Greek letters Olivier Levyne (2020) Refresher on the Black & - PowerPoint PPT Presentation

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Greek letters Olivier Levyne (2020) Refresher on the Black & Scholes model Notations C = call premium P = put premium S = spot price of the underlying asset E = options strike price t = time to expiration of the

Greek letters Olivier Levyne (2020)
Refresher on the Black & Scholes’ model • Notations • C = call premium • P = put premium • S = spot price of the underlying asset • E = option’s strike price • t = time to expiration of the option • s = volatility of the underlying asset • r = risk free rate • Black & Scholes formula: C= 𝑇. Φ 𝑒 1 − 𝐹𝑓 −𝑠𝜐 Φ 𝑒 2 𝐹 + 𝑠 + 𝜏 2 ln 𝑇 . 𝜐 2 𝑦 𝑦 𝑓 − 𝑢 2 1 2 𝑒𝑢 = න 𝑒 1 = , 𝑒 2 = 𝑒 1 − 𝜏 𝜐 , Φ 𝑦 = න 𝑔(𝑢)𝑒𝑢 𝜏 𝜐 2𝜌 −∞ −∞ • Call-put parity: 𝑄 = 𝐷 − 𝑇 + 𝐹𝑓 −𝑠𝜐
Greek letters ’ formulas • Delta • Sensitity of the option’s premium to a slight change in the spot price of the underlying asset 𝜀𝐷 • Delta of a call = 𝜀𝑇 = Φ 𝑒 1 𝜀𝑄 • Delta of a put = 𝜀𝑇 = Φ 𝑒 1 -1= −Φ −𝑒 1 • Vega • Sensitity of the option’s premium to a slight change in the volatility of the underlying asset 𝜀𝐷 𝜀𝑄 • Vega of a call = 𝜀𝜏 = 𝑇. 𝑔(𝑒 1 ) 𝜐 𝜀𝜏 = Vega of a put = • Rho • Sensitity of the option’s premium to a slight change in the risk-free rate 𝜀𝐷 𝜀𝑠 = 𝜐𝐹𝑓 −𝑠𝜐 Φ 𝑒 2 • Rho of a call = 𝜀𝑄 • Rho of a put = 𝜀𝑠 = −𝜐𝐹𝑓 −𝑠𝜐 Φ −𝑒 2 • Theta • Sensitity of the option’s premium to a slight change in the option’s time to expiration 𝜀𝐷 𝜏 𝜀𝜐 = 𝑠𝐹𝑓 −𝑠𝜐 Φ 𝑒 2 + 𝑇𝑔(𝑒 1 ) • Theta of a call = 2 𝜐 𝜀𝑄 𝜏 𝜀𝜐 = −𝑠𝐹𝑓 −𝑠𝜐 Φ −𝑒 2 + 𝑇𝑔(𝑒 1 ) • Theta of a put = 2 𝜐
Numerical example Spot price of the underlying asset : S 120 121 120 120 120 Greek letters Strike price : E 100 100 100 100 100 Delta Valuation date : t' 01/01/2019 01/01/2019 01/01/2019 02/01/2019 01/01/2019 Call Expiration date : t" 01/11/2019 01/11/2019 01/11/2019 01/11/2019 01/11/2019 D = F (d1) 0,91 0,92 Volatility : s 20% 20% 21% 20% 20% Change in delta 0,01 Risk free rate in discrete time : r' 6,00% 6,00% 6,00% 6,00% 7,07% Time to expiration (in years) : t =(t''-t')/365 0,833 0,83 0,83 0,83 0,83 Put D = - F (-d1) Risk free rate in continuous time : r = ln (1+r') 5,83% 5,83% 5,83% 5,83% 6,83% -0,09 -0,08 Change in delta 0,01 1,36 1,40 1,30 1,36 1,40 Vega: call and put 𝑔 𝑒 1 . 𝑇. 𝜐 V for 100% = 17,42 𝑒 2 = 𝑒 1 − 𝜏 𝜐 1,17 1,22 1,11 1,17 1,22 F (d1) 0,9125 0,9195 0,9033 0,9126 0,9195 Véga for 1% = V / 100 0,17 F (d2) 0,8797 0,8886 0,8662 0,8800 0,8886 Theta Rhô Call premium: C = S. F (d 1 )-E.exp(-rt) .F (d 2 ) 25,69 26,61 25,87 25,67 26,39 Call Call Put premium: P = C-S+E.exp(-rt) 0,95 0,87 1,14 0,95 0,86 r for 100% T for 1 year 6,97 69,80 Rhô for 1% = r /100 Théta for 1 day = / 365 0,02 0,70 Gap on call premium 0,92 0,18 -0,02 0,70 Gap on put premium -0,08 0,18 0,00 -0,09 Put Put r for 100% T for 1 year 1,42 -9,54 0,16 0,15 0,17 0,16 0,15 Rhô for 1% = r /100 Théta for 1 day = / 365 0,00 -0,10

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