Dimension reduction numerical methods for Bermudan options Scott - PowerPoint PPT Presentation

Dimension reduction numerical methods for Bermudan options Scott Sues Probability, Numerics, and Finance, Le Mans 29 Jun-1 Jul 2016

Outline 1. Dimension reduction approach 2. Fourier cosine method 3. Bermudan option pricing 1 / 25

Abstract ◮ a dimension reduction approach, built on a combination of Monte Carlo and Fourier Cosine-based methods, for options in high-dimensional models ◮ realistic jump-diffusion models with stochastic variance and multi-factor stochastic interest rates applicable to foreign exchange options 2 / 25

FX model ◮ spot price with jumps d S ( t ) � � S ( t − ) = r − λδ d t + σ d W S ( t ) + d J ( t ) r : instantaneous interest rate W S ( t ) : Wiener process for spot price J ( t ) : jump process, J ( t ) = � N ( t ) j =0 ( y j − 1) N ( t ) : number of jumps � � y : jump amplitude, δ = E y − 1 λ : jump intensity 3 / 25

FX model ◮ spot price with jumps d S ( t ) � � � S ( t − ) = r − λδ d t + ν ( t ) d W S ( t ) + d J ( t ) ◮ stochastic volatility (CIR) � � � d ν ( t ) = κ ν ν − ν ( t ) ¯ d t + σ ν ν ( t ) d W ν ( t ) κ ν : mean-reversion rate ν : mean volatility ¯ σ ν : volatility of volatility W ν ( t ) : volatility Wiener process 3 / 25

FX model ◮ spot price with jumps d S ( t ) � � � S ( t − ) = r ( t ) − λδ d t + ν ( t ) d W S ( t ) + d J ( t ) ◮ multi-factor stochastic interest rates � t m r ( t ) = r (0) e − κ r t + κ r r ( s ) e − κ r ( t − s ) d s + � ¯ X i ( t ) 0 i =1 d X i ( t ) = − κ r i X ( t ) d t + σ r i d W r i ( t ) κ r i : mean-reversion rate ¯ r ( t ) : mean interest rate σ r i : volatility W r i ( t ) : interest rate Wiener process 3 / 25

FX model ◮ spot price with jumps d S ( t ) � � � S ( t − ) = r ( t ) − q ( t ) − λδ d t + ν ( t ) d W S ( t ) + d J ( t ) ◮ stochastic volatility � � � d ν ( t ) = κ ν ν − ν ( t ) ¯ d t + σ ν ν ( t ) d W ν ( t ) ◮ multi-factor stochastic interest rates m � r ( t ) = γ r ( t ) + X i ( t ) i =1 d X i ( t ) = − κ r i X ( t ) d t + σ r i d W r i ( t ) n � q ( t ) = γ q ( t ) + Y j ( t ) j =1 d Y j ( t ) = − κ q j Y ( t ) d t + σ q j d W q j ( t ) 3 / 25

Abstract ◮ Early exercise: Bermudan and American options, barriers ◮ Put option with exercise payoff Φ = ( K − S t ) + 0 ≤ t ≤ T ◮ Optimal exercise boundary � τ � � D (0 , τ ) ( K − S τ ) + � � V 0 = sup E , D (0 , τ ) = exp − r ( s ) d s τ ∈T 0 4 / 25

Abstract The approach involves 1. applying conditional Monte Carlo on the variance factor 2. solving the conditional value using the Fourier Cosine method 3. enforcing the optimality condition at each early exercise date using the bundling technique Numerical results indicate that the approach offers very efficient computation of the prices and hedging parameters. 5 / 25

Outline 1. Dimension reduction approach 2. Fourier cosine method 3. Bermudan option pricing 6 / 25

Dimension reduction approach 1. Conditional expectation � � V ( S 0 , 0) = E D (0 , T ) Φ( S T ) � � � � G T � � = E D (0 , T ) Φ( S T ) E ◮ filtration {G T , 0 ≤ t ≤ T } generated by the processes � � W ν , W r 1 , . . . , W r m , W q 1 , . . . , W q n (all except W S ) 2. Inner expectation: PDE methods ◮ Feynman-Kac link between E and PDE ◮ analytical solution using Fourier transforms ◮ dimension reduction to 1 3. Outer expectation: Monte Carlo simulation 7 / 25

Conditional PIDE � �� � G T � � � � � � V S (0) , 0 = E E D (0 , T ) Φ S ( T ) ◮ Using Feynman-Kac formula � � �� � � V S (0) , 0 = E U S (0) , 0; G T where U ( · ) is the unique solution to the conditional PIDE ∂t + a 2 2 ν ( t ) S 2 ∂S 2 ∂U S ∂S 11 � � ∂ 2 U + r ( t ) − q ( t ) − λδ ∂U � ∞ � � − r ( t ) + λ U + λ U ( Sy ) g ( y ) d y = 0 0 with terminal condition � � � � U S ( T ) , T ; G T = Φ S ( T ) 8 / 25

Dimension reduction ◮ z ( t ) = log S ( t ) , u ( z, · ) = U ( S, · ) , v ( z, · ) = V ( S, · ) ◮ Fourier transform ˆ u ( ξ ) of conditional PIDE ∂t + a 2 r ( t ) − q ( t ) − λδ − a 2 ∂ ˆ u 11 2 ν ( t )( iξ ) 2 ˆ 11 � � u + ( iξ )ˆ u 2 � � − r ( t ) + λ ˆ u + λ Γ( ξ )ˆ u = 0 ◮ ODE can be solved in 1 timestep � T � T a 2 � � r ( t ) − q ( t ) − λδ − ν ( t ) � � ˆ − ξ 2 11 v ( ξ, t ) = E ˆ Φ( ξ ) exp 2 ν ( t ) d t + iξ d t 2 0 0 � T � T �� � � � � + iξ ν ( t ) d W j ( t ) − r ( t ) + λ d t + λT Γ( ξ ) a 1 j 0 0 j �� T � �� ◮ Dimension reduction, E exp 0 r ( t ) d t − Gξ 2 + iFξ + H + λT Γ( ξ ) � �� ˆ � � � ˆ = E Φ( ξ ) exp v ξ, t 9 / 25

Dimension reduction 2   � T h − 1 � T G = a 2 m l ν ( t ) d t +1 1 , 1 � � � � a ( j +1) ,k β d j ( t ) − a ( j + m +1) ,k β f j ( t ) + a 1 ,k ν ( t ) d t   2 2 0 0 j =1 j =1 k =2 � T � T F = − 1 � ν ( t ) d t + a 1 ,h ν ( t ) d W ν ( t ) 2 0 0 � T � T m l � � + a ( j +1) ,h β d j ( t ) d W ν ( t ) − a ( j + m +1) ,h β f j ( t ) d W ν ( t ) 0 0 j =1 j =1 � T h − 1 � T l m � � � � � + ρ s,f j β f j ( t ) ν ( t ) d t − a 1 ,k a ( j +1) ,k β d j ( t ) ν ( t ) d t 0 0 j =1 k =2 j =1   h − 1 � T m m l � � � �  d t − a ( j +1) ,k β d j ( t ) a ( j +1) ,k β d j ( t ) − a ( j + m +1) ,k β f j ( t )  0 k =2 j =1 j =1 j =1 � T + ( γ d ( t ) − γ f ( t )) d t − λδT 0 2   m � T � T h − 1 � T m γ d ( t ) d t +1 � � � H = − β d j ( t ) d W ν ( t ) − a ( j +1) ,k β d j ( t ) d t − λT a ( j +1) ,h   2 0 0 0 j =1 j =1 k =2 10 / 25

European options � − Gξ 2 + iFξ + H + λT Γ( ξ ) �� ˆ � � � ˆ v ξ, t = E Φ( ξ ) exp � � Φ( ξ ) ˆ ˆ = E L ( ξ ) ◮ Inverse transform (convolution theorem) � + ∞ � 1 � � � √ v z (0) , 0 = E φ ( x ) L ( z − x ) d x 2 π −∞ ◮ Solution: integrate L , expand e λT Γ( ξ ) in Taylor series � ∞ �� ( λT ) n � S (0) e F + G + H + W n N ( d 1 ,n ) − K e H N ( d 2 ,n ) � V ( S (0) , 0) = E n ! n =0 11 / 25

Bermudan options t n : exercise date timesteps n = 0 , . . . , N, t N = T z ( t ) = log S ( t ) K φ ( z ) = [ K (1 − e z )] + , exercise payoff v m ( z, t n ) : value conditional on variance path m = 1 , . . . , M c m ( z, t n ) : continuation value ◮ At maturity, v m � � = φ ( z ) ∀ m z, t N ◮ At prior exercise dates, � � � � �� = max φ ( z ) , c m v m z, t n z, t n � + ∞ � � � � c m z, t n = v m x, t n +1 L m ( z, x ) d x −∞ 12 / 25

Outline 1. Dimension reduction approach 2. Fourier cosine method 3. Bermudan option pricing 13 / 25

Fourier cosine series If f ( x ) is an even function periodic on [ − π, π ] ∞ � ′ f ( x ) = A k cos ( kx ) k =0 � π A k = 2 f ( x ) cos ( kx ) d x π 0 If f ( x ) decays rapidly as x → ±∞ , for a period [ a, b ] : � kπ ∞ � � ′ f ( x ) = A k cos b − a ( x − a ) k =0 � kπ � b � 2 A k = f ( x ) cos b − ax d x b − a a 14 / 25

Fourier cosine method (Fang, Oosterlee, 2008) Risk-neutral value � v ( z, t n ) = e − r ∆ t v ( x, t n +1 ) f ( x | z ) d x R Apply Fourier cosine expansion to density f ( x | z ) � kπ ∞ � � ′ f ( x | z ) = A k ( z ) cos b − a ( x − a ) k =0 � kπ ∞ �� b � � � ′ v ( z, t n ) ≈ e − r ∆ t A k ( z ) v ( x, t n +1 ) cos b − a ( x − a ) d x a k =0 ∞ � ′ = e − r ∆ t A k ( z ) C k ( t n ) k =0 15 / 25

Density coefficient � kπ � b � 2 A k ( z ) = f ( x | z ) cos b − a ( x − a ) d x b − a a � b 2 � � � � i kπ a e − ikπ = b − a Re f ( x | z ) exp b − ax d x b − a a � kπ 2 � �� a e − ikπ b − a ˆ ≈ b − a Re f b − a ; z ◮ Integral involving density f replaced with characteristic function ˆ f 16 / 25

Value coefficient � kπ � b � C k ( t n ) = v ( x, t n +1 ) cos b − a ( x − a ) d x a At maturity, � kπ � b � C k ( T ) = φ ( x ) cos b − a ( x − a ) d x a 17 / 25

Outline 1. Dimension reduction approach 2. Fourier cosine method 3. Bermudan option pricing 18 / 25

Value coefficient At prior exercise dates, separate at optimal exercise barrier B ( t n ) � � � � �� v m z, t n = max φ ( z ) , c m z, t n v ( z, t n ) = φ ( z ) for z ≤ B ( t n ) � kπ � kπ � B ( t n ) � b � � C k ( t n ) = φ ( x ) cos b − a ( x − a ) d x + v ( x, t n +1 ) cos b − a ( x − a ) d x B ( t n ) a 19 / 25

Value coefficient � kπ � kπ � B ( t n ) � b � � C k ( t n ) = φ ( x ) cos b − a ( x − a ) d x + v ( x, t n +1 ) cos b − a ( x − a ) d x a B ( t n ) � kπ � b � = I PAY ,k ( B ( t n )) + v ( x, t n +1 ) cos b − a ( x − a ) d x B ( t n ) ≈ I PAY ,k ( B ( t n ))+ � kπ � kπ � � b � J � e i jπ b − a x cos � C j ( t n +1 ) ˆ Re f b − a ; 0 b − a ( x − a ) d x B ( t n ) j =0 � kπ J � � � C j ( t n +1 ) ˆ � = I PAY ,k ( B ( t n )) + Re f b − a ; 0 I CONT ,k ( B ( t n )) j =0 ◮ Solve C k ( t n ) backwards in time using C k ( t n +1 ) 20 / 25

Simulation and bundling 21 / 25