Break Even Volatilities

1
Break Even Volatilities Dr Bruno Dupire Dr Arun
Verma Quantitative Research, Bloomberg LP
2
Theoretical Skew from Prices
? gt

• 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 gt converges quickly to same
  volatility for all strike/maturity breaks
  auto-correlation and vol/spot dependency.

3
Theoretical Skew from Prices (2)

1. Discounted average of the Intrinsic Value from
   re-centered 3 month histogram.
2. ?-Hedging compute the implied volatility
   which makes the ?-hedging a fair game.

4
Theoretical Skewfrom 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

5
Zero-finding of PL
6
Strike dependency

• Fair or Break-Even volatility is an average of
  returns, weighted by the Gammas, which depend on
  the strike

7

8

9

10

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)

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.

13
Alternate approaches Fit the best log-normal
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 PLs over the different time
  windows. In other words, in the first approach,
  we average the volatilities that cancel each PL
  whilst in the second approach, we seek the
  volatility that cancel the average PL. 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 PL 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.

15
Strike dependency for multiple paths
16
SPX Index BEVL ltGOgt
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!

18
Discrete Local Volatility Or Regional Volatility
19
Local Volatility Model
GOOD
Given smooth, arbitrage free ,
there is a unique
Given by
(r0)
BAD

• Requires a continuum of strikes and maturities
• Very sensitive to interpolation scheme
• May be compute intensive

20
Market facts
21
SP Strikes and Maturities
K
Oct 07
Dec 07
Aug 07
Sept 07
Mar 08
Jun 08
Jun 09
Dec 08
Mar 09
T
22
Discrete Local Volatilities
23
Discrete Local Volatilities
24
Taking a position

• Local vol 5
• User thinks it should be ?10

25
PL at T1

• Buy , Sell

26
PL at T2

• Buy , Sell

27
Link Discrete Local Vol / Local Vol
28
Numerical example
29
Price stripping

• Finite difference approximation

Crude approximation for instance constant
volatility (Bachelier model) does not give
constant discrete local volatilities
K
T
30
Cumulative Variance

• Naïve idea

• Better approximation

31
Vol stripping

• The approximation leads to
• Better following geodesics

where
where
Anyway, still first order equation
32
Vol stripping

• The exact relation is a non linear PDE

• Finite difference approximation
• Perfect if

K
T
33
Numerical examples
BS prices (S0100 s20, T1Y) stripped with
Bachelier formula ? sths.K
Price Stripping Vol Stripping
K
34
Accuracy comparison
1
3
2
T
K
1
(linearization of )
2
3
3
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
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

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.