Option Pricing with Long Range Dependence
Megh Shah
Thesis Supervised by Dr. Andriy OlenkoDepartment of Mathematics and Statistics
La Trobe University
Masters in Statistical Science, 2011
Megh Shah Option Pricing with Long Range Dependence
Long Range Dependence
Definition of Long Range Dependence
Long range dependency for a stationary process is defined as
∞∑l=1
γl =∞.
Long range dependency means that events that happened a longtime ago would still have an impact on the present or future valuesof the process.
In contrast, short range dependency presupposes that theautocovariance decays fast enough to be summable.
Megh Shah Option Pricing with Long Range Dependence
Autocorrelation in Stock Returns
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
ACF plot of S&P 500 Returns from 4/1/1990 to 31/8/2011
Megh Shah Option Pricing with Long Range Dependence
Long Range Dependence in Squared Stock Returns
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
ACF plot of S&P 500 Squared Returns from 4/1/1990 to 31/8/2011
Megh Shah Option Pricing with Long Range Dependence
Call Option Payoff
Call Option: The option contract that gives the right but not theobligation to buy the underlying contract (currency, stocks, interestrates, commodity, bonds etc) is termed a call option.The payoff for a European call option C with a given strike price Kand stock price s at expiry is given as
C = Max (s − K , 0) .
60 80 100 120 140
010
2030
4050
Payoff of the European Call Option at expiry
Strike price
Cal
l opt
ion
pric
e
Stock price=100In the money CallsOut of the Money Calls
At the money Call
Megh Shah Option Pricing with Long Range Dependence
Fractional Brownian Motion
Fractional Brownian motion is capable of capturing long rangedependence.
Properties of fractional Brownian motion
BH0 = 0
E(BHt
)= 0 ∀ t∈R.
E(BHt BH
s
)= 1
2
[| t |2H + | s |2H − | t − s |2H
], ∀ t,s∈R.
When H = 12 the process has independent increments and corresponds to
Brownian motion. But when 12 < H ≤ 1 the process is said to have long
range dependence or long memory.
Megh Shah Option Pricing with Long Range Dependence
Arbitrage
Arbitrage is a strategy such that you make a “riskless profit” beyondthe risk free rate.
This strategy must be self-financing. The change in the portfolio isbecause of the change in the value of the asset without money beingwithdrawn or added to the portfolio.
Arbitrage strategy for a portfolio Vt
1 V0 = 0, the initial value of this strategy is 0.
2 ∃ t such that
P(Vt ≥ 0) = 1 which states that the portfolio would have a valuegreater than 0 almost surely.
P(Vt > 0) > 0, which means that we win with non zero probability.
Megh Shah Option Pricing with Long Range Dependence
Arbitrage in Fractional Brownian Markets
Simulation of Shiryayev’s Arbitrage
Time
Por
tfol
io v
alue
0 0.04 0.098 0.16 0.218 0.28 0.338 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
02
46
810
Megh Shah Option Pricing with Long Range Dependence
Los and Jaimdee Model
The option price for stock price s, strike price K , time left for maturity t,volatility σ and Hurst exponent H as is
C0 = sSD(
d′
1
)− ke−rtSD
(d′
2
),
where
d′
1 =ln(
sK
)+ rt + 1
2σ2t2H
σtH,
d′
2 =ln(
sK
)+ rt − 1
2σ2t2H
σtH.
In the expression above SD() is the cumulative distribution function ofStable distribution.
Megh Shah Option Pricing with Long Range Dependence
Hu and Øksendal Model
For stock price s, strike price K , time left for maturity t, volatility σ andHurst exponent H the European call option price is given as
C0 = sN(
d′
1
)− ke−rtN
(d′
2
)where
d′
1 =ln(
sK
)+ rt + 1
2σ2t2H
σtH,
d′
2 =ln(
sK
)+ rt − 1
2σ2t2H
σtH.
Megh Shah Option Pricing with Long Range Dependence
Long Range Dependencies in Asset Prices using FractalActivity Time Model (FATGBM)
The subordinator model describes stock price St dynamics as
St = S0eµt+θTt+σB(Tt),
where Tt is a positive non-decreasing random process with stationary butnot necessarily independent increments, denoted over unit time byτt = Tt − Tt−1. µ, θ and σ > 0 are all constants.
Features of FATGBM Model
Skewess and leptokurtosis in returns.
ACF for returns would not display long memory but squared or absolute returnswould.
Stochastic volatility in returns.
Returns can be modelled using heavy tailed or semi-heavy tailed distribution.
Aggregational gaussianity in real returns.
Arbitrage would not be possible under an appropriate change of probabilitymeasure.
Megh Shah Option Pricing with Long Range Dependence
FATGBM Models
The distribution of stock returns Xt in FATGBM model is
Xt = log (St)− log (St−1)d= µ+ θτt + στ
12t B (1) .
Student t FATGBM Model
If τt is Inverse Gamma (RΓ) distributed with parameters (α, β) then thisresults in Xt having marginal (skew) t distribution with v degrees offreedom where v = 2α.
Variance Gamma FATGBM Model
If τt is gamma (Γ) distributed with parameters (α, λ) then this results inXt having a marginal (skew) variance gamma distribution.
Megh Shah Option Pricing with Long Range Dependence
FATGBM Models
The distribution of stock returns Xt in FATGBM model is
Xt = log (St)− log (St−1)d= µ+ θτt + στ
12t B (1) .
Student t FATGBM Model
If τt is Inverse Gamma (RΓ) distributed with parameters (α, β) then thisresults in Xt having marginal (skew) t distribution with v degrees offreedom where v = 2α.
Variance Gamma FATGBM Model
If τt is gamma (Γ) distributed with parameters (α, λ) then this results inXt having a marginal (skew) variance gamma distribution.
Megh Shah Option Pricing with Long Range Dependence
Option Pricing in FATGBM model
Option pricing in Student t FATGBM Model
C (t,K ) =∫∞
0
[StN
(ln( St
K )+rt+ 12 σ
2u
σ√u
)− Ke−rtN
(ln( St
K )+rt− 12 σ
2u
σ√u
)]×
t−H fRΓ
(u−t+t−H
tH ; v2 ,
v−22
)du.
Option pricing in Variance Gamma FATGBM Model
C (t,K ) =∫∞
0
[StN
(ln( St
K )+rt+ 12 σ
2u
σ√u
)− Ke−rtN
(ln( St
K )+rt− 12 σ
2u
σ√u
)]×
t−H fΓ
(u−t+t−H
tH ; v2 ,
v2
)du.
Megh Shah Option Pricing with Long Range Dependence
Option Pricing in FATGBM model
Option pricing in Student t FATGBM Model
C (t,K ) =∫∞
0
[StN
(ln( St
K )+rt+ 12 σ
2u
σ√u
)− Ke−rtN
(ln( St
K )+rt− 12 σ
2u
σ√u
)]×
t−H fRΓ
(u−t+t−H
tH ; v2 ,
v−22
)du.
Option pricing in Variance Gamma FATGBM Model
C (t,K ) =∫∞
0
[StN
(ln( St
K )+rt+ 12 σ
2u
σ√u
)− Ke−rtN
(ln( St
K )+rt− 12 σ
2u
σ√u
)]×
t−H fΓ
(u−t+t−H
tH ; v2 ,
v2
)du.
Megh Shah Option Pricing with Long Range Dependence
Calibrating Option Prices
Loss functions compute the difference in the model price andobserved market price of the option.
$RMSE (θ) =
√√√√1
n
n∑k=1
ek(θ)2 where ek = Ck − C (θ).
By minimizing these loss functions using an optimization routine wecan calibrate the pricing model.
Megh Shah Option Pricing with Long Range Dependence
Calibrated Option Prices in Black Scholes Model
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BS calibrated Price vs Market prices
Strike prices
Opt
ion
pric
es
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
x BS PriceMarket Price
November contracts
December contracts
January contracts
April contracts
$RMSE=$6.29
Megh Shah Option Pricing with Long Range Dependence
Calibrated Option Prices in Hu and Øksendal Model
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Hu and Oksendal's model calibrated Price vs Market prices
Strikeprices
Opt
ion
pric
es
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
x Hu and Oksendal PriceMarket Price
November contracts
December contracts
January contracts
April contracts
$RMSE=2.25
Megh Shah Option Pricing with Long Range Dependence
Calibrated Option Prices in Student t FATGBM Model
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2030
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Student t FATGBM model calibrated Price vs Market prices
Strikeprices
Opt
ion
pric
es
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
x Student t FATGBM PriceMarket Price
November contracts
December contracts
January contracts
April contracts
$RMSE=$2.24
Megh Shah Option Pricing with Long Range Dependence
Calibrated Option Prices in Variance Gamma FATGBMModel
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Variance Gamma FATGBM model calibrated Price vs Market prices
Strikeprices
Opt
ion
pric
es
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
x Variance Gamma FATGBM PriceMarket Price
November contracts
December contracts
January contracts
April contracts
$RMSE=$2.23
Megh Shah Option Pricing with Long Range Dependence
Calibrated Parameters and $RMSE Values
Parameters
Models σ H vBlack Scholes 0.7669898
Hu and Øksendal 0.424934 0.51000Student t FATGBM 0.4102277 0.8781472 44.57739
Variance Gamma FATGBM 0.422940 0.851147 53.028287
Model $RMSE Error
Black Scholes 6.290929Hu and Øksendal 2.250464
Student t FATGBM 2.244872Variance Gamma FATGBM 2.236839
Megh Shah Option Pricing with Long Range Dependence
Contribution
My contribution in this thesis:
Applied the modified ITo’s formula to develop a portfolio strategywhich demonstrates arbitrage in fractional Brownian motion settingwith derivation and simulation.
Critically reviewed Jamdee,S. & Los, C. (2007) Long memoryoptions: LM evidence and simulations. Research in InternationalBusiness and Finance 21(2), Pages 260-280.
Justification for the measure change from real world measure toskew corrected martingale measure is given for FATGBM modelsalong with detailed proof for pricing European style options in theFATGBM models.
R codes to calibrate and compare four models versus market prices.
Megh Shah Option Pricing with Long Range Dependence