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 Black & Scholes : Vega υ
Issued on July 20 2011 par Strategies-options.com
There is no vega in the Black & Scholes model. Implied Volatility is taken as a constant. Traders modified the rules.
As for interest rates, volatility is said to be constant in the Black & Scholes model. Hence, if it's constant, there is no variation. If there is no variation, vega is nil.
I - Modification by traders But traders were aware of that assumption, and modified the rule by letting implied volatility varying in order to price how its variation would impact option prices. That's vega. II - Maths If we have : t as the starting date S as the spot K as the strike r the continuously compounded annualized interest rate q the continuously compounded annualized dividend rate T as the maturity σ the annualized volatility It follows that : For the call ν = ∂C / ∂σ ν = S √(T-t) exp(-q(T-t))N’(d2) For the put ν = ∂P / ∂σ ν = ∂P / ∂σ = S √(T-t) exp(-q(T-t))N’(d2) Put and call vegas are the same ! III - Graph  Vega is maximal away from expiry. Vega is maximal near the strike price. There is no vega at all at expiry. Next : Black & Scholes : Rhô ρ Previous : Black & Scholes : Gamma Г Pdf connexes : - Understanding N(d1) and N(d2) : Risk-Adjusted Probabilities in the Black-Scholes Model - Black-Scholes Option Pricing Model OPTIONS PRICING MODEL - INDEX OPTIONS PRICING MODEL - INDEX OPTIONS PRICING MODEL - CHAPTER I OPTIONS PRICING MODEL - CHAPTER II OPTIONS PRICING MODEL - CHAPTER III Strategies-options.com |
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