IMPLEMENTING THE SPECTRAL CALIBRATION OF EXPONENTIAL LÉVY MODELS

Transcription

1 IMPLEMENTING THE SPECTRAL CALIBRATION OF EXPONENTIAL LÉVY MODELS DENIS BELOMESTNY AND MARKUS REISS 1. Introduction The aim of this report is to describe more precisely how the spectral calibration method for eponential Lévy models from option price data, as studied in 1], is implemented and to show some results for finite samples. In Section 2 the whole method is described step by step and the choice of certain parameters is discussed. Then in the first part of Section 3 we show simulation results for the performance of the method in the cases of the Merton 6] and Kou 5] models and different accuracy of the option price data. Section 3 closes with the calibration of real option prices for the German DAX stock inde on two different dates with three different times to maturity. Finally, some conclusions on the practical performance of the method are drawn in Section Description of the algorithm Based on the basic procedure presented in 1] we give a step by step algorithmic eplanation of the implementation Data transformation. Let the prices C 1,..., C N of N call options at strikes K 1 <... < K N be observed. We transform the observations (K j, C j ) to ( j, ) by := C j /S (1 K j e rt /S) +, j := log(k j /S) rt, where T is the time to maturity, r is the short rate and S is the spot price Estimation of O. We find the function Õ among all functions O with two continuous derivatives as the minimizer of the penalized residual sum of squares (1) N+1 RSS(O, κ) = (O i O( i )) 2 + κ i=0 N+1 0 O (u)] 2 du, Date: December 23,

2 2 DENIS BELOMESTNY AND MARKUS REISS where 0 1 and N+1 N are two etrapolated points with artificial values O N+1 = O 0 = 0. The first term in (1) measures closeness to the data, while the second term penalizes curvature in the function, and κ establishes a tradeoff between the two. The two special cases are κ = 0 when Õ interpolates the data and κ = when a straight line using ordinary least squares is fitted. It can be shown that (1) has an eplicit, finite dimensional, unique minimizer which is a natural cubic spline with knots at the values of i, i = 1,..., N. By general cross validation a data-driven choice of κ is obtained, see e.g. 4]. Note that due to the non-smooth behavior of O() at 0 (compare 1]) it is preferable to fit the splines separately for > 0 and 0 and to combine them afterwards Estimation of FO. Since the solution of (1) is a natural cubic spline, we can write Õ() = N θ j β j () j=1 where β j (), j = 1,..., N, is a set of basis function representing the family of natural cubic splines. We estimate FO(v + i) by N FÕ(v + i) = θ j Fe β j ()](v). j=1 Although Fe β j ()] can be computed in closed form, we just use the Fast Fourier Transform (FFT) and compute FÕ(v +i) on a fine dyadic grid Estimation of ψ. Calculate (2) ψ(v) := 1 T log ( 1 + v(v + i)fõ(v + i) ), v R, where log( ) is taken in such a way that ψ(v) is continuous with ψ( i) = Estimation of σ, γ and λ. With an estimate ψ of ψ at hand, we obtain estimators for the parametric part (σ 2, γ, λ) by an averaging procedure taking into account the polynomial structure of ψ. Upon

3 IMPLEMENTING SPECTRAL CALIBRATION 3 fiing the spectral cut-off value U, we set (3) (4) (5) ˆσ 2 := ˆγ := ˆλ := U U U U U U Re( ψ(u))w U σ (u) du, Im( ψ(u))w U γ (u) du, Re( ψ(u))w U λ (u) du, where the weight functions w U σ, w U γ and w U λ are given eplicitly by wσ U r+3 (u) = U (r+3) u r ( /(r+1) { u >2 U}), u U, U], 1/(r+1) wγ U (u) := r + 2 2U r+2 u r sgn(u), u U, U], r+1 w λ (u) := U (r+1) u r ( (2 2/(r+3) 1) { u <2 U}), u U, U]. 1/(r+3) The order r > 0 of the kernel functions reflects the maimal regularity of the Lévy density. It turns out that r = 3 is usually a good choice Estimation of ν. Define ( (6) ˆν(u) := F 1 ψ( ) + ˆσ iˆγ + ˆλ ) ] K() (u), u R, where K() is a compactly supported kernel. In all simulations we take K() = ( 1 (/U ν ) 2) +, R. This kernel or filter has the advantage of decreasing smoothly the impact of the information given in high frequencies, which is much more robust than a hard spectral cut-off by an indicator function kernel. The tuning parameter U ν may differ from U, which often improves the quality of ˆν significantly, although this is asymptotically negligeable Correction of ν. Due to the estimation error and as a result of the cut-off procedure in (6) the estimate ˆν can take negative values and needs correcting. Our aim is therefore to find ˆν + such that subject to ˆν + ˆν 2 L 2 (R) min, R ˆν + () d = R inf R ˆν+ () 0 ˆ d. It follows that the solution of the above problem is given by ˆν + (; ξ) := ma{0, ˆ ξ}, R

4 4 DENIS BELOMESTNY AND MARKUS REISS where ξ is chosen to satisfy the equation ˆν + (u; ξ) du = ˆν(u) du. R 2.8. Choice of parameters. Theoretically appealing approaches to choose the critical tuning parameter U include stagewise aggregation discussed in 1] and risk hull minimization as developed in 2]. In practical situations, however, the following heuristic criterion for choosing the spectral cut-off parameter U is employed (7) U d = argmin U du ˆσ U, where ˆσ U is calculated on a grid of values U and the derivative is approimated by a difference quotient. U corresponds to the flattest region of the curve U ˆσ U or in other words to the region where ˆσ U stabilizes. For the nonparametric part the tuning parameter Uν is found similarly as the value where ˆν U stabilizes: (8) Uν d = argmin Uν ˆν Uν du ν. L2 In the case of the real data eamples below the following criterion proved to be useful because it puts more weight on fitting the observed option prices by the model prices. Here, U is obtained as a solution of the minimization problem N ] (9) C(K i ; T U ) Y i 2 + α ˆν U() 2 d min! i=1 for some weight α > 0 with C(K; T U ) denoting the price at strike K for the triplet T U = (ˆσ U, ˆγ U, ˆν U ). In all eamples below the cut-off values U, U ν lie in the range 20, 50]. R R 3. Simulated and Real Data Eamples We now present simulated as well as real data eamples demonstrating the performance of the calibration algorithm. In our simulations different data designs, sample sizes and noise levels are used in order to investigate the behaviour of the procedure in settings mimicking real ones. The data design ( i ) is chosen to be normally distributed with zero mean and variance 1/3 and reflects the structure at the market where much more contracts are settled at the money than in or out of

5 IMPLEMENTING SPECTRAL CALIBRATION 5 money. Once the design is simulated, the option prices O( i ) are computed from the underlying model and then perturbed by i.i.d. Gaussian noise with zero mean and standard deviation δ O( i ). In the simulations below we always present the outcome of one typical calibration with a data-driven choice of the tuning parameters U and U ν, as discussed above. The simulations are ordered with respect to decreasing accuracy of the observations such that the quality of calibration worsens. Although the results are usually satisfactory, in certain cases (for eample, when the noise level is high) the choices of the tuning parameters can be bad and the calibrated Lévy triplet is way off the true triplet. This instability with respect to the tuning parameters has already been realized in 2] for an idealized linear inverse problem. A more robust algorithm for selecting the tuning parameters also in the present calibration problems remains still a challenge. For this reason, we present in addition for each simulation the root mean square error (RMSE), using a Monte Carlo estimate, for an optimal choice of the parameters U and U ν. This oracle-type approach shows the best performance the calibration algorithm could possibly attain. This will serve as a future benchmark. First the jump diffusion model by Merton 6] is considered where jumps are normally distributed with mean η and variance υ 2 : = λ υ 2π ep ( ( η)2 2υ 2 ), R. The parameter γ in the Lévy triplet (σ 2, γ, ν) is uniquely determined by the martingale condition Our choice of parameters γ = (σ 2 /2 + λ(ep(υ 2 /2 + η) 1)). σ = 0.1, λ = 5, η = 0.1, υ = 0.2 implies γ = Note that under such a choice the mean jump size is negative in agreement with empirical findings. More interesting is the second eample in terms of the Kou 5] model where the Lévy density ν has a double-eponential structure: ( ) = λ pλ + e λ + 1 0, ) () + (1 p)λ e λ 1 (,0) (), R. The parameters are λ +, λ 0, p 0, 1]. Again the drift γ is defined implicitly and evaluates to γ = (σ 2 /2 + λ(p/(λ + 1) (1 p)/(λ + 1))).

6 6 DENIS BELOMESTNY AND MARKUS REISS Under the choice σ = 0.1, λ = 5, λ = 4, λ + = 8, p = 1/3 we have γ = The Kou model permits to model different tail behaviour for positive and negative jumps which makes the modelling more realistic. Finally, two real data eamples are presented. Both data sets consist of options (put and call) on the German DAX stock inde for different maturities and strikes. An eponential Lévy model is calibrated separately for each maturity. Moreover, the implied model option prices are presented together with the empirical prices as a model validation criterion.

7 IMPLEMENTING SPECTRAL CALIBRATION 7 Merton Model. sample size: N = 100 noise level: δ = 0.05 O Figure 1. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 1. Parameters estimates for the sample shown in Fig. 1 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 2. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

8 8 DENIS BELOMESTNY AND MARKUS REISS Merton Model. sample size: N = 100 noise level: δ = Figure 2. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 3. Parameters estimates for the sample shown in Fig. 2 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 4. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

9 IMPLEMENTING SPECTRAL CALIBRATION 9 Merton Model. sample size: N = 50 noise level: δ = Figure 3. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 5. Parameters estimates for the sample shown in Fig. 3 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 6. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

10 10 DENIS BELOMESTNY AND MARKUS REISS Merton Model. sample size: N = 50 noise level: δ = Figure 4. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 7. Parameters estimates for the sample shown in Fig. 4 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 8. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

11 IMPLEMENTING SPECTRAL CALIBRATION 11 Kou Model. sample size: N = 100 noise level: δ = Figure 5. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 9. Parameters estimates for the sample shown in Fig. 5 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 10. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

12 12 DENIS BELOMESTNY AND MARKUS REISS Kou Model. sample size: N = 100 noise level: δ = Figure 6. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 11. Parameters estimates for the sample shown in Fig. 6 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 12. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

13 IMPLEMENTING SPECTRAL CALIBRATION 13 Kou Model. sample size: N = 50 noise level: δ = Figure 7. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 13. Parameters estimates for the sample shown in Fig. 7 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 14. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν.

14 14 DENIS BELOMESTNY AND MARKUS REISS Kou Model. sample size: N = 50 noise level: δ = Figure 8. Left: Sample ( ). Right: True ν (dashed) and estimated ˆν (solid) Lévy densities. ˆσ ˆγ ˆλ Table 15. Parameters estimates for the sample shown in Fig. 8 obtained using U and U ν given by (7) and (8). E(ˆσ σ) E(ˆγ γ) E(ˆλ λ) E ˆν ν 2 L 2 ] 1/ Table 16. RMSE estimated using 500 Monte Carlo simulations under optimal (oracle) choices of U and U ν

15 IMPLEMENTING SPECTRAL CALIBRATION 15 Real Data Eample I. DAX options, 22 March T=28 T=91 T= Figure 9. Left: Observed (triangles) and fitted (solid line) put ( j < 0) and call ( j 0) prices for different maturities. Right: Estimated Lévy densities. T N ˆσ ˆγ ˆλ Table 17. Parameters of Lévy triple estimated from DAX data shown in Fig. 9 with U chosen via the criterion (9) with α = 1.0e 8. All options data used here are publicly available in MDBase at

16 16 DENIS BELOMESTNY AND MARKUS REISS Real Data Eample II. DAX options, 21 June T=28 T=91 T= Figure 10. Left: Observed (triangles) and fitted (solid line) put ( j < 0) and call ( j 0) prices for different maturities. Right: Estimated Lévy densities. T N ˆσ ˆγ ˆλ Table 18. Parameters of Lévy triple estimated from DAX data shown in Fig. 10 with U chosen via the criterion (9) with α = 1.0e 8. All options data used here are publicly available in MDBase at

17 IMPLEMENTING SPECTRAL CALIBRATION Conclusions The eperience gained from the implementation of the calibration algorithm shows that: The performance of the procedure depends strongly on the noise level and the number of observations. Usually, the dependence on the former dominates. The smoother the Lévy density the better is the performance of the procedure. The difference is more pronounced for smaller noise levels. The main features of Lévy measures (mode, skewness, different tail behaviour, negative mean jump size) are preserved during the reconstruction. The data-driven choice of the tuning parameter is not yet very stable. In the real data eamples the algorithm produces estimates for λ which decrease with the time to maturity, while the volatility is calibrated in a very stable manner. The jump distributions have negative mean jump sizes. This agrees also with the findings in 3]. Given the rather small compleity of the Lévy measures, the model option prices fit the empirical prices remarkably well. References 1] Belomestny, D., and M. Reiß (2005) Optimal calibration of eponential Levy models, Preprint 1017, Weierstraß Institute (WIAS) Berlin. 2] Cavalier, L. and Y. Golubev (2005) Risk hull method and regularization by projections of ill-posed inverse problems, Ann. Stat., to appear. 3] Cont, R., and P. Tankov (2004) Nonparametric calibration of jump-diffusion option pricing models, Journal of Computational Finance 7(3), ] Green, P. and Silverman, B. (1994). Nonparametric Regression and Generalized linear Models: A Roughness Penalty Approach, Chapman and Hall, London. 5] Kou, S. (2002) A jump diffusion model for option pricing, Management Science 48(4), ] Merton, R. (1976) Option Pricing When Underlying Stock Returns Are Discontinuous, J. Financial Economics 3(1), Weierstraß Institute for Applied Analysis and Stochastics (WIAS), Mohrenstraße 39, Berlin, Germany address: belomest@wias-berlin.de Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, Heidelberg, Germany address: reiss@statlab.uni-heidelberg.de