1. Option Pricing with Long Range Dependence
Megh Shah
Thesis Supervised by Dr. Andriy Olenko
Department of Mathematics and Statistics
La Trobe University
Masters in Statistical Science, 2011
Megh Shah Option Pricing with Long Range Dependence

2. Long Range Dependence
Definition of Long Range Dependence
Long range dependency for a stationary process is defined as
∞
γl = ∞.
l=1
Long range dependency means that events that happened a long
time ago would still have an impact on the present or future values
of the process.
In contrast, short range dependency presupposes that the
autocovariance decays fast enough to be summable.
Megh Shah Option Pricing with Long Range Dependence

3. Autocorrelation in Stock Returns
ACF plot of S&P 500 Returns from 4/1/1990 to 31/8/2011
1.0
0.8
0.6
ACF
0.4
0.2
0.0
0 50 100 150
Lag
Megh Shah Option Pricing with Long Range Dependence

4. Long Range Dependence in Squared Stock Returns
ACF plot of S&P 500 Squared Returns from 4/1/1990 to 31/8/2011
1.0
0.8
0.6
ACF
0.4
0.2
0.0
0 50 100 150
Lag
Megh Shah Option Pricing with Long Range Dependence

5. Call Option Payoff
Call Option: The option contract that gives the right but not the
obligation to buy the underlying contract (currency, stocks, interest
rates, commodity, bonds etc) is termed a call option.
The payoff for a European call option C with a given strike price K
and stock price s at expiry is given as
C = Max (s − K , 0) .
Payoff of the European Call Option at expiry
50
Stock price=100
In the money Calls
40
Out of the Money Calls
Call option price
30
At the money Call
20
10
0
60 80 100 120 140
Strike price
Megh Shah Option Pricing with Long Range Dependence

6. Fractional Brownian Motion
Fractional Brownian motion is capable of capturing long range
dependence.
Properties of fractional Brownian motion
H
B0 = 0
E BtH = 0 ∀ t∈R.
1
E BtH Bs =
H
2 | t |2H + | s |2H − | t − s |2H , ∀ t,s∈R.
When H = 1 the process has independent increments and corresponds to
2
Brownian motion. But when 1 < H ≤ 1 the process is said to have long
2
range dependence or long memory.
Megh Shah Option Pricing with Long Range Dependence

7. Arbitrage
Arbitrage is a strategy such that you make a “riskless profit” beyond
the risk free rate.
This strategy must be self-financing. The change in the portfolio is
because of the change in the value of the asset without money being
withdrawn or added to the portfolio.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 value
greater 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

8. Arbitrage in Fractional Brownian Markets
Simulation of Shiryayev’s Arbitrage
10
8
6
Portfolio value
4
2
0
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
Time
Megh Shah Option Pricing with Long Range Dependence

9. 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 d1 − ke −rt SD d2 ,
where
s
ln K + rt + 1 σ 2 t 2H
2
d1 = ,
σt H
s
ln + rt − 1 σ 2 t 2H
K 2
d2 = .
σt H
In the expression above SD() is the cumulative distribution function of
Stable distribution.
Megh Shah Option Pricing with Long Range Dependence

10. Hu and Øksendal Model
For stock price s, strike price K , time left for maturity t, volatility σ and
Hurst exponent H the European call option price is given as
C0 = sN d1 − ke −rt N d2
where
s
ln K + rt + 1 σ 2 t 2H
2
d1 = ,
σt H
s
ln K + rt − 1 σ 2 t 2H
2
d2 = .
σt H
Megh Shah Option Pricing with Long Range Dependence

11. Long Range Dependencies in Asset Prices using Fractal
Activity Time Model (FATGBM)
The subordinator model describes stock price St dynamics as
St = S0 e µt+θTt +σB(Tt ) ,
where Tt is a positive non-decreasing random process with stationary but
not necessarily independent increments, denoted over unit time by
τt = 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 returns
would.
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 probability
measure.
Megh Shah Option Pricing with Long Range Dependence

12. FATGBM Models
The distribution of stock returns Xt in FATGBM model is
1
d
Xt = log (St ) − log (St−1 ) = µ + θτt + στt2 B (1) .
Student t FATGBM Model
If τt is Inverse Gamma (RΓ) distributed with parameters (α, β) then this
results in Xt having marginal (skew) t distribution with v degrees of
freedom where v = 2α.
Variance Gamma FATGBM Model
If τt is gamma (Γ) distributed with parameters (α, λ) then this results in
Xt having a marginal (skew) variance gamma distribution.
Megh Shah Option Pricing with Long Range Dependence

13. FATGBM Models
The distribution of stock returns Xt in FATGBM model is
1
d
Xt = log (St ) − log (St−1 ) = µ + θτt + στt2 B (1) .
Student t FATGBM Model
If τt is Inverse Gamma (RΓ) distributed with parameters (α, β) then this
results in Xt having marginal (skew) t distribution with v degrees of
freedom where v = 2α.
Variance Gamma FATGBM Model
If τt is gamma (Γ) distributed with parameters (α, λ) then this results in
Xt having a marginal (skew) variance gamma distribution.
Megh Shah Option Pricing with Long Range Dependence

14. Option Pricing in FATGBM model
Option pricing in Student t FATGBM Model
∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 1 σ 2 u
C (t, K ) = 0
St N K
√ 2
σ u
− Ke −rt N K
√ 2
σ u
×
u−t+t −H v v −2
t −H fRΓ tH
; 2, 2 du.
Option pricing in Variance Gamma FATGBM Model
∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 2 σ 2 u
1
C (t, K ) = 0
St N K
√ 2
σ u
− Ke −rt N K
√
σ u
×
u−t+t −H v v
t −H fΓ tH
; 2, 2 du.
Megh Shah Option Pricing with Long Range Dependence

15. Option Pricing in FATGBM model
Option pricing in Student t FATGBM Model
∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 1 σ 2 u
C (t, K ) = 0
St N K
√ 2
σ u
− Ke −rt N K
√ 2
σ u
×
u−t+t −H v v −2
t −H fRΓ tH
; 2, 2 du.
Option pricing in Variance Gamma FATGBM Model
∞ ln( St )+rt+ 1 σ 2 u ln( St )+rt− 2 σ 2 u
1
C (t, K ) = 0
St N K
√ 2
σ u
− Ke −rt N K
√
σ u
×
u−t+t −H v v
t −H fΓ tH
; 2, 2 du.
Megh Shah Option Pricing with Long Range Dependence

16. Calibrating Option Prices
Loss functions compute the difference in the model price and
observed market price of the option.
n
1
$RMSE (θ) = ek (θ)2 where ek = Ck − C (θ).
n
k=1
By minimizing these loss functions using an optimization routine we
can calibrate the pricing model.
Megh Shah Option Pricing with Long Range Dependence

17. Calibrated Option Prices in Black Scholes Model
BS calibrated Price vs Market prices
50
x BS Price
Market Price
November December
40
$RMSE=$6.29
contracts contracts
January April
contracts contracts
x
x
30
x x
Option prices
x x
x
x
x x x
x x
20
x x
x x
x x x
x x
x x x
xx x
x x x
x x
10
xx x x
x
xx x x
x x
xx x xx
xx xx xx
xx x x xx xx
0
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
Strike prices
Megh Shah Option Pricing with Long Range Dependence

18. Calibrated Option Prices in Hu and Øksendal Model
Hu and Oksendal's model calibrated Price vs Market
prices
50
x Hu and Oksendal Price
Market Price
November December $RMSE=2.25
40
contracts contracts
January April
contracts contracts
30
x
Option prices
x x
x
x x
x
20
x x
x x
x x
x x
xx x x
10
x x x
x
xx x x x
x x x
xx x xx
xx x xx xx
xx xxx x
x x xx x x x x xx x xx x
0
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
Strike
prices
Megh Shah Option Pricing with Long Range Dependence

19. Calibrated Option Prices in Student t FATGBM Model
Student t FATGBM model calibrated Price vs Market prices
50
x Student t FATGBM Price
Market Price
November December
40
$RMSE=$2.24
contracts contracts
January April
contracts contracts
30
x
Option prices
x x
x
x x
x
20
x x
x
x x
x
x x
xx x x
10
x x x
xx x x
x x
x x x
xx x xx
xx x xx xx
xx xxx x
x x xx x x x x xx x x xx
0
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
Strike
prices
Megh Shah Option Pricing with Long Range Dependence

20. Calibrated Option Prices in Variance Gamma FATGBM
Model
Variance Gamma FATGBM model calibrated Price vs Market prices
50
x Variance Gamma FATGBM Price
Market Price
November December
40
$RMSE=$2.23
contracts contracts
January April
contracts contracts
30
x
Option prices
x x
x
x x
x
20
x x
x x
x x
x x
xx x x
10
x x x
x
xx x x x
x x x
xx x xx
x xx xx
xx xx x x xx x
x x xx x x x xx x xxx
0
95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150
Strike
prices
Megh Shah Option Pricing with Long Range Dependence

21. Calibrated Parameters and $RMSE Values
Parameters
Models σ H v
Black Scholes 0.7669898
Hu and Øksendal 0.424934 0.51000
Student t FATGBM 0.4102277 0.8781472 44.57739
Variance Gamma FATGBM 0.422940 0.851147 53.028287
Model $RMSE Error
Black Scholes 6.290929
Hu and Øksendal 2.250464
Student t FATGBM 2.244872
Variance Gamma FATGBM 2.236839
Megh Shah Option Pricing with Long Range Dependence

22. Contribution
My contribution in this thesis:
Applied the modified ITo’s formula to develop a portfolio strategy
which demonstrates arbitrage in fractional Brownian motion setting
with derivation and simulation.
Critically reviewed Jamdee,S. & Los, C. (2007) Long memory
options: LM evidence and simulations. Research in International
Business and Finance 21(2), Pages 260-280.
Justification for the measure change from real world measure to
skew corrected martingale measure is given for FATGBM models
along with detailed proof for pricing European style options in the
FATGBM models.
R codes to calibrate and compare four models versus market prices.
Megh Shah Option Pricing with Long Range Dependence