The equilibrium price of the call option (C; European on a nondividend paying stock) is shown by Black and Scholes to be: are given by. p T t.


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1 THE GREEKS BLACK AND SCHOLES (BS) FORMULA The equilibrium price of the call option (C; European on a nondividend paying stock) is shown by Black and Scholes to be: C t = S t N(d 1 ) Xe r(t t) N(d ); Moreover d 1 and d are given by d 1 = ln(s t X ) + (r + 1 )(T t) p ; T t d = ln(s t X ) + (r 1 )(T t) p T t Note that d = d 1 p T
2 Delta of a (European; nondividend 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 nondividend paying stock) call option is given by = C t = N(d 1 ) + S t N(d 1 ) + Xe r(t t)n(d ) (1) where we have applied the product rule: [S t N(d 1 )] = N(d 1 ) + S t N(d 1 ) = = N(d 1 ) + S t N(d 1 )
3 Next we apply the chain rule Since it follows that N(d 1 ) = N(d 1) d 1 d 1 () N(d 1 ) = Z d1 1 1 p e X dx By using N(d 1 ) d 1 = N 0 (d 1 ) = 1 p e d 1 (3) we have d 1; = ln(s t X ) + (r 1 )(T t) p T t d 1 = d = 1 S t p T t (4)
4 Using equations ()(4) it can be shown that or S t N(d 1 ) = Xe r(t t)n(d ) S t N 0 (d 1 ) = Xe r(t t) N 0 (d ) Thus equation (1) reduces to = C t = N(d 1 ) > 0 (5)
5 A delta neutral position: Consider the following portfolio: A short position in one call and a long position in C S = stocks: = C C S S ) S = C C S S = 0: 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 s position 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.
6 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 C t S t Recall that from equation () we have = = N(d 1) (6) where and N(d 1 ) = N(d 1) d 1 d 1 N(d 1 ) d 1 = N 0 (d 1 ) = 1 p e d 1 = d = 1 S t p T t d 1
7 and the time to maturity for an at the money call option: Recall that = C t = N(d 1 ) Thus applying the chain rule gives where = N 0 (d 1 ) d 1 ; N 0 (d 1 ) = 1 p e d 1 > 0 For an at the money call option (S t = X), since ln( S t X ) = 0, we have Thus d 1 = (r + 1 ) p d 1 = (r + 1 ) p = (r + 1 ) p > 0 It can be shown that < 0
8 BLACK AND SCHOLES (BS) FORMULA The equilibrium price of the call option (C; European on a nondividend paying stock) is shown by Black and Scholes to be: C t = S t N(d 1 ) Xe r(t t) N(d ); where d 1 and d are given by or d 1 = ln(s t X ) + (r + 1 )(T t) p ; T t d = ln(s t X ) + (r 1 )(T t) p T t d = d 1 p
9 Theta: Theta is the rate of change of the price of the call with respect to time to maturitywith all else remaining the same: C t = S t N 0 (d 1 ) d 1 + rxe r N(d ) Xe r(t t) N 0 (d ) d Utilizing the fact that we write Since d = d 1 p C t = S tn 0 (d 1 ) d 1 + rxe r N(d ) Xe r(t t) N 0 (d ) {z } S t N 0 (d 1 ) d 1 + Xe r(t t) N 0 (d ) p S t N 0 (d 1 ) = Xe r(t t) N 0 (d )
10 C simpli es to C t = +rxe r N(d )+Xe r(t t) N 0 (d ) p > 0 or = C t t = C t < 0 which is negative. This is because as the time to maturity decrease the option 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.
11 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 nondividend paying stock) call option is given by C t = S tn 0 (d 1 ) d 1 Utilizing the fact that Xe r(t t) N 0 (d ) d : we write C t = S tn 0 (d 1 ) {z } d = d 1 Xe r(t t) N 0 (d ) d 1 p +Xe r(t t) N 0 (d ) p : Xe r(t t) N 0 (d ) d 1
12 C t = S tn 0 (d 1 ) {z } Xe r(t t) N 0 (d ) d 1 +Xe r(t t) N 0 (d ) p : Xe r(t t) N 0 (d ) d 1 Since C simpli es to S t N 0 (d 1 ) = Xe r(t t) N 0 (d ) C t = Xe r(t t) N 0 (d ) p = = S t N 0 (d 1 ) p > 0:
13 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 nondividend paying stock) call option is given by C t r = S t N 0 (d 1 ) {z } d 1 r Xe r(t t) N 0 {z} (d ) d r +(T t)xe r(t t) N(d ): Xe r(t t) N 0 (d ) d r Utilizing the fact that d = d 1 r r ; and S t N 0 (d 1 ) = Xe r(t t) N 0 (d ) C t r simpli es to C t r = (T t)xe r(t t) N(d ) > 0
14 PutCall Parity: Recall the putcall parity: P t = C t + Xe r(t t) S t Utilizing the Black and Scholes formula for the call we write P t = S t N(d 1 ) Xe r(t {z t) N(d ) } + Xe r(t t) C t = S t [N(d 1 ) {z 1] } N( d 1 ) + Xe r(t t) [1 N(d )] {z } N( d ) = N( d 1 )S t + Xe r(t t) N( d ): S t
15 Delta of a (European; nondividend paying stock) put option Rewrite the put call parity P t = C t + Xe r(t t) S t It follows that the Delta of the put option is given by p = P t = C t 1 = = N(d 1 ) 1 = N( d 1 ) < 0
16 Gamma of a (Europ.; nondividend paying stock) put option The Gamma of the put option is P t = C t 1 ) p = P t St = 1 p e = C t S t d 1 = N 0 (d 1 ) d 1 1 S t p T {z } {z } N 0 (d 1 ) d 1 Since the Gamma of the call option is t > 0 c = C t S t = N 0 (d 1 ) d 1
17 Theta of a (Europ.; nondividend paying stock) put option Rewrite the put call parity P t = C t + Xe r(t t) S t It follows that the theta of the put option is given by P t Thus = C t rxe r = + rxe r N(d ) + Xe r(t t) N 0 (d ) p {z } C t rxe r = rxe r [N(d ) 1] + Xe r(t t) N 0 (d {z } ) {z } p N( d ) S t N 0 (d 1 ) = rxe r N( d ) + S t N 0 (d 1 ) p p = P t t = P t = rxe r N( d ) S t N 0 (d 1 ) p
18 Vega: For a European (on a nondividend paying stock) put option the vega is given by p = P t = c = C t = S tn 0 (d 1 ) p > 0 since according to the putcall parity P t = C t + Xe r(t t) S t
19 Rho: Recall the putcall parity P t = C t + Xe r(t t) S t For a European (on a nondividend paying stock) put option the rho is given by P t r since = C t r (T t)xe r(t t) = (T t)xe r(t t) N(d ) {z } C t r = (T t)xe r(t t) [N(d ) {z 1] } N( d ) = (T t)xe r(t t) N( d ) < 0 C t r = (T t)xe r(t t) N(d ) (T t)xe r(t t)
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