Skew Hedging. Szymon Borak Matthias R. Fengler Wolfgang K. Härdle. CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

Transcription

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

2 Motivation 1-1 Barrier options Knock-out options are financial options that become worthless as soon as the underlying reaches a prespecified barrier. asset price 1 Figure 1: 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 When the price hits the barrier (red) the option is no longer valid regardless further evolution of the price.

3 Motivation 1-2 Barrier options In BS world prices of barrier options are given analytically, all greeks can be calculated directly. The price doesn t need to be a increasing function of the volatility parameter σ. Marking to BS model is precluded due to the σ choice BS is not a good choice for handling barrier options!!!

4 down-and-out up-and-out price price BS sigma BS sigma Figure 2: Price of the call knock-out barrier options as a function of BS-σ. Asset value S = 9, strike price K = 8 time to maturity τ =.1 interest rate r =.3. Left panel: barrier B = 8. Right panel: barrier B = 12.

5 Motivation 1-4 Pricing Barrier Options For pricing barrier options a local volatility (LV) model is employed. The asset price dynamics are governed by the stochastic differential equation: ds t S t = µdt + σ(s t, t)dw t (1) 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.

6 Motivation 1-5 Pricing Barrier Options Price depends on the entire surface (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 (K, t) = 2 K 2 2 C t(k,t ) K 2 gives the LV surface σ(s t, t). For practical implementation see Andersen and Brotherthon-Ratcliffe (1997).

7 Motivation 1-6 Dynamics of the IVS The IVS reveals highly dynamic behavior, which influences the prices of the barrier options. Example Consider two one year knock-out put options with strike 11 and barrier 8, when the current spot level is 1. Take the IVS from 213 and The prices of these options are respectively 1.91 and This is a 25% difference.

8 Motivation smile on different days Figure 3: Observed smile on 213 and 2112 for the maturity.25.

9 Motivation 1-8 Vega Hedging In LV model the usual vega cannot be used because the whole IVS is an input The standard approach is to build vega hedging on the sensitivity of the up-and-down shifts. The skew changes, which may cause significant pricing differences, become unhedged.

10 Motivation 1-9 DSFM A complex dynamics of the IVS is explained in terms of a dynamic semiparametric factor model (DSFM) for the (log)-ivs Y i,j (i = day, j = intraday): Y i,j = m (X i,j ) + L β i,l m l (X i,j ) + ε i,j. (2) l=1 Here m l (X i,j ) are smooth factor functions and β i,l is a multivariate (loading) time-series.

11 Motivation 1-1 Aims to apply DSFM for identification of key factors of the IVS dynamics to improve the vega hedge by hedging against most common changes of the IVS

12 Motivation 1-11 Overview 1. Motivation 2. Dynamic Semiparametric Factor Model 3. Hedging Approach 4. Results 5. Conclusion

13 DSFM 2-1 DSFM Consider DSFM for the log-ivs: Y i,j = m (X i,j ) + L β i,l m l (X i,j ) + ε i,j, (3) l=1 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 =,..., L are basis functions in covariables X i,j (, time to maturity), and β i are time dependent factors.

14 DSFM 2-2 DSFM estimation Define estimates of m l and β i,l with β i, 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, (4) l= 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). See Fengler et al. (25), Fengler (25).

15 results 3-1 Model parameters We fit our model: L = 3 dynamic basis functions grid covering [.6, 1.3] and time to maturity [.5, 1] fix bandwidths in direction and increasing bandwidths in maturity direction on the daily IVS data from 213 till 21122

16 mhat mhat maturity maturity Figure 4: Invariant basis function m and dynamic basis function m 1 (level)

17 mhat2 mhat maturity maturity Figure 5: Dynamic basis functions m 2 (skew) and m 3 (term structure)

18 1.5 β Jan Jan1 Jan2 Jan3 Jan4 Jan5 σ ATM Jan Jan1 Jan2 Jan3 Jan4 Jan5 Figure 6: time series of weights β 1 and ATM IVS for the fixed maturity.25.

19 Jan Jan1 Jan2 Jan3 Jan4 Jan5.2 β Jan Jan1 Jan2 Jan3 Jan4 Jan5 Figure 7: Time series of weights and β 3

20 4 β Figure 8: Typical shape of the smile for different levels of β1. Changes of the β 1 influence mainly the surface s level.

21 Figure 9: Typical shape of the smile for different levels of β2. Changes of the influence the smile s skew.

22 hedging 4-1 Greeks In order to implement β-hedging one has to calculate β-greeks. They are obtained by shifting the IVS in the m direction. option β option(ivse β m ) option(ivse β m ) 2 β (5)

23 4 vega spot Figure 1: vega greek for down-and-out put option with barrier 54 and strike 7425 as a function of spot

24 6 β spot Figure 11: β 1 greek for down-and-out put option with barrier 54 and strike 7425 as a function of spot

25 spot Figure 12: greek for down-and-out put option with barrier 54 and strike 7425 as a function of spot

26 hedging 4-5 Example In the BS model the hedge portfolio (HP) for hedging plain vanilla options consists of a stocks - HP = as. The hedge ratio a (delta) is obtained from: dhp ds = a = option. S The hedge is financed by buying/selling bonds.

27 hedging 4-6 How to compute the hedge ratios Take two hedge portfolios HP 1 and HP 2. Compute the sensitivities of the hedge portfolios and the up-and-out call option (C KO ) with respect to β 1 and. Solve HP 1 HP 2 β 1 β 1 HP 1 HP 2 ( a1 a 2 ) = C KO β 1 C KO ( ) vega skew for the hedge ratios a 1, a 2. For the down-and-out put option (P KO ) the procedure is analogous.

28 hedging 4-7 Choice of the hedge portfolio Idea: choose HP 1 and HP 2 with maximum exposure to β 1 and, 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 and HP 2 β 1.

29 hedging 4-8 Risk reversal payoff payoff asset price Figure 13: The payoff of the risk reversal. It is composed from a long call with strike K 1 = 12 and a short put with strike K 2 = 8.

30 hedging 4-9 As in standard vega hedging we apply final delta hedge. In our case we apply delta hedge to C KO + a 1 HP 1 + a 2 HP 2 by calculating the number of underlying as: (C KO + a 1 HP 1 + a 2 HP 2 ) S

31 4 down and out put

32 4 down and out put, long risk reversals, long ATM call

33 4 down and out put, long risk reversals, long ATM call, short spot

34 results 5-1 Empirical Study For each of 885 days ( ) we start one long position in one year C KO and P KO. Option barrier strike maturity knock-outs in-the-money C KO 14 % 8 % 1 year 1 % 39 % P KO 8 % 11 % 1 year 81 % 5 % Table 1: barrier and strike are given as a percentage of the spot at the starting day We keep the position until maturity or knock-out.

35 results 5-2 Empirical Study We compare the β 1 β2 (skew) hedging approach with: β 1 hedging - no risk reversal (a 2 = ) and a 1 = vega hedging - no risk reversal (a 2 = ) and a 1 = C KO β 1 / HP 1 β 1 C KO σ / HP 1 σ

36 results 5-3 Aims of Hedging We define the profit and loss of the strategy at the maturity as a portfolio s value divided by notional at the starting day. C KO T + HP T + money T S The aim of the hedging is possibly large reduction of the profit and loss variation around zero.

37 results 5-4 Results Profit and loss of the strategy at the maturity. C KO min max mean median std med. abs. vega β β Table 2: all values as percentage of the underlying

38 C KO days min max mean median std med. abs. vega β β Table 3: Descriptive statistics for the hedging strategies, 1, 1 and 25 days before the knock-out or expiration - delta hedging effect (gap risk).

39 results 5-6 Results Profit and loss of the strategy at the maturity. P KO min max mean median std med. abs. vega β β Table 4: Descriptive statistics for the hedging strategies of the downand-out put

40 P KO days min max mean median std med. abs. vega β β Table 5: Descriptive statistics for the hedging strategies, 1, 1 and 25 days before the knock-out or expiration - delta hedging effect (gap risk).

41 results std. of the hedge portfolios in time (up and out call) vega β 1 std. of Cumlative HedgingError / Notional β days to expiry Figure 14: The standard deviation of the portfolios as a function of the days left to the maturity (call).

42 results std. of the hedge portfolios in time (down and out put) vega β 1 std. of Cumlative HedgingError / Notional β days to expiry Figure 15: The standard deviation of the portfolios as a function of the days left to the maturity (put).

43 conclusion 6-1 Conclusion the β hedge improves the hedging gap risk is still unhedged. better strategy might be to mix static and dynamic hedges

44 bibliography 7-1 Reference Andersen L. and Brotherton-Ratcliffe R. The equity option volatility smile: An implicit finite difference approach Journal of Computational Finance 1(2) 5-37, Borak, S., Fengler, M. and Härdle, W. DSFM fitting of Implied Volatilty Surfaces Conference proceedings of the Fifth International Conference on Intelligent Systems Design and Applications, 25. Dupire, B. Pricing with a smile, RISK, 7(1):18 2, 1994.

45 bibliography 7-2 Reference Fengler, M. Semiparametric Modelling of Implied Volatility Heildelberg: Springer Verlag, 25 Fengler, M. Arbitrage-free smoothing of the surface SFB 649 Discussion Paper, 25. Fengler, M., Härdle, W. and Mammen, E. A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics SFB 649 Discussion Paper, 25.

46 delta * mean( ds ) 15 1 PL contribution maturity spot Figure 16: Profit-and-loss contribution of the delta for down-and-out put option with barrier 54 and strike The mean value of absolute underlying changes ds = 66.1.

47 theta 25 2 PL contribution maturity spot Figure 17: Profit-and-loss contribution of the theta for down-and-out put option with barrier 54 and strike 7425.

48 vega*mean( dσ ) 5 PL contribution maturity spot Figure 18: Profit-and-loss contribution of the vega for down-and-out put option with barrier 54 and strike σ is taken as at-themoney IV for maturity.25. The mean value of absolute changes dσ =.67 in vol. points.

49 beta1*mean( dβ 1 ) PL contribution maturity spot Figure 19: Profit-and-loss contribution of the β 1 for down-and-out put option with barrier 54 and strike 7425.

50 beta2 * mean( d ) 2 1 PL contribution maturity spot Figure 2: Profit-and-loss contribution of the for down-and-out put option with barrier 54 and strike 7425.

51 Appendix in the money PL contribution / price of the option delta vega beta1 theta beta maturity Figure 21: Profit-and-loss contribution divided by down-and-out put price for ITM option.

52 Appendix at the money.2 PL contribution / price of the option delta vega beta1 theta beta maturity Figure 22: Profit-and-loss contribution divided by down-and-out put price for ATM option.

53 Appendix out of the money.5 PL contribution / price of the option delta vega beta1 theta beta maturity Figure 23: Profit-and-loss contribution divided by down-and-out put price for OTM option.

54 Appendix delta spot Figure 24: Delta as a function of the spot for the half year downand-out put option with strike price 7425 and barrier 54