Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing

Transcription

1 Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing Liuren Wu, Baruch College Joint work with Peter Carr and Xavier Gabaix at New York University Board of Governors of the Federal Reserve System June 7, 2011 Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

2 Link between interest-rate term structure and dynamics The variation of the interest-rate term structure is fully determined by the risk-neutral dynamics of the interest rates. Reverse engineering: One often uses the term-structure variation to identify the risk-neutral interest-rate dynamics. Given the identified dynamics, one can in principle price any interest-rate related securities (such as caps, floors, swaptions...). Problems: Large portions of interest-rate option variation cannot be explained by the interest-rate term structure variation. Collin-Dufresne and Goldstein (2002) and Heidari and Wu (2003) Dynamic term structure models (DTSM) fitted to the interest-rate term structure do not price interest-rate options well. Dai and Singleton (2003), Li and Zhao (2006), and Heidari and Wu (2008) Bottom line: Volatility movements are not that sensitive to the term structure variation, even if they are theoretically spanned by the term structure. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

3 Lack of linkage between term structure and volatility How to handle the lack of linkage between the term structure and volatility? Current practice (HJM type): Ignore each other. Use dynamic term structure models to fit the term structure only. Volatility specifications do not matter much simplify amap. Option pricing takes the existing term structure as given (no term structure modeling) and only models volatility dynamics. Heidari & Wu (2008): m + n orthogonal factors. The n factors have small impacts on the term structure (treated as transient error), but large impacts on option implied volatility. This paper explores the linearity-generating framework of Gabaix (2007): Bond prices are linear (instead of exponential affine) in factors. Stochastic volatilities in the linear factors do not affect bond price, but affect option pricing truly unspanned. Sequential identification: Identify the factor transition matrix from the term structure; and volatility dynamics from options. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

4 A one-factor example: a partial specification Start with the instantaneous interest rate, r t = θ r + x t, with the following risk-neutral dynamics for the short-rate gap x t : dx t = x t (κ x t ) dt + dn t. Time-varying mean-reversion speed (κ x t ): high when the rate level is low, and low when the rate level is high. n t denotes the (unspecified) martingale component It could be jump, diffusion, or nothing (deterministic). The zero-coupon bond value is affine (not exponential affine) in x t : [ P (t, T ) = E Q t e ] ( ) T r t s ds = e θr τ 1 1 e κτ κ x t, τ = T t. Bond pricing does not depend on the specification of the martingale component n t. Full specification is only needed for pricing options, thus a true separation of term structure modeling and option pricing. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

5 A full specification with an alternative representation Start with the state price deflator M t such that bond pricing is given by P(t, T ) = E t [M T /M t ]. Alternative (expandable) representation wrt a positive martingale Z t : M t = ν Y t (, ) ν = [1, ( 1] ) 1 (α + β) Y t = e At θr 0, A =, α + βz t 0 κ + θ r M t > 0 for all t dictates β 0 and α 0 + α 0. Normalize Z 0 = 1 so that M 0 = 1. The vector Y t is linked to the original factor x t by ( ) 1 xt /κ F t = x t /κ = Y t M t x t = κe κt (α + βz t ) 1 α β + e κt (α + βz t ). [ ] The same bond pricing result: P (t, T ) = E MT t M t = 1 M t E t [ν Y T ] = ( ) ν e Aτ Y t M t = ν e Aτ F t = e θr τ 1 (1 e κτ ) κ x t. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

6 Fully-specified factor dynamics Assume Z t follows a GBM, dz t /Z t = σdw t. The risk-neutral factor dynamics for x t, dx t = x t (κ x t )dt + (x max x t )(x t x min ) x max x min σdw t, The volatility of x t is parabolic and becomes zero at the two boundaries, [x min, x max ], with x max = κ, and x min = κ αe κt 1 α β+αe κt. Setting α = θr κ (1 β) leads to x min = θ r and r min = 0 at t = 0. The coefficient β determines the market price of risk: with γ t = dm t M t = r t dt γ t σdw t, e κt βz t 1 α β+e κt (α+βz. t) The instantaneous risk premium on x t is given by µ t = (κ x t )γ 2 t σ 2. as dn t can also be written as (κ x t )γ t σdw t. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

7 Option pricing It is convenient to price options with the M t = ν Y t representation. State-price deflated bond prices are linear in Z t : P(t, T ) = E t [ MT M t ] = 1 M t e θr T ( 1 + e κt β(z t 1) ). So are the prices of deflated bond portfolios (or coupon bonds): P T = 1 s P(T, s) = M T e θ r s (1 + e κs β(z T 1)). Options on bond portfolios can be valued as: [ ] MT [ O t = E t (DP T K) + = E t (F t + G t Z T ) +]. M t Many specifications on Z t generate tractable pricing for both caps/floors and swaptions. If dz t /Z t = σdw, the Black-Scholes call formula applies: O t = BSC ( G t Z t, F t, σ τ ). Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

8 Multi-factor generalization The M t = ν Y t representation can( be readily generated ) to multiple 1 factors: M t = ν Y t, Y t = e At (αj + β j ) {α j + β j Z t } m, F t = Y t /M t. j=1 with an (m + n) factor structure extension: The transition matrix A R (m+1) (m+1) and the loading vector ν R (m+1) determine the m-dimensional interest rate structure. The innovation Z t can have an n-dimensional stochastic volatility structure, the specification of which is independent of the term structure. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

9 Interest-rate transition dynamics and bond pricing Bond price: P (t, T ) = E t [ MT M t ] = ν e Aτ F t. If the transition matrix A has distinct real eigenvalues, one can diagonalize the transition matrix A d while setting ν d = (1,, 1) without losing any generality. A = θ r + 0, κ 1,, κ m with 0 < κ 1 <... < κ m. F t = (1 m x i i=1 κ i, x1 κ 1,, xm κ m ). ( Bond price: P(t, T ) = e θr τ 1 ) m i=1 (1 e κ i τ ) x i t κ i. The short rate: r t = θ r + m i=1 x i t. This diagonalization is not possible in exponential-affine models: One can either diagonalize the transition matrix or the covariance matrix, but not both. Here, the covariance matrix does not matter for bond pricing. Caveat: Not sure whether this is a good or bad: It makes the specification more parsimonious, but does it hurt the performance? Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

10 Statistical dynamics Bond pricing only needs the transition matrix A and the factor levels X t = {x j t } m j=1. Model estimation needs the full statistical dynamics of X t. Assume for now dz t /Z t = σdw t. The coefficients β j determine the market price and statistical dynamics: dx j t = x j t (κ j i x i t)dt Risk-neutral drift κ j β j e κ j t ( i β i e κ i t )Z 2 t (α σ 2 Risk premium 0+ i α i +β i e κ i t Z t) 3 κ j β j e κ j t Z t with α 0 = 1 i (α i + β i ). + (α σdw 0+ i α i +β i e κ i t Z t) 2 t Volatility Parsimonious, dimension-invariant specification: κ j = b j 1 κ, b > 1, β j = β, power law scaling for frequency distribution Identical risk premium Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

11 Estimation Estimate parameters (θ r, κ, b, β, σ) and extract states {x j t } m i=1 using LIBOR/swap rates. Regard X t as hidden state and LIBOR/swap rates as noisy observations. Euler approximate the statistical dynamics for state propagation. Given parameters, apply nonlinear filtering to extract the states X t. Define the likelihood on the forecasting error of LIBOR/swap rates. Maximize the likelihood to estimate the model parameters. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

12 Interest-rate volatility dynamics Z t is driven by an n-dimensional stochastic volatility dynamics: dz t /Z t = vt n dw t, ( ) dvt j = κ vj vt j 1 vt j dt + σ vj vt j dwt vj, j = 1,, n; v 0 = θ v γ(w vj ) = γ vj vt j, ρdt = E[dW t dwt vn ], zero correlation among other Brownian pairs. Identification: Reset each date t = 0 and Z 0 = 1. Parsimonious, dimension-invariant specification: κ vj = bv j 1 κ v, b v > 1, power law scaling for frequency distribution γ vj = γ v, Identical risk premium σ vj = σ v, iid for shocks of all frequencies The choices for the interest-rate factor dimension m and the volatility dimension n are mutually independent. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

13 Pricing interest-rate options [ M Option pricing: O t = E T t M t (D P ) ] + [ T K = E t (F t + G t Z T ) +]. The Fourier transform of ln Z T /Z t is exponential affine in v: [ ] n ( φ (u) E t e iu ln Z T /Z t = exp ) a j (τ) b j (τ) vt j. j=1 Option value can be computed numerically via FFT. Estimate volatility dynamics parameters (θ v, κ v, b v, ρ, σ v, γ v ) and extract volatility states {vt j } n j=1 using caps/floors or swaptions. The estimation procedure is similar to the first-stage estimation on interest rates. Since options are written on observed rates, not on model values, we ll need a high-dimensional interest-rate model (m is high) as the basis for option pricing. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

14 Data Interest-rate term structure estimation uses 1,2,3,6,9,12-month LIBOR and 2,3,4,5,7,10,15,20,30-year swap rates. Volatility term structure estimation uses 70 swaption implied volatility series, from a matrix of 7 swap maturities: 1,2,3,4,5,7,10 years 10 option maturities: 1,3,6 months and 1,2,3,4,5,7,10 years The implied volatility is plugged into the Black formula to generate the invoice price. 10 years of weekly-sampled data from August 19, 1998 to August 20, 2008, 523 weeks. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

15 Interest rate time series and term structure 8 6 LIBOR/swap rates, % months 2 years 30 years LIBOR/swap rates, % Mean term structure 13 Aug Sep Oct Maturity, years The sample spans a few different interest-rate regimes. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

16 Swaption average implied volatility and variation Mean implied volatility, % Implied volatility change Std, % Option maturity, years Rate maturity, years Option maturity, years Rate maturity, years 10 Implied volatility levels are higher at short option and swaption maturities. Volatilities of implied volatility movements are also higher at shorter option and swaption maturities. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

17 PCA on the implied volatility surface Factor loading on P Factor loading on P Option maturity, years Rate maturity, years 10 5 Option maturity, years Rate maturity, years 10 Second and third principal components reflect variations in the slope and curvature along the option maturity dimension. There is little slope/curvature variation along the swap maturity. All interest-rate factors may be driven by the same volatility, but the dynamics of this volatility contain multiple dimensions of variations. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

18 Pricing errors on LIBOR/swap rates with 3 factors Exponential-affine Linearity-generating M Mean Rmse Max Auto VR Mean Rmse Max Auto VR 1 m m m m m y y y y y y y y Avg A 3-factor LG model performs similarly to a 3-factor exponential-affine model. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

19 Pricing errors on LIBOR/swap with high-dim LG models m 3 7 M Mean Rmse Max Auto VR Mean Rmse Max Auto VR 1 m m m m m y y y y y y y Avg A high-dimensional LG model generates negligible pricing errors, and can thus be used as the basis curve for pricing options. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

20 Interest-rate factor dynamics m θ r κ b σ β 3 [ (1266.4) ] [ (166.6) ] [ (163.1) ] [ (63.2) ] [ (206.9) ] 7 [ (169.2) ] [ (242.8) ] [ (750.1) ] [ (89.9) ] [ (66.0) ] With the dimension-invariant specification, all parameters are estimated with high statistical significance. With more factors, the frequency components are distributed more closely with each other. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

21 Interest-rate factor dynamics and the term structure Under the LG model, spot rates are given by ( y(x t, τ) = ln P(X t, τ) = θ r 1 τ τ ln 1 i ) (1 e κ i τ ) xt i. κ i The instantaneous factor loadings at the mean factor level x t = 0 are: y(x t, τ) = (1 e κ j τ ) κ j τ 1 3-factor x j t x=0 1 7-factor Factor Loading Factor Loading Maturity, Years Maturity, Years Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

22 Transition matrix diagonalization revisited Factor Loading Diagonal κ κb κb m 1 Factor Loading Cascade κb m 1 κb m bκ bκ 0 κ Maturity, Years Maturity, Years Under the cascade structure, the loadings of the different states have more separation along the term structure. Hence, identification could be easier at high dimensions. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

23 To be continued on cascades... The current term structure estimation is on the diagonalized additive structure, r t = θ r + j x j t. While the cascade structure is equivalent theoretically, it can perform better in terms of factor identification. We need more exploration of the term structure cascade dynamics and its estimation. Currently, an additive volatility dynamics of the following form is estimated: vt j dwt j, j = 1,, n dz t /Z t = j Compared to the cascade volatility structure, the additive structure is more tractable analytically. The cascade structure may make more economic sense. New estimation is needed to explore its empirical performance. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

24 Better dynamics or pure tricks? The literature: Simpler factor dynamics specification: AR(1), VAR(1). Tractable bond pricing in exponential-affine/quadratic forms. Tractable pricing in caps/floors (options on zero-coupon bonds). Not tractable for pricing swaptions (options on coupon bonds). Numerical approximation is needed. Linearity-generating dynamics: More complex factor dynamics, with time-varying mean reversion. Tractable bond pricing in affine forms. Pricing caps/floors and swaptions are equally tractable. Questions for future analysis: How does the new dynamics compare with AR(1) in matching the time-series behaviors? Does the time-varying mean reversion tell us anything? Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25

25 Concluding remarks The linearity-generating dynamics are intriguing, at least. With such dynamics, we can Achieve a true separation of the interest rate term structure and option pricing. Equal tractability for pricing caps and swaptions. Explore some nonlinear interest-rate dynamics. Pricing interest-rate options are easier said than done: Complex quoting conventions. Numerical errors can overwhelm the calculation. It remains difficult to accommodate multiple interest rate innovations. Liuren Wu (Baruch) Linearity-Generating Processes Fed, 6/7/ / 25