Break Even Volatilities Quantitative Research, Bloomberg LP Dr Bruno Dupire Dr Arun Verma Quantitative Research, Bloomberg LP 13th Nov, 2007 King’s 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. Discounted average of the historical Intrinsic Values. Bad : depends on bull/bear, no call/put parity. 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. 13th Nov, 2007 King’s College, London

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

Zero-finding of P&L 13th Nov, 2007 King’s College, London

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

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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) 13th Nov, 2007 King’s 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. 13th Nov, 2007 King’s College, London

Alternate approaches: Fit the best log-normal 13th Nov, 2007 King’s 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. 13th Nov, 2007 King’s College, London

Strike dependency for multiple paths 13th Nov, 2007 King’s College, London

SPX Index BEVL <GO> 13th Nov, 2007 King’s 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! 13th Nov, 2007 King’s College, London

Discrete Local Volatility Or Regional Volatility 13th Nov, 2007 King’s College, London

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

Market facts 13th Nov, 2007 King’s College, London

S&P Strikes and Maturities Aug 07 Sept 07 Oct 07 Dec 07 Mar 08 Jun 08 Jun 09 Dec 08 Mar 09 T 13th Nov, 2007 King’s College, London

Discrete Local Volatilities Price at T1 of : Can be replicated by a PF of T1 options: of known price 13th Nov, 2007 King’s College, London

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

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

P&L at T1 Buy , Sell 13th Nov, 2007 King’s College, London

P&L at T2 Buy , Sell 13th Nov, 2007 King’s College, London

Link Discrete Local Vol / Local Vol is a weighted average of with the restriction of the Brownian Bridge density between T1 and T2 Assume real model is:  Market prices tell us about some averages of local volatilities - Regional Vols 13th Nov, 2007 King’s College, London

Numerical example 13th Nov, 2007 King’s College, London

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

Cumulative Variance Naïve idea: Better approximation: 13th Nov, 2007 King’s College, London

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

Vol stripping K T The exact relation is a non linear PDE : Finite difference approximation: Perfect if K T 13th Nov, 2007 King’s College, London

Numerical examples BS prices (S0=100; s=20%, T=1Y) stripped with Bachelier formula  sth=s.K Price Stripping Vol Stripping K 13th Nov, 2007 King’s College, London

Accuracy comparison 1 3 2 T K 1 2 3 3 (linearization of ) 13th Nov, 2007 King’s College, London

Local Vol Surface construction Finite difference of Vol PDE gives averages of s2, which we use to build a full surface by interpolation. Interpolate from with (where ) 13th Nov, 2007 King’s 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 13th Nov, 2007 King’s 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. 13th Nov, 2007 King’s College, London