Proxy Simulation Schemes for Generic Robust Monte-Carlo Sensitivities and High Accuracy Drift Approximation

Transcription

1 Proxy Simulation Schemes for Generic Robust Monte-Carlo Sensitivities and High Accuracy Drift Approximation with Applications to the LIBOR Market Model Christian Fries (Version 1.5) 1

2 Agenda Monte Carlo Method: A Review of Challenges & Solutions Temporal Discretization Error Sensitivity Calculation in Monte-Carlo Finite Differences Pathwise Differentiation Pathwise Differentiation (alternative view) Likelihood Ratio Method Malliavin Calculus Proxy Simulations Scheme Method Pricing & Sensitivity Calculation Implementation Densities & Weak Schemes A Note on Degenerate Diffusion Matrix / Measure Equivalence Summary Summary: Requirements, Properties Example: LIBOR Market Model Numerical Results Drift Approximations Sensitivity Calculations Appendix A Quadratic WKB Expansion for Transition Probability of the LIBOR Market Model References 2

3 Monte Carlo Method A Short Review of Challenges and Solutions 3

4 Discretization Error Drift Approximations 4

5 Monte Carlo Method: Discretization Error Consider for SDE dx(t) = µ(t, X(t))X(t)dt + σ(t, X(t))X(t)dW(t), e.g. the Log-Euler Scheme X(t + t) = X(t) exp ( µ(t, X(t)) t 1 2 σ2 (t, X(t)) t + σ(t, X(t)) W(t) ). If σ is constant on [t, t] (Black Model, LIBOR Market Model) but µ is stochastic and/or non-linear (LIBOR Market Model), then the discretization error is given by a drift approximation error, e.g. here t+ t t µ(τ, X(τ))dτ µ(t, X(t)) t. Solutions Predictor Corrector Method(s) (= alternative integration rule) Proxy Simulation Scheme / Weak Scheme (discussed later) 5

6 Sensitivities in Monte Carlo Partial Derivative with respect to Model Parameters 6

7 Monte Carlo Method: Sensitivities Let Z denote a random variable depending on realizations Y := (X(T 1 ),... X(T m )) of our simulated (Numéraire relative) state variables Z = f (Y) = f (X(T 1 ),... X(T m )) e.g. the Numéraire relative path values of a financial product. Then the (Numéraire relative) price is given by E Q (Z F T0 ) = E Q ( f (X(T 1 ),... X(T m )) F T0 ). Challenge: Let θ denote a parameter of the model SDE (e.g. its initial condition X(0), volatility σ or any other complex function of those). We are interested in θ EQ (Z F T0 ) = f (X(T θ 1, ω, θ),... X(T m, ω, θ)) dq(ω) Ω = θ IR f (x 1,... x m ) m } {{ } φ (X(T1,θ),...,X(T m,θ))(x 1,... x m ) d(x } {{ } 1,... x m ) payoff density - in general smooth in θ may be discontinuouse Problem: Monte-Carlo approximation inherits regularity of f not of φ: E Q (Z F T0 ) Ê Q (Z F T0 ) := 1 n n i=1 f (X(T 1, ω i, θ),... X(T m, ω i, θ)) } {{ } payoff on path - may be discontinuouse 7

8 A Note on Sensitivies in Monte Carlo 8

9 Example: AutoCap Sensitivities: Cap Products Caplet: Single option on forward rate. Payoff profile: max(l i (T i ) K, 0) (T i+1 T i ) paid in T i+1 Cap: Portfolio (series) of n options on forward rates (Caplets). Value = Sum of Caplets. Chooser Cap: Cap, where only some (k < n) options may be exercised. Holder may choose upon each excersie date. Value is given by optimal exercise strategy. Value depends continuously on model & product parameters. Auto Cap: Cap, where only some (k < n) options may be exercised. Excercise is triggered is Caplet payout is positive. Payoff profile: max(l i (T i ) K, 0) (T i+1 T i ) if { j : L j (T j ) K > 0 and j < i } < k 0 else paid in T i+1. Auto Cap Features: On a single (fixed) path the product depends discontinuously on the input data (e.g. todays interest rate level). Note: Chooser Cap depends continuously on model & product parameters. Thus: Using Monte-Carlo, numerical evaluation of partial derivatives (greeks) is terrible inaccurate. 9

10 Example: AutoCap Sensitivities The value of an Auto Cap conditioned to a single path ω is a discontinuous function of the interest rate curve. Example: Auto Cap pays the first 2 positive Caplet payouts out of 3 (a) Interest rate curve on a single path (scenario) ω Payment Strike T 0 T 1 T 2 T 3 10

11 Example: AutoCap Sensitivities The value of an Auto Cap conditioned to a single path ω is a discontinuous function of the interest rate curve. Example: Auto Cap pays the first 2 positive Caplet payouts out of 3 (a) (b) Interest rate curve on a single path (scenario) ω Payment Strike T 0 T 1 T 2 T 3 10

12 Example: AutoCap Sensitivities The value of an Auto Cap conditioned to a single path ω is a discontinuous function of the interest rate curve. Example: Auto Cap pays the first 2 positive Caplet payouts out of 3 (a) (c) (b) Interest rate curve on a single path (scenario) ω Payment Strike T 0 T 1 T 2 T 3 10

13 Example: AutoCap Sensitivities The value of an Auto Cap conditioned to a single path ω is a discontinuous function of the interest rate curve. Example: Auto Cap pays the first 2 positive Caplet payouts out of 3 (a) (c) (b) Interest rate curve on a single path (scenario) ω Payment Strike T 0 T 1 T 2 T 3 10

14 Example: AutoCap Sensitivities: 1 bp shift 11

15 Example: AutoCap Sensitivities: 1 bp shift Jump due to change in exercise times on some path Price dependence of the caplets, without a change in exercise times. 11

16 Example: AutoCap Sensitivities 12

17 Example: AutoCap Sensitivities: 10 bp shift 13

18 Example: AutoCap Sensitivities 14

19 Example: AutoCap Sensitivities: 100 bp shift 15

20 Example: AutoCap Sensitivities: Delta 100 bp 16

21 Example: AutoCap Sensitivities: Gamma 100 bp 17

22 A Note on Generic Sensitivities 18

23 Monte Carlo Methods: Sensitivities A Note on Generic Sensitivities What a mathematician considers as the delta of an option is not what a trader considers as the delta. After a change in market data a model has to be recalibrated. Example: Given the assumption of a certain volatility modeling (e.g. sticky strike versus sticky moneyness), a change in the underlying might also imply a change in the whole volatility surface. We have to distinguish (generic) market sensitivities and model sensitivities. Market Data Set Calibration Procedure Model Parameter Set Pricing Model Product Price + Szenario difference = sensitivity Market Data Set 2 Calibration Procedure Model Parameter Set 2 Pricing Model Product Price 2 19

24 Monte Carlo Methods: Sensitivities A Note on Generic Sensitivities What a mathematician considers as the delta of an option is not what a trader considers as the delta. After a change in market data a model has to be recalibrated. Example: Given the assumption of a certain volatility modeling (e.g. sticky strike versus sticky moneyness), a change in the underlying might also imply a change in the whole volatility surface. We have to distinguish (generic) market sensitivities and model sensitivities. Model Sensitivity Market Data Set Calibration Procedure Model Parameter Set Pricing Model Product Price + Szenario difference = sensitivity Market Data Set 2 Calibration Procedure Model Parameter Set 2 Pricing Model Product Price 2 19

25 Monte Carlo Methods: Sensitivities A Note on Generic Sensitivities What a mathematician considers as the delta of an option is not what a trader considers as the delta. After a change in market data a model has to be recalibrated. Example: Given the assumption of a certain volatility modeling (e.g. sticky strike versus sticky moneyness), a change in the underlying might also imply a change in the whole volatility surface. We have to distinguish (generic) market sensitivities and model sensitivities. Market Sensitivity Market Data Set Calibration Procedure Model Parameter Set Pricing Model Product Price + Szenario difference = sensitivity Market Data Set 2 Calibration Procedure Model Parameter Set 2 Pricing Model Product Price 2 19

26 Monte Carlo Methods: Sensitivities A Note on Generic Sensitivities Methods for calculating generic sensitivities: Finite Differences Problem: May be numerically unstable. Chain rule and finite differences for market data / calibration some other method (see below) for model sensitivities Problem: May require full set of model sensitivities. Finite Differences on a Proxy Simulation Scheme Methods for calculating model sensitivities: Finite Differences Pathwise Differentiation Likelihood Ration Method Malliavin Calculus 20

27 Sensitivities in Monte Carlo Overview 21

28 Monte Carlo Methods: Sensitivities Finite Differences: θ EQ ( f (Y(θ)) F T0 ) θêq ( f (Y(θ)) F T0 ) 1 (ÊQ ( f (Y(θ + h)) F 2h T0 ) Ê Q ( f (Y(θ h)) F T0 ) ) = 1 n 1 ( f (Y(ωi, θ + h) f (Y(ω n 2h i, θ h) ) i=1 Requirements Requires no additional information from the model sde dx =... Requires no additional information from the simulation scheme X(T i+1 ) =... Requires no additional information from the payout f Requires no additional information on the nature of θ ( generic sensitivities) Properties Generic sensitivities (market sensitivies) Biased derivative for large h due to finite difference of order h Large variance for discontinuous payouts and small h (order h 1 ) 22

29 Monte Carlo Methods: Sensitivities: Pathwise Differentiation Pathwise Differentiation: θ EQ ( f (Y(θ)) F T0 ) θêq ( f (Y(θ)) F T0 ) = 1 n f (Y(ωi, θ) n θ( ) = 1 n i=1 n i=1 f (Y(ω i, θ)) Y(ω i, θ) θ Requirements Requires additional information on the model sde dx =... Requires no additional information on the simulation scheme X(T i+1 ) =... Requires additional information on the payout f (derivative of f must be known) Requires additional information on the nature of θ ( restricted class of model parameters) Properties No generic gensitivities (model sensitivies only) Unbiased derivative Requires smoothness of payout? (in this formulation) 23

30 Monte Carlo Methods: Sensitivities: Pathwise Differentiation Pathwise Differentiation (alternative interpretation): θ EQ ( f (Y(θ)) F T0 ) = f (Y(ω, θ)) dq(ω) = f (Y(ω, θ)) dq(ω) θ Ω Ω θ = f Y(ω, θ) (Y(ω, θ)) dq(ω) = E Q ( f (Y(θ)) Y(θ) F Ω θ θ T0 ) Ê Q ( f (Y(θ)) Y(θ) F θ T0 ) = 1 n f (Y(ω n i, θ)) Y(ω i, θ) θ Note: See Joshi & Kainth [JK] or Rott & Fries [RF] for an example on how use pathwise differentiation with discontinuous payouts (there in the context of Default Swaps, CDOs). Requirements Requires additional information on the model sde dx =... Requires no additional information on the simulation scheme X(T i+1 ) =... Requires additional information on the payout f (derivative of f must be known) Requires additional information on the nature of θ ( restricted class of model parameters) Properties No generic gensitivities (model sensitivies only) Unbiased derivative Discontinuous payouts may be handled (interpret f as distribution, for applications see e.g. [JK, RF]) i=1 24

31 Monte Carlo Methods: Sensitivities: Likelihood Ratio Likelihood Ratio: θ EQ ( f (Y(θ)) F T0 ) = = θ Ω IR m f (y) f (Y(ω, θ)) dq(ω) = θ φ Y(θ)(y) φ Y(θ) (y) Ê Q ( f (Y) w(θ) F T0 ) = 1 n θ IR m f (y) φ Y(θ)(y) dy φ Y(θ) (y) dy = E Q ( f (Y) w(θ) F T0 ) n f (Y(ω i )) w(θ, ω i ) i=1 Requirements Requires additional information on the model sde dx =... ( φ Y(θ) ) Requires no additional information on the simulation scheme X(T i+1 ) =... Requires no additional information on the payout f Requires additional information on the nature of θ ( restricted class of model parameters) Properties No generic gensitivities (model sensitivies only) Unbiased derivative Discontinuous payouts may be handled. 25

32 Monte Carlo Methods: Sensitivities: Malliavin Calculus Malliavin Calculus: θ EQ ( f (Y(θ)) F T0 ) = E Q ( f (Y(θ)) w(θ) F T0 ) Ê Q ( f (Y(θ)) w(θ) F T0 ) = 1 n n i=1 f (Y(θ, ω i )) w(θ, ω i ) Note: Benhamou [B01] showed that the Likelihood Ratio corresponds to the Malliavin weights with minimal variance and may be expressed as a conditional expectation of all corresponding Malliavin weights (we thus view the Likelihood Ratio as an example for the Malliavin weighting method). Requirements Requires additional information on the model sde dx =... ( w) Requires no additional information on the simulation scheme X(T i+1 ) =... Requires no additional information on the payout f Requires additional information on the nature of θ ( restricted class of model parameters) Properties No generic gensitivities (model sensitivies only) Unbiased derivative Discontinuous payouts may be handled. 26

33 Proxy Simulation Scheme 27

34 Proxy Simulation Scheme Pricing / Sensitivities 28

35 Proxy Scheme Simulation: Pricing Proxy Scheme: Consider three stochastic processes X t X(t) t IR model sde X T i X (T i ) i = 0, 1, 2,... time discretization scheme of X target scheme X T i X (T i ) i = 0, 1, 2,... any other time discrete stochastic process (assumed to be close to X ) proxy scheme Pricing: Let Y = (X(T 1 ),..., X(T m ), Y = (X (T 1 ),..., X (T m ), Y = (X (T 1 ),..., X (T m ). We have E Q ( f (Y(θ)) F T0 ) E Q ( f (Y (θ)) F T0 ) and furthermore E Q ( f (Y (θ)) F T0 ) = f (Y (ω, θ)) dq(ω) = Ω IR f (y) φ m Y (θ)(y) dy = IR f (y) φy (θ)(y) φ m φ Y (y) Y (y) dy = E Q ( f (Y ) w(θ) F T0 ) where w(θ) = φ Y (θ) (y) φ Y (y). Note: For X = X we have w(θ) = 1 ordinary Monte Carlo. Y is seen as beeing independent of θ. implications on sensitivities. Requirement: y : φ Y (y) = 0 φ Y (y) = 0 29

36 Proxy Scheme Simulation: Sensitivities Proxy Scheme Sensitivities: θ EQ ( f (Y (θ)) F T0 ) = f (Y (ω, θ)) dq(ω) = θ Ω θ IR f (y) φ m Y (θ)(y) dy = IR f (y) θ φ Y (θ)(y) φ m φ Y (y) Y (y) dy = E Q ( f (Y ) w (θ) F T0 ) Ê Q ( f (Y ) w (θ) F T0 ) = 1 n f (Y (ω n i )) w (θ, ω i ) i=1 Requirements Requires no additional information on the model sde dx =... Requires additional information on the simulation scheme X (T i+1 ), X (T i+1 ) Requires no additional information on the payout f Requires additional information on the nature of θ ( restricted class of model parameters) Properties No generic gensitivities (model sensitivies only) Unbiased derivative (biased if finite differences are used for w) Discontinuous payouts may be handled. 30

37 Proxy Scheme Simulation: Sensitivities Proxy Scheme Sensitivities: θ EQ ( f (Y (θ)) F T0 ) Requirements Requires no additional information on the model sde dx =... Requires additional information on the simulation scheme X (T i+1 ), X (T i+1 ) Requires no additional information on the payout f Requires no additional information on the nature of θ ( generic sensitivities) Properties Generic gensitivities (market sensitivies) Biased derivative (but small shift h possible!) Discontinuous payouts may be handled. 1 ( E Q ( f (Y (θ + h)) F 2h T0 ) E Q ( f (Y (θ h)) F T0 ) ) = θ IR f (y) 1 m 2h (φ Y (θ+h)(y) φ Y (θ h)(y)) dy 1 = IR f (y) 2h (φ Y (θ+h)(y) φ Y (θ h)(y)) φ m φ Y (y) Y (y) dy 1 n f (Y 1 (ω n i )) 2h (w(θ + h, ω i) w(θ h, ω i )) i=1 31

38 Proxy Scheme Simulation: Sensitivities Proxy Scheme Sensitivities: θ EQ ( f (Y (θ)) F T0 ) 1 ( E Q ( f (Y (θ + h)) F 2h T0 ) E Q ( f (Y (θ h)) F T0 ) ) = θ IR f (y) 1 m 2h (φ Y (θ+h)(y) φ Y (θ h)(y)) dy 1 = IR f (y) 2h (φ Y (θ+h)(y) φ Y (θ h)(y)) φ m φ Y (y) Y (y) dy 1 n f (Y 1 (ω n i )) 2h (w(θ + h, ω i) w(θ h, ω i )) i=1 Finite difference applied to the pricing results in a finite difference approximation of the Likelihood Ratio thus We have all the nice properties of the Likelihood Ratio combined with the genericity of Finite Differences 32

39 Proxy Simulation Scheme Implementation 33

40 Implementation Standard Monte Carlo Simulation: Pricing InputData Market Data Model Calibrated Model Parameters Simulation Equally weighted Paths of Simulation Scheme Product! f(y(" i )) #n Price 34

41 Implementation Standard Monte Carlo Simulation: Sensitivities InputData Market Data!+h InputData Market Data!-h Model Calibrated Model Parameters Simulation Equally weighted Paths of Simulation Scheme Product! f(y(" i )) #n Price Price Sensitivity as Finite Difference 35

42 Implementation Proxy Simulation Method: Pricing InputData Market Data Model Calibrated Model Parameters Proxy Model Model Parameters Simulation Monte Carlo weights Paths of Proxy Scheme Product! f(y(" i )) w i Price 36

43 Implementation Proxy Simulation Method: Sensitivities InputData Market Data!+h InputData Market Data!-h Model Calibrated Model Parameters Proxy Model Model Parameters Simulation Monte Carlo weights Paths of Proxy Scheme Price Product! f(y(" i )) w i Price LR like Sensitivity as Finite Difference 37

44 Proxy Simulation Scheme A Note on Denstities and Weak Schemes 38

45 Proxy Scheme Simulation: Densities / Weak Schemes Proxy Scheme: Consider three stochastic processes X t X(t) t IR model sde X T i X (T i ) i = 0, 1, 2,... time discretization scheme of X target scheme X T i X (T i ) i = 0, 1, 2,... any other time discrete stochastic process (assumed to be close to X ) proxy scheme Pricing: Let Y = (X(T 1 ),..., X(T m ), Y = (X (T 1 ),..., X (T m ), Y = (X (T 1 ),..., X (T m ). where Note: E Q ( f (Y(θ)) F T0 ) E Q ( f (Y (θ)) F T0 ) = E Q ( f (Y ) w(θ) F T0 ) w(θ) = φ Y (θ)(y) φ Y (y) (calculated numerically). From the scheme X we need the realizations (to generate the path) Need something explicit (Euler-Scheme, Predictor Corrector, etc.) From the scheme X we need the transition probability only (weaker requirement) May use complex implicit schemes or expansions of the the transition probability of the (true) model sde. Kampen derived a quadratic WKB expansion for the LIBOR Market Model (see appendix) 39

46 Summary: Requirements / Implementation Proxy Scheme Weights: w(t i+1 ) FTk = i j=k φ K (T j, K j ; T j+1, K j+1 ) φ K (T j, K j ; T j+1, K j+1 ) Implementation: The transition densities φ K and φ K are densities from the numerical schemes K and K. They may be calculated numerically (on the fly together with the (proxy) schemes paths)! Requirement: φ K (T i, K i ; T i+1, K i+1 ) = 0 = φk (T i, K i ; T i+1, K i+1 ) = 0 This requirement corresponds to the non-degeneracy condition imposed on the diffusion matrix in the continuouse case (e.g. Malliavin Calculus). However: Here, this requirement may be achieved even for a degenerate diffusion matrix, e.g. by a non-linear drift. Moreover: Since we are free to choose the proxy sheme, it may choosen such that the condition holds. 40

47 Summary: Note on the non-degeneracy condition (1/2) A note on the requirement y : φ Y (y) = 0 φ Y (y) = 0 (*) The condition ensures that calculating an expectation on (weighted) paths Y may be equivalent to calculation expectation on paths Y. No Y -path is missing. Question: Is it possible to fulfill this condition in general? What happens if the condition is violated? Observation 1: While for Malliavin Calculus one would expect some non-degeneracy condition imposed on the diffusion matrix. Here, condition (*) is much weaker. Since we may choose the (time-discrete) simulation scheme we may make (*) hold. Either add artificial diffusion or use multiple euler steps: Example: Consider a model on two state variable (here an LMM) with a degenerate (rank 1) diffusion matrix (red) and a stochastic drift term (like in LMM). Then a single Euler step will span a line (blue). Using this as a proxy scheme will not allow drift corrections outside that 1-dim hypersurface. However, two subsequent Euler steps of half the size, generate diffusion perpendicular to the 1-dim hypersurface (green). See [F06]. Logarithm of LIBOR(5.5,6.0) LMM Euler Scheme with and witout Drift Logartihm of LIBOR(5.0,5.5) 41

48 Summary: Note on the non-degeneracy condition (2/2) A note on the requirement y : φ Y (y) = 0 φ Y (y) = 0 (*) Observation 2: Since we use the proxy scheme to generate the paths Y (ω) we trivially have φ Y (Y (ω)) 0 on all paths ω generated. Thus the implementation will never suffer from a division by zero error. So how about neglecting condition (*). Observation 3: If the requirement (*) does not hold, then the expectation E Q( f (Y ) φy (Y ) φ Y (Y ) F T 0 ) will leave out some mass. If the two schemes are close, this missed mass is small. In addition one may numerically correct for the missed mass. Note: If we are in the setup of sensitivities and φ Y is a scenario perturbation of φ Y, then a violation of (*) means that the scenario is impossible under the original model. Either the relevance of the scenario or the explanatory power of the model should be put into question. 42

49 Summary 43

50 Summary: Properties / Achivements Requirements: Requires no additional information on the model sde dx =... Requires additional information on the simulation scheme X (T i+1 ), X (T i+1 ) Requires no additional information on the payout f Requires no additional information on the nature of θ ( generic sensitivities) Stable for small shifts h Discontinuous payouts may be handled. Achievements: Stable Generic Sensitivities: Finite Differences result in numerical Likelihood Ratios Weak Schemes: Allows to correct for an improper transition density. 44

51 Example: LIBOR Market Model 45

52 Example: Proxy Scheme Simulation for a LIBOR Market Model LIBOR Market Model: dl i = L i µ L i dt + L i σ i dw i, i = 1,... n, with µ L i = i< j n L j δ j 1 + L j δ j σ i σ j ρ i, j, dw = Σ Γ du, where dw = (dw 1,..., dw n ), dw i dw j = ρ i, j dt, Σ = diag(σ 1,..., σ n ), Γ Γ T = (ρ i, j ). Log-normal model (common extensions: local vol., stoch. vol., jump) Non-linear drift High dimensional (no low dimensional Markovian state variable) Driving factors may be low dimensional (parsimonious model) Γ is an n m matrix. LIBOR Market Model & Numerical Schemes in Log-Coordinates: model sde: dk = µ K dt + Σ Γ du K := log(l), µ K := µ L 2 1Σ2 proxy scheme: K (T i+1 ) = K (T i ) + µ K (T i ) T i + Σ (T i ) Γ (T i ) U(T i ) target scheme: K (T i+1 ) = K (T i ) + µ K (T i ) T i + Σ(T i ) Γ(T i ) U(T i ) 46

53 Example: Proxy Scheme Simulation for a LIBOR Market Model LIBOR Market Model & Numerical Schemes in Log-Coordinates: model sde: dk = µ K dt + Σ Γ du K := log(l), µ K := µ L 1 2 Σ2 proxy scheme: K (T i+1 ) = K (T i ) + µ K (T i ) T i + Σ (T i ) Γ (T i ) U(T i ) sample path target scheme: K (T i+1 ) = K (T i ) + µ K (T i ) T i + Σ(T i ) Γ(T i ) U(T i ) Tansition Probabilites T i T i+1 : Assume for simplicity that µ K (T i ) depends on K (T i ), K (T i+1 ) only (and same for ) ( true for, e.g. Euler Scheme, Predictor Corrector), then φ K (T i, K i ; T i+1, K i+1 ) = 1 (2Π T i ) n/2 exp ( 1 2 T i ( Λ 1/2 F T Σ 1( K i+1 K i µ K (T i ) T i )) 2 ) φ K (T i, K i ; T i+1, K i+1 ) = 1 (2Π T i ) n/2 exp ( 1 2 T i ( Λ 1/2 F T Σ 1( K i+1 K i µk (T i ) T i )) 2 ) Proxy Scheme Weights: w(t i+1 ) FTk = i j=k φ K (T j, K j ; T j+1, K j+1 ) φ K (T j, K j ; T j+1, K j+1 ) Note: We used the factor decomposition (PCA) Γ = F Λ where Λ = diag(λ 1,..., λ m ) are the non-zero Eigenvalues of Γ Γ T. A change of market data / calibration enters into transition probabilities only. monte carlo weights 47

54 Examples and Numerical Results 48

55 Numerical Results Proxy Scheme: Consider three stochastic processes X t X(t) t IR model sde X T i X (T i ) i = 0, 1, 2,... time discretization scheme of X target scheme X T i X (T i ) i = 0, 1, 2,... any other time discrete stochastic process (assumed to be close to X ) proxy scheme Test Case: X LIBOR Market Model X Target Scheme: Some standard discretization of LMM. X Proxy Scheme: Log-normal scheme without drift (LMM drift zero) (extrem test case). Check for: Bond prices ( can we correct for the drift) Sensitivities of Trigger Products (Digitals, Auto Caps) 49

56 Example 1: Correcting the Drift 50

57 Numerical Results: Monte Carlo Bond Price Distributions Comparison of Bond Deviations of Simulation Schemes Frequency 0,050 0,025 better worse 0,000 0,000 0,005 0,010 0,015 0,020 0,025 L1 deviation of Bond prices Zero drift Euler Scheme Euler Scheme Shown: Absolute Bond price Monte Carlo error distribution for Euler Scheme with drift zero (red) and Euler Scheme with Euler drift (yellow): Neglecting drift results in large Bond price errors and even higher Monte Carlo variance (since here drift would generate mean reversion). Next: Use zero-drift Euler Scheme as proxy scheme and correct drift towards Euler Scheme with drift (target scheme). 51

58 Numerical Results: Monte Carlo Bond Price Distributions Comparison of Bond Deviations of Simulation Schemes Frequency 0,050 0,025 0,000 0,000 0,005 0,010 0,015 0,020 0,025 Zero drift Euler Scheme Euler Scheme L1 deviation of Bond prices Zero drift Proxy Scheme corrected for Euler Drift Shown: Use zero-drift Euler Scheme as proxy scheme (red) and correct drift towards Euler Scheme with drift (target scheme, blue) Next: Take a closer look. Compare proxy scheme simulation with direct simulation 52

59 Numerical Results: Monte Carlo Bond Price Distributions Comparison of Bond Deviations of Simulation Schemes Frequency 0,050 0,025 0,000 0, , , , , ,00125 Zero drift Euler Scheme Euler Scheme L1 deviation of Bond prices Zero drift Proxy Scheme corrected for Euler Drift Shown: Monte Carlo Error of Bond Prices for Proxy-Scheme Method (using zero-drift Euler Scheme as proxy scheme) (blue) and direct simulation of target scheme (yellow) Next: Refine target scheme by more accurate transition probabilities 53

60 Numerical Results: Monte Carlo Bond Price Distributions Comparison of Bond Deviations of Simulation Schemes Frequency 0,050 0,025 0,000 0, , , , , ,00125 Euler Scheme L1 deviation of Bond prices Zero drift Proxy Scheme corrected for Euler Drift Zero drift Proxy Scheme corrected for Trapezoidal Drift Shown: Direct Euler Scheme simulation (yellow), Proxy Sheme simulation with Euler Scheme as target scheme (blue), Proxy Scheme simulation with transition probablities derived from trapezoidal integration rule for the drift (green). 54

61 Example 2: Robust Generic Sensitivities 55

62 Numerical Results: Monte Carlo Sensitivities Digital Caplet delta (maturity 5.0, paths) 0,9% 0,8% 0,7% delta 0,6% 0,5% 0,4% 0,3% 0,2% 0,1% 0,0% shift in basis points analytic Proxy Scheme Sensitivity shows an increase of variance for large shift (well known effect for Likelihood Ratio / Malliavin Calculus) Proxy Scheme Sensitivity remains stable for small shifts 56

63 Numerical Results: Monte Carlo Sensitivities Digital Caplet delta (maturity 5.0, paths) 0,9% 0,8% 0,7% delta 0,6% 0,5% 0,4% 0,3% 0,2% 0,1% 0,0% shift in basis points analytic Proxy Scheme Sensitivity shows an increase of variance for large shift (well known effect for Likelihood Ratio / Malliavin Calculus) Proxy Scheme Sensitivity remains stable for small shifts 57

64 Numerical Results: Monte Carlo Sensitivities Digital Caplet gamma (maturity 5.0, paths) 100,00% 75,00% 50,00% gamma 25,00% 0,00% -25,00% -50,00% -75,00% -100,00% shift in basis points Proxy Scheme Sensitivity shows an increase of variance for large shift (well known effect for Likelihood Ratio / Malliavin Calculus) Proxy Scheme Sensitivity remains stable for small shifts 58

65 Numerical Results: Monte Carlo Sensitivities Digital Caplet gamma (maturity 5.0, paths) 100,00% 75,00% 50,00% gamma 25,00% 0,00% -25,00% -50,00% -75,00% -100,00% shift in basis points Proxy Scheme Sensitivity shows an increase of variance for large shift (well known effect for Likelihood Ratio / Malliavin Calculus) Proxy Scheme Sensitivity remains stable for small shifts 59

66 Appendix 60

67 Appendix: Quadratic WKB Expansion for the LMM Transition Probability Density Three assumptions. First (A) The operator L is uniformly parabolic in R n, i.e. there exists 0 < λ < Λ < such that for all ξ R n \ {0} n 0 < λ a ij (x)ξ i ξ j Λ. (1) i,j=1 (B) The coefficients of L are bounded functions in R n which are uniformly Hölder continuous of exponent α (α (0, 1)). guarantee that fundamental solution exists and is is strictly positive. The third assumption (C) the growth of all derivatives of the smooth coefficients functions x a ij (x) and x b i (x) is at most of exponential order, i.e. there exists for each multiindex α a constant λ α > 0 such that for all 1 i, j, k n α a jk, α b i ( exp λα x α x α x 2), (2) guarentees (pointwise) convergence of coefficient functions x c y k (x) := c k(x, y) and x d y k (x) := d k(x, y) in the standard WKB-expansion 1 p(δt, x, y) = n exp d2 (x, y) + c i (x, y)δt i. (3) 2πδt 2δt i 0 and in the new WKB expansion (we call it the quadratic WKB expansion), which is From the target scheme only the transition probability is needed. Kampen [KF] derived a quadratic WKB expansion for the LIBOR Market Model (see left). This enables us to construct a proxy scheme simulation with almost arbitrary small time discretization error - even for a large time steps ΔT. p(δt, x, y) = 1 2πδt n exp ( (P i 0 d iδt i ) 2 2δt + ( i 0 (cy i (y) + c y i ) ) i 1 l=1 dy l dy i l (y) (x y))δt i. (4) (This is from the ansatz ( ) 2 1 i 0 d iδt i p(δt, x, y) = n exp 2πδt 2δt + (α y i + βy i i 0 (x y))δti. (5) where denotes the scalar product. Here α y i and βy are affine terms depending on y (compensation terms). 61

68 References 62

69 References (1/2) A detailed discussion of proxy simulation schemes may be found in [FK], a short introduction in [F05], an in depth discussion of the LIBOR Market Model in [FF]. For an overview on other methods for sensitivies in Monte-Carlo see [BG96], [F05] and [G03]. For an application of the pathwise method to discontinuous payouts see [JK] and [RF]. For an overview on Malliavin calculus and/or its application to sensitivities in Monte-Carlo see [FLLLT], [M97] and [B01]. [B01] BENHAMOU, ERIC: Optimal Malliavin Weighting Function for the Computation of the Greeks [BG96] BROADIE, MARK; GLASSERMAN, PAUL: Estimating Security Price Derivatives using Simulation. Management Science, 1996, Vol. 42, No. 2, [F05] FRIES, CHRISTIAN P.: Bumping the Model [F06] FRIES, CHRISTIAN P.: A Short Note on the Regularization of the Diffusion Matrix for the Euler Scheme of an SDE [FF] FRIES, CHRISTIAN P.: Mathematical Finance. Lecture Notes [FK] [FLLLT] [G03] FRIES, CHRISTIAN P.; KAMPEN, JÖRG: Proxy Simulation Schemes for generic robust Monte-Carlo sensitivities and high accuracy drift approximation (with applications to the LIBOR Market Model) FOURNIÉ, ERIC; LASRY JEAN-MICHEL; LEBUCHOUX, JÉRÔME; LIONS, PIERRE-LOUIS; TOUZI, NIZAR: Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stochastics. 3, (1999). Springer- Verlag GLASSERMAN, PAUL: Monte Carlo Methods in Financial Engineering. (Stochastic Modelling and Applied Probability). Springer, ISBN

70 References (2/2) [JK] [KF] [M97] [RF] JOSHI, MARK S.; KAINTH, DHERMINDER: Rapid computation of prices and deltas of n th to default swaps in the Li Model. Quantitative Finance, volume 4, issue 3, (June 04), p KAMPEN, JÖRG; FRIES, CHRISTIAN: A Quadratic WKB Expansion for the Transition Probability of the LIBOR Market Model. in preparation MALLIAVIN, PAUL: Stochastic Analysis (Grundlehren Der Mathematischen Wissenschaften). Springer Verlag, ISBN ROTT, MARIUS G.; FRIES, CHRISTIAN P.: Fast and Robust Monte Carlo CDO Sensitivities and their Efficient Object Oriented Implementation please check for updates 64