THE GREEKS BLACK AND SCHOLES (BS) FORMULA The equilibrium price of the call option (C; European on a non-dividend paying stock) is shown by Black and Scholes to be: r(T t) Ct = StN(d ) Xe N(d ); 1 2 Moreover d1 and d2 are given by ln(St) + (r + 12)(T t) d = X 2 ; 1 pT t ln(St) + (r 12)(T t) d = X 2 2 pT t Note that d = d pT 2 1 Delta of a (European; non-dividend paying stock) call option: The delta of a derivative security, , is de…ned as the rate of change of its price with respect to the price of the underlying asset. For a European (on a non-dividend paying stock) call option is given by #Ct #N(d1) = = N(d1) + St + #St #St r(T t)#N(d2) Xe (1) #St where we have applied the product rule: #[StN(d1)] #St #N(d1) = N(d1) + St = #St #St #St #N(d1) = N(d1) + St #St Next we apply the chain rule #N(d ) #N(d )#d 1 = 1 1 (2) #St #d1 #St Since d1 1 X2 N(d1) = e 2 dx p2 Z1 it follows that d2 #N(d1) 1 1 = N0(d1) = e 2 (3) #d1 p2 By using St 1 2 ln(X ) + (r 2 )(T t) d ; = 1 2 pT t we have #d #d 1 1 = 2 = (4) #St #St StpT t Using equations (2)-(4) it can be shown that #N(d1) r(T t)#N(d2) St = Xe #St #St or r(T t) StN0(d1) = Xe N0(d2) Thus equation (1) reduces to #Ct = = N(d1) > 0 (5) #St A delta neutral position: Consider the following portfolio: #C A short position in one call and a long position in #S = stocks: #C = C S #S ) # #C #C = = 0: #S #S #S The delta of the investor’s hedge position is therefore zero. The delta of the asset position o¤sets the delta of the option position. A position with a delta of zero is referred to as being delta neutral. It is important to realize that the investor’sposition only remains delta hedged (or delta neutral) for a relatively short period of time. This is because delta changes with both changes in the stock and the passage of time. The Gamma of a call option: The second derivative of the call option with respect to the price of the stock is called the Gamma of the option and is given by 2 # Ct # #N(d1) 2 = = (6) #St #St #St Recall that from equation (2) we have #N(d ) #N(d )#d 1 = 1 1 #St #d1 #St where d2 #N(d1) 1 1 = N0(d1) = e 2 #d1 p2 and #d #d 1 1 = 2 = #St #St StpT t and the time to maturity for an at the money call option: Recall that #Ct = = N(d1) #St Thus applying the chain rule gives # #d = N (d ) 1; # 0 1 # where d2 1 1 N0(d1) = e 2 > 0 p2 St For an at the money call option (St = X), since ln(X ) = 0, we have 1 2 1 2 (r + 2 ) (r + 2 )p d1 = = p Thus #d (r + 12) 1 = 2 > 0 # 2p 2 It can be shown that # < 0 # 2 BLACK AND SCHOLES (BS) FORMULA The equilibrium price of the call option (C; European on a non-dividend paying stock) is shown by Black and Scholes to be: r(T t) Ct = StN(d ) Xe N(d ); 1 2 where d1 and d2 are given by ln(St) + (r + 12)(T t) d = X 2 ; 1 pT t ln(St) + (r 12)(T t) d = X 2 2 pT t or d = d p 2 1 Theta: Theta is the rate of change of the price of the call with respect to time to maturity-with all else remaining the same: #Ct #d1 r = StN (d ) + rXe N(d ) # 0 1 # 2 #d Xe r(T t)N (d ) 2 0 2 # Utilizing the fact that #d #d 2 = 1 # # 2p we write #Ct #d1 r = StN (d ) + rXe N(d ) # 0 1 # 2 r(T t) #d1 r(T t) Xe N0(d2) + Xe N0(d2) # 2p StN0(d1) Since | {z } r(T t) StN0(d1) = Xe N0(d2) #C # simpli…es to #Ct r r(T t) = +rXe N(d2)+Xe N0(d2) > 0 # 2p or #C #C = t = t < 0 #t # which is negative. This is because as the time to maturity decrease the op- tion tends to become less valuable. Theta is not the same type of hedge parameter as delta and gamma. This is because although there is some uncertainty about the future stock price there is no uncertainty about the passage of time. It does not make sense to hedge against the e¤ect of the passage of time on an option portfolio. Vega: The vega of a derivative security, , is de…ned as the rate of change of its price with respect to the volatility of the underlying asset. For a European (on a non-dividend paying stock) call option is given by #Ct #d1 r(T t) #d2 = StN (d ) Xe N (d ) : # 0 1 # 0 2 # Utilizing the fact that #d #d 2 = 1 p # # we write #Ct #d1 r(T t) #d1 = StN (d ) Xe N (d ) # 0 1 # 0 2 # r(T t) Xe N0(d2) | {zr(T }t) +Xe N0(d2)p: #Ct #d1 r(T t) #d1 = StN (d ) Xe N (d ) # 0 1 # 0 2 # r(T t) Xe N0(d2) | {zr(T }t) +Xe N0(d2)p: Since r(T t) StN0(d1) = Xe N0(d2) #C # simpli…es to #C t = Xe r(T t)N (d )p = # 0 2 = StN0(d1)p > 0: Rho: The rho of a derivative security is de…ned as the rate of change of its price with respect to the interest rate. For a European (on a non-dividend paying stock) call option is given by #Ct #d1 r(T t) #d2 = StN (d ) Xe N (d ) #r 0 1 #r 0 2 #r Xe r(T t)N (d ) 0 2 #d2 | {z } #r |{z} +(T t)Xe r(T t)N(d ): 2 Utilizing the fact that #d #d 2 = 1; and #r #r r(T t) StN0(d1) = Xe N0(d2) #Ct #r simpli…es to #C t = (T t)Xe r(T t)N(d ) > 0 #r 2 Put-Call Parity: Recall the put-call parity: r(T t) Pt = Ct + Xe St Utilizing the Black and Scholes formula for the call we write r(T t) r(T t) Pt = StN(d ) Xe N(d ) + Xe St 1 2 Ct r(T t) = S| t[N(d ) 1]{z+ Xe }[1 N(d )] 1 2 N( d ) N( d ) 1 2 r(T t) = N| ( d{z)St +} Xe N|( d {z): } 1 2 Delta of a (European; non-dividend paying stock) put option Rewrite the put call parity r(T t) Pt = Ct + Xe St It follows that the Delta of the put option is given by #Pt #Ct p = = 1 = #St #St = N(d ) 1 = N( d ) < 0 1 1 Gamma of a (Europ.; non-dividend paying stock) put option The Gamma of the put option is #P #C t = t 1 #St #St ) 2 2 # Pt # Ct #d1 p = 2 = 2 = N0(d1) #St #St #St d2 1 1 1 = e 2 > 0 p2 StpT t #d N0(d1) 1 #St | {z }| {z } Since the Gamma of the call option is 2 # Ct #d1 c = 2 = N0(d1) #St #St Theta of a (Europ.; non-dividend paying stock) put option Rewrite the put call parity r(T t) Pt = Ct + Xe St It follows that the theta of the put option is given by #P #C t = t rXe r = # # r r(T t) + rXe N(d2) + Xe N0(d2) 2p #Ct # | {z } rXe r r r(T t) = rXe [N(d2) 1] + Xe N0(d2) 2p N( d ) StN (d ) 2 0 1 r = rXe | N({zd2) +} St|N0(d1) {z } 2p Thus #Pt #Pt p = = #t # r = rXe N( d2) StN0(d1) 2p Vega: For a European (on a non-dividend paying stock) put option the vega is given by #Pt #Ct p = = c = = StN (d )p > 0 # # 0 1 since according to the put-call parity r(T t) Pt = Ct + Xe St Rho: Recall the put-call parity r(T t) Pt = Ct + Xe St For a European (on a non-dividend paying stock) put option the rho is given by #P #C t = t (T t)Xe r(T t) #r #r r(T t) = (T t)Xe r(T t) N(d ) (T t)Xe 2 #Ct #r | {z } = (T t)Xe r(T t)[N(d ) 1] 2 N( d ) 2 = (T t)Xe r(T |t)N({zd ) <} 0 2 since #C t = (T t)Xe r(T t)N(d ) #r 2.