Transcript Slide 1

Break Even Volatilities
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.
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.
13th Nov, 2007
King’s 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 Δhedging 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?
S
Example:
K
ST
3 month daily data
t
T1
1 strike K  k ST1
T2
– a) price and delta hedge for a given  within Black-Scholes
1
–
–
–
–
model
b) compute the associated final Profit & Loss: PL 
c) solve for  k / PL  k  0
d) repeat a) b) c) for general time period and average
e) repeat a) b) c) and d) to get the “theoretical Skew”
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
13th Nov, 2007
King’s College, London
13th Nov, 2007
King’s College, London
13th Nov, 2007
King’s College, London
13th Nov, 2007
King’s 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)
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 BreakEven 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
2
• Find break-even vols for the power payoffs S , S
• 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
3
Discrete Local Volatility
Or
Regional Volatility
13th Nov, 2007
King’s College, London
Local Volatility Model
GOOD


Given smooth, arbitrage free CK ,T 0 K ,T , there is a unique  S, t  :
1) dS   S , t dW


2) E ST  K   CK ,T 0
Given by
BAD

(r=0)
C
K , T 
2
2
 K , T   2 T
C
K , T 
2
K
•
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
13th Nov, 2007
King’s College, London
Jun 09
Mar 09
Dec 08
Jun 08
Mar 08
Dec 07
Aug 07
Sept 07
Oct 07
K
T
Discrete Local Volatilities
C 0  
Ki ,T1
Price at T1 of
CK ,T2
i
T1 ,T2
 CK ,T2 0
:
Ki
 T ,T
1
2
K
S0 ,T0 
ST1
K
Can be replicated by a PF of T1 options:
K
13th Nov, 2007
T2
T1
i  CKi ,T1of known pricef  

ST1
King’s College, London
Discrete Local Volatilities
f  
CK ,T2 0

D
K ,T

Discrete local vol:  KD,T 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 CK ,T2, Sell
13th Nov, 2007
 5%C
i
Ki ,T1
King’s College, London
P&L at T2
•
Buy CK ,T2, Sell i 10%CKi ,T1
13th Nov, 2007
King’s College, London
Link Discrete Local Vol / Local
Vol
Assume real model is:
dS   S , t dW
K
 KD,T is a weighted average of 
with the restriction of the Brownian
Bridge density between T1 and T2
S0 ,T0 
T1
 Market prices tell us about some averages of local volatilities Regional Vols
13th Nov, 2007
King’s College, London
T2
Numerical example
13th Nov, 2007
King’s College, London
Price stripping
Finite difference approximation:
C
 C K ,T
C
K , T  2 C
2 K ,T T
T
 2 K , T   2 T
 T 
 C
 KK C C K K ,T  C K K ,T  2C K ,T
K , T 
2
K
K 2
2
Crude approximation:
for instance constant volatility
(Bachelier model)
dS   dW
does not give constant discrete local
K
volatilities:
13th Nov, 2007
King’s College, London
T
Cumulative Variance
2
VK ,T   K ,T T
•
Naïve idea:
K
K
T2
VK ,T2  VK ,T1    2 K , t dt
S0 ,T0 
T1
T1
•
Better approximation:
VK ,T
t




    S0  K  S0 , t dt
T


0
T
VK ,T2  VK ',T1
13th Nov, 2007
K
2


t  T1
K  K ', t dt
    K '
T2  T1


T1
T2
2
King’s College, London
T2
K'
S0 ,T0 
T1
T2
Vol stripping
• The approximation
t


VK ,T    2  S0  K  S0 , t dt
T


0
T
V V K  S 0 V
 K , T    

u T
T K
2
where
leads to
  1 
u  K  S0


 T 
T
• Better: following geodesics: VK ,T
   2  f K ,T t , t dt
0
V V
V
'
 K , T    
 f K ,T T 
u T
K
2
  1 

where u   '
 f K ,T T 
Anyway, still first order equation
13th Nov, 2007
King’s College, London
Vol stripping
The exact relation is a non linear PDE :
2


S  K V   S0  K 
V
1
  2 K , T 1  0
 


  2V 

T
V

K
4V


• Finite difference approximation:
 K , T  
 TV
2
•
Perfect if
2
 S K

S

K
1
 0

1  0

 KV  
 


V
  2V  4V



1
2
 KV    KK V 


2


dS   dW :  KFD,T  
K
13th Nov, 2007
 V  2 1  2V 




2
 K 

2

K


King’s College, London
T
Numerical examples
BS prices (S0=100; =20%, T=1Y) stripped with Bachelier formula
 th=.K
 estimated   th
Price Stripping
Vol Stripping
K
13th Nov, 2007
King’s College, London
Accuracy comparison
 th   estimated
1
3
2
T
2
1  K , T    T V 
2
2  K , T  
2
K , T  

3
13th Nov, 2007
K
K  S0
 KV
T
 TV
S K
1 0
 KV
V
(linearization of 3 )
 TV
 S 0  K  2
S0  K
1 
1
2
1
 KV  
 
  KV    KK V
V
4V 
2
 2V 
King’s College, London
Local Vol Surface construction
Finite difference of Vol PDE gives averages of 2, which we use to build
a full surface by interpolation.
Interpolate  2 K, T  from

 2  K i ,

Ki  2
T j  T j 1 
 
2

Ki 1
 TV
 S  K 
S0  K
1 
1
2
 KV   0


  KV    KK V
V
4V 
2
 2V 
2
1
with
 TV 
Ki
Ki 1
Ki  2
Vi , j 1  Vi , j
T
1 Vi 1, j  Vi 1, j Vi 1, j 1  Vi 1, j 1 


2K
2K


T j 1
Tj
 KV  
2
1 Vi 1, j  Vi 1, j  2Vi , j Vi 1, j 1  Vi 1, j 1  2Vi , j 1 


2
K 2
K 2

 KK V  
13th Nov, 2007
King’s College, London
(where Vi, j  V Ki , Tj  )
T j 1
Reconstruction accuracy
• Use FWD PDE
option prices
C  2  2C to recompute

T
2 K 2
• 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