Implied Volatility String Dynamics

Transcription

1 Szymon Borak Matthias R. Fengler Wolfgang K. Härdle Enno Mammen CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin and Universität Mannheim

2 aims and generic challenges 1-1 Aims Model and estimate implied volatility surfaces (IVS) for trading hedging of derivative positions risk management. In these contexts the IVS acts as a very high-dimensional state variable. Practice requires a low-dimensional representation of the IVS

3 aims and generic challenges 1-2 Challenges Large number of observations (> 2 million contracts, > observations per day). Data appear in strings. Strings are not locally fixed, but move through the observation space (expiry effect). In the moneyness dimension observations may be missing in certain sub-regions for some dates i

4 aims and generic challenges 1-3 Degenerated Design IVS Ticks Data Design Time to maturity Moneyness Figure 1: Left panel: call and put implied volatilities observed on Right panel: data design on ; ODAX, difference-dividend correction according to Hafner and Wallmeier (2001) applied

5 aims and generic challenges 1-4 Purpose A modelling strategy in terms of a dynamic semiparametric factor model (DSFM) for the (log)-ivs Y i,j (i = day, j = intraday): Y i,j = m 0 (X i,j ) + L β i,l m l (X i,j ). (1) Here m l (X i,j ) are smooth factor functions and β i,l is a multivariate (loading) time-series. l=

6 aims and generic challenges 1-5 Traditional model fit Model fit Figure 2: Traditional model (Nadaraya-Watson estimator) and semiparametric factor model fit for Bandwidths for both estimates h 1 = 0.03 for the moneyness and h 2 = 0.08 for the time to maturity dimension

7 2-1 Overview 1. aims and generic challenges 2. implied volatilities 3. short literature review 4. model 5. algorithm 6. results 7. application 8. outlook 0 0.4

8 implied volatilities 3-1 Implied volatilities Black and Scholes (1973) (BS) formula prices European options under the assumption that the asset price S t follows a geometric Brownian motion with constant drift and constant volatility coefficient σ: C BS t = S t Φ(d 1 ) Ke rτ Φ(d 2 ), where d 1,2 = ln(st/k)+(r± 1 2 σ2 )τ σ. Φ(u) is the CDF of the standard τ normal distribution, r a constant interest rate, τ = T t time to maturity, K the strike price

9 implied volatilities 3-2 Implied volatilities Volatility ˆσ as implied by observed market prices C t : ˆσ : C t C BS t (S t, K, τ, r, ˆσ) = 0. Unlike assumed in the BS model, ˆσ t (K, τ) exhibits distinct, time-dependent functional patterns across K (smile or smirk), and a term-structure T t: Thus ˆσ t (K, τ) is interpreted as a random surface: the implied volatility surface (IVS)

10 short literature review 4-1 Related work One strand of literature models IVS slices using PCA: Alexander (2001) analyzes fixed strike deviations, Skiadopoulos et al. (1999) explore the smile in different maturity buckets, Avellaneda and Zhu (1997); Fengler et al. (2002) investigate the term structure

11 short literature review 4-2 Related work Recently, a more comprehensive surface perspective is adopted: Fengler et al. (2003) propose a simultaneous decomposition of maturity groups in a common principal components framework. Cont and da Fonseca (2002) employ the Karhunen und Loève decomposition. This literature does not properly cope with the degenerated design. Estimates are necessarily biased

12 model 5-1 The semiparametric factor model Consider DSFM for the IVS: Y i,j = m 0 (X i,j ) + L β i,l m l (X i,j ), (2) Y i,j is log IV,i denotes the trading day (i = 1,..., I ), j = 1,..., J i is an index of the traded options on day i. m l ( ) for l = 0,..., L are basis functions in covariables X i,j, and β i are time dependent factors. l=

13 model 5-2 For m l ( ), l = 0,..., L consider two different set-ups in X i,j : (A) X i,j is a two-dimensional vector containing time to maturity τ i,j and forward moneyness, κ i,j = K F (t i,j ), i.e. strike K divided by futures price F (t i,j ) = S ti,j exp(r τi,j τ i,j ) (B) as in (A) but with one-dimensional X i,j that only contains κ i,j. Here, we focus on (A)

14 model 5-3 Space and time smoothing Define estimates of m l and β i,l with β i,0 def = 1, as minimizers of: I J i { Y i,j i=1 j=1 2 L β i,l m l (u)} K h (u X i,j ) du, (3) l=0 where K h denotes a two dimensional product kernel, K h (u) = k h1 (u 1 ) k h2 (u 2 ), h = (h 1, h 2 ) with a one-dimensional kernel k h (v) = h 1 k(h 1 v)

15 model 5-4 Replace in (3) m l by m l + δg and β i,l by β i,l + δ. Take derivatives wrt δ, (1 l L, 1 i I ): I J i βi,l q i (u) = i=1 q i (u) m l (u) du = I i=1 l=0 J i L β i,l βi,l p i (u) m l (u), (4) l=0 L β i,l p i (u) m l (u) m l (u) du, (5) p i (u) = 1 J i K h (u X i,j ), J i j=1 q i (u) = 1 J i K h (u X i,j )Y i,j. J i j=

16 model 5-5 Model characteristics Consider the case L = 0: the log-implied volatilities Y i,j are approximated by a surface m 0 not depending on day i. Then, i,j m 0 (u) = K h(u X i,j )Y i,j i,j K, h(u X i,j ) m 0 is equal to the Nadaraya-Watson estimate based on the pooled sample of all days

17 model 5-6 Model characteristics Consider a fixed day i and L = 0:, Ji 0 (u) = j=1 K h(u X i,j )Y i,j Ji j=1 K, h(u X i,j ) m (i) Traditional model fit

18 model 5-7 Model characteristics IVS s are fitted in neighborhoods of the observed design points X i,j, i.e. we do not fit the surface on the whole design space on each day (as in a functional PCA (fpca), Ramsay and Silverman (1997)). we circumvent global fits and thus avoid large bias effects caused by the degenerated string design

19 model 5-8 Model characteristics In fpca factors are eigenfunctions of a covariance operator. Here, the norm: f 2 (u)ˆp i (u)du, changes each day i, where ˆp i (u) = J 1 Ji i j=1 K h(u X i,j ). Eigenfunctions m l may not be nested for increasing L: Hence, the m l cannot be calculated iteratively, i.e. by moving from L 1 components to L components, and so forth

20 model 5-9 Model characteristics In the DSFM framework the IVS s are approximated by surfaces moving in the function space { m 0 + L α l m l : α 1,..., α L R}. l=1 The estimates m l are not uniquely defined: they can be replaced by estimates that span the same affine space. Natural choice: orthogonalize m l in an appropriate function space. Order the resulting functions according to maximum variance in β l

21 model 5-10 Orthogonalization Replace: where: m 0 by m new 0 = m 0 γ Γ 1 m m by m new = Γ 1/2 m β i by βnew i = Γ 1/2 ( β i + Γ 1 γ) m = ( m 1,..., m L ), β i = ( β i,1,..., β i,l ), p(u) = 1 I I i=1 p i(u) Γ is (LxL) matrix with Γ l,l = m l (u) m l (u) p(u)du γ is (Lx1) vector with γ l = m 0 (u) m l (u) p(u)du

22 Average density Figure 3: The average density p(u)

23 model 5-12 Ordering Define matrix B with B l,l = I β i=1 i,l βi,l and Z = (z 1,..., z L ) where z 1,...,z L eigenvectors of B. Replace: m by m new = Z m β i by βnew i = Z βi The orthonormal basis m 1,..., m L is chosen such that I β i=1 i,1 2 is maximal and given β i,1, m 0, m 1 the quantitiy I β i=1 i,2 2 is maximal and so forth

24 algorithm 6-1 Algorithm The algorithm exploits equations (4) and (5) iteratively: 1. for an appropriate initialization of β (0) l,i, i = 1,..., I, l = 1,..., L get an initial estimate of m (0) = ( m 0,..., m L ) 2. update β (1) i, i = 1,..., I, 3. estimate m (1). 4. go to step 2. until minor changes occur during the cycle. Optimization implemented in XploRe, DSFM.xpl, Härdle et al. (2000)

25 results 7-1 Data Overview Min. Max. Mean Median Stdd. Skewn. Kurt. T. to mat Moneyness IV Table 1: Summary statistics from to Source: EU- REX, ODAX, stored in the SFB 649 FEDC. J i observations per day total time series has I 1000 days. N = IJ i 2.8 million contracts, 0 0.4

26 results 7-2 Model selection For a data-driven choice of bandwidths we propose a weighted AIC since the distribution of observations is very unequal: Ξ AIC1 = 1 {Y i,j N i,j L l=0 alternatively (computationally easier): Ξ AIC2 = 1 {Y i,j N i,j β i,l m l (X i,j )} 2 w(x i,j ) exp{ 2L N K h(0) w(u)du}, L β i,l m l (X i,j )} 2 exp{ 2L w(u)du N K h(0) }. w(u)p(u)du l=0 w is a given weight function. Putting w(u) = 1 delivers common AIC, putting w(u) = 1 give equal weight everywhere. p(u) 0 0.4

27 results 7-3 Model selection For the model size (L) selection use the: RV (L) = I Ji i I i j {Y i,j L β l=0 i,l m l (X i,j )} 2 Ji j (Y i,j Ȳ ) 2 where Ȳ denotes the overall mean of the observations. L 1-RV(L) RV Table 2: Explained variance for the model size

28 results 7-4 Estimation Results We fit the model for L = 3, i.e. there are one invariant basis function m 0 and 3 dynamic basis functions m 1, m 2, m 3 3 time series of {β l,i } I i=1 with l = 1, 2, 3 The bandwidths were chosen according to AIC 2 criterion: h 1 = 0.03, h 2 =

29 results Figure 4: Ξ AIC2 dependence on the bandwidths

30 mhat 0 mhat Figure 5: Invariant basis function m 0 and dynamic basis function m 1

31 mhat 2 mhat Figure 6: Dynamic basis functions m 2 and m 3

32 mhat Figure 7: The dynamic basis function m 1

33 DAX 1000+Y*E X beta1 Y X Figure 8: DAX and time series of weights β 1

34 beta2 Y X beta3 Y X Figure 9: Time series of weights β 2 and β 3

35 lag acf lag lag lag results 7-11 Correlogram for β 1, β 2 and β 3 Sample autocorrelation function (acf) Sample autocorrelation function (acf) Sample autocorrelation function (acf) acf acf lag lag pacf Sample partial autocorrelation function (pacf) pacf Sample partial autocorrelation function (pacf) pacf Sample partial autocorrelation function (pacf) Figure 10: acf and pacf of β 1, β 2 and β 3 respectively 0 0.4

36 results 7-12 Testing for random walk coeff. lag suggested differences break date test-value β β β * Table 3: Unitroot test in the presence of structural break. Critical values for rejecting the hypothesis of unit root are at 5% significance level and at 1% significance level. (*) indicate significance at 5% level. Lane et al. (2002) 0 0.4

37 results 7-13 We model first differences of β 1, β 2 and level β 3 in the form Y t = ( β 1, β 2, ˆβ 3 ) Y t = υ + A 1 Y t 1 + A 2 Y t 2 +,..., +A p Y t p + ε t Y t = (Y 1t,..., Y kt ) are vectors of the k = 3 endogenous variables υ = (υ 1,..., υ k ) is a vector of intercept terms, A i are (K K) coefficient matrices ε t is a white noise with covariance matrix Σ ε >

38 results 7-14 Order Selection Criteria Lag ln(fpe) AIC SC HQ * * * * Table 4: VAR Lag Order Selection. * indicates lag order selected by the criterion up to a maximum order 8. We chose to apply a VAR(2) as indicated by the SC criterion

39 results 7-15 β 1t β 2t β 3t = β 1t,t 1 β 2t,t 1 β 3t,t β 1t,t 2 β 2t,t 2 β 3t,t 2 + û 1,t û 2,t û 3,t VAR model for first difference levels, ( β 1, β 2, ˆβ 3) 0 0.4

40 results 7-16 Model Stability Time invariance of the model has been evaluated through the roots of the characteristic polynomial for the VAR(2) model as well as coefficient stability through the cumulative sum of squares of the residuals. roots modulus ± 0.4i ± 0.2i Table 5: Roots of characteristic polynomial for the VAR(2): stability condition is satisied since no root lies outside the unit circle

41 results 7-17 Model Stability CUSUM-square statistic: W 2 r S t = t r=k+1 W 2 r T r=k+1 W 2 r (recursive residuals) is the square one-period ahead prediction error. r = k + 1,..., T ( k, the number of regressors including a constant and T, sample size. We plot S r together with significance level lines E[S r ] ± C 0, the statistical boundaries. C 0 depends on T k and the significance level desired, see Harvey (1990)

42 results 7-18 Y CUSUM - Square Test: (5%) Y CUSUM - Square Test: (5%) Time: 1999:4-2003: Time: 1999:4-2003:3 Figure 11: CUSUM-square statistics for z 1 and z 2 equation 0 0.4

43 results 7-19 Y CUSUM - Square Test: (5%) Time: 1999:4-2003:3 Figure 12: CUSUM-square statistics for z 3 equation Coefficient stability is not rejected as all plots lies within the critical boundaries

44 application 8-1 Hedging exotic options Knock-out options are financial options that become worthless as soon as the underlying reaches a prespecified barrier. asset price Figure 13: Example of two possible paths of asset s price. When the price hits the barrier (red) the option is no longer valid regardless further evolution of the price

45 Figure 14: Newspaper advertisement of Sal. Oppenheim s knock-out options (source: Frankfurter Allgemeine Zeitung, November 2004)

46 Figure 15: Bid-/Ask information of Sal. options Oppenheim s knock-out

47 application 8-4 Hedging exotic options In BS world prices of barrier options are given analytically, all greeks can be calculated directly. There exists static replication for some barrier option if: the underlying has no drift the IV on the market only depends on time not on strike 0 0.4

48 down-and-out up-and-out price price BS sigma BS sigma Figure 16: Price of the call knock-out barrier options as a function of BS-σ. Asset value S 0 = 90, strike price K = 80 time to maturity τ = 0.1 interest rate r = Left panel: barrier B = 80. Right panel: barrier B = 120.

49 application 8-6 Example Consider a short position in a knock-out call option (C KO ) with strike 100 and barrier 90. Consider also one long position in a European call with strike 100 and a short position in 100/90 European puts with strike 81. if spot is at the barrier level 90 call and put would be worth the same if barrier was not reached before maturity the payoff of C KO is equal to the payoff of the call C KO is replicated with vanilla options

50 application 8-7 Value at time t Value at time T Position hits barrier doesn t hit barrier C BS call (K = 100) (S T 100) + 100/90P BS put(k = 81) 0 C KO 0 (S T 100) + Sum 0 0 For each time t and each value of σ if r = 0 and S t = 90 then BS call (K = 100) = BS put(k = 81) 0 0.4

51 application 8-8 Dynamic hedging Use approximation of the option value changes and adjust constantly the hedge portfolio. C KO C KO ( S, σ) S S + C KO σ σ The changes in the asset price (delta risk) can be hedge the asset itself. The changes in volatility (vega risk) can be hedge with at-the-money plain vanilla call option (C)

52 application 8-9 Dynamic hedging The sensitivity of the hedge portfolio HP = a 1 S + a 2 C w.r.t. S and σ should be equal to the sensitivity of the C KO. The hedge coefficients a1, a2 are given by the equation: ( 1 C S 0 C σ ) ( a1 a 2 ) = ( C KO S C KO σ ) 0 0.4

53 application 8-10 Local Volatility Model In local volatility (LV) models the asset price dynamics are governed by the stochastic differential equation: ds t S t = µdt + σ(s t, t)dw t (6) where W t is a Brownian motion, µ the drift and σ(s t, t) the local volatility function which depends on the asset price and time only

54 application 8-11 For pricing the options the partial differential equation (6) is solved. Price depends on the entire IVS. From the IVS one can calculate C t (K, T ). Dupire formula: C t(k,t ) σ 2 T + rk Ct(K,T ) K (S t, t) = 2 K 2 2 C t(k,t ) K 2 gives the local volatility surface σ(s t, t)

55 application 8-12 Hedging exotic options Most greeks can be calculated: C KO Delta, S, gamma, 2 C KO C KO and theta, S 2 t, can be read from the grid of the finite difference scheme; C KO C KO rho, r, and dividend-rho, δ, are typically computed via a difference quotient assuming a flat term structure. What about the vega?? The usual vega, cannot be used since the entire IVS is input. C KO σ 0 0.4

56 application 8-13 Classical vega hedging Classical vega hedging corresponds to parallel move of IVS In BS there is only one volatility number In LV it protects only of parallel move of the smile (β 1 effect)

57 application 8-14 Bucket hedging With term structure of the IVS one may compute a bucket vega hedging. It provides a sensitivity measure of parallel movements over each maturity string. The procedure indicates which European option maturities should be used for hedging Sensitivity related to strike is not given

58 application 8-15 Superbucket hedging In superbucket analysis one has to compute sensitivity of exotics w.r.t. a move of each individual implied volatility. Sensitivity by strike and maturity is obtained The calculation needs to be done for each single point

59 application 8-16 Vega-hedging of the two DSFM factors In DSFM the IV decomposition is given only by L + 1 factors: ( L ) σ i = exp β i,l m l. We can compute the sensitivities w.r.t. the factor loadings β l! From the interpretations, we receive an immediate understanding of the sensitivities: l=0 β 1 β 3 is an up-and-down shift vega of the IVS; is a slope shift vega of the IVS

60 application 8-17 How to compute the hedge ratios Take two hedge portfolios HP 1 and HP 2. Compute the sensitivities of the hedge portfolios and the knock-out option with respect to β 1 and β 3. Solve HP 1 HP 2 β 1 β 1 HP 1 HP 2 β 3 β 3 for the hedge ratios a 1, a 2. ( a1 a 2 ) = C KO β 1 C KO β

61 application 8-18 Choice of the hedge portfolio Idea: choose HP 1 and HP 2 with maximum exposure to β 1 and β 3, respectively: HP 1 should be most sensitive to up-and-down shifts: use a portfolio of at-the-money plain vanilla options; HP 2 should be most sensitive to slope changes: use a portfolio of vega-neutral risk reversals. Then HP 1 β 3 0 and HP 2 β

62 application 8-19 Risk reversal payoff payoff asset price Figure 17: The payoff of the risk reversal. It is compounded from long call with strike K 1 = 120 and short put with strike K 2 =

63 outlook 9-1 Outlook Agenda: - local-linear smoothing - data driven choice of L (number of m), and bandwidth h - forecasting exercise (almost done) - investigate obvious relations to Kalman Filtering, Fengler et al. (2005): Y i,j = m 0 (X i,j ) + L β i,l m l (X i,j ) + ɛ i (7) l=1 β i = β i (θ) + η i (8) 0 0.4

64 outlook 9-2 Outlook Agenda: - hedging empirical studies - estimation of state price density (SPD) f T t (K) = e r(t t) 2 C t (K, T ) K 2 (9) where f T t (K) is SPD of the time T taken in the time t 0 0.4

65 bibliography 10-1 Reference Alexander, C. Principles of the Skew RISK, 2001, 14(1):S29 S32. Avellaneda, M. and Zhu, Y. An E-ARCH Model for the Term-Structure of Implied Volatility of FX Options Applied Mathematical Finance, 4:81 100, Black, F. and Scholes, M. The pricing of options and corporate liabilities Journal of Political Economy, 81: ,

66 bibliography 10-2 Reference Brown, R.L., Durbin, J. and Evans, J.M. Techniques for testing the constancy of regresion relationship over time Journal of the Royal Statistical Society B, 37: , Cont, R. and da Fonseca, J. The Dynamics of Implied Volatility Surfaces Quantitative Finance, 2(1):45 602, Dupire, B. Pricing with a smile, RISK, 7(1):18 20,

67 bibliography 10-3 Reference Fengler, M., Härdle, W. and Villa, C. The dynamics of implied volatilities: A common principle components approach Review of Derivatives Research, 6: , Fengler, M., Härdle, W. and Mammen, E. A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics SFB 649 Discussion Paper, Fengler, M., Härdle, W. and Schmidt, P. Common Factors Govering VDAX Movements and the Maximum Loss Journal of Financial Markets and Portfolio Management, 16(1):16 29,

68 bibliography 10-4 Reference Hafner, R. and Wallmeier, M. The Dynamics of DAX Implied Volatilities International Quarterly Journal of Finance, 1(1):1 27, Härdle, W., Klinke, S. and Müller, M. XploRe - Learning Guide Heildelberg: Springer Verlag, 2000 Lane, M., Lütkepohl, H. and Saikkonen, P. Comparing of unit root tests for time series with level shifts Journal of Time Series Analysis, 23: ,

69 bibliography 10-5 Reference Ramsay, J. O. and Silverman, B. W. Functional Data Analysis Springer, 1997 Skiadopoulos, G., Hodges, S. and Clewlow, L. The Dynamics of S&P 500 Implied Volatility Surface Review of Derivatives Research, 3: ,