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Probability Trading
Oliver BLASKOWITZ
Wolfgang HÄRDLE
Center for Applied Statistics and
Economics (CASE)
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Motivation
1-1
Motivation
Recall from option pricing theory
European put: Pt (K, τ )
−rτ
Z
∞
max(K − ST , 0)q(ST )dST
= e
0
European call: Ct (K, τ )
= e−rτ
Z
∞
max(ST − K, 0)q(ST )dST ,
0
with time to maturity τ = T − t, strike price K and risk–free interest
rate r.
What is q(ST )? – A state price density (SPD) of the underlying!
Black–Scholes world: q(ST ) lognormal and unique.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Motivation
1-2
Illustration of SPD estimation
DAX
2150
Density Estimation Illustration
3m DAX TimeSeries
Estimate Diffusion
3m Monte Carlo
Simulation
DAX-implied
g*:
2100
f*:
Option-implied
OptionData
1 tick day
2050
19/12/94
3m Mixture of
3 LogNormals
20/03/95
Time
16/06/95
Figure 1: Illustration of SPD estimation
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Motivation
1-3
Illustration of Probability Trade
• Suppose there are two
SPDs, f ∗ , g ∗ .
Probability Trade
f*
g*
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Figure 2: Probability Trade
• f ∗ : estimated from calls
and puts.
• g ∗ : estimated from a DAX
time series.
• We take the position of g ∗ .
• Consider a butterfly with
payoff function h(ST ).
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Motivation
1-4
The fair price for a ’European’ butterfly spread is given by
Z ∞
Butterfly spread: B = e−rτ
h(ST )q(ST )dST .
0
This density situation implies for the butterfly spread with strikes K1 , K2 , K3
B(f ∗ )
>
B(g ∗ )
price computed with f ∗
>
price computed with g ∗
If the butterfly is priced using f ∗ but one regards g ∗ as a better
approximation of the underylings’ SPD, one would sell it and hedge it
wrt to g ∗ . The price difference (B(f ∗ )-B(g ∗ )) is the gain.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Motivation
1-5
In this work, f ∗ is an option implied SPD and g ∗ is a (historical) time
series SPD.
To compare implied to (historical) time series SPD’s, we use a a mixture
of 3 log–normal densities to estimate f ∗ whereas g ∗ is inferred from a
combination of a non–parametric estimation from a historical time series
of the DAX and a forward Monte Carlo simulation.
Within such a framework, is it profitable to trade ATM probabilities?
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Motivation
1-6
Outline of the talk
X
1.
Motivation
2.
Software and Data
3.
Estimation of the Option Implied SPD
4.
Estimation of the Time Series SPD
5.
Strategy Performance
6.
TO DO
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-1
Software and Data
This study was accomplished using XploRe (http://www.xplore-stat.de).
Daily closing prices of DAX performance index.
DAX 1pm prices at 3rd Friday of March, June, September, December
(to approximate portfolio payoff).
EUREX DAX–option tick prices.
MD*BASE (http://www.mdtech.de) Oracle database.
Time period in this study: 03/95 – 03/01.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-2
Option Data
1.
index
2075.27
2.
strike
2050
3.
risk free interest rate
4.
time to maturity
0.05
5.
option price
19.5
6.
call (1) or put (0)
7.
moneyness (strike/future price)
8.
time (sec after midnight, centiseconds)
9.
implied volatility
10.
number of contracts
11.
date
0.05153
1
0.978
37179.31
0.17696
100
19950102
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-3
Option Data (2)
95.00
71.25
47.50
23.75
2000
2025
2050
2075
2100
2125
2150
2200
Scatterplot of the option prices against strike price (right) on 16-th
January 1995.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-4
Option Data (3)
Inverting the BS formula, implied volatilities of calls and puts can differ
significantly, thus violating the put-call-parity and general market
efficiency considerations.
Hafner & Wallmeier (2001) propose a methodology to derive an
‘adjusted’ DAX index from future prices such that Put–Call–Parity is
satisfied.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
150
Prices
2-5
100
5
0.1+IV*E-2
Software and Data
0
50
Option Data (4)
5
10
0.9+Moneyness*E-2
15
20
5
15
20
Call Prices nach PCP - NbTransacts C:117 P:137
100
Prices
0
50
5
0.1+IV*E-2
150
10
IV nach PCP: Call-K:12 Put-K:15
10
0.9+Moneyness*E-2
5
10
0.9+Moneyness*E-2
15
20
5
10
0.9+Moneyness*E-2
15
20
Figure 3: Hafner, Wallmeier (2000) corrected implied volatility smile on
March 20, 1995.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-6
Option Data (5)
Note: The DAX index is a capital weighted performance index, i.e.
dividends less corporate tax are reinvested into the index.
Hafner & Wallmeier (2001) argue that the marginal investor’s individual
tax scheme is different from the one assumed to compute the DAX index.
Consequently, the net dividend for this investor can be higher or lower
than the one used for the index computation.
This discrepancy, which the authors call ‘difference dividend’ drives a
wedge into the option prices and hence into implied volatilities.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-7
Option Data (6)
Let ∆Dt,T denote the time T value of this difference dividend incurred
between t and T . Let Ft be the futures price with maturity TF and
Ct , Pt be the prices of call and put options with maturity TO .
The ‘adjusted’ index level is given by
S̃t = Ft e−r( TF −t) + ∆Dt,TO ,TF ,
(1)
where ∆Dt,TO ,TF = ∆Dt,TF e−r(TF −t) − ∆Dt,TO e−r(TO −t) is the
desired difference dividend.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Software and Data
2-8
Option Data (7)
The ’adjusted’ index is that index level, which ties put and call implied
volatilities exactly to the same levels when used in the inversion of the
BS formula.
While dividend information is publicly available, the marginals investors
tax rate is unknown.
Assuming Put–Call–Parity holds, ∆Dt,TO ,TF is estimated by :
Ct − Pt = St − ∆Dt,TO e−rH (TO −t) − Ke−rH (TO −t) .
(2)
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-1
Mixture of Log–Normals
We assume that the SPD be a mixture of Log–normal (MLN) densities,
τ
τ
τ
qi,t ST ; αi,t , βi,t :
τ
qM
LN (ST )
=
I
X
τ τ
θi,t
qi,t
τ
τ
ST ; αi,t
, βi,t
i=1
τ
qi,t
(ln(ST ) −
√ exp −
τ
τ
2βi,t
ST βi,t 2π
1
=
τ 2
αi,t
)
!
,
τ
where the log–normal densities qi,t
are independent and
I
X
τ
θi,t
=1 ,
τ
θi,t
> 0,
i
τ
αi,t
= ln(ST ) +
(µτi,t
−
τ
0.5σi,t
)
,
τ
βi,t
=
τ
σi,t
√
τ.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-2
Minimize over parameters (α, β, θ)T squared sum of deviations (SSD)
SSD
=
Call
NX
Ctobs (Kj , τ )
−
2
M LN
Ct
(Kj , τ )
−
2
M LN
Pt
(Kj , τ )
j=1
+
P ut
N
X
Ptobs (Kj , τ )
j=1
+
2
τ
Ftobs (τ ) − EqM
(ST )
LN,t
of observed option/future prices and theoretical option/future prices.
Estimate Pf ∗ (ST ∈ [K1 , K3 ]) =
l=L−1
X
l=0
R K3
K1
τ
qM
LN (ST )dST by
τ
qM
LN (ST,l )(ST,l+1 , −ST,l ),
ST,l
K3 − K 1
= K1 +
l.
L
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-3
Pro/Cons
• Weighted sum of Black–Scholes prices
τ
τ
τ
CtM LN (K, τ ) = θ1,t
C1,t (K, τ ) + θ2,t
C2,t (K, τ ) + θ3,t
C3,t (K, τ )
and
τ
τ
τ
PtM LN (K, τ ) = θ1,t
P1,t (K, τ ) + θ2,t
P2,t (K, τ ) + θ3,t
P3,t (K, τ ).
Thus
– easy and fast to compute,
– ensures no-arbitrage (mon. decreasing and convex in strike K).
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-4
• Flexible
– MLN captures numerous different density shapes, (f. e. bi–modal)
– able to produce smile features such as skewness and kurtosis.
• A SPD is consistent with many different stochastic processes, but
the converse is not true.
• Limited number of observable options → limited number of
parameters.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-5
Application to EUREX DAX–Options
24 non overlapping periods from March 1995 to March 2001
(τ ≈ 88/250 fixed)
period from Monday following 3rd Friday to 3rd Friday 3 months later
Using ≈ 1 day of option data (10am–2pm) to estimate 3m ahead SPD
In this work: J = 3 (Trade–off between flexibility and numerical issues).
Example: on Monday, 20/03/95, we estimate f ∗ of Friday, 16/06/95
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-6
Illustration of the Estimation Procedure
DAX
Density Estimation Illustration
2150
f*
2100
OptionData
1 day
2050
19/12/94
MLN
3 months
20/03/95
Time
16/06/95
Figure 4: Procedure to estimate option implied SPD of Friday, 16/06/95,
estimated on Monday, 20/03/95, by means of 1 day of option tick data.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-7
Estimated SPD
10 15
0
5
Y*E-4
20 25
3 LN Mix: 3, Mo, 19950320 TTM: 88
5
10
15
20
25
500+Underlying*E2
30
35
10 15 20 25
0
5
Y*E-4
7.60, 0.08, 7.57, 0.09, 7.62, 0.07, 0.34, 0.33,
5
10
15
20
25
500+Underlying*E2
30
35
Figure 5: Option implied SPD of Friday, 16/06/95, estimated on Monday,
20/03/95.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Option–Implied SPD
3-8
Comparison of Theoretical and Observed Prices
100
0
50
Y
150
3 LN Mix CallPrices(K):
1
2
3
1800+Strike*E2
4
-0.3 -0.2 -0.1
Y
0
0.1
3 LN Mix: 3, Mo, 19950320 TTM: 88
1
2
3
1800+Underlying*E2
4
Figure 6: Call price function over K and price deviations in percentage.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-1
Estimation of the Time Series SPD
The estimation of the (historical) time series SPD is based on
Aı̈t–Sahalia, Wang & Yared (2001).
S follows a diffusion process
dSt
= µ(St )dt + σ(St )dWt .
Further assume a flat yield curve and the existence of a risk–free asset B
which evolves according to
Bt
= B0 ert .
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-2
Estimation of the Time Series SPD (2)
Then the risk–neutral process follows from Itô’s formula and Girsanov’s
theorem (giving a SPD g ∗ which will later be compared to the SPD f ∗ ):
dSt∗
= rSt∗ dt + σ(St∗ )dWt∗
Drift adjusted but diffusion function is identical in both cases !
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-3
Estimation of the Diffusion Function
Florens–Zmirou (1993), Härdle & Tsybakov (1997) estimator for σ
PN ∗ −1
σ̂ 2 (S)
=
i=1
Kσ (
Si/N ∗ −S
)N ∗ {S(i+1)/N ∗
hσ
PN ∗
Si/N ∗ −S
K
(
)
σ
i=1
hσ
− Si/N ∗ }2
Kσ kernel (in our simulation: Gaussian), hσ bandwidth, N ∗ number of
observed index values (N ∗ ≈ 65) in the time interval [0, 1]
σ̂ consistent estimator of σ as N ∗ → ∞
σ̂ estimated using a 3 month time series of DAX prices
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-4
3200
3400
3600
DAX
3800
0.3
0.2
0.25
DiffusionFct
0.3
DiffusionFct
0.25
0.2
0.2
0.25
DiffusionFct
0.3
0.35
Est. Diffusion for 06-09/01
0.35
Est. Diffusion for 03-06/01
0.35
Est. Diffusion for 03-06/97
5400
5600
5800
DAX
6000
6200
3900
4200
4500
4800 5100
DAX
5400
5700
6000
Figure 7: Estimated diffusion functions (”standardized” by σ̂/S0 ) for
the periods 24/03/97–20/06/97 (hσ = 469.00), 19/03/01–15/06/01
(460.00), 18/06/01–21/09/01 (1597.00) respectively.
regxest.xpl
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-5
Simulation of the Time Series SPD
• To simulate a path use Milstein scheme given by
∗
Si/N
∗
∗
∗
∗
∗
= S(i−1)/N
∗ + rS(i−1)/N ∗ ∆t + σ̂(S(i−1)/N ∗ )∆Wi/N ∗ +
n
o
∂
σ̂
1
∗
∗
∗
2
σ̂(S(i−1)/N
(S
∗)
∗ ) (∆W(i−1)/N ∗ ) − ∆t ,
(i−1)/N
2
∂S ∗
∗
where ∆Wi/N
∗ ∼ N (0, ∆t) with ∆t =
∆σ
∗
approximated by ∆S
∗ , i = 1, . . . , N
1
N∗ ,
drift set equal to r,
∂σ
∂S ∗
∗
• Simulate m = 1, . . . , M = 10000 paths Sm,N
∗ /N ∗ for time to
maturity τ =
N∗
252
• Estimate Pg∗ (ST ∈ [K1 , K3 ]) by
∗
#{m, Sm,N
∗ /N ∗ ∈ [K1 , K3 ] , m = 1, . . . , M }
M
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-6
Application to DAX
24 periods from March 1995 to March 2001 (τ ≈ 65/252 fixed)
period from Monday following 3rd Friday to 3rd Friday 3 months later
Example: on Monday, 20/03/95, we estimate g ∗ of Friday, 16/06/95
• Friday, December 16, 1994, is the 3rd Friday
• σ̂ estimated using DAX prices from Monday, December 19, 1994, to
Friday, March 17, 1995
• Monte–Carlo simulation with parameters
– DAX on Monday March 20, 1995, S0 = 1984.99
– time to maturity τ = 65/250 and interest rate r = 5.10
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
4-7
Comparison of SPD
3m SPD est. on Mo, 18/09/1995
3m SPD est. on Mo, 18/12/2000
DAX 1pm on 15/12/1995
DAX 1pm on 16/03/2001
g*
SPD
SPD
f*
g*
0
0
f*
1800
2300
Underlying
2800
3500
5500
7500
9500
Underlying
Figure 8: Option- & DAX-implied SPD of Fr, 15/12/95, & Fr, 16/03/01.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Estimation of the Time Series SPD
DAX
2150
4-8
Density Estimation Illustration
3m DAX TimeSeries
Estimate Diffusion
3m Monte Carlo
Simulation
DAX-implied
g*:
2100
f*:
Option-implied
2050
19/12/94
OptionData
1 tick day
3m Mixture of
3 LogNormals
20/03/95
Time
16/06/95
Figure 9: Comparison of procedures to estimate time series and option
implied SPD of Friday, 16/06/95. SPD’s estimated on Monday, 20/03/95,
by means of 3 months of index data respectively 1 day of option tick data.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-1
Strategy Performance
24 periods of 3 months length from March 1995 to March 2001.
We start with a bank account deposit of 100 units (=index points).
• t=0:
– Monday following 3rd Friday of Mar, Jun, Sep, Dec
– Compare SPD’s (Signal)
– Set up butterfly portfolio (Long/Short)
• 0 ≤ t ≤ T : Hedge option portfolio daily (Black–Scholes delta)
• t=T :
– 3rd Friday 3 months later (i.e. of Mar, Jun, Sep, Dec)
– Compute payoff from butterfly
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-2
Unhedged Positions – Signals
Buy–Signal: Pf ∗ (ST ∈ [K1 , K3 ]) < Pg∗ (ST ∈ [K1 , K3 ])
correct prediction if ST ∈ [K1 , K3 ] (payoff at T positive)
Sell–Signal: Pf ∗ (ST ∈ [K1 , K3 ]) > Pg∗ (ST ∈ [K1 , K3 ])
correct prediction if ST ∈
/ [K1 , K3 ] (payoff 0 and not negative)
Moneyness
# correct signals
0.90–0.95
0.95–1.00
1.00–1.05
1.05–1.10
ITM (t=T)
10
9
7
14
15
Table 1: Number of correct signals (out of 24) for different moneyness
(strike/future price) intervals.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-3
Unhedged Positions – Option Cash Flows
Cash flows:
• Cash flow in t = 0,
CF0 : Buying/selling options
• Cash flow in t = T,
CFT : Payoff of options in portfolio
• Total cash flow: T CF = CF0 + CFT
Strategies:
(a) In each period consider only butterflies that are ATM in t = 0.
(b) In each period consider only butterflies that end up ITM in t = T .
Performance:
Compare conditional on signal and unconditional long/short.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-4
Unhedged Positions – Option Cash Flows (2)
ATM (in t = 0)
ITM (in t=T)
Conditional
-24.21
128.83
Uncond. Long
-219.41
1179.37
Uncond. Short
219.41
-1179.37
Table 2: Total Cash Flows in Index Points
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-5
Unhedged Positions – Option Cash Flows (3)
0
-40
-20
CashFlow
20
40
OptionCashFlows
0
5
10
15
20
25
Time
Figure 10: Cash flows from buying/selling and terminal payoffs of butterfly
portfolio conditional on density comparison.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-6
Hedged Positions - Portfolio Value
ATM (in t = 0)
ITM (in t=T)
Conditional
6.13/-42.79%
266.99/21.70%
Uncond. Long
-7.58/-100%
444.47/34.76%
Uncond. Short
257.73/20.85%
-194.32/-100%
Table 3: Portfolio Value at t=T in Index Points/Yield (annualized)
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-7
Portfolio Value
0
100
Pf-Value
200
PortfolioValue
0
5
10
15
Time*E2
Figure 11: Portfolio value conditional on density comparison.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
Strategy Performance
5-8
DAX Performance
DAX 4.1994-4.2001
DAX
8000
7000
6000
5000
4000
3000
2000
1/95
1/96
1/97
1/98
1/99
1/00
1/01
Time
12/01
Figure 12: DAX Performance.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
TO DO
6-1
TO DO
Estimate diffusion function from DAX tick data.
Compute delta hedge numerically using g ∗ .
Improve Black–Scholes delta hedge by smoothing implied volatility curve.
Improve trading signal by testing for significant inequality of densities.
Consider transaction costs for option and stock transactions.
Strategy performance conditional for all butterfly portfolios (not only
subset).
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
References
7-1
References
Aı̈t–Sahalia, Y., Wang, Y. & Yared, F. (2001). Do Option Markets
correctly Price the Probabilities of Movement of the Underlying
Asset?, Journal of Econometrics 102: 67–110.
Bahra, B. (1995). Implied risk-neutral probability density functions from
option prices: theory and application, Bank of England Working
Papers 66.
Black, F. & Scholes, M., (1998). The Pricing of Options and Corporate
Liabilities, Journal of Political Economy 81: 637–659.
Breeden, D. & Litzenberger, R., (1978). Prices of State Contingent
Claims Implicit in Option Prices, Journal of Business, 9, 4: 621–651.
Florens–Zmirou, D. (1993). On Estimating the Diffusion Coefficient from
Discrete Observations, Journal of Applied Probability 30: 790–804.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3
References
7-2
Franke, J., Härdle, W. & Hafner, C. (2003). Einführung in die Statistik
der Finanzmärkte, 2nd edition, Springer Verlag, Heidelberg.
Hafner, R. & Wallmeier, M., (2001). The Dynamics of DAX Implied
Volatilities, International Quarterly Journal of Finance,1 1:1 –27.
Härdle, W. & Tsybakov, A., (1997). Local Polynomial Estimators of the
Volatility Function in Nonparametric Autoregression, Journal of
Econometrics, 81: 223–242.
Kloeden, P., Platen, E. & Schurz, H. (1994). Numerical Solution of SDE
Through Computer Experiments, Springer Verlag, Heidelberg.
Rubinstein, M. (1994). Implied Binomial Trees, Journal of Finance
49: 771–818.
Yatchew, A. & Härdle, W. (2003). Dynamic Nonparametric State Price
Density Estimation using Constrained Least Squares and the
Bootstrap, Journal of Econometrics, accepted.
Probability Trade
f*
g*
Probability Trading
Sell Butterfly
h(S_T)
K_1
K_2
K_3