Hedging under Model Uncertainty

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Hedging under Model Uncertainty Efficient Computation of the Hedging Error using the POD 6th World Congress of the Bachelier Finance Society June, 24th 2010 M. Monoyios, T. Schröter, Oxford University M. Rometsch, K. Urban, Ulm University RTG 1100 Ulm University Oxford University

Page 2 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Hedging under Model Uncertainty Considered Models Reduced Model and POD Results

Page 3 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Hedging under Model Uncertainty Hedging under Model Uncertainty Considered Models Reduced Model and POD Results

Page 4 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Hedging under Model Uncertainty Hedging a exotic Option under Model Uncertainty Exotic Option: Asian Call C A pt, S T q s S T K Ÿ Ÿ Model uncertainty: true model hedge model Relation between true model and hedge model: Ÿ Vanilla Options for the calibration of the hedge model parameters

Page 4 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Hedging under Model Uncertainty Hedging a exotic Option under Model Uncertainty Exotic Option: Asian Call C A pt, S T q s S T K Ÿ Ÿ Ÿ Model uncertainty: true model hedge model Relation between true model and hedge model: Ÿ Vanilla Options for the calibration of the hedge model parameters Hedging Approach: Ÿ Delta-Hedge with bank account and underlying Ÿ Delta- and Vega-Hedge with bank account, underlying and vanilla option

Page 5 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Considered Models Hedging under Model Uncertainty Considered Models Reduced Model and POD Results

Page 6 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Considered Models True Models 3-Factor Model (3FM) ps t, v t, ρ t q driven by Wt 1, Wt 2, Wt 3 Extended Heston Model with correlated jumps (SVJJ) S t S 0 e xt where dx t µ 1 2 v t dt? vt dw 1 t ξ x dn t dv t α pβ v t q dt σ v? vt ρdw 1 t a 1 ρ2 dw 2 t Extended CGMY Model (CGMYe)! ) S t S 0 exp µ ω η {2 2 t Xt CGMY ηw t ξ v dn t

Page 7 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Considered Models Hedge Models Black-Scholes Model (BS) ds t rs t dt σs t dw t Stochastic Alpha, Beta, Rho Model (SABR) ds t rs t dt σ t S t dwt 1 a dσ t ασ t ρdwt 1 1 ρ2 dwt 2 Heston Model (SV)? ds t rs t dt vt S t dwt 1? dv t α pβ v t q dt σ v vt ρdwt 1 a 1 ρ2 dwt 2

Page 8 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Considered Models Model Properties True Models Ÿ Models represent various stylized facts. Ÿ Calibrated parameter are available for P and for Q. Ÿ Vanilla Option prices available via Ÿ Monte-Carlo (3FM) Ÿ Analytic Formula (SVJJ) Ÿ PIDE (CGMYe) Ÿ Valuation of the Asian Option by Monte-Carlo. Hedge Models Ÿ Ÿ (Semi-)analytic formulas available for the Vanilla Option prices. Valuation of the Asian Option and hedging weights via PDE.

Page 9 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Hedging under Model Uncertainty Considered Models Reduced Model and POD Results

Page 10 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD PDE in Heston Model Idea: Vecer 02, Shreve 08 ñ C A pt, S t, v t q S t gpt, Y t, v t q where 2-dimensional parabolic PDE in Heston Model B gpt, y, vq Bt ν B φ v Bv gpt, y, vq 1 2 v pq t yq 2 B 2 gpt, y, vq By 2 1 2 ϕ2 v B2 gpt, y, vq Bv 2 ϕρv pq t yq B 2 gpt, y, vq 0, ByBv gpt, y, vq y and Y t 1» 1 t S u du K S t T 0 on Ω p 1, 1q p0, 8q. Further Advantage: Greeks are directly computable from the PDE.

Page 11 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD PDE in Black-Scholes Model Again we have C A pt, S t q S t gpt, Y t q where this time 1-dimensional parabolic PDE in Black-Scholes Model B σ2 gpt, yq Bt 2 pq t yq 2 B 2 gpt, yq 0, By 2 gp0, yq y on Ω p 1, 1q Similar PDE for Λ (Vega) B σ2 Λpt, yq Bt 2 pq t yq 2 B 2 By 2 Λpt, yq σ pq t yq 2 B 2 gpt, yq By 2

Page 12 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Methods and POD Problem Many solutions of the PDE are necessary for the Calculation of the Hedge weights in the simulation (N 63), each with different parameters. ùñ Approximate the solution with a reduced model. Instead of the classical hat basis (FE) 1 Hat basis on Ω p 1, 1q with Ny 8 0.8 0.6 0.4 0.2 0-1 -0.5 0 0.5 1 try to use a reduced basis consisting of empirical eigenfunctions.

Page 13 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis for the Black-Scholes Model Store snapshots of the solution gpt i, y j ; θq M,N i,j1 of the PDE in a matrix Z pgpt i, y j ; θ l qq M,N,l and calculate via the SVD the reduced basis. i,j,l1 Example: Calculation of gpt, yq with N y 801, M 400 1 gpt, yq gp0, yq 100000 Singular values of Z 0.8 1 0.6 1e-05 0.4 1e-10 0.2 1e-15 0-1 -0.5 0 0.5 1 y 1e-20 1 10 100 1000 0.15 ψ1 Offline: Calculate the reduced basis. Online: Use N! N degrees of freedom for the actual computation. 0.1 0.05 0-0.05-0.1-0.15 ψ2 ψ3 ψ4 ψ5-0.2-1 -0.5 0 0.5 1

Page 14 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Efficiency in Black-Scholes Model FEM-calculation with 801 basis functions: 35 seconds σ C A p0, 100q Delta Vega 0.1 10.5969 0.989909 0.32507 0.188889 10.8205 0.926435 5.15053 0.277778 11.4764 0.845893 9.20365 0.366667 12.4004 0.787475 11.3734 0.455556 13.4674 0.747672 12.5557 0.544444 14.6143 0.720255 13.2377 0.633333 15.8083 0.701023 13.6474 0.722222 17.0308 0.687362 13.8967 0.811111 18.2704 0.677634 14.0426 0.9 19.5192 0.670807 14.1108

Page 14 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Efficiency in Black-Scholes Model FEM-calculation with 801 basis functions: 35 seconds Computation of the POD-basis: 39 seconds σ C A p0, 100q Delta Vega 0.1 10.5969 0.989909 0.32507 0.188889 10.8205 0.926435 5.15053 0.277778 11.4764 0.845893 9.20365 0.366667 12.4004 0.787475 11.3734 0.455556 13.4674 0.747672 12.5557 0.544444 14.6143 0.720255 13.2377 0.633333 15.8083 0.701023 13.6474 0.722222 17.0308 0.687362 13.8967 0.811111 18.2704 0.677634 14.0426 0.9 19.5192 0.670807 14.1108

Page 14 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Efficiency in Black-Scholes Model FEM-calculation with 801 basis functions: 35 seconds Computation of the POD-basis: 39 seconds POD-calculation with 15 basis functions: 1.2 seconds σ C A p0, 100q Delta Vega Price error Delta error Vega error 0.1 10.5968 0.989825 0.34534 8.3006e 05 8.38586e 05 0.0202638 0.188889 10.8198 0.926762 5.12669 0.000685423 0.000326805 0.0238353 0.277778 11.4769 0.845723 9.21924 0.000444494 0.000169424 0.0155903 0.366667 12.4005 0.787226 11.3620 0.000149486 0.000249543 0.0114599 0.455556 13.4670 0.748043 12.5556 0.000365294 0.000371058 0.000164472 0.544444 14.6142 0.720466 13.2412 0.000161619 0.000211238 0.00358165 0.633333 15.8085 0.700741 13.6519 0.000218763 0.000282165 0.00455001 0.722222 17.0313 0.686913 13.8967 0.000465986 0.000449245 7.45315e 05 0.811111 18.2704 0.677614 14.0334 4.05614e 05 2.066e 05 0.00913167 0.9 19.5182 0.671855 14.0983 0.00100108 0.00104796 0.0125393

Page 15 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis for the Heston Model Example: Calculate gpt, y, vq with N y 61, N v 41, M 625 gpt, y, vq 10000 100 Singular values of Z 1 0.8 0.6 0.4 0.2 0-0.2 1 0.01 0.0001 1e-06 2 1e-08 v 1.5 1 0.5-1 -0.5 0 y 0.5 1 1e-10 1e-12 1e-14 1 10 100 1000 ψ1 ψ2 ψ3 0.05 0.04 0.03 0.02 0.01 0-0.01 0.08 0.06 0.04 0.02 0-0.02 0.1 0.08 0.06 0.04 0.02-0.02 0-0.04-0.06 2 v 1.5 1 0.5-1 -0.5 0 y 0.5 1 2 v 1.5 1 0.5-1 -0.5 0 y 0.5 1 2 v 1.5 1 0.5-1 -0.5 0 y 0.5 1 ψ 1 ψ 2 ψ 3

Page 16 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Efficiency in Heston Model FEM-calculation with 81 61 basis functions: 123 seconds θ i C A p0, 100, 0.022q Delta Vega 1 11.421172 0.885409 4.287797 2 11.832293 0.850130 5.537867 3 12.015476 0.851747 4.963191 4 11.716659 0.852718 5.177694 5 10.913815 0.948827 3.548435 6 11.540681 0.863221 4.433649 7 11.465448 0.883098 6.339542 8 11.307237 0.888459 8.173694 9 11.147410 0.914950 5.961920 10 11.098593 0.900337 3.640245 11 10.985362 0.936262 5.988487 12 11.286456 0.904651 7.177647 13 10.858524 0.958111 1.823538 14 10.945517 0.941020 4.182806 15 11.559169 0.877870 5.840867

Page 16 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Efficiency in Heston Model FEM-calculation with 81 61 basis functions: 123 seconds Computation of the POD-basis: 132 seconds θ i C A p0, 100, 0.022q Delta Vega 1 11.421172 0.885409 4.287797 2 11.832293 0.850130 5.537867 3 12.015476 0.851747 4.963191 4 11.716659 0.852718 5.177694 5 10.913815 0.948827 3.548435 6 11.540681 0.863221 4.433649 7 11.465448 0.883098 6.339542 8 11.307237 0.888459 8.173694 9 11.147410 0.914950 5.961920 10 11.098593 0.900337 3.640245 11 10.985362 0.936262 5.988487 12 11.286456 0.904651 7.177647 13 10.858524 0.958111 1.823538 14 10.945517 0.941020 4.182806 15 11.559169 0.877870 5.840867

Page 16 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Reduced Basis Efficiency in Heston Model FEM-calculation with 81 61 basis functions: 123 seconds Computation of the POD-basis: 132 seconds POD-calculation with 42 basis functions: 1.7 seconds θ i C A p0, 100, 0.022q Delta Vega Price error Delta error Vega error 1 11.402936 0.885104 4.485977 0.018236 0.000305 0.198180 2 11.842398 0.851246 5.364510 0.010105 0.001115 0.173357 3 12.022748 0.853884 4.436634 0.007271 0.002136 0.526557 4 11.715899 0.853218 5.118843 0.000761 0.000500 0.058851 5 10.910492 0.947674 3.534090 0.003323 0.001153 0.014345 6 11.539249 0.862584 4.491026 0.001432 0.000637 0.057377 7 11.457013 0.884826 6.363838 0.008435 0.001728 0.024296 8 11.312174 0.887855 8.136989 0.004937 0.000604 0.036705 9 11.150456 0.916820 5.748191 0.003046 0.001870 0.213729 10 11.090177 0.903810 3.242317 0.008416 0.003473 0.397928 11 10.986913 0.936491 6.183714 0.001551 0.000229 0.195227 12 11.296781 0.907058 6.747697 0.010326 0.002407 0.429950 13 10.849729 0.959915 1.469688 0.008794 0.001804 0.353850 14 10.949302 0.939686 4.321778 0.003785 0.001334 0.138972 15 11.547142 0.879573 5.907704 0.012027 0.001704 0.066837

Page 17 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Approach Scenario Generation 1. Choose a true model as well as its P-and Q parameters. 2. Generate N 50000 trajectories each with 63 days under P with daily observation of S t and calculate daily Q-prices of C E,1,..., C E,15.

Page 17 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Reduced Model and POD Approach Scenario Generation 1. Choose a true model as well as its P-and Q parameters. 2. Generate N 50000 trajectories each with 63 days under P with daily observation of S t and calculate daily Q-prices of C E,1,..., C E,15. Hedge Calculation and Evaluation 1. Choose the maturity of C A as t21, 126, 189u days. 2. Choose the hedge model. 3. For each trajectory the hedge model gets calibrated to S t and (a subset of) C E,1,..., C E,15 on a daily basis. 4. Calculate the hedge portfolio. 5. At the end of the path the hedging error is evaluated. 6. Build up the hedging error distribution.

Page 18 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results Hedging under Model Uncertainty Considered Models Reduced Model and POD Results

Page 19 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results 3FM, Local Calibration, Delta-Hedge BS in 3FM(21d) BS in 3FM(126d) BS in 3FM(189d) 0 10 20 0 10 20 0 10 20 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 SABR in 3FM(21d) SABR in 3FM(126d) SABR in 3FM(189d) 0 10 20 0 10 20 0 10 20 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5

Page 20 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results 3FM, Local Calibration, Delta- and Vega-Hedge BS(vega) in 3FM(21d) BS(vega) in 3FM(126d) BS(vega) in 3FM(189d) 0 10 20 0 10 20 0 10 20 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 SABR(vega) in 3FM(21d) SABR(vega) in 3FM(126d) SABR(vega) in 3FM(189d) 0 10 20 0 10 20 0 10 20 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5

Page 21 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results SVJJ, Local Calibration, Delta-Hedge BS in SVJJ(21d) BS in SVJJ(126d) BS in SVJJ(189d) HEST in SVJJ(21d) HEST in SVJJ(126d) HEST in SVJJ(189d)

Page 22 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results SVJJ, Local Calibration, Delta- and Vega-Hedge BS(vega) in SVJJ(21d) BS(vega) in SVJJ(126d) BS(vega) in SVJJ(189d) HEST(vega) in SVJJ(21d) HEST(vega) in SVJJ(126d) HEST(vega) in SVJJ(189d)

Page 23 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results CGMYe, Local Calibration, Delta-Hedge SABR in CGMYe(21d) SABR in CGMYe(126d) SABR in CGMYe(189d) 6 4 2 0 2 6 4 2 0 2 6 4 2 0 2 HEST in CGMYe(21d) HEST in CGMYe(126d) HEST in CGMYe(189d) 6 4 2 0 2 6 4 2 0 2 6 4 2 0 2

Page 24 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results CGMYe, Local Calibration, Delta- and Vega-Hedge SABR(vega) in CGMYe(21d) SABR(vega) in CGMYe(126d) SABR(vega) in CGMYe(189d) 6 4 2 0 2 6 4 2 0 2 6 4 2 0 2 HEST(vega) in CGMYe(21d) HEST(vega) in CGMYe(126d) HEST(vega) in CGMYe(189d) 6 4 2 0 2 6 4 2 0 2 6 4 2 0 2

Page 25 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results 3FM, Global Calibration, Delta-Hedge BS in 3FM(21d) BS in 3FM(126d) BS in 3FM(189d) 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 HEST in 3FM(21d) HEST in 3FM(126d) HEST in 3FM(189d) 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5

Page 26 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results 3FM, Global Calibration, Delta- and Vega-Hedge BS(vega) in 3FM(21d) BS(vega) in 3FM(126d) BS(vega) in 3FM(189d) 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 HEST(vega) in 3FM(21d) HEST(vega) in 3FM(126d) HEST(vega) in 3FM(189d) 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5

Page 27 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results SVJJ, Global Calibration, Delta-Hedge SABR in SVJJ(21d) SABR in SVJJ(126d) SABR in SVJJ(189d) Adj. Rel. Frequency Adj. Rel. Frequency Adj. Rel. Frequency HEST in SVJJ(21d) HEST in SVJJ(126d) HEST in SVJJ(189d) Adj. Rel. Frequency Adj. Rel. Frequency Adj. Rel. Frequency

Page 28 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results SVJJ, Global Calibration, Delta- and Vega-Hedge SABR(vega) in SVJJ(21d) SABR(vega) in SVJJ(126d) SABR(vega) in SVJJ(189d) Adj. Rel. Frequency Adj. Rel. Frequency Adj. Rel. Frequency HEST(vega) in SVJJ(21d) HEST(vega) in SVJJ(126d) HEST(vega) in SVJJ(189d) Adj. Rel. Frequency Adj. Rel. Frequency Adj. Rel. Frequency

Page 29 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results Wrap up Ÿ Ÿ Ÿ Ÿ Hedging under model uncertainty POD for parameter dependent (parabolic) PDEs Analysis of the hedging error distribution Simple models (BS) seem to be preferable in unknown markets

Page 29 Hedging under Model Uncertainty June, 24th 2010 M. Monoyios, M. Rometsch, T. Schröter, K. Urban Results Wrap up Ÿ Ÿ Ÿ Ÿ Hedging under model uncertainty POD for parameter dependent (parabolic) PDEs Analysis of the hedging error distribution Simple models (BS) seem to be preferable in unknown markets Contact RTG 1100 Ulm University M. Monoyios, T. Schröter, Oxford University M. Rometsch, K. Urban, Ulm University {roman.rometsch, karsten.urban}@uni-ulm.de {monoyios, schroter}@maths.ox.ac.uk Oxford University Thank you for your attention

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