(d2)N(-0.04)0.484Value of CallSN(d1)-Ee-rft(Nd2)0.110221American Call0.110221 The four time-step binomial tree, gives a price of 0.0709 for the American Call option. Using Black-Scholes model, the price of this option is 0.1102. The difference is due to time-discrete approximation which used for four time-intervals only which have only five possible prices at the expiry time. The binomial model would give the same value as the Black-Scholes model as the number of time-steps approach infinity. A nine-month up-and-out barrier call option on the currency with a strike of 0.9100 and knockout barrier of 1.1000. An up-and-out call option gives the holder the right to buy the underlying asset at the strike price on the expiration date so long as the price of that asset did not go above a pre-determined barrier during the options lifetime. When the price of the underlying asset is above the barrier, the option is knocked-out
