# A price a european call written on this stock with an

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a) Price a European call written on this stock with an exercise price of \$100 and a time tomaturity of one year.b) Price an American call with an exercise price of \$100 and a time to maturity of one year usingthe pseudo-American call pricing method.c) How accurate is the price you calculated in (b)? Explain.d) If the dividend were \$8 instead of \$1, would the pseudo-American (Black approximation) callprice obtained (please estimate) be as accurate? Explain.
SolutionsNTln)
19
202.a) Calculate the price of a call option, using the Black Scholes formula with the followinginputsS=44.375,K=45,r=7%=0.07,σ2=0.0961,σ= 0.31, T = 156/365= 0.427Therefore:?1=ln (44.37545) + (0.07 +120.0961) × 0.4270.31 × √0.427≅ 0.1798?2= ?1− 0.31 × √0.427= −0.02278N(d1) = 0.57134N(d2) = 0.49091c=SN(d1)KerTN(d2) =3.913b) Usingthe Put-Call parity:P = C + PV(K)S = 2.3075 + 50×e-0.05×1/450 = 1.68639Van also be done directly using BSM formula.c) Options with longer maturity and no dividends will have a higher value.c=SN(d1)KerTN(d2) =50×0.74245×e-0.06/4×0.665 = 7.62p=KerTN (−d2)S×N (−d1) = 45×e-0.06/4×(1-0.655)− 50×(1-0.742) = 1.954Stock Priced1N(d1)d2N(d2)\$500.650.7420.430.655
21Putcall parity:p + S=C+ KerT1.954 + 50 = 7.62 + 44.33 = 51.953.a)c = S.N(d1)Ke-rTN(d2)p = K e-rTN(−d2)S. N(−d1)Therefore:c − p = S[N(d1) + N(−d1)] − Ke-rT[N(d2) +N(−d2)] = S − Ke-rTNote: N(d)+N(−d) = 1. Done.b) When S is much larger than K, d1 and d2 are very large => N(d1) and N(d2) are very close to1. Therefore:c = S.N(d1) − Ke-rTN(d2) = S − Ke-rTThe value of a forward contract on the stock with delivery price K and time to maturity of T.4.a) First adjust for the dividend => S* = S − PV(D) = 95 − 0.95123 = 94.0488.

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Term
Fall
Professor
ATAMAZAHERI
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