1a) Let S = $55, K = $50, r = 6% (continuously compounded), d = 2%, s = 40%, T =
ID: 1175951 • Letter: 1
Question
1a)
Let S = $55, K = $50, r = 6% (continuously compounded), d = 2%, s = 40%, T = 0.5, and n = 5. In this situation, the appropriate values of u and d are 1.13939 and 0.88471, respectively. What is the value of p*, the risk-neutral probability of an upward movement in the stock price at any node of the binomial tree?
0.5316
0.4998
0.3738
0.4146
0.4684
1b)
Let S = $65, r = 3% (continuously compounded), d = 5%, s = 30%, T = 2. In this situation, the appropriate values of u and d are 1.32313 and 0.72615, respectively. Using a 2-step binomial tree, calculate the value of a $55-strike European call option.
$14.416
$14.291
$13.458
$13.868
$14.519
1c)
Let S = $40, r = 5% (continuously compounded), d = 4%, s = 20%, T = 1.5. In this situation, the appropriate values of u and d are 1.19806 and 0.84730, respectively. Using a 2-step binomial tree, calculate the value of a $30-strike American call option.
$10.327
$10.983
$10.579
$10.273
$10.190
1d)
Suppose the exchange rate is $1.51/€. Let r$ = 6%, r€ = 7%, u = 1.34, d = 0.73, and T = 2. Using a 2-step binomial tree, calculate the value of a $1.45-strike European call option on the euro.
$0.2425
$0.2361
$0.2151
$0.2284
$0.1960
1e)
Suppose the exchange rate is $1.14/C$. Let r$ = 7%, rC$ = 4%, u = 1.33, d = 0.79, and T = 1.5. Using a 2-step binomial tree, calculate the value of a $1.20-strike American put option on the Canadian dollar.
$0.1434
$0.1621
$0.1823
$0.1592
$0.1511
Let S = $55, K = $50, r = 6% (continuously compounded), d = 2%, s = 40%, T = 0.5, and n = 5. In this situation, the appropriate values of u and d are 1.13939 and 0.88471, respectively. What is the value of p*, the risk-neutral probability of an upward movement in the stock price at any node of the binomial tree?
Answers: a.0.5316
b.0.4998
c.0.3738
d.0.4146
e.0.4684
Explanation / Answer
1a) Let S = $55, K = $50, r = 6% (continuously compounded), d = 2%, s = 40%, T = 0.5, and n = 5. In this situation, the appropriate values of u and d are 1.13939 and 0.88471, respectively. What is the value of p*, the risk-neutral probability of an upward movement in the stock price at any node of the binomial tree?
p = (e^rt - d) / (u-d) = (e^0.06*0.1 -0.88471)/(1.13939-0.88471) = 0.47
Ans is e) 0.4684
1b) Let S = $65, r = 3% (continuously compounded), d = 5%, s = 30%, T = 2. In this situation, the appropriate values of u and d are 1.32313 and 0.72615, respectively. Using a 2-step binomial tree, calculate the value of a $55-strike European call option.
Cuu = Suu - 55 = 65*1.32313*1.32313 - 55 = 58.7937
Cud = Sud - 55 = 65*1.32313*0.72615 - 55 = 7.45
Sdd = 65*0.72615*0.72615 = 34.274 < 55 ====> Cdd = 0
C = (58.79375*p^2 + 2*7.45*p*(1-p) + 0)*e^(-0.03*2)
p = (e^rT - d) / (u -d) = (e^(0.03*1) - 0.72615)/ (1.32313 - 0.72615) = 0.50974
C =
1c) Let S = $40, r = 5% (continuously compounded), d = 4%, s = 20%, T = 1.5. In this situation, the appropriate values of u and d are 1.19806 and 0.84730, respectively. Using a 2-step binomial tree, calculate the value of a $30-strike American call option
Su = 47.9224
Suu = 57.4139
Sd = 33.892
Sud = Sdu = 40.6046
Sdd = 28.71669
1d) Suppose the exchange rate is $1.51/€. Let r$ = 6%, r€ = 7%, u = 1.34, d = 0.73, and T = 2. Using a 2-step binomial tree, calculate the value of a $1.45-strike European call option on the euro
Euu = 1.51*1.34*1.34 - 1.45 = 2.711356 - 1.45 = 1.26
Eud = 1.51*1.34*0.73 - 1.45 = 0.027082
Edd = 0
p = (e^(0.06-0.07)-0.73)/(1.34-0.73) = 0.426
C = (p^2*Euu + 2*p*(1-p)*Eud)*e^(-0.06*2) = 0.2151
Ans is
$0.2151
1e) Suppose the exchange rate is $1.14/C$. Let r$ = 7%, rC$ = 4%, u = 1.33, d = 0.79, and T = 1.5. Using a 2-step binomial tree, calculate the value of a $1.20-strike American put option on the Canadian dollar.
Eu = 1.14*1.33 = 1.5162
Euu = 1.5162*1.33 = 2.016546
Eud = 1.14*1.33*0.79 = 1.197798
Ed = 1.14*0.79 = 0.9006
Edd = 0.711474
p = (e^(0.07 - 0.04)*1.5/2 - d ) /(u -d) = 0.431
at T= 1
Put option has value Pu = 1.188*10^-3 with p=0.431 & Pd = 0.2994 with pd=1-0.431
P = (0.4631*1.188*10^-3 + 0.537*0.299)*e^(-0.07*0.75) = 0.1621
Ans is b) 0.1621
c.$0.2151
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