1. Break Even Volatilities Quantitative Research, Bloomberg LP
Dr Bruno Dupire
Dr Arun Verma
Quantitative Research, Bloomberg LP
13th Nov, 2007
King’s College, London

2. 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

3. 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

4. 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

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

6. 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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King’s College, London

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King’s College, London

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King’s College, London

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King’s College, London

11. 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

12. 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

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

14. 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

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

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

17. 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

18. Discrete Local VolatilityOr
Regional Volatility
13th Nov, 2007
King’s College, London

19. 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

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

21. 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

22. 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

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

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

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

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

27. 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

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

29. 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

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

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

32. 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

33. 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

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

35. 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

36. 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

37. 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