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Quantitative Finance by Erik Schlogl

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Chapter 6

Implied volatility and volatility smiles

Recall the Black and Scholes (1973) formula for a European call option in its simplest form,

C(t)=X(t)N(h1)Ker(Tt)N(h2)(6.1)             h1,2=lnX(t)Kr(Tt)±12σX2(Tt)σXTt

on a non-dividend–paying asset X(t) with constant (scalar) volatility σX, strike K, time to maturity T − t and constant continuously compounded risk– free interest rate r. K and T − t are specified in the option contract, and X(t) and r can be assumed to be observable in the market.1 Only the volatility σX is not directly observable. In the absence of any other market information, a simple approach would be to estimate it as the annualised standard deviation of past logarithmic returns of the underlying asset. However, ...

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