13 th nov, 2007kings college, london break even volatilities dr bruno dupire dr arun verma...
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13th Nov, 2007 King’s College, London
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 Δ-hedging a fair game.
13th Nov, 2007 King’s College, London
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”
1TSkK
PL 0/ kPLk
t
S
1T 2T
K1TS
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 approachesShifting the returnsA 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 detailsTime 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!
32 ,SS
13th Nov, 2007 King’s College, London
Discrete Local Volatility
Or
Regional Volatility
13th Nov, 2007 King’s College, London
Local Volatility Model
Given smooth, arbitrage free , there is a unique :
Given by
tS , TKTKC ,, 0
0
,
,TKT CKSE
dWtSdS
2)
1)
TKKC
TKTC
TK,
,2,
2
22
GOOD
BAD• Requires a continuum of strikes and maturities
• Very sensitive to interpolation scheme
• May be compute intensive
(r=0)
13th Nov, 2007 King’s College, London
Market facts
13th Nov, 2007 King’s College, London
S&P Strikes and Maturities
T
KS
ept 0
7
Oct
07
Dec
07
Mar
08
Jun
08
Dec
08
Mar
09
Jun
09
Aug
07
13th Nov, 2007 King’s College, London
Discrete Local Volatilities
002211 ,,, TKTTiTK CC
i
1TSK
Price at T1 of :2,TK
C
Can be replicated by a PF of T1 options: of known price 1,TKi i
C
K 1TS
f
iK
K
1T 2T
00 ,TS
21 ,TT
13th Nov, 2007 King’s College, London
Discrete Local Volatilities
f
02,TK
C
DTK ,
Discrete local vol: that retrieves market priceDTK ,
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
• Buy , Sell
P&L at T1
1,
%5 TKi iC
2,TKC
13th Nov, 2007 King’s College, London
P&L at T2
• Buy , Sell 1,
%10 TKi iC
2,TKC
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: dWtSdS ,
DTK ,
1T
K
00 ,TS
2T
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
Crude approximation:
for instance constant volatility
(Bachelier model)
does not give constant discrete local
volatilities:
Price stripping
Finite difference approximation:
2,,,
,,
2
22
2
22
,
,2,
K
CCCT
CC
C
C
TKKC
TKTC
TKTKTKKTKK
TKTTK
KK
T
dWdS
TK
13th Nov, 2007 King’s College, London
Cumulative Variance
TV TKTK2,,
• Naïve idea:
dttKVVT
T
TKTK 2
1
12,2
,,
1T 2T
00 ,TS
K K
1T 2T
00 ,TS
K'K
• Better approximation:
dttKKTT
TtKVV
dttSKT
tSV
T
T
TKTK
T
TK
2
1
12,''
,
12
12,',
0
002
,
13th Nov, 2007 King’s College, London
Vol stripping
• The approximation leads to
• Better: following geodesics:
dttSKT
tSV
T
TK
0
002
, ,
K
V
T
SK
T
V
u
VTK
02 ,
T
SKu 0
1where
dtttfVT
TKTK 0
,2
, ,
K
VTf
T
V
u
VTK TK
',
2 , where
Tfu
TK',
1
Anyway, still first order equation
13th Nov, 2007 King’s College, London
Vol stripping
The exact relation is a non linear PDE :
2
222
002
2
1
4
1
21,
K
V
K
V
VV
KS
K
V
V
KSTK
T
V
• Finite difference approximation:
• Perfect if
VVVV
KSV
VKS
VTK
KKKK
T
21
41
21
,2
2
00
2
FDTKdWdS ,:
TK
13th Nov, 2007 King’s College, London
Price Stripping Vol Stripping
BS prices (S0=100; =20%, T=1Y) stripped with Bachelier formula th=.K
Numerical examples
thestimated
K
13th Nov, 2007 King’s College, London
estimatedth
1
2
3
VT
SKVTK KT 02 ,
1
2
3 VV
VVKS
VVKS
VTK
KKKK
T
21
41
21
,2
2
00
2
V
V
KSV
TK
K
T
0
2
1,
KT
Accuracy comparison
(linearization of )3
13th Nov, 2007 King’s College, London
Interpolate from
with
Local Vol Surface constructionFinite difference of Vol PDE gives averages of 2, which we use to build
a full surface by interpolation.
TK ,2
jT 1jT
iK
1iK
2iK
1iK
2iK
1jT
2
1,1,11,1
2
,,1,1
1,11,1,1,1
,1,
22
2
1
222
1
K
VVV
K
VVVV
K
VV
K
VVV
T
VVV
jijijijijijiKK
jijijijiK
jijiT
VVVV
KSV
VKS
V
TTK
KKKK
T
jji
21
41
21
2,
22
00
12
jiji TKVV ,, (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
2
22
2 K
C
T
C
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.
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