Efficient Derivative Pricing by the Extended Method of Moments Patrick GAGLIARDINI University of Lugano and Swiss Finance Institute Joint work with Christian GOURIEROUX 1 and Eric RENAULT 2 1 CREST (Paris), CEPREMAP and University of Toronto 2 CIRANO-CIREQ (Montreal) and University of North Carolina at Chapel Hill Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 1 / 20
Introduction The goal of the paper: To estimate the pricing operator at a given date by using cross-sectional data on option prices and historical data on underlying asset returns The pricing problem: The investor at date t 0 wants to estimate the current price c t 0 ( h , k ) of a European derivative with time-to-maturity h and moneyness strike k that is not actively traded on the market The investor has data on a cross-section of n current option prices c t 0 ( h j , k j ) , j = 1 , ..., n a time series of T daily returns of the underlying asset Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 2 / 20
Contributions of the paper Introduce the Extended Method of Moments (XMM): An extension of the Generalized Method of Moments (GMM) to accommodate a more general set of moment restrictions for option pricing Account for the fact that the trading activity on each single derivative is much smaller than on the underlying index, and the characteristics of actively traded options vary over time Provide a semi-parametric estimator of the pricing operator at a given day The XMM estimators of risk premia and option prices are consistent for a large number T of historical observations on underlying asset returns and a finite number n of cross-sectionally observed option prices The XMM estimator outperforms the traditional cross-sectional calibration approach for S&P 500 option data Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 3 / 20
Activity on the S&P 500 index option market The Chicago Board Options Exchange (CBOE) enhances the market of derivatives on the S&P 500 index by periodic issuing of new option contracts Jan Feb Mar Apr Mai Jun Jul Aug Sep Oct Nov Dec Mar · · · · · · Jan 1 m 2 m 3 m 6 m 9 m 12 m Feb 1 m 2 m 3 m 5 m 8 m 11 m Mar 1 m 2 m 3 m 4 m 7 m 10 m Apr 1 m 2 m 3 m 6 m 9 m 12 m Mai 1 m 2 m 3 m 5 m 8 m 11 m Jun 1 m 2 m 3 m 4 m 7 m 10 m Jul 1 m 2 m 3 m 6 m 9 m Aug 1 m 2 m 3 m 5 m 8 m Sep 1 m 2 m 3 m 4 m 7 m Oct 1 m 2 m 3 m 6 m Nov 1 m 2 m 5 m Dec 1 m 4 m New 12m options are issued when the old ones attain 9m-to-maturity For any time-to-maturity, options are issued for a limited number of strikes Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 4 / 20
Highly traded S&P 500 options: times-to-maturity We consider S&P 500 options with daily trading volume larger than 4000 contracts in June 2005 400 300 time−to−maturity (days) Long times-to-maturity 200 are rarely actively traded! 100 0 Jun 01 Jun 06 Jun 11 Jun 16 Jun 21 Jun 26 Jul 01 date Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 5 / 20
Highly traded S&P 500 options: moneyness strikes 1.2 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.0 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Moneyness strike 0.8 + call 0.6 o put + + + 0.4 Jun 01 Jun 06 Jun 11 Jun 16 Jun 21 Jun 26 Jul 01 date The number of highly traded options in a given trading day is rather small! The number of options, the times-to-maturity and the moneyness strikes vary from one trading day to the other! Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 6 / 20
Calibration based on current derivative prices The issuing cycle and the trading activity imply that options’ returns are rarely observable and potentially nonstationary! Standard methodology to circumvent these difficulties: daily calibration of the pricing operator using the cross-section of current option prices n � ˆ [ c t 0 ( h j , k j ) − c t 0 ( h j , k j ; θ )] 2 At date t 0 : θ t 0 = arg min θ j = 1 [Bakshi et al. (1997), Jackwerth (2000), Ait-Sahalia, Duarte (2003), Bondarenko (2003), Stutzer (1996), Jackwerth, Rubinstein (1996)] Drawbacks: Estimated parameters ˆ θ t 0 are erratic over time c t 0 ( h j , k j ) = c t 0 ( h j , k j ; ˆ Approximated prices ˆ θ t 0 ) differ from observed prices for highly traded options Estimates are not very accurate when n is small Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 7 / 20
Semi-parametric pricing X t is the vector of observable state variables: a Markov process in X ⊂ R d Semi-parametric specification: the historical transition pdf h ( x t | x t − 1 ) of process X t is left unconstrained the stochastic discount factor M t , t + 1 = m ( X t + 1 ; θ ) is parameterized by θ ∈ R p The price at date t of a European call option with time-to-maturity h and moneyness strike k is � � M t , t + h ( θ ) ( exp R t , h − k ) + | X t c t ( h , k ) = E The goal is to estimate the pricing operator ( h , k ) → c t 0 ( h , k ) at a given date t 0 Data consists in Cross-section of n derivative prices c t 0 ( h j , k j ) , j = 1 , ..., n observed at t 0 Time-series of T observations of the state variables X t before t 0 Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 8 / 20
The moment restrictions The no-arbitrage constraints concerning the underlying asset: E [ M t , t + 1 ( θ ) exp r t + 1 | X t = x ] = 1 ∀ x ∈ X ⇔ : E [ g ( Y ; θ ) | X = x ] = 0 ∀ x ∈ X (1) The no-arbitrage constraints concerning the n observed derivatives at t 0 : � � M t , t + h j ( θ )( exp R t , h j − k j ) + | X t = x t 0 c t 0 ( h j , k j ) = E j = 1 , . . . , n ⇔ : E [˜ g ( Y ; θ ) | X = x 0 ] = 0 x 0 ≡ x t 0 (2) The conditional moment restrictions (1) are uniform since they hold for all values x ∈ X The conditional moment restrictions (2) are local since they hold for the given value x 0 only → the key difference compared to standard GMM! g ′ ) ′ Total set of n + 1 local moment restrictions at t 0 given by g 2 = ( g ′ , ˜ Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 9 / 20
Information-based GMM We need an estimator of both the sdf parameter θ and the historical transition pdf f ( y | x ) to estimate the state price density! Related to the information-based GMM [Kitamura, Stutzer (1997), Imbens, Spady, Johnson (1998)] The kernel estimator of the historical transition pdf is � y t − y � � x t − x � � x t − x � T T � � f ( y | x ) = 1 ˆ K K / K h d h T h T h T T t = 1 t = 1 where K is the kernel and h T is the bandwidth Select the conditional pdf that is the closest to ˆ f ( y | x ) and satisfies the moment restrictions (1) and (2) Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 10 / 20
The XMM estimator � � f ∗ ( ·| x 0 ) , ˆ f ∗ ( ·| x 1 ) , ..., ˆ f ∗ ( ·| x T ) , � ˆ The XMM estimator θ consists of the functions f 0 , f 1 , ..., f T and the parameter value θ that minimize � � � 2 � � � � T f ( y | x t ) − f t ( y ) � f 0 ( y ) L T = 1 dy + h d f 0 ( y ) dy log T T � � f ( y | x t ) f ( y | x 0 ) t = 1 subject to the constraints � � f t ( y ) dy = 1 , t = 1 , ..., T f 0 ( y ) dy = 1 � � g ( y ; θ ) f t ( y ) dy = 0 , t = 1 , ..., T g 2 ( y ; θ ) f 0 ( y ) dy = 0 The chi-square criterion evaluated at the sample points allows for computation of ˆ θ by parametric optimization The KLIC criterion evaluated at x 0 ensures a positive estimated state price density at t 0 Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 11 / 20
The XMM estimator XMM estimator of the historical conditional pdf given x 0 � � ˆ ′ g 2 ( y ; ˆ λ θ ) exp � �� f ∗ ( y | x 0 ) = � � � f ( y | x 0 ) , y ∈ Y ˆ ′ g 2 (ˆ � exp λ θ ) | x 0 E � � � � λ ∈ R n + 1 is s.t. � where the Lagrange multiplier ˆ g 2 (ˆ ˆ ′ g 2 (ˆ E θ ) exp λ θ ) | x 0 = 0 XMM estimator of the derivative price c t 0 ( h , k ) � θ ) ( exp R t 0 , h − k ) + ˆ f ∗ ( y | x 0 ) dy M t 0 , t 0 + h (ˆ ˆ c t 0 ( h , k ) = for any time-to-maturity h and moneyness strike k Estimator ˆ c t 0 ( h , k ) is equal to the observed price c t 0 ( h j , k j ) for h = h j and k = k j Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 12 / 20
Application to S&P 500 options Compare XMM estimation and cross-sectional calibration on S&P 500 options with daily trading volume larger than 4000 contracts in June 2005 Cross-sectional calibration Parametric pricing formula from the risk-neutral distribution of a stochastic volatility model Stochastic volatility σ 2 t follows a discrete-time Heston (1993) model XMM estimation t ) ′ where σ 2 X t = ( r t , σ 2 State variable vector: t is the realized volatility of the S&P 500 index � � − θ 1 − θ 2 σ 2 t + 1 − θ 3 σ 2 Parametric sdf: M t , t + 1 ( θ ) = exp t − θ 4 r t + 1 XMM estimator is computed for each trading day t 0 using current option data and previous T = 1000 daily observations of the state variables Patrick Gagliardini (UNI Lugano and SFI) Derivative Pricing by the XMM 13 / 20
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