Statistics 441 (Fall 014) November 14, 014 Prof. Michael Kozdron Lecture #9: he Greeks Recall that if V (0,S 0 )denotesthefairrice(attime0)ofaeuroeancallotionwithstrike rice E and exiry date, then the Black-Scholes otion valuation formula is log(s0 /E)+(r + 1 1 ) V (0,S 0 )=S 0 Ee r log(s0 /E)+(r ) where = S 0 (d 1 ) Ee r (d ) d 1 = log(s 0/E)+(r + 1 ) and d = log(s 1 0/E)+(r ) = d 1. We see that this formula deends on the initial rice of the stock S 0, the exiry date, the strike rice E, the risk-free interest rate r, and the stock s volatility. he artial derivatives of V = V (0,S 0 )withresecttothesevariablesareextremelyimortant in ractice, and we will now comute them; for ease, we will write S = S 0. In fact, some of these artial derivatives are given secial names and referred to collectively as the Greeks : = V S (delta), = V S = S (gamma), = V r (rho), = V (theta), vega = V. Note. Vega is not actually a Greek letter. Sometimes it is written as (which is the Greek letter nu). 18 1
Remark. On age 80 of [11], Higham changes from using V (0,S 0 )todenotethefairrice at time 0 of a Euroean call otion with strike rice E and exiry date to using C(0,S 0 ). Both notations seem to be widely used in the literature. he financial use of each of he Greeks is as follows. Delta measures sensitivity to a small change in the rice of the underlying asset. Gamma measures the rate of change of delta. Rho measures sensitivity to the alicable risk-free interest rate. heta measures sensitivity to the assage of time. Sometimes the financial definition of is V. With this definition, if you are long an otion, then you are short theta. Vega measures sensitivity to volatility. Aarently, there are even more Greeks. Lambda, the ercentage change in the otion value er unit change in the underlying asset rice, is given by = 1 V V S = log V S. Vega gamma, or volga, measures second-order sensitivity to volatility and is given by V. Vanna measures cross-sensitivity of the otion value with resect to change in the underlying asset rice and the volatility and is given by V S =. It is also the sensitivity of delta to a unit change in volatility. Delta decay, or charm, given by V S =, measures time decay of delta. (his can be imortant when hedging a osition over the weekend.) 18
Gamma decay, or colour, given by 3 V S, measures the sensitivity of the charm to the underlying asset rice. Seed, given by 3 V S, 3 measures third-order sensitivity to the underlying asset rice. In order to actually erform all of the calculations of the Greeks, we need to recall that 0 (x) = 1 e x /. Furthermore, we observe that which imlies that S 0 (d 1 ) log =0 (18.1) Ee r 0 (d ) S 0 (d 1 ) Ee r 0 (d )=0. (18.) Exercise 18.1. Verify (18.1) and deduce (18.). Since d 1 = log(s/e)+(r + 1 ) we find d 1 S = 1 S, d 1 r =, d 1 = log(s/e)+(r + 1 ) = d, and Furthermore, since we conclude d 1 = log(s/e)+(r + 1 ). 3/ d = d 1, d S = 1 S, d r =, d = d 1 = d, and d = d 1 = log(s/e)+(r + 1 ) 3/ = log(s/e)+(r 1 ). 3/ 18 3
Delta. Since V = S (d 1 ) Ee r (d ), we find = V S = (d 1)+S (d 1) S = (d 1 )+S 0 (d 1 ) d 1 S = (d 1 )+ where the last ste follows from (18.). Gamma. Since = (d 1 ), we find Ee r (d ) S 0 (d 1 ) Ee r 0 (d ) S Ee r 0 (d ) d S = (d 1 )+ 1 S S 0 (d 1 ) Ee r 0 (d ) = (d 1 ) = V S = S = 0 (d 1 ) d 1 S = 0 (d 1 ) S. Rho. Since V = S (d 1 ) Ee r (d ), we find = V r = S (d 1) + Ee r (d ) Ee r (d ) r r = S 0 (d 1 ) d 1 r + Ee r (d ) Ee r 0 (d ) d r = S 0 (d 1 )+Ee r (d ) Ee r 0 (d ) = S 0 (d 1 ) Ee r 0 (d ) + Ee r (d ) = Ee r (d ) where, as before, the last ste follows from (18.). heta. Since V = S (d 1 ) Ee r (d ), we find = V = S (d 1) + Ere r (d ) Ee r (d ) = S 0 (d 1 ) d 1 + Ere r (d ) Ee r 0 (d ) d = S 0 (d 1 ) d 1 + Ere r (d ) Ee r 0 (d ) ale d1 = S 0 (d 1 ) Ee r 0 (d ) d 1 + Ere r (d )+ Ee r 0 (d ) = Ere r (d )+ Ee r 0 (d ) 18 4
where, as before, the last ste follows from (18.). However, (18.) also imlies that we can write as =Ere r (d )+ S 0 (d 1 ). (18.3) Vega. Since V = S (d 1 ) Ee r (d ), we find vega = V = S (d 1) = S 0 (d 1 ) d 1 = d S 0 (d 1 ) Ee r (d ) Ee r 0 (d ) d d Ee r 0 (d ) = d S 0 (d 1 ) Ee r 0 (d ) + Ee r 0 (d ) = Ee r 0 (d ) where, as before, the last ste follows from (18.). However, (18.) also imlies that we can write vega as vega = S 0 (d 1 ). Remark. Our definition of is slightly di erent than the one in Higham [11]. We are di erentiating V with resect to the exiry date as oosed to an arbitrary time t with 0 ale t ale.hisaccountsforthediscreancyintheminussignsin(10.5)of[11]and(18.3). Exercise 18.. Comute lambda, volga, vanna, charm, colour, and seed for the Black- Scholes otion valuation formula for a Euroean call otion with strike rice E. We also recall the ut-call arity formula for Euroean call and ut otions from Lecture #: V (0,S 0 )+Ee r = P (0,S 0 )+S 0. (18.4) Here P = P (0,S 0 ) is the fair rice (at time 0) of a Euroean ut otion with strike rice E. Exercise 18.3. Using the formula (18.4), comute the Greeks for a Euroean ut otion. hat is, comute = P S, = P S, = P r, =P P, and vega =. Note that gamma and vega for a Euroean ut otion with strike rice E are the same as gamma and vega for a Euroean call otion with strike rice E. 18 5