ARMA, GARCH and Related Option Pricing Method

ARMA, GARCH and Related Option Pricing Method Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook September 21, 2012

Outline 1 Introduction Time Series Model Difference Equation 2 Autoregressive Model ACF and PACF 3 Trends and Volatility ARCH Process The GARCH Model 4 Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model

Time Series Model Time Series Model Difference Equation Time Series Models Objective: forcasting, interpreting and testing hypotheses data Method: decompose a series into a trend, seasonal, cyclical and irregular component these components can be represented as difference equation a difference equation expresses the value of a variable as a function of its own lagged values, time and other variables usually trend and seasonal term are both functions of time irregular term is a function of its own lagged value and stochastic variable many economic and finance theories have natural representation as stochastic differential equation

Time Series Model (Cont.) Time Series Model Difference Equation Figure: The Austrilian Red Wine Sales, Jan. 1980-Oct. 1991

Difference Equation Time Series Model Difference Equation The common type of difference equation is n y t = a 0 + a i y t i + x t where x t is the forcing process. And a special case for {x t} is i=1 x t = β i ɛ t i i=0 where β i are constants and the individual elements {ɛ t} are independent of y t.

Solution by Iteration Time Series Model Difference Equation first-order difference equation y t = a 0 + a 1 y t 1 + ɛ t with initial condition y 0 have solution t 1 y t = a 0 a1 i + at 1 y t 1 0 + a1 i ɛ t i i=0 i=0 the case without initial condition, substitute y 0 = a 0 + a 1 y 1 + ɛ 0 t 1 y t = a 0 a1 i + at 1 y t 1 0 + a1 i ɛ t i = a 0 i=0 i=0 t i=0 a i 1 + at+1 1 y 1 + t+m t+m = a 0 a1 i + at+m+1 1 y m 1 + a1 i ɛ t i i=0 i=0 t a1 i ɛ t i If a 1 < 1, then a t+m+1 1 y m 1 approaches 0 as m approaches. Thus y t = a 0 1 a 1 + a1 i ɛ t i However, the general solution is y t = Aa t 1 + a 0 1 a 1 + i=0 ai 1 ɛ t i i=0 i=0

Lag Operators Time Series Model Difference Equation Definition The lag operator L is defined to be a linear operator such that for any value y t L i y t = y t i. the lag of a constant is a constant Lc = c distributive law ( L i + L j) y t = L i y t + L j y t = y t i + y t j associative law L i L j y t = L i ( L j y t ) = L i y t j = y t i j L rasied to a negative power is a lead factor L i y t = y t+i For a < 1, ( 1 + al + a 2 L 2 + ) y t = yt 1 al the pth-order equation y t = a 0 + a 1y t 1 + + a py t p + ɛ t can be written (1 a 1L a pl p ) y t = A (L) y t = a 0 + ɛ t

White Noise Autoregressive Model ACF and PACF Definition The sequence {ɛ t } is a white-noise process if for each time period t and for all j and all s E (ɛ t) = E (ɛ t 1) = = 0 ( ) ( ) E ɛ 2 t = E ɛ 2 t 1 = = σ 2 ( ( ) ( ) or var ɛ 2 t = var ɛ 2 t 1 = = σ 2) E (ɛ tɛ t s) = E (ɛ t j ɛ t j s ).

Autoregressive Model ACF and PACF Moving Average and Autoregressive Model Definition The moving average model of order q denoted as MA (q) can be written as x t = q β i ɛ t i i=0 where β 0,, β q are the parameters of the model and {ɛ t} are white noise. Definition The autoregressive model of order p denoted as AR (p) which can be written as y t = a 0 + p a i y t i + ɛ t i=1 where a 1,, a p are parameters, a 0 is a constant and the random variable {ɛ t} are white noise.

Autoregressive Model ACF and PACF Autoregressive Moving Average Model Definition Combine moving average model with autoregressive model, we get autoregressive moving average (ARMA) model denoted as ARMA (p, q) y t = a 0 + p a i y t i + ɛ t + i=1 q β i ɛ t i if the characteristic roots of the equation are all in unit circle. If one or more characteristic roots is greater than or equal to unity, the sequence {y t} is an integrated process and is called an autoregressive integrated moving average (ARIMA) model. Rewrite the model with lag operators i=1 p q 1 a i L i y t = a 0 + β i ɛ t i i=1 i=0 then the particular solution is y t = a 0 + q ( i=0 β i ɛ t i 1 p ). i=1 a i Li

Stationary Autoregressive Model ACF and PACF Definition A stochastic process {y t} is stationary if for all t and t s E E (y t) = E (y t s) = µ [ (y t µ) 2] = E [ (y t s µ) 2] = σ 2 E [(y t µ) (y t s µ)] = E [(y t j µ) (y t j s µ)] = γ s where µ, σ 2, γ s are all constants. γ s is autocovariance function (ACF). For stationary process {y t}, the mean, variance and autocorrelations can be well approximated by sufficiently long time averages based on the single set of realization.

Autoregressive Model ACF and PACF Stationary Restrictions for ARMA(p,q) (Cont.) Consider MA ( ) model x t = i=0 β iɛ t i Expectation E (x t) = E (ɛ t + β 1ɛ t 1 + β 2ɛ t 2 + ) = 0 constant and independent of t. Variance [ Var (x t) = E (ɛ t + β 1ɛ t 1 + β 2ɛ t 2 + ) 2] ( ) = σ 2 1 + β1 2 + β2 2 + constant and independent of t. Covariance E (x tx t s) = E [(ɛ t + β 1ɛ t 1 + ) (ɛ t s + β 1ɛ t s 1 + )] = σ 2 (β s + β 1β s+1 + β 2β s+2 + ) only depend on s and the value should be finite. By constraint i=0 β2 i <, ARMA (p, q) can be transformed to MA ( ).

Autocorrelation Function Autoregressive Model ACF and PACF Definition The autocorrelation function (ACF) of y t and y t s is defined by autocovariance function γ s as ρ s = γs /γ 0. Consider ARMA (1, 1) case y t = a 1y t 1 + ɛ t + β 1ɛ t 1, then Yule-Walker equations are: Ey ty t = a 1Ey t 1y t +Eɛ ty t +β 1Eɛ t 1y t γ 0 = a 1γ 1 +σ 2 +β 1 (a 1 + β 1) σ 2 Ey ty t 1 = a 1Ey t 1y t 1 + Eɛ ty t 1 + β 1Eɛ t 1y t 1 γ 1 = a 1γ 0 + β 1σ 2 Ey ty t s = a 1Ey t 1y t s+eɛ ty t s+β 1Eɛ t 1y t s γ s = a 1γ s 1, s = 2, 3,

Autocorrelation Function (Cont.) Autoregressive Model ACF and PACF Solve first two equations, we get γ 0 = 1 + β2 1 + 2a 1β 1 σ 2, γ 1 a1 2 1 = hence, we get autocorrelation function ρ 1 = (1 + a1β1) (a1 + β1) 1 + β 2 1 + 2a1β1 For all s 2, given γ s = a 1γ s 1, we have ρ s = a 1ρ s 1. (1 + a1β1) (a1 + β1) σ 2 1 a1 2 If 0 < a 1 < 1, ACF converges directly; if 1 < a 1 < 0, ACF will oscillate.

Autocorrelation Function (Cont.) Autoregressive Model ACF and PACF Figure: The ACF of AR (2), a 1 = 2, a 2 = 5 and AR (2), a 1 = 10/9, a 2 = 2

Autocorrelation Function (Cont.) Autoregressive Model ACF and PACF ( Figure: The ACF of AR (2), a 1 = 10/9, a 2 = 2 and AR (2), a 1 = 2 1+i ) ( 3 /3, a 2 = 2 1 i ) 3 /3

Partial Autocorrelation Function Autoregressive Model ACF and PACF Definition The partial autocorrelations function (PACF) can be obtained from ACF ( φ 11 = ρ 1, φ 22 = ρ 2 ρ 2 ) 1 / ( 1 ρ 2 ) 1 φ ss = ρ s s 1 j=1 φ s 1ρ s j 1 s 1 j=1 φ s 1ρ j, s = 3, 4, 5 where φ sj = φ s 1,j φ ssφ s 1,s j, j = 1, 2,, s 1. partial autocorrelation ignores the effect of the intervening values PACF plays an important role in identifying the lag in an AR model direct method for estimating PACF: form series {y t } by subtracting the mean of {yt}: y t = y t µ form autoregression equation y t = φ s1 y t 1 + φ s2y t 2 + + φss y t s + et φ ss is the partial autocorrelation coefficient between y t and y t s

Autoregressive Model ACF and PACF Partial Autocorrelation Function (Cont.) Figure: ACF and corresponding PACF of specific ARMA model

Empirical Facts Trends and Volatility ARCH Process The GARCH Model Most of the series contains a clear trend. Any shock to a series displays a high degree of persistence. The volatility of many series is not constant over time. A stochastic variable with a constant variance is called homoskedastic as oppoesd to heteroskedastic conditionally heteroskedastic: unconditional (or long-term) variance is constant but there are periods with relative high variance Some series share comovements with other series.

Empirical Facts Trends and Volatility ARCH Process The GARCH Model Four figures denote: GDP and actual Consumption, Federal funds and 10-year bond, NYSE stock index and Exchange rate of US, Canada and UK

ARCH Process Trends and Volatility ARCH Process The GARCH Model For investor plan to buy at t and sell at t + 1, forecasts of the rate of return and variance over the holding period is more important than unconditional variance. Simple example y t+1 = ɛ t+1x t where y t+1 is variable of interest, ɛ t+1 is white-noise with variance σ 2 and x t is an independent variable observed at period t. If x t = x t 1 = = constant, {y t} is a white-noise process. If {x t} are not all equal, then Var (y t+1 x t ) = x 2 t σ 2 If xt 2 is large (small), the variance of y t+1 will be large (small) Introduction of {x t} can explain periods of volatility in {y t}

ARCH Process (Cont.) Trends and Volatility ARCH Process The GARCH Model Preliminary idea of Engle: conditional forcasts are vastly superior to unconditional forecasts Given stationary ARMA model y t = a 0 + a 1y t 1 + ɛ t, the conditional forecasts are [ E ty t+1 = a 0 + a 1y t, E t (y t+1 a 0 a 1y 1) 2] = E tɛ 2 t+1 = σ 2 the unconditional forecasts are [ ( ) ] 2 Ey t+1 = a0, E y t+1 a0 = σ2 1 a 1 1 a 1 1 a1 2 Since σ2 > σ 2, unconditional forecast has greater variance than 1 a1 2 conditional forecast. Conditional forecasts consider both current and past realization of series are preferable.

ARCH Process (Cont.) Trends and Volatility ARCH Process The GARCH Model Definition Suppose the conditional variance is not a constant but an AR (q) process, we obtain an autoregressive conditional heteroskedastic (ARCH) model for the square of the estimated residuals: ˆɛ 2 t = α 0 + α 1ˆɛ 2 t 1 + α 2ˆɛ 2 t 2 + + α qˆɛ 2 t q + v t where v t is a white noise process. If α 1, α 2,, α n all equal zero, the estimated variance is constant. There are many possible applications for ARCH model, since the residuals can come from an autoregression, an ARMA model or a standard regression model. The linear ARCH model is not most convenient. The model for {y t} and the conditional variance are best estimated simultaneously using maximum likelihood techniques. A better way is to specify v t as a multiplicative disturbance.

ARCH Process (Cont.) Trends and Volatility ARCH Process The GARCH Model The simplest example from the class of multiplicative conditionally heteroskedastic models proposed by Engle (1982) is ɛ t = v t α 0 + α 1ɛ 2 t 1 where v t is white-noise process with σ 2 v = 1 and ɛ t 1 are mutually independent, α 0 > 0 and 0 < α 1 < 1 are constants. Given Ev t = 0, the unconditional expectation is Given Ev tv t i = 0 ( Eɛ t = E [v t α 0 + α 1 ɛ 2 ) ] 1/2 ( t 1 = Ev t E α 0 + α 1 ɛ 2 1/2 t 1) = 0 Eɛ t ɛ t i = 0, i 0 The unconditional variance is Eɛ 2 t [v = E 2 ( t α 0 + α 1 ɛ 2 )] t 1 = Ev 2 ( t E α 0 + α 1 ɛ 2 ) t 1 since Ev 2 t = σ 2 = 1 and unconditional variance of ɛ t is identical to ɛ t 1, we have Eɛ 2 t = α 0 1 α 1

ARCH Process (Cont.) Trends and Volatility ARCH Process The GARCH Model The conditional mean of ɛ t is E (ɛ t ɛ t 1, ɛ t 2, ) = Ev te ( α 0 + α 1ɛ 2 t 1) 1/2 = 0 The conditional variance of ɛ t is ( ) E ɛ 2 t ɛ t 1, ɛ t 2, = α 0 + α 1ɛ 2 t 1 The multiplicative form has no influence on the unconditional expectation and variance, conditional mean. The influence falls entirely on conditional variance which is a first-order autoregressive process denoted by ARCH (1). The higher order ARCH (q) processes q ɛ t = v t α0 + α i ɛ 2 t i since ɛ t 1 to ɛ t q have a direct effect on ɛ t, thus conditional variance acts like an autoregressive process of order q. i=1

GARCH Model Trends and Volatility ARCH Process The GARCH Model Definition Bollerslev (1986) extended Engle s work by introducing both autoregressive and moving average components (ARMA) in heteroskedastic variance which is called generalized ARCH (p, q) model or GARCH (p, q) model ɛ t = v t ht where σ 2 v = 1 and q p h t = α 0 + α i ɛ 2 t i + β i h t i i=1 i=1 The key feature of GARCH model is that the conditional variance of the disturbances of {y t} sequence constitues an ARMA process. The conditional variance is q p E t 1 ɛ 2 t = α 0 + α i ɛ 2 t i + β i h t i i=1 i=1

Expanded GARCH Model Trends and Volatility ARCH Process The GARCH Model Definition The Nonlinear GARCH (NGARCH) model is also known as Nonlinear Asymmetric GARCH (1, 1) was introduced by Engle and Ng in 1993 h t = ω + α ( ) 2 2 ɛ t 1 θh t 1 + βh t 1, ω, α, β 0. Definition Integrated GARCH (IGARCH) is a restricted version of GARCH model, with condition p q β i + α i = 1. i=1 i=1 Definition The GARCH-in-mean (GRACH-M) model adds a heteroskedasticity term to mean q y t = βx t + δh t + ɛ t, ɛ t = v t ht, h t = α 0 + α i ɛ 2 t i. i=1

Preliminary Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model GARCH option pricing model has three distinctive features: The GARCH option pricing is a function of risk premium embedded in the underlying asset. standard preference-free option GARCH option pricing model is non-markovian. the underlying asset value is a diffusion process standard approach is Markovian GARCH option pricing can potentially explain some well-documented systematic biases associated with Black-Scholes model. underpricing of short-maturity option implied volatility smile underpricing of options on low-volatility securities

Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model Consider discrete-time economy and let X t be the asset price at time t One-period of return under probability measure P is ln Xt = r + λ h t 1 ht + ɛt X t 1 2 where ɛ t has mean zero and conditional variance h t, r is the constant one-period risk-free rate of return and λ the constant unit risk premium. Assume that ɛ t follows a GARCH (p, q) under measure P h t = α 0 + ɛ t φ t 1 N (0, h t) q α i ɛ 2 t i + i=1 p β i h t i. If p = 0 and q = 0, this is exactly the Black-Scholes case. i=1

Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model (Cont.) The conventional risk-neutral valuation relationship has to be generalized to accomodate heteroskedasticity of the asset return process. Definition A pricing measure Q is said to satisfy the locally risk-neutral valuation relationship (LRNVR) if measure Q is mutually absolutely continuous with respect to measure P, Xt /X t 1 φ t 1 distributes lognormally (under Q) E Q ( Xt /X t 1 φ t 1 ) = e r, Var Q (ln ( Xt /X t 1 ) φ t 1 ) = Var P (ln ( Xt /X t 1 ) φ t 1 ) almost surely with respect to measure P.

Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model (Cont.) Theorem The LRNVR implies that, under pricing measure Q where h t = α 0 + ln q i=1 Xt = r 1 ht + ξt X t 1 2 ξ t φ t 1 N (0, h t) α i ( ξ t i λ h t i ) 2 + p β i h t i. i=1 The asset price can be denoted as [ X T = X t exp (T t) r 1 2 ] T T h s + ξ s s=t+1 s=t+1 The discounted asset price process e rt X t is Q-martingale.

Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model (Cont.) Theorem Under the GARCH (p, q) specification, a European option with exercise price K maturing at time T has time t value to Corollary C GH t = e (T t)r E Q [max (X T K, 0) φ t ]. The delta of an option is the first partial derivative of the option price with respect to the underlying asset price. The GARCH option delta at time t by GH t is [ ] GH t = e (T t)r E Q XT 1 {XT K} φ t. X t

Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model Generalized NGARCH Option Pricing Model Definition The log-returns of asset are assumed to follow the following dynamic under the objective measure P ( ) Xt ( ) log = r t d t + λ t ht g ht; θ + ɛ t X t 1 ɛ t = h tv t, h t = α 0 + α 1 h t 1 (ɛ t 1 γ) 2 + β 1 h t 1 where X t is closing price of underlying asset at date t, r t and d t denote continuously compounded risk-free rate of return and dividend rate for [t 1, t]. v t, with parameter θ, is a random variable with zeros mean and unit variance. g (x; θ) = log (E [e xvt ]) is the log-laplace-transform of v t. If the distribution of v t is standard normal distribution, then this is Duan s option pricing model. For α 1 = β 1 = 0, this is discrete-time Black-Scholes log-normal model with constant volatility.

Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model Generalized NGARCH Option Pricing Model (Cont.) Theorem For any standardized probability distribution ξ t with parameters θ which is equivalent to the marginal distribution of v t, then the distribution of following process is equivalent to the NGARCH stock price model described above ( ) Xt ( ) log = r t d t g ht; θ + h tξ t X t 1 h t = α 0 +α 1 h t 1 (ξ t 1 λ t + 1 ( ) ( )) 2 (g ht 1 ; θ g ht 1 ; θ γ) +β 1 h t 1. ht 1 This theorem allows for different distribution forcing process under the objective measure and the risk-neutral measure. A discrete-time financial market with a continuous distribution for the return is inevitably incomplete. This is just a possible risk-neutral measure.

References Duan s Option Pricing Model A Generalized NGARCH Option Pricing Model Walter Enders, Applied Econometric Time Series, John Wiley & Sons. Inc, Nov. 2009 Christian Menn, Svetlozar T. Rachev, Soomthly Truncated Stable Distributions, GARCH-Models and Option Pricing, Journal of Banking and Finance, Nov. 2007 Svetlozar T. Rachev, Young Shin Kim, Michele L. Bianchi, Frank J. Fabozzi, Financial Models with Lévy Processes and Volatility Clustering, John Wiley and Sons, ISBN 0470482354, 9780470482353, 2011 Jin-Chuan Duan,, Mathematical Finance, Vol. 5, No. 1, pp 13-32, Jan. 1995