Abstract
Black-Scholes option pricing model (1973) assumes that all option prices on the same underlying asset with the same expiration date, but different exercise prices should have the same implied volatility. However, instead of a flat implied volatility structure, implied volatility (inverting the Black-Scholes formula) shows a smile shape across strikes and time to maturity. This paper compares parametric volatility models with stochastic volatility models in capturing this volatility smile. Results show empirical evidence in favor of parametric volatility models. Keywords: smile volatility, parametric, stochastic, Black-Scholes. JEL Classification: C14 C68 G12 G13
Highlights
Obtaining the implied volatility with the Newton-Raphson algorithmThe Black-Scholes European call option formula is the following one:. where S0 is the underlying asset (in this case, the future contract of IBEX-35 index), q is the expected dividends paid over the option’s life, X is the option’s strike price, (T-t) is the time to expiration, r is the risk-free interest rate, d1 ln(S0 / K ) (r q V 2 / 2)(T t) , V T t (2)
There is a concern about how modelling the volatility, and how modelling the volatility in option pricing
There is a concern about how modelling the volatility in option pricing
Summary
The Black-Scholes European call option formula is the following one:. where S0 is the underlying asset (in this case, the future contract of IBEX-35 index), q is the expected dividends paid over the option’s life, X is the option’s strike price, (T-t) is the time to expiration, r is the risk-free interest rate, d1 ln(S0 / K ) (r q V 2 / 2)(T t) , V T t (2). The Black-Scholes European call option formula is the following one:. Where S0 is the underlying asset (in this case, the future contract of IBEX-35 index), q is the expected dividends paid over the option’s life, X is the option’s strike price, (T-t) is the time to expiration, r is the risk-free interest rate, d1 ln(S0 / K ) (r q V 2 / 2)(T t) , V T t (2). D2 d1 V T t , ı is the volatility rate, and N(d) is the cumulative unit normal density function with upper integral limit d. The implied Black-Scholes volatility can be found individually from traded option prices: wBS wV. The Newton-Raphson algorithm provides a numerical way to invert the Black-Scholes formula in order to recover ı from the market prices of the call option C (or Put option P).
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