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  • 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

    l =.

    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

    ACF

    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

    ACF

    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

    Call o

    ptio

    n pr

    ice

    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 tR.E(BHt B

    Hs

    )= 12

    [| t |2H + | s |2H | t s |2H] , t,sR.When H = 12 the process has independent increments and corresponds toBrownian motion. But when 12 < H 1 the process is said to have longrange 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 Vt1 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 Shiryayevs Arbitrage

    Time

    Portf

    olio

    va

    lue

    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 and Hurst exponent H as is

    C0 = sSD(

    d1

    ) kertSD

    (d2

    ),

    where

    d1 =

    ln(sK

    )+ rt + 12

    2t2H

    tH,

    d2 =

    ln(sK

    )+ rt 122t2HtH

    .

    In the expression above SD() is the cumulative distribution function ofStable distribution.

    Megh Shah Option Pricing with Long Range Dependence

  • Hu and ksendal Model

    For stock price s, strike price K , time left for maturity t, volatility andHurst exponent H the European call option price is given as

    C0 = sN(

    d1

    ) kertN

    (d2

    )where

    d1 =

    ln(sK

    )+ rt + 12

    2t2H

    tH,

    d2 =

    ln(sK

    )+ rt 122t2HtH

    .

    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 = S0et+Tt+B(Tt),

    where Tt is a positive non-decreasing random process with stationary butnot necessarily independent increments, denoted over unit time byt = Tt Tt1. , and > 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 (St1) d= + t + 12t B (1) .

    Student t FATGBM Model

    If t is Inverse Gamma (R) distributed with parameters (, ) then thisresults in Xt having marginal (skew) t distribution with v degrees offreedom where v = 2.

    Variance Gamma FATGBM Model

    If t is gamma () distributed with parameters (, ) 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 (St1) d= + t + 12t B (1) .

    Student t FATGBM Model

    If t is Inverse Gamma (R) distributed with parameters (, ) then thisresults in Xt having marginal (skew) t distribution with v degrees offreedom where v = 2.

    Variance Gamma FATGBM Model

    If t is gamma () distributed with parameters (, ) 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( StK )+rt+

    12

    2u

    u

    ) KertN

    (ln( StK )+rt 122u

    u

    )]

    tH fR(ut+tH

    tH ;v2 ,

    v22

    )du.

    Option pricing in Variance Gamma FATGBM Model

    C (t,K ) =

    0

    [StN

    (ln( StK )+rt+

    12

    2u

    u

    ) KertN

    (ln( StK )+rt 122u

    u

    )]

    tH f(ut+tH

    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( StK )+rt+

    12

    2u

    u

    ) KertN

    (ln( StK )+rt 122u

    u

    )]

    tH fR(ut+tH

    tH ;v2 ,

    v22

    )du.

    Option pricing in Variance Gamma FATGBM Model

    C (t,K ) =

    0

    [StN

    (ln( StK )+rt+

    12

    2u

    u

    ) KertN

    (ln( StK )+rt 122u

    u

    )]

    tH f(ut+tH

    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 () =

    1n

    nk=1

    ek()2 where ek = Ck C ().

    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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    2030

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    BS calibrated Price vs Market prices

    Strike prices

    Opt

    ion pr

    ices

    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 ksendal Model

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    4050

    Hu and Oksendal's model calibrated Price vs Market prices

    Strikeprices

    Opt

    ion pr

    ices

    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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    4050

    Student t FATGBM model calibrated Price vs Market prices

    Strikeprices

    Opt

    ion pr

    ices

    95 105 115 125 80 95 115 135 85 105 125 145 165 90 110 130 150

    x Student t F