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1 Probability Trading Oliver BLASKOWITZ Wolfgang HÄRDLE Center for Applied Statistics and Economics (CASE) Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Motivation 1-1 Motivation Recall from option pricing theory European put: Pt (K, τ ) −rτ Z ∞ max(K − ST , 0)q(ST )dST = e 0 European call: Ct (K, τ ) = e−rτ Z ∞ max(ST − K, 0)q(ST )dST , 0 with time to maturity τ = T − t, strike price K and risk–free interest rate r. What is q(ST )? – A state price density (SPD) of the underlying! Black–Scholes world: q(ST ) lognormal and unique. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Motivation 1-2 Illustration of SPD estimation DAX 2150 Density Estimation Illustration 3m DAX TimeSeries Estimate Diffusion 3m Monte Carlo Simulation DAX-implied g*: 2100 f*: Option-implied OptionData 1 tick day 2050 19/12/94 3m Mixture of 3 LogNormals 20/03/95 Time 16/06/95 Figure 1: Illustration of SPD estimation Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Motivation 1-3 Illustration of Probability Trade • Suppose there are two SPDs, f ∗ , g ∗ . Probability Trade f* g* Sell Butterfly h(S_T) K_1 K_2 K_3 Figure 2: Probability Trade • f ∗ : estimated from calls and puts. • g ∗ : estimated from a DAX time series. • We take the position of g ∗ . • Consider a butterfly with payoff function h(ST ). Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Motivation 1-4 The fair price for a ’European’ butterfly spread is given by Z ∞ Butterfly spread: B = e−rτ h(ST )q(ST )dST . 0 This density situation implies for the butterfly spread with strikes K1 , K2 , K3 B(f ∗ ) > B(g ∗ ) price computed with f ∗ > price computed with g ∗ If the butterfly is priced using f ∗ but one regards g ∗ as a better approximation of the underylings’ SPD, one would sell it and hedge it wrt to g ∗ . The price difference (B(f ∗ )-B(g ∗ )) is the gain. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Motivation 1-5 In this work, f ∗ is an option implied SPD and g ∗ is a (historical) time series SPD. To compare implied to (historical) time series SPD’s, we use a a mixture of 3 log–normal densities to estimate f ∗ whereas g ∗ is inferred from a combination of a non–parametric estimation from a historical time series of the DAX and a forward Monte Carlo simulation. Within such a framework, is it profitable to trade ATM probabilities? Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Motivation 1-6 Outline of the talk X 1. Motivation 2. Software and Data 3. Estimation of the Option Implied SPD 4. Estimation of the Time Series SPD 5. Strategy Performance 6. TO DO Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-1 Software and Data This study was accomplished using XploRe (http://www.xplore-stat.de). Daily closing prices of DAX performance index. DAX 1pm prices at 3rd Friday of March, June, September, December (to approximate portfolio payoff). EUREX DAX–option tick prices. MD*BASE (http://www.mdtech.de) Oracle database. Time period in this study: 03/95 – 03/01. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-2 Option Data 1. index 2075.27 2. strike 2050 3. risk free interest rate 4. time to maturity 0.05 5. option price 19.5 6. call (1) or put (0) 7. moneyness (strike/future price) 8. time (sec after midnight, centiseconds) 9. implied volatility 10. number of contracts 11. date 0.05153 1 0.978 37179.31 0.17696 100 19950102 Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-3 Option Data (2) 95.00 71.25 47.50 23.75 2000 2025 2050 2075 2100 2125 2150 2200 Scatterplot of the option prices against strike price (right) on 16-th January 1995. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-4 Option Data (3) Inverting the BS formula, implied volatilities of calls and puts can differ significantly, thus violating the put-call-parity and general market efficiency considerations. Hafner & Wallmeier (2001) propose a methodology to derive an ‘adjusted’ DAX index from future prices such that Put–Call–Parity is satisfied. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 150 Prices 2-5 100 5 0.1+IV*E-2 Software and Data 0 50 Option Data (4) 5 10 0.9+Moneyness*E-2 15 20 5 15 20 Call Prices nach PCP - NbTransacts C:117 P:137 100 Prices 0 50 5 0.1+IV*E-2 150 10 IV nach PCP: Call-K:12 Put-K:15 10 0.9+Moneyness*E-2 5 10 0.9+Moneyness*E-2 15 20 5 10 0.9+Moneyness*E-2 15 20 Figure 3: Hafner, Wallmeier (2000) corrected implied volatility smile on March 20, 1995. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-6 Option Data (5) Note: The DAX index is a capital weighted performance index, i.e. dividends less corporate tax are reinvested into the index. Hafner & Wallmeier (2001) argue that the marginal investor’s individual tax scheme is different from the one assumed to compute the DAX index. Consequently, the net dividend for this investor can be higher or lower than the one used for the index computation. This discrepancy, which the authors call ‘difference dividend’ drives a wedge into the option prices and hence into implied volatilities. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-7 Option Data (6) Let ∆Dt,T denote the time T value of this difference dividend incurred between t and T . Let Ft be the futures price with maturity TF and Ct , Pt be the prices of call and put options with maturity TO . The ‘adjusted’ index level is given by S̃t = Ft e−r( TF −t) + ∆Dt,TO ,TF , (1) where ∆Dt,TO ,TF = ∆Dt,TF e−r(TF −t) − ∆Dt,TO e−r(TO −t) is the desired difference dividend. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Software and Data 2-8 Option Data (7) The ’adjusted’ index is that index level, which ties put and call implied volatilities exactly to the same levels when used in the inversion of the BS formula. While dividend information is publicly available, the marginals investors tax rate is unknown. Assuming Put–Call–Parity holds, ∆Dt,TO ,TF is estimated by : Ct − Pt = St − ∆Dt,TO e−rH (TO −t) − Ke−rH (TO −t) . (2) Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-1 Mixture of Log–Normals We assume that the SPD be a mixture of Log–normal (MLN) densities, τ τ τ qi,t ST ; αi,t , βi,t : τ qM LN (ST ) = I X τ τ θi,t qi,t τ τ ST ; αi,t , βi,t i=1 τ qi,t (ln(ST ) − √ exp − τ τ 2βi,t ST βi,t 2π 1 = τ 2 αi,t ) ! , τ where the log–normal densities qi,t are independent and I X τ θi,t =1 , τ θi,t > 0, i τ αi,t = ln(ST ) + (µτi,t − τ 0.5σi,t ) , τ βi,t = τ σi,t √ τ. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-2 Minimize over parameters (α, β, θ)T squared sum of deviations (SSD) SSD = Call NX Ctobs (Kj , τ ) − 2 M LN Ct (Kj , τ ) − 2 M LN Pt (Kj , τ ) j=1 + P ut N X Ptobs (Kj , τ ) j=1 + 2 τ Ftobs (τ ) − EqM (ST ) LN,t of observed option/future prices and theoretical option/future prices. Estimate Pf ∗ (ST ∈ [K1 , K3 ]) = l=L−1 X l=0 R K3 K1 τ qM LN (ST )dST by τ qM LN (ST,l )(ST,l+1 , −ST,l ), ST,l K3 − K 1 = K1 + l. L Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-3 Pro/Cons • Weighted sum of Black–Scholes prices τ τ τ CtM LN (K, τ ) = θ1,t C1,t (K, τ ) + θ2,t C2,t (K, τ ) + θ3,t C3,t (K, τ ) and τ τ τ PtM LN (K, τ ) = θ1,t P1,t (K, τ ) + θ2,t P2,t (K, τ ) + θ3,t P3,t (K, τ ). Thus – easy and fast to compute, – ensures no-arbitrage (mon. decreasing and convex in strike K). Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-4 • Flexible – MLN captures numerous different density shapes, (f. e. bi–modal) – able to produce smile features such as skewness and kurtosis. • A SPD is consistent with many different stochastic processes, but the converse is not true. • Limited number of observable options → limited number of parameters. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-5 Application to EUREX DAX–Options 24 non overlapping periods from March 1995 to March 2001 (τ ≈ 88/250 fixed) period from Monday following 3rd Friday to 3rd Friday 3 months later Using ≈ 1 day of option data (10am–2pm) to estimate 3m ahead SPD In this work: J = 3 (Trade–off between flexibility and numerical issues). Example: on Monday, 20/03/95, we estimate f ∗ of Friday, 16/06/95 Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-6 Illustration of the Estimation Procedure DAX Density Estimation Illustration 2150 f* 2100 OptionData 1 day 2050 19/12/94 MLN 3 months 20/03/95 Time 16/06/95 Figure 4: Procedure to estimate option implied SPD of Friday, 16/06/95, estimated on Monday, 20/03/95, by means of 1 day of option tick data. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-7 Estimated SPD 10 15 0 5 Y*E-4 20 25 3 LN Mix: 3, Mo, 19950320 TTM: 88 5 10 15 20 25 500+Underlying*E2 30 35 10 15 20 25 0 5 Y*E-4 7.60, 0.08, 7.57, 0.09, 7.62, 0.07, 0.34, 0.33, 5 10 15 20 25 500+Underlying*E2 30 35 Figure 5: Option implied SPD of Friday, 16/06/95, estimated on Monday, 20/03/95. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Option–Implied SPD 3-8 Comparison of Theoretical and Observed Prices 100 0 50 Y 150 3 LN Mix CallPrices(K): 1 2 3 1800+Strike*E2 4 -0.3 -0.2 -0.1 Y 0 0.1 3 LN Mix: 3, Mo, 19950320 TTM: 88 1 2 3 1800+Underlying*E2 4 Figure 6: Call price function over K and price deviations in percentage. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-1 Estimation of the Time Series SPD The estimation of the (historical) time series SPD is based on Aı̈t–Sahalia, Wang & Yared (2001). S follows a diffusion process dSt = µ(St )dt + σ(St )dWt . Further assume a flat yield curve and the existence of a risk–free asset B which evolves according to Bt = B0 ert . Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-2 Estimation of the Time Series SPD (2) Then the risk–neutral process follows from Itô’s formula and Girsanov’s theorem (giving a SPD g ∗ which will later be compared to the SPD f ∗ ): dSt∗ = rSt∗ dt + σ(St∗ )dWt∗ Drift adjusted but diffusion function is identical in both cases ! Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-3 Estimation of the Diffusion Function Florens–Zmirou (1993), Härdle & Tsybakov (1997) estimator for σ PN ∗ −1 σ̂ 2 (S) = i=1 Kσ ( Si/N ∗ −S )N ∗ {S(i+1)/N ∗ hσ PN ∗ Si/N ∗ −S K ( ) σ i=1 hσ − Si/N ∗ }2 Kσ kernel (in our simulation: Gaussian), hσ bandwidth, N ∗ number of observed index values (N ∗ ≈ 65) in the time interval [0, 1] σ̂ consistent estimator of σ as N ∗ → ∞ σ̂ estimated using a 3 month time series of DAX prices Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-4 3200 3400 3600 DAX 3800 0.3 0.2 0.25 DiffusionFct 0.3 DiffusionFct 0.25 0.2 0.2 0.25 DiffusionFct 0.3 0.35 Est. Diffusion for 06-09/01 0.35 Est. Diffusion for 03-06/01 0.35 Est. Diffusion for 03-06/97 5400 5600 5800 DAX 6000 6200 3900 4200 4500 4800 5100 DAX 5400 5700 6000 Figure 7: Estimated diffusion functions (”standardized” by σ̂/S0 ) for the periods 24/03/97–20/06/97 (hσ = 469.00), 19/03/01–15/06/01 (460.00), 18/06/01–21/09/01 (1597.00) respectively. regxest.xpl Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-5 Simulation of the Time Series SPD • To simulate a path use Milstein scheme given by ∗ Si/N ∗ ∗ ∗ ∗ ∗ = S(i−1)/N ∗ + rS(i−1)/N ∗ ∆t + σ̂(S(i−1)/N ∗ )∆Wi/N ∗ + n o ∂ σ̂ 1 ∗ ∗ ∗ 2 σ̂(S(i−1)/N (S ∗) ∗ ) (∆W(i−1)/N ∗ ) − ∆t , (i−1)/N 2 ∂S ∗ ∗ where ∆Wi/N ∗ ∼ N (0, ∆t) with ∆t = ∆σ ∗ approximated by ∆S ∗ , i = 1, . . . , N 1 N∗ , drift set equal to r, ∂σ ∂S ∗ ∗ • Simulate m = 1, . . . , M = 10000 paths Sm,N ∗ /N ∗ for time to maturity τ = N∗ 252 • Estimate Pg∗ (ST ∈ [K1 , K3 ]) by ∗ #{m, Sm,N ∗ /N ∗ ∈ [K1 , K3 ] , m = 1, . . . , M } M Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-6 Application to DAX 24 periods from March 1995 to March 2001 (τ ≈ 65/252 fixed) period from Monday following 3rd Friday to 3rd Friday 3 months later Example: on Monday, 20/03/95, we estimate g ∗ of Friday, 16/06/95 • Friday, December 16, 1994, is the 3rd Friday • σ̂ estimated using DAX prices from Monday, December 19, 1994, to Friday, March 17, 1995 • Monte–Carlo simulation with parameters – DAX on Monday March 20, 1995, S0 = 1984.99 – time to maturity τ = 65/250 and interest rate r = 5.10 Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD 4-7 Comparison of SPD 3m SPD est. on Mo, 18/09/1995 3m SPD est. on Mo, 18/12/2000 DAX 1pm on 15/12/1995 DAX 1pm on 16/03/2001 g* SPD SPD f* g* 0 0 f* 1800 2300 Underlying 2800 3500 5500 7500 9500 Underlying Figure 8: Option- & DAX-implied SPD of Fr, 15/12/95, & Fr, 16/03/01. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Estimation of the Time Series SPD DAX 2150 4-8 Density Estimation Illustration 3m DAX TimeSeries Estimate Diffusion 3m Monte Carlo Simulation DAX-implied g*: 2100 f*: Option-implied 2050 19/12/94 OptionData 1 tick day 3m Mixture of 3 LogNormals 20/03/95 Time 16/06/95 Figure 9: Comparison of procedures to estimate time series and option implied SPD of Friday, 16/06/95. SPD’s estimated on Monday, 20/03/95, by means of 3 months of index data respectively 1 day of option tick data. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-1 Strategy Performance 24 periods of 3 months length from March 1995 to March 2001. We start with a bank account deposit of 100 units (=index points). • t=0: – Monday following 3rd Friday of Mar, Jun, Sep, Dec – Compare SPD’s (Signal) – Set up butterfly portfolio (Long/Short) • 0 ≤ t ≤ T : Hedge option portfolio daily (Black–Scholes delta) • t=T : – 3rd Friday 3 months later (i.e. of Mar, Jun, Sep, Dec) – Compute payoff from butterfly Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-2 Unhedged Positions – Signals Buy–Signal: Pf ∗ (ST ∈ [K1 , K3 ]) < Pg∗ (ST ∈ [K1 , K3 ]) correct prediction if ST ∈ [K1 , K3 ] (payoff at T positive) Sell–Signal: Pf ∗ (ST ∈ [K1 , K3 ]) > Pg∗ (ST ∈ [K1 , K3 ]) correct prediction if ST ∈ / [K1 , K3 ] (payoff 0 and not negative) Moneyness # correct signals 0.90–0.95 0.95–1.00 1.00–1.05 1.05–1.10 ITM (t=T) 10 9 7 14 15 Table 1: Number of correct signals (out of 24) for different moneyness (strike/future price) intervals. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-3 Unhedged Positions – Option Cash Flows Cash flows: • Cash flow in t = 0, CF0 : Buying/selling options • Cash flow in t = T, CFT : Payoff of options in portfolio • Total cash flow: T CF = CF0 + CFT Strategies: (a) In each period consider only butterflies that are ATM in t = 0. (b) In each period consider only butterflies that end up ITM in t = T . Performance: Compare conditional on signal and unconditional long/short. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-4 Unhedged Positions – Option Cash Flows (2) ATM (in t = 0) ITM (in t=T) Conditional -24.21 128.83 Uncond. Long -219.41 1179.37 Uncond. Short 219.41 -1179.37 Table 2: Total Cash Flows in Index Points Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-5 Unhedged Positions – Option Cash Flows (3) 0 -40 -20 CashFlow 20 40 OptionCashFlows 0 5 10 15 20 25 Time Figure 10: Cash flows from buying/selling and terminal payoffs of butterfly portfolio conditional on density comparison. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-6 Hedged Positions - Portfolio Value ATM (in t = 0) ITM (in t=T) Conditional 6.13/-42.79% 266.99/21.70% Uncond. Long -7.58/-100% 444.47/34.76% Uncond. Short 257.73/20.85% -194.32/-100% Table 3: Portfolio Value at t=T in Index Points/Yield (annualized) Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-7 Portfolio Value 0 100 Pf-Value 200 PortfolioValue 0 5 10 15 Time*E2 Figure 11: Portfolio value conditional on density comparison. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 Strategy Performance 5-8 DAX Performance DAX 4.1994-4.2001 DAX 8000 7000 6000 5000 4000 3000 2000 1/95 1/96 1/97 1/98 1/99 1/00 1/01 Time 12/01 Figure 12: DAX Performance. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 TO DO 6-1 TO DO Estimate diffusion function from DAX tick data. Compute delta hedge numerically using g ∗ . Improve Black–Scholes delta hedge by smoothing implied volatility curve. Improve trading signal by testing for significant inequality of densities. Consider transaction costs for option and stock transactions. Strategy performance conditional for all butterfly portfolios (not only subset). Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 References 7-1 References Aı̈t–Sahalia, Y., Wang, Y. & Yared, F. (2001). Do Option Markets correctly Price the Probabilities of Movement of the Underlying Asset?, Journal of Econometrics 102: 67–110. Bahra, B. (1995). Implied risk-neutral probability density functions from option prices: theory and application, Bank of England Working Papers 66. Black, F. & Scholes, M., (1998). The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81: 637–659. Breeden, D. & Litzenberger, R., (1978). Prices of State Contingent Claims Implicit in Option Prices, Journal of Business, 9, 4: 621–651. Florens–Zmirou, D. (1993). On Estimating the Diffusion Coefficient from Discrete Observations, Journal of Applied Probability 30: 790–804. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3 References 7-2 Franke, J., Härdle, W. & Hafner, C. (2003). Einführung in die Statistik der Finanzmärkte, 2nd edition, Springer Verlag, Heidelberg. Hafner, R. & Wallmeier, M., (2001). The Dynamics of DAX Implied Volatilities, International Quarterly Journal of Finance,1 1:1 –27. Härdle, W. & Tsybakov, A., (1997). Local Polynomial Estimators of the Volatility Function in Nonparametric Autoregression, Journal of Econometrics, 81: 223–242. Kloeden, P., Platen, E. & Schurz, H. (1994). Numerical Solution of SDE Through Computer Experiments, Springer Verlag, Heidelberg. Rubinstein, M. (1994). Implied Binomial Trees, Journal of Finance 49: 771–818. Yatchew, A. & Härdle, W. (2003). Dynamic Nonparametric State Price Density Estimation using Constrained Least Squares and the Bootstrap, Journal of Econometrics, accepted. Probability Trade f* g* Probability Trading Sell Butterfly h(S_T) K_1 K_2 K_3