Megh's slides

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.


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


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


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


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.


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.


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


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.


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.


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.


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.


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.


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.


Calibrated Option Prices in Black Scholes Model

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xxxxx

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010

2030

4050

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


Calibrated Option Prices in Hu and Øksendal Model

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xxxxxxx

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2030

4050

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


Calibrated Option Prices in Student t FATGBM Model

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xxxxxx

010

2030

4050

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


Calibrated Option Prices in Variance Gamma FATGBMModel

x

xx

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xxxxxx

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xxxxxxx

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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


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


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.


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