By Sebastien Bossu, Peter Carr
In Advanced fairness Derivatives: Volatility and Correlation, Sébastien Bossu stories and explains the complex suggestions used for pricing and hedging fairness unique derivatives. Designed for monetary modelers, choice investors and complex traders, the content material covers crucial theoretical and functional extensions of the Black-Scholes model.
Each bankruptcy comprises a variety of illustrations and a quick number of difficulties, masking key issues reminiscent of implied volatility floor versions, pricing with implied distributions, neighborhood volatility versions, volatility derivatives, correlation measures, correlation buying and selling, neighborhood correlation versions and stochastic correlation.
The writer has a twin specialist and educational historical past, making Advanced fairness Derivatives: Volatility and Correlation the proper reference for quantitative researchers and mathematically savvy finance pros seeking to gather an in-depth knowing of fairness unique derivatives pricing and hedging.
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Additional info for Advanced Equity Derivatives: Volatility and Correlation
3 Sticky True Delta Rule Consider a one-year vanilla call with strike K = 1, and let ???? ∗ (S) be its implied volatility at various spot price assumptions S. Assume zero rates and dividends and denote the call price: ( c(S) = cBS (S, ???? ∗ (S)) = SN ln S + 12 ???? ∗2 (S) ???? ∗ (S) ) ( −N ln S − 12 ???? ∗2 (S) ???? ∗ (S) ) ADVANCED EQUITY DERIVATIVES 32 (a) Show that the option’s delta Δ is dc Δ(S) = =N dS ( ln S + 12 ???? ∗2 (S) ???? ∗ (S) ) ( ∗′ + ???? (S)N ′ ln S − 12 ???? ∗2 (S) ) ???? ∗ (S) (b) Assume that ???? ∗ is a linear function of Δ: ???? ∗ (S) = a + bΔ(S).
B) Find a replicating portfolio for the “free” option using vanilla and exotic options. (c) Calculate the fair value of the “free” option premium using the BlackScholes model with 20% volatility, S0 = K = $100, one-year maturity, zero interest and dividend rates. 7 × S0 the option pays off max(1 + 3C, S3 ∕S0 ); Otherwise, the option pays off S3 /S0 . Assuming S0 = $100, zero interest and dividend rates, and 25% volatility, estimate the level of C so that the option is worth 1 using Monte Carlo simulations.
In the presence of the smile, we may want to use a different implied volatility to reprice the call, and we must make an assumption on the behavior of the smile curve: ■ ■ If we assume that the smile curve does not change at all, we should use the same implied volatility to reprice the call. This is known as the sticky-strike rule and produces the same delta as Black-Scholes. 91 call to reprice the call. This is known as the sticky-moneyness rule and produces a higher delta than Black-Scholes. ADVANCED EQUITY DERIVATIVES 18 ■ If we assume that the smile curve does not change with respect to delta, we should use the implied volatility corresponding to the new delta to reprice the call.