Question

2. Consider the Black-Scholes prices of a European Futures call and put options: C(F,t) = (FN(d1f) – E N(d2f))e -- (T-1), P(F,t) = (E N(-d2f) – FN(-d1f))e-r(T-1), In(F/E) + ļoʻ(T – t) _In(F/E) – } 0?(T – t) dif = a OVT-t ? 25 OVT-t. Compute and simplify these expressions for at-the-money options (where F = E). Use the identity N(-2) = 1-N2), and, since F = E, the at-the-money prices can be expressed in terms of E, t,r, o, T. Comment on the difference between the price of an at-the-money Futures call option and the price of an at-the-money Futures put option. d2f =-

Answer 1

The prices of both at-the-money Future Call Option and the Future Put Option are equal. The derivation is provided in the below images.

using the Black-Scholes prices of a Gurdon butires call and put options ft) CCF, t) = (EN(als) -EN(dar) e P(Est) (E N-das) - FN (-dis) - dig: bn (F/F) + 6 (It) a ott dag a bn (P/E) - 28²Cr-t) On of it I for at thermony options , F=G i dy ln CEE) 6 (T-t) Gji - ln (1) + 48 (T-t) offt 0 8 8² (I-t GT-t C I Because I (1) >o-

dis2 1 x GXJTE - das en (F/F) - 18 LT-t) - ost-t Sirices FE dag - ln (G/E) - 66²21-ts Gjze = lof 1)- / 6² (T-to - 5174 = 6-128 (J-9 EjTt das - XE XITE. - - به روه ر 009 فلل & N(dig) NEWT-t) N (021) 3 NE- Live at t) -NC-det)

Sinces NL-2 - N(2) الملحات (1-آل عم . FETE ..NCCl4)2N Colas) - I NCON © CB, E) - (F nland) - 6 N(096) 2017 Since, F E and N(dat) 1-N(dit) . Clfyty2 CE N(as) - Excl-Neolit) Jets = LEN(due) - 6 EN(Me pelo Curry (QEN (04) -E) OUT- PCF,+)2 (E N(Olot) - FNC-0.45ETETE - Corpi - Nildas) - Exft-Nicorus) estany - CE* {1-1 tald19) - E*$l-N0193) ELBL 74

USB) - LE XN_Glut) -E PEN(dit) Pf103 (QEN QH) - E) é um from equation and I we con - Conclude that the price of the Call and put future option which are at the money are agud CEM -6) e su - BEN 4 CSF:) - 6) ett é (an(4011) -1) 107-t