Recent Developments of Statistical Application in. Finance. Ruey S. Tsay. Graduate School of Business. The University of Chicago

Transcription

1 Recent Developments of Statistical Application in Finance Ruey S. Tsay Graduate School of Business The University of Chicago Guanghua Conference, June 2004

2 Summary Focus on two parts: Applications in Finance: Four (4) illustrative examples Recent Developments: Three (3) topics 1

3 Examples of Application January effect on small-cap stocks VaR for risk management Arbitrage opportunities in index futures trading Market inefficiency in the presence of daily price limit

4 Recent developments Estimation of stochastic diffusion equation (SDE) models High-frequency finance: realized volatility Credit risk modeling: bankruptcy prediction

5 Application: Nature of Seasonality Data: Monthly simple returns of CRSP Decile 1 index Time span: January 1960 to December 2003 with 528 data points 2

6 Feature: Sample autocorrelation shows strong seasonality Aim: What is the nature of the seasonality?

7 Monthly simple returns of Decile 1: return year Sample ACF acf Lag Fig: Time plot and Sample ACF of Decile 1 returns 3

8 Model for the decile returns (R t ): (1 φ 1 B)(1 φ 12 B 12 )R t = c + (1 θ 12 B 12 )a t where B is the back-shift (or lag) operator such that BR t = R t 1. Conditional maximum likelihood estimate: (1 0.25B)(1 0.99B 12 )R t =.0004+(1 0.92B 12 )a t, σ a =

9 Exact MLE (Hillmer & Tiao, 1979) (1.264B)(1.996B 12 )R t =.0002+(1.999B 12 )a t, σ a = Cancellation in the model! Deterministic seasonality

10 January indicator: Jan t = A simple linear regression 1 if t is January, 0 otherwise, R t = Jan t + e t, R 2 = 0.23

11 Monthly Decile 1 returns: January effect removed Adj rtn Year ACF Lag Time plot and Sample ACF of Adjusted Decile 1 returns

12 Value at Risk (VaR) for risk management VaR: Minimum amount of potential loss of a financial position over a given time period for a small tail probability In statistics: VaR: quantile of a predictive distribution of loss 5

13 Several statistical methods available Econometric modeling: volatility models Extreme value theory: traditional & Peaks over threshold

14 Example: Daily log returns of IBM stock from 1962/7/2 to 2003/12/31/ Long on the stock: $1 million dollars Given p = 0.01, what is the VaR for the next trading day? Longer horizon & other tail probability can be used if needed.

15 Daily log returns of IBM stock: ln rtn Year Time plot of daily IBM stock log returns: 62-03

16 Two approaches: Econometric modeling: GARCH(1,1) with Student-t dist r t = a t, a t = σ t ɛ t σ 2 t = a 2 t σ 2 t 1 with estimated degrees of freedom 6.18.

17 1-step ahead prediction: ˆr = , ˆσ = Quantile is [ (3.1123/ 6.18/4.18)] = VaR = $24,807.

18 Extreme value theory: Over a high threshold IBM log returns have heavy tails. Focus on negative returns Tail behavior: QQ-plot against exponential

19 IBM negative returns Exponential Quantiles Ordered Data QQ-plot vs exponential distribution for negative daily IBM log returns: 62-03

20 Daily negative log returns of IBM stock: Mean Excess Threshold Mean excess plot for daily negative log returns of IBM stock

21 Use threshold of 2.5% (dropped 2.5% or more): 456 exceedances (about 4.3% of the data) Estimated tail parameters: ξ = (0.0555), α = (0.0007) Quantile is VaR: $41,738. { [(10446/456)(1 0.99)] ξ 1 }.

22 Fu(x u) x (on log scale) Residuals Ordering 1 F(x) (on log scale) x (on log scale) Exponential Quantiles Ordered Data Model checking for negative IBM log returns: threshold 0.025

23 Does arbitrage opportunity exist? High-frequency (minute by minute) data of S&P 500 index futures and cash price Data span: May 1993 contracts that mature in June 1993 Two time series must move closely together Large deviations signify arbitrage opportunity. 6

24 Intraday data of SP 500 index futures: May 1993 ln(f) Cash price ln(c) Log prices of S&P 500 index futures and stocks: May 1993

25 Theory: Cost-of-carrying model f t s t = (r t,t q t,t )(T t) + z t t and T : current and expiration time f t & s t : log prices of index futures and index r t,t : risk-free interest rate q t,t : dividend yield

26 Concept: z t should be stationary Statistical tools: z t should follow a 3-regime threshold model f t and s t should be threshold co-integrated Thresholds determined by transaction cost and risk premium Tsay (1989, JASA) and Dwyer, Locke & Yu (1996, RFS).

27 Difference: ln(f) ln(c) diff Adjusted series: cost of carrying z star Deviation between index futures and cash market

28 Results Data confirm that z t follows a 3-regime TAR model f t and s t are threshold co-integrated Arbitrage opportunity exists in a short time period (1 minute) Past values of f t are more informative than those of s t Thresholds are close to being symmetric

29 Impact of daily price limits Reference: Cho, Russell, Tsay & Tiao (2003, J. Emp. F) Data: Intraday 5-minute returns of stocks from Taiwan Stock Exchange Price limit: 7% of previous closing price 345 stocks used (relatively large-cap) 7

30 Data span: January 3, 1998 to March 20, 1999 Validation period: March 21, 1999 to April 29, 2000.

31 Concept: Price limits lead to market inefficiency resulting in magnet effect Magnet effect: Price has a tendency to approach the limits once it crosses a middle point Can be exploited for profits, among other adverse effects

32 Illustration of Magnet Effect Upper limit change Upper threshold Prior closing price time Illustration of magnet effect for the upper limit

33 Statistical tools used: Regression with correlated residuals & conditional heteroscedasticity Use ceiling and floor variables r t = α i=1 α i r t i + γ 1 C t 1 + γ 2 F t 1 + e t Use generalized method of moments (GMM)

34 to handle correlated residuals: average ˆγ 1 = 0.124(.011). Trading strategies also considered, including taxes and costs.

35 Some recent developments Estimation of stochastic diffusion equation (SDE) models Analysis of high-frequency financial data: realized volatility, jumps Credit risk: predicting bankruptcy Inference when many tests are used: Bennett s inequality 8

36 Estimation of SDE joint with J.C. Artigas 1. Stochastic volatility (SV) models 2. Algorithm to estimate SDE models (a) Parameters (b) Jumps (c) Volatility 3. Results and Extensions 4. Concluding Remarks 9

37 1. Stochastic Diffusion Model Let X t := (Y t, Z t ) be a 2-dimensional diffusion process s.t. dx t = µ(x t ; θ)dt + σ(x t ; θ)dw t + ξ t dn t. Discretized squared-root process (Eraker 10

38 et al. (2003)) Y t = µ + Z t 1 ε t + ξ y t Jy t, Z t = κ(θ Z t 1 ) + σ z Zt 1 η t + ξ z t J z t. Discretized log-volatility process (Chib et al. (2002)) Y t = θ r + κ r Y t 1 + e 1 2 Z t Y β t 1 ε t + ξ y t Jy t, Z t = θ z + κ z Z t 1 + σ z η t + ξ z t J z t.

39 The model used in our study is Y t = θ r + κ r Y t 1 + e 1 2 Z t 1 Y β t 1 ε t + ξ t J t,(1) Z t = θ z + κ z Z t 1 + σ z η t + (ξ z t J z t ) (2) where (ε t, η t ) are bivariate Normal with mean zero, unit variance and correlation ρ, J t Ber(λ); and ξ t N(µ J, σ 2 J ). 11

40 The likelihood function is given by L(θ) = n t=1 = n t=1 = n t=1 p(x t X t 1, J t, ξ t, θ) p(z t X t 1, Y t, J t, ξ t, θ)p(y t X t 1, J t, ξ t, θ) p(y t X t 1, Z t, J t, ξ t, θ)p(z t Z t 1, θ)

41 First, we have X t X t 1, J t, ξ t, θ, θ J N 2 ( µ, DΣD ) where µ = D = Σ = µ y µ z = σ y 0 0 σ z 1 ρ ρ 1. θ r + (1 + κ r )Y t 1 + ξt y J t θ z + (1 + κ z )Z t 1 = e 1 2 Z t 1 Y β t σ z,, 11-1

42 Secondly, with Z t X t 1, Y t, θ, θ J N(µ z y, σ 2 z y ), µ z y = θ z + (1 + κ z )Z t 1 + ρ σ z σ y (Y t µ y ), σ 2 z y = σ2 z (1 ρ 2 ).

43 (Conditional distribution continued.) Finally, with Y t X t 1, Z t, θ, θ J N(µ y z, σ 2 y z ), µ y z = θ r + (1 + κ r )Y t 1 + ξ y t J t + ρ σ y σ z (Z t µ z ) σ 2 y z = ez t 1Y 2β t 1 (1 ρ2 ).

44 2. Estimation Algorithm 0. Initialize any needed variables 1. Sample each element of θ from its posterior distribution 2. Sample from the log-volatility path Z 1,..., Z n 3. Sample the jump times, and the intensity parameter 12

45 4. Sample from the posterior of the remaining elements of θ J 5. Sample from the posterior distribution of the jump sizes 6. Go to 1

46 (a) Parameters: with notation Y n = (Y 1,, Y n ). Posterior of (θ r, κ r ) p(θ r, κ r Y n, Z n, ξ n, J n, θ, θ J ) N 2 (ˆµ r, Sr) with ˆµ r = (X X) 1 X y and Sr = (1 ρ 2 )(X X) 1. Posterior of (θ z, κ z ) p(θ z, κ z Y n, Z n, ξ n, J n, θ, θ J ) N 2 (ˆµ z, Sz) 13

47 with ˆµ z = (X z X z) 1 X z y z and Sz = σ 2 z (1 ρ 2 )(X z X z) 1. Posterior of (σ z, ρ) (Jacquier et al. (2002)) Ω = 1 ρσ z ρσ z σ 2 z = 1 ψ ψ φ + ψ 2 & B = then X t X t 1, J t, ξ t, θ, θ J N 2 ( µ, BΩB ). σ y ;

48 Posterior of β (Eraker (2001)) Using M-H with proposal N β l (β) l (β), (l (β)) 1. (b) Jumps Jump Times and Intensity Parameter 14

49 The posterior probability of jump time is P (J t = 1 X n, ξ n, θ, θ J ) λφ Y t µ y z σ y z P (J t = 0 X n, ξ n, θ, θ J ) (1 λ)φ J t = 1 Y t µ y z σ y z The posterior distribution of λ is p(λ J n ) Beta(α 0 + n n t=1 J t, β 0 + n t=1 J t )., J t = 0.

50 Posterior of the Jump Sizes and µ J The posterior of µ J is p(µ J X n, J n, θ, θ J ) N(ˆµ J, S J ), where S J = n t=1 J t σ 2 y z +σ2 J J t 1, ˆµ J = S J n t=1 ( Y t θ r κ r Y t 1 ρ σ z σ 2 y z +σ2 J J t σy (Z t µ z ))J t. 15

51 Jump sizes: if there is no jump, jump size comes from prior; if a jump is present at time t, we draw from p(ξ t X t, X t 1, J t = 1, θ, θ J ) N ( τ 1 t α t, τ t ),

52 where α t = ( Y t θ r κ r Y t 1 ρσ z σ y (Z t µ z ))/σ 2 y z + µ J /σ 2 J, 1 = 1 τ t σy z σj 2.

53 (c) Volatility Recursive method for drawing volatility path (Eraker; Jaquier, Polson, and Rossi.) Draw from posterior Z t Z(t) n, Y n, θ. Due to Markovian property, equivalent to draw from p(z t Z t 1, Z t+1, Y n, θ) p(z t+1 Z t, Y n, θ)p(z t Z t 1, Y n, θ)p (Y t Z t, θ). 16

54 Drawbacks (in our experience): Computationally intensive (hours, even days) Sensitive to volatility initialization Performance highly dependent on M-H algorithm used.

55 Joint method for drawing volatility path Draw from joint posterior distribution p(z 1,..., Z n Y n, θ). Linearize equation (1) and use Kalman filter 17

56 Thus, we work with the system Y t = Z t 1 + w t (3) Z t = θ z + (1 + κ z )Z t 1 + σ z η t (4) where w t (s t = i) N(m i, v 2 i ) and Y t := log(y t µ y ) 2 βlog(y t 1 ) 2. If ρ = 0, then Cov(ε t, η t) = 0. ρ 0? What if

57 Recall Y t = θ r + κ r Y t 1 + e 1 2 Z t 1 Y β t 1 ε t + ξ t J t, Z t = θ z + κ z Z t 1 + σ z η t. Let Y t := log(y t µ y ) 2 βlog(y t 1 ) 2 and ε t = log(ε 2 t ). Then, Y t = Z t 1 + ε t Z t = θ z + (1 + κ z )Z t 1 + σ z η t 17-1

58 where p(ε t ) = ( ) 7 ε i=1 q i φ t m i v. i

59 Non-zero correlation in errors Let ρ := Corr(ε, η). Then, ρ. = Unconditional Given ε > Corr(ε*,η) ρ Correlation between ε and η. 18

60 The joint distribution of ε t and η t is given by f ε t,η t (x, y) = 1 2π 1 ρ 2 { exp 1 2(1 ρ 2 ) cosh ( ρ 1 ρ 2 e 1 2 x y ( e x (1 ρ 2 )x + y 2)} where cosh(z) := 1 2 (ez +e z ) = k=0 z 2k /(2k)! ), 19

61 The marginal distribution of ε t is f ε t (x) = 1 2π exp 1 2 (ex x).

62 Table 1: Results from a simulation study using 10,000 samples of 100,000 observations from the joint distribution of (ε t, η t ), from which the distribution of ε t was obtained. ρ M SD Sk Ku

63 Conditional distribution of η t ε t by = x is given 2π(1 ρ 2 ) { exp 1 2(1 ρ 2 ) f ηt ε t (y x) = 1 cosh ( ρ 1 ρ 2 e 1 2 x y ( y 2 + ρ 2 e x)} ). 21

64 η t ρ Cond. dist. of η t ε = x as a function of ρ when ε t =

65 Solutions work with non-linear system Y t = Z t 1 + ε t Z t = θ z + (1 + κ z )Z t 1 + ρσ z (Y t µ y )Y β t 1 e 1 2 Z t 1 + σ z 1 ρ 2 η t using Taylor expansion, or mixture of normals; 22

66 compute mean and variance directly from conditional distribution.

67 Consider the non-linear system z k+1 = f k + G k w k+1 y k+1 = H k z k + v k+1. where f k (z k ) := (1+κ z )z k +ρσ v (y k+1 µ y )y β and set F k = z k f k z k =ẑ k k. k e z k/2, 22-1

68 The time-update recursions, for k 0, are ẑ k+1 k = f k (ẑ k k ), P k+1 k = F k P k k F k + G kq k+1 G k. where ẑ 0 0 = µ 0, and P 0 0 = P 0.

69 The measurement update recursions, for k 0, are ẑ k+1 k+1 = ẑ k+1 k + K k (y k+1 H kẑk k ), P k+1 k+1 = P k+1 k K k (F k P k k H k + G k z k+1 ), K k = (F k P k k H k +G k z k+1 )(H k P k k H k+r k+1 ) 1.

70 3. Results: (Eracker, 2001) Data: 2,288 weekly yields of the 3-m TB from 1/1/54 to 10/5/ yield observation 23

71 Simulation Table 2: Simulation results for the SV model Posterior means and standard errors of the SV model from simulated data with 2,000 observations. Part I display results by proposed methodology using 10,000 (3,000 burn-in.) Part II shows results by recursive methodology using 30,000 iterations (10,000 burn-in.) Computational time (1,000 iterations): (I) 25 min; (II) 5 hr. 24

72 I θ r κ r θ z κ z σ z β True Mean Std II θ r κ r θ z κ z σ z β True Mean Std

73 Table 3: Simulation results for the SV model with negative correlation Posterior means and standard errors of the SV model with correlation. Part I shows the results by proposed algorithm; simulated data (10,000 observations); using 10,000 (3,000 burn-in.) Part II corresponds to the results using recursive method; simulated data (2,500 observations); run for 30,000 (10,000 burn-in.) Computational time (1,000 iterations): (I) 2.2 hr; (II) 6.7 hr. 25

74 I θ r κ r θ z κ z σ z β ρ True Mean Std II θ r κ r θ z κ z σ z β ρ True Mean Std

75 θ r θ z x 10 3 κ r κ z σ z β ρ Posterior draws against iteration. 25-1

76 log volatility Observed Estimated Observed log-volatility vs. estimated log-volatility using proposed methodology. 25-2

77 Table 4: Simulation results for the SVJ model with Z 1,..., Z n given. Posterior means and standard errors using 10,000 (3,000 burn-in). θ r κ r θ z κ z σ z β λ µ J True Mean Std Table 5: Simulation results for the SVJ model. Posterior means and standard errors using 10,000 (3,000 burn-in). θ r κ r θ z κ z σ z β λ µ J True Mean Std

78 1 Mean Probability of Jump at J t = Mean Probability of Jump at J t = Mean probabilities of jump of the SVJ model given volatility. 26-1

79 6 True jump size at J t = Estimated jump size at J t = True and estimated jump size at J t =

80 0.8 Mean probability of jump at J t = Mean probability of jump at J t = Mean probabilities of jump of the SVJ model 26-3

81 Treasury bill data Table 6: Estimation of the SV for the Treasury bill data. Posterior means and standard errors for the SV model fitted to the weekly (Wednesday) three-month t-bill rate from 1/1/1954 to 10/5/1997. Part I presents the results in Eraker (2001) using t=1 and t=1/8; it also shows results obtained by proposed methodology using 10,000 iterations (3,000 burn-in.) Part II displays the results obtained by using the recursive method to sample from the volatility path; the algorithm was run for 30,000 iterations with (5,000 burn-in.) Computational time (1,000 iterations): (I) 30 min; (II) 5.7 hr. 27

82 I θ r κ r θ z κ z σ z β ρ t: t : Mean Std II θ r κ r θ z κ z σ z β ρ Mean Std

83 Joint distribution of e y and e z Probability Probability Normality plot for e y Normality plot for e z Residual analysis of the SV model for the t-bill data. 27-1

84 Extensions A logical next step is to consider the so-called Stochastic Volatility with Conditional Jumps (SVCJ) model. Y t = θ r + κ r Y t 1 + e 1 2 Z t 1 Yt 1 β ε t + ξt y J t, (5) Z t = θ z + κ z Z t 1 + σ z η t + ξ z t J t (6) where (ε t, η t ) are bivariate Normal with mean zero, unit variance and correlation ρ; J t 28

85 Ber(λ); log( ξt z ) N(µz, σj 2 z ); and log( ξt y ) log( ξz t ) N(µ y J + ρ J log( ξz t ), σ2 J y ). In other words, the jump sizes ξt y and ξt z are formed of a sign component and a magnitude component.

86 Table 7: Estimation of the SVCJ model. This table presents the posterior means and standard errors of the SVCJ model by the algorithm using 10,000 iterations from which the first 3,000 were discarded. θ r κ r θ z κ z σ z β True Mean Std λ µ y J σ J y µ z J σ J z ρ J True Mean Std

87 4. Concluding Remarks on SDE estimation Fast, stable and reliable algorithm using joint distribution of volatility path When correlation is present, Kalman filter needs to be modified If volatility is drawn recursively, computation time is ten fold 30

88 Room for improvement with jump component Implications and results in option pricing