d) European gap put option:
Stricke price = K1
Payment trigger = K2
Expiry date = T
Optimal payment level of trigger K2 =
GapPut(S,K1,K2,T) = K1e^-rT *N (-d2) –Se ^–dt N (-d1)
Suitable upper bond = [p <= K1*exp(-rt)]
e) Current price S0 = $ 20
Risk free force of interest = 3%
Continuously compounded real world drift = 7%
Time = 2 years
30, if S1 > 23 and S2 < 1.25S1,
15, if S1 < 23 and S2 < 1.25S1,
0, otherwise .
Current value of option using the risk neutral sequential pricing approach
Since the stock does not pay no dividend = So-Ke^-rT
C (So) = $ 20
Volatility = 0.15
d - a = .07
To seek the number S such that Pr( S0.5<St0.5) = 0.95
The random variable (s0.5/.20) is normally distributed with
Mean = (.07-0.5*0.15^2)*0.5 = 0.029375
Standard deviation = 0.15 * sqrt0.5 = 0.106066
Because N^-1 (0.95) = 1.645, we have
0.029375 + 0.106066 N^-1 (0.95) = 0.203854
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