## Status:

## Personal website:

+44 1865 270500

## ORCID iD:

https://orcid.org/0000-0001-7938-370X## Research groups:

## Address

University of Oxford

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

Oxford

OX2 6GG

## Recent Books:

Practical Applied Mathematics: Modelling, Analysis, Approximation

(2005)

Applied Partial Differential Equations

(1999)

Mathematical models in finance

ISBN-13: 9780412630705
(1995)

Mathematical Models of Financial Derivatives: a Student Introduction

ISBN-10: 0521497892
(1993)

Option Pricing: Mathematical Models and Computation

(1993)

## Recent Publications:

Stem Cell Differentiation as a Non-Markov Stochastic Process.

Cell systems
issue 3
volume 5
page 268-282.e7
(September 2017)

Quasi-centralized limit order books

QUANTITATIVE FINANCE
issue 6
volume 17
page 831-853
(2017)
Full text available

It's Harder to Splash on Soft Solids.

Physical review letters
issue 18
volume 117
page 184502-
(26 October 2016)

It’s harder to splash on soft solids

Physical Review Letters
(26 October 2016)

The Long Memory of Order Flow in the Foreign Exchange Spot Market

Market Microstructure and Liquidity
(16 June 2016)
Full text available

## Research interests:

I am an Associate Professor in Applied Mathematics with the title of Professor. I have in the past served as Director of OCIAM and of the Nomura Centre for Mathematical Finance, and as head of Department for the Mathematical Institute (2011-15). I am currently a member of Coulcil of the LMS. My research interests in physical applied mathematics include applications of differential equations. free and moving boundary problems in heat and mass transfer, and fluid dynamics. In mathematical finance, I work on derivatives pricing, asymptotic methods, models in energy markets and models for optimal production of exhaustible resources. A recent interst is models using networks, and community detection in networks.

## Major / recent publications:

Under review: *Generative Benchmark Models for Mesoscale Structure in Multilayer Networks*, available at https://arxiv.org/abs/1608.06196 (with Marya Bazzi, Lucas G. S. Jeub, Alex Arenas and Mason A. Porter).

82. *Stem cell differentiation as a non-Markov stochastic process*, in press, Cell Systems; available at http://biorxiv.org/content/early/2017/01/17/101048 (with Patrick S. Stumpf, Rosanna C. G. Smith, Michael Lenz, Andreas Schuppert, Franz-Josef Müller, Ann Babtie, Thalia E. Chan, Michael P. H. Stumpf, Colin P. Please, Fumio Arai and Ben D. MacArthur);

81. Soft substrates suppress droplet splashing, accepted by Phys. Rev. Lett. 2016, current version at http://arxiv.org/abs/1511.06212 (with CJ Howland, A Antowniak, JR Castrejon-Pita, JM Oliver, RW Style & AA Castrejon-Pita).

80. *Quasi-centralized limit order books*, accepted by Quantitative Finance, preliminary version on arxiv.org/abs/1502.00680v1 (with MD Gould & MA Porter);

79. *The long memory of order flow in the foreign exchange spot market*, Market Microstructure and Liquidity **2**(1), 1650001, 2016 (with MD Gould and MA Porter).

78, *Community detection in temporal multilayer networks, and its application to correlation networks*, SIAM Multiscale Modelling & Simulation **14**, 1-41, 2016 (with M Bazzi, MA Porter, S Williams, M McDonald & DJ Fenn).

77. *Risk-Neutral Pricing of Financial Instruments in Emission Markets: A Structural Approach,* SIAM Review **57**, 95-127, 2015 (with D Schwarz; SIGEST paper, extended version of [71]).

76. *Limit Order Books*, Quantitative Finance **13**, 1709-1742, 2013 (with D Fenn, M Gould, M MacDonald, MA Porter & S Williams).

75. *The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options*, SIAM J. Financial Mathematics **4**, 539-574, 2013 (with Christoph Reisinger and Jan Hendrik Witte).

74. *Three-dimensional oblique water entry problems at small deadrise angle*, J. Fluid Mech **711**, 259-280, 2012 (with JR Ockendon, MR Moore & JM Oliver),

73. *A note on oblique water entry*, J. Eng. Math. **81**, 67-74, 2013 (with JR Ockendon, MR Moore & JM Oliver)

72. *Partial Differential Equations that are hard to classify*, J. Part. Diff. Eq. **25**, 41-65, 2012 (with AA Lacey & JR Ockendon)

71. *Risk-Neutral Pricing of Financial Instruments in Emission Markets: A Structural Approach*, SIAM J. Financial Math. **3**, 709-739, 2012 (with D Schwarz).

70. Asymptotic approximations for Asian, European and American options with discrete averaging, or discrete dividend/coupon payments, SIAM J. Financial Math. 3, 215-241, 2012.

69. *Games with exhaustible resources*, SIAM J. Appl. Math. **70**, 2556-2581, 2010 (with Chris Harris & Ronnie Sircar).

68. *Modelling spikes and pricing swing options in electricity markets*, Quantitative Finance **9**, 937-949, 2009 (with B Hambly & T Kluge).

67. *Option pricing with Levy-stable processes generated by Levy-stable integrated variance*, Quantitative Finance **9**, 397-409, 2009 (with A Cartea).

66. *Stochastic behavior of the electricity bid stack: from fundamental drivers to power prices*. J. Energy Markets **2**, 2009 (with M Coulon).

65. *The mirage of triangular arbitrage in the spot foreign exchange market*, IJTAF **12**, 1105-1123, 2009 (with D Fenn, S Williams & NF Johnson).

64. *Impact of unexpected events, shocking news, and rumors on foreign exchange market dynamics*, Phys. Rev. E *77*, 46110-46122, 2008 (with M McDonald, O Suleman, S Williams & NF Johnson).

63. *Non-classical shallow water flows*, IMA J. Appl. Math. **73**, 137-157, 2008 (with Carina Edwards, H Ockendon & JR Ockendon)

62. *The motion of a viscous filament in a porous medium or Hele-Shaw cell: a physical realisation of the Cauchy-Riemann Equations*, Applied Math. Letters **19**, 356-361, 2006 (with CL Farmer).

61. *A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 2: Bermudan options*. Applied Mathematical Finance **14**, 91-104, 2007.

60. *A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 1: Barrier options,* Applied Mathematical Finance **14**, 63-89, 2007 (with Mario Steinberg).

59.* A class of exactly solvable free boundary inhomogeneous porous medium flows*, Applied Math. Letters **20**, 93-97, 2007 (with I Loutsenko & JR Ockendon).

58. *Detecting a currency's dominance or dependence using foreign exchange network trees*, Phys. Rev. E **72**, 046106, 2005 (with M McDonald, O Suleman, S Williams & NF Johnson).

57. *Droplet impact on a thin fluid layer*, J. Fluid. Mech. **542**, 1-23, 2005 (with JR Ockendon, JM Oliver, R Purvis & FT Smith).

56. *Matched asymptotic expansions in financial engineering*, J. Eng. Math. **53**, 385-406, 2005 (James Lighthill Memorial Paper).

55. *Ray methods for free boundary problems*, Quart. Appl. Math. **64**, 41-59, 2006 (with JA Addison & JR King).

54. *A comparison of option prices under different pricing measures in a stochastic volatility model with correlation*, Rev. Deriv. Research **8**, 5-25, 2005 (with V Henderson, D Hobson & T Kluge).

53. *A free boundary problem arising in a model for shallow water entry at small deadrise angles*, Europ. J. Appl. Math. **15**, 839-851, 2004 (with JM Oliver).

52. *Pricing and hedging of volatility derivatives*, Applied Math. Finance **11**, 317-348, 2004 (with A Rafailidis & H Rasmussen).

51. *Global existence, singular solutions and ill-posedness for the Muskat problem*, Comm. Pure Appl. Math. **LVII**, 1-38, 2004 (with MJ Siegel & RE Caflisch).

50. *Using Greeks of options as liquidity protection*, Derivatives Trading, Use & Regulation **9**, 234-245, 2004 (with D Bakstein).

49. *Mushy regions in negative squeeze films*, Quart. J. Mech. Appl. Math. **56**, 361-379, 2003 (with AA Lacey & JR Ockendon).

48. *Oblique slamming, planning and skimming*, J. Eng. Math. **48**, 321-337, 2004 (with JM Oliver & JR Ockendon).

47. *Deep- and shallow-water slamming at small and zero deadrise angles*, J. Eng. Math. **42**, 373-388, 2002 (with JM Oliver & JR Ockendon).

46. *Application of multi-agent games to the prediction of financial time series*, Physica A **299**, 222-227, 2001 (with NF Johnson, D Lamper, P Jefferies & ML Hart).

45. *Predictability of large future changes in a competitively evolving population*, Phys. Rev. Lett. **88**, 017902, 2002 (with NF Johnson & D Lamper).

44. *Trading volume in models of financial derivatives*, Applied Mathematical Finance **8**, 119-135, 2001 (with David Lamper).

43. *A note on the two-phase Hele-Shaw problem*, J. Fluid Mech. **409**, 243-249, 2000.

42. *Two-dimensional Stokes flow with suction and small surface tension*, Europ. J. Appl. Math. **10**, 681-706, 1999 (with LJ Cummings).

41. *Two-dimensional Stokes and Hele-Shaw flow with free surfaces*, Europ. J. Appl. Math. **10**, 635-680, 1999 (with LJ Cummings & JR King).

40. *Two-dimensional solidification and melting in Darcy flow*, J. Fluid Mech. **378**, 1-18, 1999 (with L.J. Cummings, Yu E Hohlov & K Kornev).

39. *Computation of deterministic volatility surfaces*, J. Comp. Finance **2**, 5-32, 1999 (with E Suli & NA Jackson).

38. *Dynamics of line singularities*, Phil. Trans. Roy. Soc. **355**, 2013-2024, 1997 (with A Carpio, SJ Chapman & JR Ockendon).

37. *Numerical and asymptotic solution of a sixth-order nonlinear diffusion equation and related coupled systems*, IMA J. Appl. Math. **57**, 79-98, 1996 (with WR Smith & DF Mayers).

36. *A mathematical model for drying paint layers*, J. Eng. Math. **32**, 377-394, 1997 (with J Moriarty, JR Ockendon, L Terrill & SK Wilson).

35. *Conserved quantities in Stokes flow with free surfaces*, Phys. Fluids **9**, 477-480, 1997 (with LJ Cummings & JR King).

34. *Asymptotics of violent surface motion*, Phil. Trans. Roy. Soc. **355**, 679-685, 1997 (with SJ Chapman, KA Gillow & JR Ockendon).

33. *A class of codimension two free boundary problems*, SIAM Review **39**, 221-253, 1997 (with JD Morgan & JR Ockendon).

32. *A dynamic simulation of the conforming shell gas journal bearing*, J. Eng. Math. **29**, 557-574, 1995 (with DF Mayers & WR Smith).

31. *Cusp development in free boundaries and two-dimensional slow viscous flows*, Europ. J. Appl. Math. **6**, 441-454, 1995 (with S Richardson).

30. *Applied mathematics and finance*, Phil. Trans Roy. Soc. A **347**, 465-470, 1994.

29. *On the stability of Poiseuille flow of a Bingham Fluid*, J. Fluid Mech. **263**, 133-150, 1994 (with I Frigaard & I Sobey).

28. *An equilibrium theory of dislocation continua*, SIAM Review **35**, 580-609, 1993 (with AK Head. JR Ockendon & SP Tighe).

27. *Some mathematical results in the pricing of American Options*, Europ. J. Appl. Math. **4**, 381-398, 1993 (with JN Dewynne, I Rupf & P Wilmott).

26. *A model for Hele-Shaw flows with non-smooth free boundaries*, Quart. J. Mech. Appl. Math. **47**, 107-128, 1994 (with Yu E Hohlov, C Huntingford, AA Lacey & JR Ockendon).

25. *O klassifikatsii reshenii v zadache o techeniyakh Khele-Shou s neizvestnoi granitsei*, Dokl. Rossiiskoi Akad. Nauk. **325**, 1161-1166, 1992 (with Yu E Hohlov).

24. *Fluorescent transfer of light in dyed materials*, SIAM J. Appl. Math. **53**, 447-458, 1993 (with RJ Lawrence).

23. *On the classification of solutions to the zero-surface-tension Hele-Shaw problem*, Quart. Appl. Math **51**, 777-789, 1993 (with Yu E Hohlov).

22. *Macroscopic models for superconductivity*, SIAM Review **34**, 529-560, 1992 (with SJ Chapman & JR Ockendon).

21. *Mathematical modelling of dislocation plasticity*, Physica Scripta T**44**, 135-136, 1992 (with AK Head, JR Ockendon & SP Tighe).

20. *Complex variable methods in Hele-Shaw moving boundary problems*, Europ. J. Appl. Math. **3**, 209-224 (1992).

19. *Normal-superconducting transitions in Landau-Ginzburg theory*, Proc. Roy. Soc. Edinburgh **119**A, 117-124, 1991 (with SJ Chapman, JB McLeod & JR Ockendon).

18. *Stationary solutions to the thermistor problem*, J. Math Anal. Applics **174**, 573-588, 1993 (with M Shillor & J-F Rodrigues).

17. *Irregular morphologies in unstable Hele-Shaw free boundary problems*, Quart. J. Mech. Appl. Math. **43**, 387-405, 1990 (with AA Lacey, JR Ockendon & P Wilmott).

16. *Inviscid water entry problems at small deadrise angles*, J. Fluid Mech. **222**, 215-230, 1991 (with JR Ockendon & SK Wilson).

15. *Temperature surges in current-limiting circuit devices*, SIAM J. Appl. Math. **52**, 998-1011, 1992 (with AC Fowler & I Frigaard).

14. *Some remarks on the regularisation of supercooled Stefan problems in one space dimension*, Q. Appl. Maths **48**, 153-168, 1990 (with A Fasano, M Primicerio & JR Ockendon).

13. *A note on the thermistor problem in two space dimensions*, Q. Appl. Maths **47**, 509-512, 1989.

12. *Asymptotic behaviour of solutions to the Stefan problem with a kinetic condition at the free boundary*, J. Aust. Math. Soc. Ser. B, **31**, 81-96, 1989 (with JN Dewynne, JR Ockendon & W Xie).

11. *Explicit solutions to six free boundary problems in fluid flow and diffusion*, IMA J. Appl. Maths **42**, 155-175, 1989 (with JR King).

10. *Similarity solutions to the Stefan problem and the binary alloy problem*, IMA J. Appl. Maths **40**, 147-161, 1988.

9. *Slot suction from inviscid channel flow*, J. Fluid Mech. **200**, 265-282, 1989 (with JN Dewynne, LC Morland, JR Ockendon & EJ Watson).

8. *Hele-Shaw free boundary problems with suction*, Q. J. Mech. Appl. Maths 4**1**, 183-193, 1988 (with AA Lacey & JR Ockendon).

7. *Existence and stability for spherical crystals growing in a supersaturated solution*, IMA J. Appl. Maths **39**, 1-15, 1987 (with J Chadam & P Ortoleva).

6. *A continuum model for two-dimensional dislocation distributions*, Phil. Mag. **55**A, 617-629, 1987 (with AK Head, JR Ockendon, JB Titchener & P Wilmott).

5. *Fingering in Hele-Shaw cells*, J. Fluid Mech. **167**, 439-453, 1986.

4. *Bubble growth in porous media and Hele-Shaw cells*, Proc. Roy. Soc. Edinburgh **102**A, 141-148, 1986.

3. *Cusp development in Hele-Shaw flow with a free surface*, SIAM J. Appl. Math. **46**, 20-26, 1986.

2. *Singularity development in moving boundary problems*, Q. J. Mech. Appl. Maths **38**, 343-360, 1985 (with AA Lacey & JR Ockendon).

1. *Numerical computation of Hele-Shaw flows with free boundaries*, J. Comp. Phys. **36**, 376-390, 1985 (with JM Aitchison).

**Books**

*Option Pricing: Mathematical Models and Computation* (with JN Dewynne & P Wilmott). Oxford Financial Press 1993.

*Mathematical Models in Finance* (edited with FP Kelly and P Wilmott). Chapman & Hall 1995.

*Mathematical Models of Financial Derivatives: a Student Introduction* (with P Wilmott & JN Dewynne). Cambridge University Press 1995.

*Applied Partial Differential Equations* (with JR Ockendon, AA Lacey & AB Movchan). Oxford University Press 1999 (revised edition 2003).

*Practical Applied Mathematics: Modelling, Analysis, Approximation*, Cambridge University Press, 2005.

**Edited collections**

*Mathematical Models in Finance*, proceedings of Royal Society Discussion Meeting, Phil. Trans. Roy. Soc. **347**, 449-598, 1994 (with FP Kelly & P Wilmott).

*Violent Surface Motion*, proceedings of Royal Society Discussion Meeting, Phil. Trans. Roy. Soc. **355**, 487-689, 1997 (with JR Ockendon, DH Peregrine & DW Moore.

*Vortices, Dislocations and Line Singularities in Partial Differential Equations*, proceedings of a Royal Society Discussion Meeting, Phil. Trans. Roy. Soc. **355**, 1945-2072, 1997 (with SJ Chapman, CM Elliott, AK Head, FM Leslie & JR Ockendon).

*Mathematics in Finance*, themed issue of Phil. Trans. Roy. Soc. **357**, 2009-2133, 1999 (with JN Dewynne & P Wilmott).

**Plenary lectures and refereed conference proceedings**

*Some aspects of the Stefan model for phase changes*, in Proceedings of the NATO Symposium on Structure and Dynamics of Partially Solidified Systems, ed. DE Loper, Nijhoff 1987.

*Equilibrium distributions of screw dislocations*, in Proceedings of the Conference on Free Boundary Problems, Irsee 1987, ed. K-H Hoffman, Longman 1989.

*Some remarks concerning the mathematical theory of solidification*, in Proceedings of Solidification Processing 1987, 370-372, Institute of Metals (London) 1988 (with AB Crowley & JROckendon).

*Temperature surges in thermistors*, in Proceedings of the Third European Symposium on Mathematics in Industry, ed. J Manley, 197-204, Teubner 1990 (with AC Fowler).

*Kinetic undercooling regularisation of supercooled Stefan Problems*, in Mathematical Models for Phase Change Problems, ed. J-F Rodrigues, Birkhauser 1989 (with W Xie).

*On the singularities of one-dimensional Stefan problems with supercooling*, in Mathematical Models for Phase Change Problems, ed. J-F Rodrigues,Birkhauser 1989 (with A Fasano, M Primicerio & JR Ockendon).

*The numerical solution of a continuous casting problem*, in Free Boundary problems, eds K-H Hoffmann & J Sprekels, International Series of Numerical Mathematics 95, Birkhauser-Verlag Basel 1990 (with JN Dewynne & JR Ockendon).

*Codimension-two free boundary problems*, in Proc. Int'l. Coll. on Free Boundary Problems, ed J Chadam & H Rasmussen, Pitman 1991 (Plenary lecture}.

*Black-Scholes models for financial derivative products*, in Proc. ECMI Conference on Mathematics in industry, ed. A Fasano, Teubner, 1995 (with JN Dewynne).

*Patch cavitation in flow past a rigid body*, in Proceedings of the IUTAM Workshop on Cavitated Flows, Birmingham 1993, ed. JR Blake (with JD Morgan & JR Ockendon).

*Trading volume in models of financial derivatives*, in proceedings ECMI Conference (2000), ed. V Capasso, Springer 2001 (with David Lamper).

*Vortex velocity laws to I-V data for flat superconductors*, IEEE Trans. Appl. Superconductivity 11, 3943-3946, 2001 (with SJ Chapman, AD Grief, MD McCulloch, D Dew-Hughes, J Moore & CM Grovenor).

*P Ya Kochina i Khele-Shou v sovremennoi matematike, estestvennykh naukakh i tekhnike,* Prikl. Mat. Mech., 2002 (with JR Ockendon, transl. VM Entov) (Plenary lecture).

*A note on the pricing and hedging of volatility derivatives*, Bachelier Finance Society Proceedings 2002, with A Rafailidis & HO Rasmussen.

*Monte-Carlo techniques for American options*, in Progress in Industrial Mathematics, Proc. ECMI conference 2002, Ed. A Buikis et al., Springer 2004 (with D Lamper).

*What shakes the FX tree? Understanding currency dominance, dependency and dynamics*. Proc. SPIE conference on fluctuations and noise, 2005 (with M McDonald, NF Johnson,O Suleman and S Williams).