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# Greeks

It is partice on the market to calculate the sensitivity of derivative w.r.t. various parameters such as:
**Greeks" ** because they are often denoted by Greek letters; although the volatility related "Greeks" vega, vanna and volga are not Greek letters. The are a number of reasons for calculating the Greeks. Usually traders have limits on a how high and low the "Greeks" are are allowed to be be for their current trading positions. The "Greeks" are also related to hedging portfolios (see Delta hedge, Delta-gamma hedge and Delta-vega hedge).

## Black-Scholes Greeks for the European call option

Assume that we have a standard European call option with strike K and maturity T and that the underlying asset S follows the standard Black-Scholes model. At time t<T the "Greeks" are given by (see also Black-Scholes formula):

Δ=N(d_{1}),

Γ=φ(d_{1})/(S(t)σ(T-t)^{1/2}),

ρ=K(T-t)e^{-r(T-t)}N(d_{2}),

Θ=-(S(t)σφ(d_{1}))/(2(T-t)^{1/2})-rKe^{-r(T-t)}N(d_{2}),

Ѵ=S(t)φ(d_{1})(T-t)^{1/2},

Vanna=φ(d_{1})((T-t)^{1/2}-d_{1}/σ),

Volga=S(t)d_{1}φ(d_{1})(T-t)^{1/2}(d_{1}/σ-(T-t)^{1/2}),

where N is the distribution function of the standard Gaussian distribution, φ the corresponding density function,

d_{1}=(ln(S(t)/K)+(r+σ^{2}/2)(T-t))/(σ(T-t)^{1/2}),

d_{2}=d_{1}-σ(T-t)^{1/2}.

The Black-Scholes "Greeks" are often used even in situtations where we do not believe that the underlying asset follows the Black-Scholes model. What one doeas is to calculate the Black-Scholes volatility that makes the Black-Scholes price to match the observed call option price, i.e. the so called implied volatility, then one calculates the Black-Scholes "Greeks" at this volatility. This we here denote the implied Black-Scholes "Greeks". Although it is market practice to use implied Black-Scholes "Greeks" one should be a bit careful here. In some situations these implied "Greeks" can be misleading; in the sense that they under (or over) estimate the sensitivities. We are here also ignoring the fact that we are using the wrong model. Practitioners sometimes use various "tricks" to improve the implied "Greeks" using variations on the implied volatility such as e.g. "sticky strike" or "sticky moneyness" volatilities.

- The value of the underlying asset(
**Δ**, Delta) - Second derivative w.r.t. the value of the underlying asset(
**Γ**, Gamma) - Present time (
**Θ**, Theta) - Interest rate (
**ρ**, rho) - Volatility (
**Ѵ**Vega) - Volatility and underlying assets (
**Vanna**) - Second derivative w.r.t. to volatility (
**Volga**).

Δ=N(d

Γ=φ(d

ρ=K(T-t)e

Θ=-(S(t)σφ(d

Ѵ=S(t)φ(d

Vanna=φ(d

Volga=S(t)d

where N is the distribution function of the standard Gaussian distribution, φ the corresponding density function,

d

d

The Black-Scholes "Greeks" are often used even in situtations where we do not believe that the underlying asset follows the Black-Scholes model. What one doeas is to calculate the Black-Scholes volatility that makes the Black-Scholes price to match the observed call option price, i.e. the so called implied volatility, then one calculates the Black-Scholes "Greeks" at this volatility. This we here denote the implied Black-Scholes "Greeks". Although it is market practice to use implied Black-Scholes "Greeks" one should be a bit careful here. In some situations these implied "Greeks" can be misleading; in the sense that they under (or over) estimate the sensitivities. We are here also ignoring the fact that we are using the wrong model. Practitioners sometimes use various "tricks" to improve the implied "Greeks" using variations on the implied volatility such as e.g. "sticky strike" or "sticky moneyness" volatilities.

Questions: Magnus Wiktorsson

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