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13 th Nov, 2007Kings College, London Break Even Volatilities Dr Bruno Dupire Dr Arun Verma Quantitative Research, Bloomberg LP

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13 th Nov, 2007Kings College, London Theoretical Skew from Prices Problem : How to compute option prices on an underlying without options? For instance : compute 3 month 5% OTM Call from price history only. 1)Discounted average of the historical Intrinsic Values. Bad : depends on bull/bear, no call/put parity. 2)Generate paths by sampling 1 day return re-centered histogram. Problem : CLT => converges quickly to same volatility for all strike/maturity; breaks auto-correlation and vol/spot dependency. ? =>

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13 th Nov, 2007Kings College, London Theoretical Skew from Prices (2) 3)Discounted average of the Intrinsic Value from re-centered 3 month histogram. 4) Δ-Hedging : compute the implied volatility which makes the Δ- hedging a fair game.

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13 th Nov, 2007Kings College, London Theoretical Skew from historical prices (3) How to get a theoretical Skew just from spot price history? Example: 3 month daily data 1 strike –a) price and delta hedge for a given within Black-Scholes model –b) compute the associated final Profit & Loss: –c) solve for –d) repeat a) b) c) for general time period and average –e) repeat a) b) c) and d) to get the theoretical Skew t S K

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13 th Nov, 2007Kings College, London Zero-finding of P&L

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13 th Nov, 2007Kings College, London Strike dependency Fair or Break-Even volatility is an average of returns, weighted by the Gammas, which depend on the strike

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13 th Nov, 2007Kings College, London Alternative approaches Shifting the returns A simple way to ensure the forward is properly priced is to shift all the returns,. In this case, all returns are equally affected but the probability of each one is unchanged. (The probabilities can be uniform or weighed to give more importance to the recent past)

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13 th Nov, 2007Kings College, London Alternative approaches Entropy method For those who have developed or acquired a taste for equivalent measure aesthetics, it is more pleasant to change the probabilities and not the support of the measure, i.e. the collection of returns. This can be achieved by an elegant and powerful method: entropy minimization. It consists in twisting a price distribution in a minimal way to satisfy some constraints. The initial histogram has returns weighted with uniform probabilities. The new one has the same support but different probabilities. However, this is still a global method, which applies to the maturity returns and does not pay attention to the sub period behavior. Remember, option pricing is made possible thanks to dynamic replication that grinds a global risk into a sequence of pulverized ones.

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13 th Nov, 2007Kings College, London Alternate approaches: Fit the best log-normal

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13 th Nov, 2007Kings College, London Implementation details Time windows aggregation The most natural way to aggregate the results is to simply average for each strike over the time windows. An alternative is to solve for each strike the volatility that would have zeroed the average of the P&Ls over the different time windows. In other words, in the first approach, we average the volatilities that cancel each P&L whilst in the second approach, we seek the volatility that cancel the average P&L. The second approach seems to yield smoother results. Break-Even Volatility Computation The natural way to compute Break-Even volatilities is to seek the root of the P&L as a function of. This is an iterative process that involves for each value of the unfolding of the delta-hedging algorithm for each timestep of each window. There are alternative routes to compute the Break-Even volatilities. To get a feel for them, let us say that an approximation of the Break-Even volatility for one strike is linked to the quadratic average of the returns (vertical peaks) weighted by the gamma of the option (surface with the grid) corresponding to that strike.

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13 th Nov, 2007Kings College, London Strike dependency for multiple paths

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13 th Nov, 2007Kings College, London SPX Index BEVL

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13 th Nov, 2007Kings College, London New Approach: Parametric BEVL Find break-even vols for the power payoffs This gives us the different moments of the distribution instead of strike dependent vol which can be noisy Use the moment based distribution to get Break even implied volatility. Much smoother!

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13 th Nov, 2007Kings College, London Discrete Local Volatility Or Regional Volatility

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13 th Nov, 2007Kings College, London Local Volatility Model Given smooth, arbitrage free, there is a unique : Given by GOOD BAD Requires a continuum of strikes and maturities Very sensitive to interpolation scheme May be compute intensive (r=0)

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13 th Nov, 2007Kings College, London Market facts

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13 th Nov, 2007Kings College, London S&P Strikes and Maturities T K Sept 07 Oct 07Dec 07 Mar 08 Jun 08 Dec 08Mar 09 Jun 09 Aug 07

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13 th Nov, 2007Kings College, London Discrete Local Volatilities Price at T 1 of : Can be replicated by a PF of T 1 options: of known price

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13 th Nov, 2007Kings College, London Discrete Local Volatilities Discrete local vol: that retrieves market price

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13 th Nov, 2007Kings College, London Taking a position Local vol = 5% User thinks it should be 10%

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13 th Nov, 2007Kings College, London Buy, Sell P&L at T 1

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13 th Nov, 2007Kings College, London P&L at T 2 Buy, Sell

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13 th Nov, 2007Kings College, London Link Discrete Local Vol / Local Vol is a weighted average of with the restriction of the Brownian Bridge density between T 1 and T 2 Assume real model is: Market prices tell us about some averages of local volatilities - Regional Vols

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13 th Nov, 2007Kings College, London Numerical example

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13 th Nov, 2007Kings College, London Crude approximation: for instance constant volatility (Bachelier model) does not give constant discrete local volatilities: Price stripping Finite difference approximation: T K

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13 th Nov, 2007Kings College, London Cumulative Variance Naïve idea: Better approximation:

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13 th Nov, 2007Kings College, London Vol stripping The approximation leads to Better: following geodesics: where Anyway, still first order equation

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13 th Nov, 2007Kings College, London Vol stripping The exact relation is a non linear PDE : Finite difference approximation: Perfect if T K

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13 th Nov, 2007Kings College, London Price Stripping Vol Stripping BS prices (S 0 =100; =20%, T=1Y) stripped with Bachelier formula th =.K Numerical examples K

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13 th Nov, 2007Kings College, London K T Accuracy comparison (linearization of ) 3

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13 th Nov, 2007Kings College, London Interpolate from with Local Vol Surface construction Finite difference of Vol PDE gives averages of 2, which we use to build a full surface by interpolation. (where )

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13 th Nov, 2007Kings College, London Reconstruction accuracy Use FWD PDE to recompute option prices Compare with initial market price Use a fixed point algorithm to correct for convexity bias

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13 th Nov, 2007Kings College, London Conclusion Local volatilities describe the vol information and correspond to forward values that can be enforced. Direct approaches lead to unstable values. We present a scheme based on arbitrage principle to obtain a robust surface.

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