"Pricing Exotic Options using Strong Convergence Properties

Fourth Oxford / Princeton Workshop on Financial Mathematics "Pricing Exotic Options using Strong Convergence Properties Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz Prof. Mike Giles Mike.Giles@comlab.ox.ac.uk

Outline Introduction (purpose of the talk) Strong and weak order of convergence [3] Orthogonal transformation [],[4] Pricing exotic options using ML-MC [1] Conclusions and comments [1] Michael Giles: Multi-level Monte Carlo path simulation (ML-MC). Technical Report No. NA06/03, Oxford University Computing Laboratory, Parks Road, Oxford, U.K., (006). [] A. B. Cruzeiro, P. Malliavin and A. Thalmaier: Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation. C. R. Acad. Sci. Paris, Ser. I, 338, 481-486, (004). [3] K. Schmitz-Abe and W. T. Shaw: Measure Order of Convergence without an Exact Solution, Euler vs Milstein Scheme. International Journal of Pure and Applied Mathematics, Vol. 4, No 3, 365-381, (005). [4] K. Schmitz-Abe and M. Giles: Pricing Exotic Options using Strong Convergence Properties, ECMI 006, conference proceedings, to be published by Springer Verlag (September 007). "Pricing Exotic Options using Strong Schemes, Oxford, UK 01-0

Purpose of this talk Speed-up the computation of financial instruments TIME + accuracy(time) = = $ 0-0

Standard Euler & Milstein approximation Stochastic variance model: 1, (, ) (, ), t ds = µ S, ν dt + σ S, ν dw t dν = ϖ Sν dt+ ξ Sν dw Euler approximation: S µ σ 0 = t + W + W ν ϖ ρξ 1, t ρξ, t Milstein approximation: S ( ) t+ t = EULER + ϕ W 1, t, W, t + A1, A L ( 1, ν ) t Where the Lie brackets is defined by: σ ρξ ν A, A = 1 ( A A A A 1 1 ) = ξ ρσ S and the Lévy area is defined by: t+ t t+ t k t+ t k E dw 1, t, dw, t = ρdt E dw, dw ρ L = dw dw dw dw 1, t, t = 0 = 1 ρ ( 1,) 1,,, 1, t u k u k t t t t In practice: What is the main difference between both schemes? 03-0

Standard Euler & Milstein approximation Stochastic variance model: 1, (, ) (, ), t ds = µ S, ν dt + σ S, ν dw t dν = ϖ Sν dt+ ξ Sν dw E dw 1, t, dw, t = ρdt E dw, dw 1, t, t = 0 Simulation of the asset price S for a Monte Carlo path ( t=1/0) Is Euler scheme a good approximation? 04-0

Standard Euler & Milstein approximation Strong convergence Weak convergence 1 (, ) = E ST ST (, t) = C t γ E g ST E g ST t C t β 05-0

Standard Euler & Milstein approximation Strong convergence Weak convergence 1 (, ) = E ST ST (, t) = C t γ E g ST E g ST t C t β Euler: γ=0.51 Milstein: γ=1.0 Euler: β =1.05 Milstein: β =1.06 Can we take advantage of the strong convergence properties of Milstein scheme to price options? 05-0

Convergence without an Exact Solution Theorem 1: If a time discrete approximation Ŝ(T) converges strongly with order γ>0 at time T as t 0 to the exact solution S(T), i.e. exist a positive constant "C", that does not depend on γ such that: E S( T) S ( T, t) = C t γ Then using the same Wiener process, there exists a positive constant "C1", that is not dependent of γ, such that: t γ E S(T, t) S T, =C1 ( t) Surface error using the exact solution. Surface error using Theorem 1. [3] K. Schmitz-Abe and W. T. Shaw: Measure Order of Convergence without an Exact Solution, Euler vs Milstein Scheme. International Journal of Pure and Applied Mathematics, Vol. 4, No 3, 365-381, (005). 07-0

Problem with Milstein scheme Stochastic variance model: Milstein approximation: S ( ) t+ t = EULER + ϕ W 1, t, W, t + A1, A L ( 1, ν ) t Where the Lie brackets is defined by: σ ρξ ν A, A = 1 ξ ρσ and the Lévy area is defined by: S t+ t t+ t k t+ t k L = dw dw dw dw ( 1,) 1,,, 1, t 1, (, ) (, ), t ds = µ S, ν dt + σ S, ν dw t dν = ϖ Sν dt+ ξ Sν dw u k u k t t t t E dw 1, t, dw, t = ρdt E dw, dw ρ 1, t, t = 0 = 1 ρ Can we avoid the simulation of the Lévy area? 08-0

Orthogonal Milstein approximation If we replace the Wiener process dw by an orthogonal transform, the probability distribution does not change and we obtain the set of all orthogonal transforms: The Lie brackets for the new orthogonal process: dw 1, t cos θ sin θ dw1, t = dw, t sin ( θ) cos ( θ) dw, t ρξ σ S σ θ S ρσξ θ A, A = ν 1 ξ θ θ ρσ ξ ρσξ S ν S If we do not want to deal with the Lévy area, we need the Lie brackets to be identically zero, i.e.: A, A = 0 1 So then the rotation θ has to satisfy: dw θ 1 ξ σ ρ ξ θ 1 σ ξ ρ σ = and S + ρ = + ν ξ S ν σ ρ ξ S σ ν [4] K. Schmitz-Abe and M. Giles: Pricing Exotic Options using Strong Convergence Properties, ECMI 006, conference proceedings, to be published by Springer Verlag (September 007). 09-0

Example 1: D -- θ scheme Stochastic volatility model: ds = S µ dt + Sσ dw 1, t dσ = κ ( ϖ σ) dt + βσ λ dw, t where: µ, κ, ϖ, β, λ are constants E dw, 1, t dw, t = ρ dt ρ = 1 ρ If we do not want to deal with the Lévy area, θ has to satisfy: If we assume that θ is only a function of S and σ, we can solve it if λ= and the solution is: Remark: θ λ θ βσ θ ρ = and = S ρs σ ρσ ρ () t = log ( t ) log( S() t ) σ () ρ If λ = A, A = 0 A, A L = 0 1 1 ( 1,) β ρ t+ t t [4] K. Schmitz-Abe and M. Giles: Pricing Exotic Options using Strong Convergence Properties, ECMI 006, conference proceedings, to be published by Springer Verlag (September 007). 10-0

Example : 3D -- θ scheme Stochastic variance model: Orthogonal conditions: α 1.5 = and = S S ν θ βν θ ρ ρ ρν Using the chain rule, our SVM (1) becomes a 3-dimensional Itô process: ds S µ S 0 ν dν κ ( ϖ ν ) dt dw βν α dw = + + θ θ dθ θ θ S S ν βν α µ + κ ϖ ν S ν S ν The Lie brackets using independent Wiener processes: A, A 1 = ds = Sµ dt + S νdw 1, t dν = κ ϖ ν dt + βν α dw, t 0 0 β S ν (1) where: µ, κ, ϖ, β, α are constants ( α 1.5) E dw, 1, t dw, t = ρ dt 0 1, t, t Note that: If α = 1.5 A, A = 0 1 () t ρ β = log ( v() t ) log( S() t ) How accuracy have to be θ to obtain 1 strong order convergence in S and v? θ ρ σ = ρ = 1 ρ ρ ( ν ) max,0 11-0

Strong convergence test ds = S ( µ dt + σdw 1, t ) Case 1, Case 1: Quadratic Volatility Model dσ = κ ϖ σ dt+ βσ dw, Case : 3/ Model (A. Lewis) 3/ dν = κ ϖ ν dt + βν dw 1, t Case 3: GARCH Diffusion Model dν = κ ( ϖ ν ) dt + β νdw, t Case 4: Heston Model 1 dν = κ ϖ ν dt + β νdw 3, t t Case 3 Case 4 S(0) = 1, σ(0) = 0.1, ν(0) = 0.1 Scheme Case 1, Case 3 Case 4 Euler scheme 0.49 0.49 0.48 Milstein scheme (L=0) 0.56 0.56 0.51 Milstein scheme 0.99 0.99 0.95 D θ scheme 0.98 N/A N/A 3D θ scheme (L=0) 0.98 0.87 0.5 3D θ scheme 0.98 0.98 0.91 Convergence orders for all cases T = 1, ρ = 0.3, µ = 0.05, κ = 1.8, ϖ = 0.6 S(0) = 1, ν(0) = 0.1, ν(0) = ν(0) β = 1, β = 0.5, β = 0.5 1 3 1-0

Theorem1: D Orthogonal Milstein scheme (Exact solution) If we have a -D Itô stochastic differential equation (SDE) with a -D Wiener process: X1 a1 b1,1 a1, W1 d dt d (1) X = a + b,1 a, W where we suppose ai,bi,j are sufficiently smooth functions of X & Y in [t₀...t] and satisfy the following condition: Ψ Φ = () X1 X where Hi are the coefficients of the Levy area (Lie bracket) of (1) and: H1( b,1+ b,) H( b1,1b,1+ b1,b,) H b1,1 b1, H1 b1,1b,1 b1,b, Ψ= Φ= b b b b ( 1,1, 1,,1 ) then; if we apply an orthogonal transformation to (1) described by: dw 1, t cos( θt) sin ( θt) dw1, t = where: dw, t sin ( θt) cos( θt) dw ±, t θ = Ψ dx +ΦdX t ( X, X ) 1 ( + ) ( + ) ( bb 1,1, bb 1,,1 ) 1 the new orthogonal process has 1 strong order convergence using Milstein scheme neglecting the simulation of the Lévy Area. Conversely, for Hi 0, the strong order Milstein scheme of (1) has 0.5 strong order convergence. Proof: Ph.D. Thesis; Schmitz, Klaus 13-0

Theorem: θ scheme If we have a -D Itô stochastic differential equation (SDE) with a -D Wiener process: X1 a1 b1,1 a1, W1 d dt d X = b a +,1 a, W (1) then, if θ is described using a third SDE: where: X a b b d a dt d 1 1 1,1 1, W1 X b,1 b, = + W θ Φ a1 + Ψa bθ,1 b θ, b b = Φ b + Ψb Φ b + Ψb θ,1 θ,1 1,1,1 1,, (3) then, the 3-dimensional SDE (3) can have better strong convergence than (1) using Milstein scheme neglecting the simulation of the Lévy Area. The accuracy of θ and hence in Xi depends directly in the value of the Lie bracket RL of the system: R L 0 = 0 Ψ Φ ( bb 11, bb 1,1 ) X1 X Proof: Ph.D. Thesis; Schmitz, Klaus 14-0

Pricing Exotic Options using ML-MC MC The expectation of a payoff P with maturity T is calculated by: L F E PL = E P F 0 + E PL P L 1 L = 1 [ ] where L is the level in the algorithm that simulates the scheme or time approximation with different time steps t: TT - t = 0 for: M L M L Z For each level L, we have a repeated cycle where we calculate the option price using NL paths, i.e. simulate extra samples at each level as needed for new NL. L N = ε V t Vl L L L l = 1 t l For a given ε = exact solution approximation, the algorithm will stop when it converges: { M 1 P P 1 } < ( M ) max, 1 ε L 1 L Computation cost Standard Method M SE c N 1 + c dt 1 = O ( ε -3 ) -3 ( ε ) - ε ( ε) -3 ( ε ) - ( ε ) Euler : O O log Milstein : O O ML-MC [1] Michael Giles: Multi-level Monte Carlo path simulation. Technical Report No. NA06/03, Oxford University Computing Laboratory, Parks Road, Oxford, U.K., (006). 15-0

Pricing Exotic Options using ML-MC MC Case 1: Quadratic Volatility Model ds = S ( µ dt + σ dw 1, t ) dσ = κ( ϖ σ) dt + β σ dw 1, t Monte Carlo paths Cost of simulation European Option: PCall = max S( T ) K,0 PPut = max K S( T ),0 For ε =0.00001, the ML-MC method is more efficient than the standard Euler method for: 50 times for Euler scheme. 150 times for Milstein scheme. 300 times for θ scheme. Value of θ for one random path Top left: convergence in option value with grid level. Bottom left: reduction in ML-MC variance. Top right: # of Monte Carlo paths required on each level. Bottom right: overall computational cost. θ ρ () t = log ( t ) log( S() t ) σ () ρ T = 1, ρ = 0.3, µ = 0.05, κ = 1.8, ϖ = 0.6 S(0) = 1, σ(0) = 0.1, β = 1, K = 1, put option 1 t+ t L ( 1,) = Lévy area = 0 t 16-0 β ρ

Pricing Exotic Options using ML-MC MC Case : GARCH Diffusion Model ds = S ( µ dt + σdw 1, t ) ( ) dν = κ ϖ ν dt + β νdw, t σ = ( ν ) max,0 Monte Carlo paths Digital Options P = H ST K ( ) H x = 1 if x>0 Heaviside function = H x = 0 if x 0 Cost of simulation For ε =0.0001, the ML-MC method is more efficient than the standard Euler method for: 3 times for Euler scheme. 60 times for Milstein scheme. 90 times for θ scheme. Top left: convergence in option value with grid level. Bottom left: reduction in ML-MC variance. Top right: # of Monte Carlo paths required on each level. Bottom right: overall computational cost. T = 1, ρ = 0.3, µ = 0.05, κ = 1.8, ϖ = 0.6 (0) = 1, (0) = 0.1, = 0.5 S ν β K = 1 t+ t Lévy area 0 1, t L = = 17-0

Pricing Exotic Options using ML-MC MC Case : GARCH Diffusion Model ds = S ( µ dt + σdw 1, t ) ( ) dν = κ ϖ ν dt + β νdw, t σ = ( ν ) max,0 Monte Carlo paths Asian Options P = max ST K,0 Arithmetic average: T N dt dt S = S ( t) dt S n + S n 0 ( ) n = 1 1 Cost of simulation For ε =0.00001, the ML-MC method is more efficient than the standard Euler method for: 50 times for Euler scheme. 80 times for Milstein scheme. 110 times for θ scheme. Top left: convergence in option value with grid level. Bottom left: reduction in ML-MC variance. Top right: # of Monte Carlo paths required on each level. Bottom right: overall computational cost. T = 1, ρ = 0.3, µ = 0.05, κ = 1.8, ϖ = 0.6 (0) = 1, (0) = 0.1, = 0.5 S ν β K = 1 t+ t Lévy area 0 1, t L = = 18-0

Pricing Exotic Options using ML-MC MC Case : GARCH Diffusion Model ds = S ( µ dt + σdw 1, t ) ( ) dν = κ ϖ ν dt + β νdw, t σ = ( ν ) max,0 Monte Carlo paths Variance Swaps P ( ν ) = N T K Arithmetic average: T N dt dt ν = ν ( t) dt ν n + ν n 0 Var n = 1 1 Cost of simulation For ε =0.00001, the ML-MC method is more efficient than the standard Euler method for: 150 times for Euler scheme. 380 times for Milstein scheme. 360 times for θ scheme. Top left: convergence in option value with grid level. Bottom left: reduction in ML-MC variance. Top right: # of Monte Carlo paths required on each level. Bottom right: overall computational cost. T ρ µ κ ϖ = 1, = 0.3, = 0.05, = 1.8, = 0.6 S(0) = 1, ν(0) = 0.1, β = 0.5 N = 10, K = 0.6 Var t+ t Lévy area 0 1, t L = = 19-0

Conclusions 1. θ scheme: When a specific orthogonal transformation is applied to a -Dimensional SDE and the conditions for integrability are satisfied, we can use the formula for θ to obtain the value of the rotation angle and obtain first order strong convergence without the simulation of expensive Lévy areas. Otherwise, we can use the 3-Dimensional transformation and check the magnitude of the Lie brackets to decide if it is likely to give computational savings in the solution of our system. The numerical results demonstrate considerable computational savings when the orthogonal transformation is applied to the quadratic volatility model or the 3/ model or the GARCH model. Unfortunately, similar savings are not achieved with the Heston model.. Multilevel Monte Carlo method: The ML-MC works without any problems with all schemes and does not depend on the value of the parameters of the system. In combination with θ scheme, it can reduce substantially the computational cost in pricing exotic options, reducing the cost to achieve an r.m.s. error of size ε from O(ε ³) to O(ε ²). Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz "Pricing Exotic Options using Strong Schemes, Oxford, UK 0-0

Thank you for your attention? Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz "Pricing Exotic Options using Strong Schemes, Oxford, UK

Strong Convergence DEFINITION: We shall state that a time discrete approximation Ŝ(T) converges strongly with order γ>0 at time T as t 0 to the exact solution S(T) if there exists a positive constant "C", that does not depend on γ, such that: E S( T) S ( T, t) =C t γ (1) Kloeden & Platen If (1) is true, it can be implied that the order of strong convergence is not only in the last point T, but also uniformly within the whole time interval [0,T]. REMARK: When we talk about strong convergence, we are referring to how fast our time discrete approximation converges to the exact solution.

Weak Convergence DEFINITION: We shall state that a time discrete approximation S(T) converges weakly with order β>0 at time T as t 0 to the exact solution S(T) if there exists a positive constant "C", that does not depend on β, such that: ( ) ( (, )) = ( ) E g S T E g S T t C t β Monte Carlo Expectation: ( 1) p d (, ) g C β + M 1 E g S(T) - g S k (T, ) = M ( t ) O( t) k=1 REMARK: When we talk about weak convergence, we want to know how fast the expectation of our time discrete approximation converges to the exact expectation of our system.

Convergence without an Exact Solution Theorem 1: If a time discrete approximation Ŝ(T) converges strongly with order γ>0 at time T as t 0 to the exact solution S(T), i.e. exist a positive constant "C", that does not depend on γ such that: E S( T) S ( T, t) = C t γ Then using the same Wiener process, there exists a positive constant "C1", that is not dependent of γ, such that: t γ E S(T, t) S T, =C1 ( t) Surface error using the exact solution. Surface error using Theorem 1. [3] K. Schmitz-Abe and W. T. Shaw: Measure Order of Convergence without an Exact Solution, Euler vs Milstein Scheme. International Journal of Pure and Applied Mathematics, Vol. 4, No 3, 365-381, (005).

Conclusions (Paper) 1. In finance, stochastic variance and volatility models are very important for the valuation of exotic options. We have shown that the use of the orthogonal θ scheme can achieve the first order strong convergence properties of the Milstein numerical discretisation without the expensive simulation of Lévy areas. In combination with the recently introduced Multilevel Monte Carlo method it can reduce substantially the computational cost in pricing exotic options, reducing the cost to achieve an r.m.s. error of size ε from O(ε ³) to O(e ²).. The ML-MC works without any problems with all schemes and does not depend on the value of the parameters of the system. However, when a specific orthogonal transformation (θ scheme) is applied to a -Dimensional SDE it is only possible under certain conditions to avoid calculation of the Lévy area. When the conditions for integrability are satisfied, we can use the formula for θ to obtain the value of the rotation angle and obtain first order strong convergence. Otherwise, we have to use the 3-Dimensional transformation (θ scheme) and check the magnitude of the Lie brackets to decide if it is likely to give computational savings in the solution of our system. 3. The numerical results demonstrate considerable computational savings when the orthogonal transformation is applied to either the quadratic volatility model or the stochastic variance model. Unfortunately, similar savings are not achieved with the Heston model Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz "Pricing Exotic Options using Strong Schemes London, UK (007)