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Let's say you have 5 paths leading to given strike and time. Path 1 local vol 11%, Path 2 local vol 12%, path 3 local vol 12.5%, path 4 local vol 13%, path 5 local vol 14%. The implied vol for 0 - 5th path is the avergage, therefore 12.5%. Right? So using 12.5% implied vol would give me the market price of the vanilla for a given strike and time. What if i ...


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Loosely speaking: Local volatility is the instantaneous volatility after time T if the spot is S at that time. Implied volatility is the expected integrated volatility from today up to time T if the spot ends up at S at that time.


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Gatheral and Jacquier discuss this issue in section 4 of the paper. Instead of using the raw parameterization of the SVI, they use the natural parameterization of the total implied variance: $$ w(k) = \Delta + \frac{\omega}{2} \left\{ 1 + \zeta \rho (k - \mu) + \sqrt{(\zeta (k-\mu) + \rho)^2 + (1-\rho^2)} \right\} (\text{p. 61 of the published paper}) $$ In ...


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Thanks, so the local vol model treats volatility as a function of both the current asset level S_t and of time t. How do i use in practice? lets say I build an implied volatiliy surface i extract from market prices: Call, 100% strike, 1y. price USD 130, Black-Scholes implied vol 12% Call, 100% strike, 2y, price USD 140, BS implied 13% Call, 90% strike 1y, ...


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You should not expect the local vol to be equal to the implied vol except in the trivial case where both are constant (Black-Scholes model). I haven't read the Derman articles but it is quite clear using Dupire's formula (see Gatheral's book for example). Local volatility can be computed in terms of call prices using Dupire's formula $$ \sigma^2(T,K) = ...



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