Efficient one and multiple time-step simulation of the SABR model ´ Alvaro Leitao, Lech A. Grzelak and Cornelis W. Oosterlee Delft University of Technology - Centrum Wiskunde & Informatica CWI - February 15, 2016 A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 1 / 26
“Our” definition of simulation Generate samples from (sampling) stochastic processes. The standard approach to sample from a given distribution, Z : F Z ( Z ) d z n = F − 1 = U thus Z ( u n ) , F Z is the cumulative distribution function (CDF). d = means equality in the distribution sense. U ∼ U ([0 , 1]) and u n is a sample from U ([0 , 1]). The computational cost depends on inversion F − 1 Z . A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 2 / 26
Outline SABR model 1 Distribution of the SABR’s integrated variance 2 One-step SABR simulation 3 Multiple time-step SABR simulation 4 Conclusions 5 A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 3 / 26
SABR model The formal definition of the SABR model [5] reads d f ( t ) = σ ( t ) f β ( t ) d W f ( t ) , f (0) = S 0 , d σ ( t ) = ασ ( t ) d W σ ( t ) , σ (0) = σ 0 , f ( t ) = S ( t ) e rt is forward price of the underlying asset S ( t ). σ ( t ) is the stochastic volatility. W f ( t ) and W σ ( t ) are two correlated Brownian motions SABR parameters: ◮ The volatility of the volatility, α > 0. ◮ The CEV elasticity, 0 ≤ β ≤ 1. ◮ The correlation coefficient, ρ ( W f W σ = ρ t ) A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 4 / 26
“Exact” simulation of SABR model Based on Islah [6], the conditional cumulative distribution function (CDF) of f ( t ) in a generic interval [ s , t ], 0 ≤ s ≤ t ≤ T : � t � � σ 2 ( z ) d z = 1 − χ 2 ( a ; b , c ) , f ( t ) ≤ K | f ( s ) > 0 , σ ( s ) , σ ( t ) , Pr s where � 2 � f ( s ) 1 − β 1 (1 − β ) + ρ a = α ( σ ( t ) − σ ( s )) , ν ( t ) K 2(1 − β ) c = (1 − β ) 2 ν ( t ) , b = 2 − 1 − 2 β − ρ 2 (1 − β ) , (1 − β )(1 − ρ 2 ) � t ν ( t ) = (1 − ρ 2 ) σ 2 ( z ) d z , s and χ 2 ( x ; δ, λ ) is the non-central chi-square CDF. Exact in the case of ρ = 0, an approximation otherwise. A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 5 / 26
Simulation of SABR model Simulation of the volatility process, σ ( t ) | σ ( s ): W σ ( t ) − 1 σ ( t ) ∼ σ ( s ) exp( α ˆ 2 α 2 t ) , where ˆ W σ ( t ) is a independent Brownian motion. � t s σ 2 ( z ) d z | σ ( t ) , σ ( s ). Simulation of the integrated variance process, � t s σ 2 ( z ) d z , σ ( t ) , σ ( s ) by Simulation of the forward process, f ( t ) | f ( s ) , inverting the CDF. The conditional integrated variance is a challenging part. We propose: ◮ Approximate the conditional distribution by using Fourier techniques and copulas. ◮ Marginal distribution based on COS method [3]. ◮ Conditional distribution based on copulas. ◮ Improvements for a fast computations. A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 6 / 26
Distribution of the integrated variance Not available. � t s σ 2 ( z ) d z . For notational convenience, we will use Y ( s , t ) := Discrete equivalent, M monitoring dates: � t M σ 2 ( z ) d z ≈ � ∆ t σ 2 ( t j ) =: ˆ Y ( s , t ) := Y ( s , t ) s j =1 where t j = s + j ∆ t , j = 1 , . . . , M and ∆ t = t − s M . In the logarithmic domain, where we aim to find an approximation of F log ˆ Y | log σ ( s ) : � x F log ˆ Y | log σ ( s ) ( x ) = f log ˆ Y | log σ ( s ) ( y ) d y , −∞ where f log ˆ Y | log σ ( s ) is the probability density function (PDF) of log ˆ Y ( s , t ) | log σ ( s ). A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 7 / 26
PDF of the integrated variance Equivalent: Characteristic function and inversion (Fourier pair). Recursive procedure to derive an approximated φ log ˆ Y | log σ ( s ) . We start by defining the logarithmic increment of σ 2 ( t ): � σ 2 ( t j ) � R j = log , j = 1 , . . . , M σ 2 ( t j − 1 ) σ 2 ( t j ) can be written: σ 2 ( t j ) = σ 2 ( t 0 ) exp( R 1 + R 2 + · · · + R j ) . We introduce the iterative process Y 1 = R M , Y j = R M +1 − j + Z j − 1 , j = 2 , . . . , M . with Z j = log(1 + exp( Y j )). A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 8 / 26
PDF of the integrated variance (cont.) ˆ Y ( s , t ) can be expressed: M ˆ � σ 2 ( t i )∆ t = ∆ t σ 2 ( s ) exp( Y M ) . Y ( s , t ) = i =1 And, we compute φ log ˆ Y | log σ ( s ) ( u ), as follows: iu log(∆ t σ 2 ( s )) � � φ log ˆ Y | log σ ( s ) ( u ) = exp φ Y M ( u ) . a , ˆ By applying COS method in the support [ˆ b ]: N − 1 ′ 2 � k π � � f log ˆ Y | log σ ( s ) ( x ) ≈ C k cos ( x − ˆ a ) , ˆ ˆ b − ˆ a b − ˆ a k =0 with � k π � � � − i ˆ ak π �� C k = ℜ φ log ˆ exp . Y | log σ ( s ) ˆ ˆ b − ˆ a b − ˆ a A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 9 / 26
CDF of the integrated variance The CDF of log ˆ Y ( s , t ) | log σ ( s ): � x F log ˆ Y | log σ ( s ) ( x ) = f log ˆ Y | log σ ( s ) ( y ) d y −∞ � x N − 1 ′ 2 � k π � � ≈ C k cos ( y − ˆ a ) d y . ˆ ˆ b − ˆ a b − ˆ a a ˆ k =0 The efficient computation of φ Y M ( u ) is crucial for the performance of the whole procedure (specially, one-step case). The inversion of F log ˆ Y | log σ ( s ) is relatively expensive (unafforable in the multi-step case). A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 10 / 26
� t s σ 2 ( z ) d z | σ ( t ) , σ ( s ) Copula-based simulation of In order to apply copulas, we need (logarithmic domain): ◮ F log ˆ Y | log σ ( s ) . ◮ F log σ ( t ) | log σ ( s ) . ◮ Correlation between log Y ( s , t ) and log σ ( t ). The distribution of log σ ( t ) | log σ ( s ) = z is � s log σ ( t ) � � 1 − P 2 N µ log σ ( t ) + P log σ ( t ) , log σ ( s ) ( z − µ log σ ( t ) ) , s log σ ( t ) , log σ ( t ) , log σ ( s ) s log σ ( s ) where all the quantities are known. Approximated Pearson’s correlation coefficient: t 2 − s 2 P log Y , log σ ( t ) ≈ 3 ts 3 − t 2 s 2 � . �� 1 3 t 4 + 2 2 For some copulas, like Archimedean, Kendall’s τ is required: � π � P = sin 2 τ . A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 11 / 26
� t s σ 2 ( z ) d z | σ ( t ) , σ ( s ): Steps Sampling 1 Determine F log σ ( t ) | log σ ( s ) and F log ˆ Y | log σ ( s ) . 2 Determine the correlation between log Y ( s , t ) and log σ ( t ). 3 Generate correlated uniform samples, U log σ ( t ) | log σ ( s ) and U log ˆ Y | log σ ( s ) by means of copula. 4 From U log σ ( t ) | log σ ( s ) and U log ˆ Y | log σ ( s ) invert original marginal distributions. � t 5 The samples of σ ( t ) | σ ( s ) and Y ( s , t ) = s σ 2 ( z ) d z | σ ( t ) , σ ( s ) are obtained by taking exponentials. A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 12 / 26
One time-step simulation of the SABR model s = 0 and t = T , with T the maturity time. The use is restricted to price European options up to T = 2. log σ ( s ) becomes constant. F log σ ( t ) | log σ ( s ) and F log ˆ Y | log σ ( s ) turn into F log σ ( T ) and F log ˆ Y ( T ) . The computation of φ log ˆ Y ( T ) is much simpler and very fast. The approximated Pearson’s coefficient results in a constant value: √ T 2 3 P log Y ( T ) , log σ ( T ) ≈ = 2 . � 1 2 3 T 4 A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 13 / 26
Approximated correlation P log Y ( T ) , log σ ( T ) 1 0.9 0.8 0.7 0.6 0.5 0.2 1 0.4 0.9 0.6 0.8 0.8 0.7 1 0.6 1.2 T 0.5 1.4 0.4 α 1.6 0.3 1.8 0.2 2 0.1 Figure: Pearson’s coefficient: Empirical (surface) vs. approximation (red grid). A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 14 / 26
Copula analysis Based on the one-step simulation, a copula analysis is carried out. Gaussian, Student t and Archimedean (Clayton, Frank and Gumbel). A goodness-of-fit (GOF) for copulas needs to be evaluated. Archimedean: graphic GOF based on Kendall’s processes. Generic GOF based on the so-called Deheuvels or empirical copula . S 0 T σ 0 α β ρ Set I 1 . 0 0 . 5 0 . 4 0 . 7 0 . 0 2 Set II 0 . 05 0 . 1 0 . 4 0 . 0 − 0 . 8 0 . 5 Set III 0 . 04 0 . 4 0 . 8 1 . 0 − 0 . 5 2 Table: Data sets. A. Leitao & Lech Grzelak & Kees Oosterlee Efficient SABR CWI - February 15, 2016 15 / 26
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