BENCHOP – SLV: the BENCHmarking project in Option Pricing – Stochastic and Local Volatility problems

Figures & data

Table 1. List of methods used with abbreviations, marker symbol used in figures, references, and whether the MATLAB-implementation makes use of parallellism.

Abbreviations	Symbol	Method	References	Parallellism
Methods for SDE formulation
MCA		Monte Carlo simulation with control and antithetic variables	[4,13,30]	Yes
MLMC		Multi-level Monte Carlo	[12]	No
SGBM		Stochastic grid bundling method	[10,22]	Yes
mSABR		The multi-step Monte Carlo simulation of the SABR model	[24]	Yes
Fourier method
FGL		Fourier method with Gauss–Laguerre quadrature	[1,5,6,14,23,25]	No
Methods for PDE formulation
ADI		Modified Craig–Sneyd alternating direction implicit finite difference method ($\theta =1/3$)	[17,20,21,36]	No
RBFFD		Radial basis function generated finite differences	[3,11,28,29]	Yes
RBFPUM		Radial basis function partition of unity methods	[28,31–33]	No

Figure 1. Results for the European call option under the SABR model, Parameter Set I. The reference values for $K_i=S_0\exp (0.1\times \sqrt{T}\times {\delta }_i)$, ${\delta }_i=-1.0,0.0,1.0$ are given by 0.221383196830866, 0.193836689413803, and 0.166240814653231.

Figure 2. Results for the European call option under the SABR model, Parameter Set II. The reference values for $K_i=S_0\exp (0.1\times \sqrt{T}\times {\delta }_i)$, ${\delta }_i=-1.0,0.0,1.0$ are given by 0.052450313614407, 0.046585753491306, and 0.039291470612989.

Figure 3. Results for the European call option under the Heston model. The reference values for $S_0=75$, 100, and 125 are given by 0.908502728459621, 9.046650119220969, and 28.514786399298796.

Figure 4. Results for the Double-no-touch option under the Heston model. The reference values for $S_0=75$, 100, and 125 are given by 0.834539127387590, 0.899829293208866, and 0.668399975738358.

Figure 5. Results for the European call option under the QLSV model. The reference values for $S_0=75$, 100, and 125 are given by 0.527472759419533, 8.902347915743665, and 29.159828965633729.

Figure 6. Results for the Double-no-touch option under the QLSV model. The reference values for $S_0=75$, 100, and 125 are given by 0.933800903110254, 0.914799140676374, and 0.592983062889906.

Figure 7. Results for the European call option under the HHW model. The reference values for $S_0=75$, 100, and 125 are given by 35.437896876285350, 54.728065308229503, and 75.397596834993621.

Table A1. SABR European call prices using the parameters from case I. Error $\sim 1\mathrm{e}-14$.

K	$0.5\exp (-0.1\sqrt{2})$	0.5	$0.5\exp \left(0.1\sqrt{2}\right)$
Call price	0.221383196830866	0.193836689413803	0.166240814653231

Table A2. SABR European call prices using the parameters from case II. Error $\sim 1\mathrm{e}-6$.

K	$0.07\exp (-0.1\sqrt{10})$	0.07	$0.07\exp \left(0.1\sqrt{10}\right)$
Call price	0.052450313614407	0.046585753491306	0.039291470612989

Table A3. Heston European call prices. Error $\sim 1\mathrm{e}-16$.

$S_0$	75	100	125
Call price	0.908502728459621	9.046650119220969	28.514786399298796

Table A4. Heston Double-no-touch prices with lower barrier L=50 and upper barrier U=150. Error $\sim 1.1\mathrm{e}-7$.

$S_0$	75	100	125
Call price	0.834539127387590	0.899829293208866	0.668399975738358

Table A5. Case quadratic stochastic-Local-Volatility European call prices. Error $\sim 2.4\mathrm{e}-6$.

$S_0$	75	100	125
Call price	0.527472759419533	8.902347915743665	29.159828965633729

Table A6. quadratic stochastic-Local-Volatility Double-no-touch prices with lower barrier L=50 and upper barrier U=150. Error $\sim 7.3\mathrm{e}-8$.

$S_0$	75	100	125
Call price	0.933800903110254	0.914799140676374	0.592983062889906

Table A7. Heston European call prices. Error $\sim 1\mathrm{e}-16$.

$S_0$	75	100	125
Call price	35.437896876285350	54.728065308229503	75.397596834993621

Table A8. Name and contribution of authors.

Name	Contributions
Lina von Sydow	Project leader
	Wrote Sections 1, 3, 4, 5, and Appendix 2
	Took part in writing the rest of the paper
	One of two behind code for RBFFD (SABR and QLSV models)
	Took part in meetings A, B, and C
Slobodan Milovanović	Performed all numerical experiments
	Created all plots in Section 4
	Took part in writing the paper
	One of two who wrote unit tests for contributed codes
	Took part in the computation of reference values
	One of two behind code for RBFFD (SABR and QLSV models)
	Took part in meetings A and B
Elisabeth Larsson	Deputy project leader
	Took part in the writing of Appendix 1
	Took part in writing the rest of the paper
	Took part in the computation of reference values
	One of two behind code for RBFPUM (all models)
	Took part in meetings A, B, and C
Karel In 't Hout	Lead the project of defining benchmark problems
	Took part in the writing of Section 2
	Took part in the computation of reference values
	One of two behind code for ADI (all models)
	Took part in meetings A, B, and C
Magnus Wiktorsson	Took part in the writing of Appendix 1
	Took part in the computation of reference values
	Behind code for FGL (Heston European call option and HHW model)
	Behind code for MCA (all models)
	Took part in meeting A
Cornelis W. Oosterlee	Took part in defining benchmark problems
	Took part in the writing of Section 2
	One of two behind code for mSABR (SABR model)
	One of two behind code for SGBM (QLSV model)
	Took part in meetings A, B, and C
Victor Shcherbakov	One of two who wrote unit tests for contributed codes
	One of two behind code for RBFPUM (all models)
	Took part in meetings A and B
	Took part in the computation of reference values
Maarten Wyns	One of two behind code for ADI (SABR and QLSV models)
	Took part in meeting B
Alvaro Leitao	One of two behind code for mSABR (SABR model)
	Behind code for MLMC (SABR model)
	Took part in meeting C
Shashi Jain	One of two behind code for SGBM (QLSV model)
Tinne Haentjens	One of two behind code for ADI (HHW model)
Johan Waldén	Took part in the writing of Sections 1 and 2
	Took part in meeting A

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