Articles

• Schrödinger bridges, optimal price bounds and Sinkhorn convergence for the FX cross-smiles problem
• Rough Bergomi revisited - exact and minimal-variance hedging for VIX and European options, and exact calibration to multiple smiles
• Optimal trade execution with unknown drift
• Small-time VIX smile and the stationary distribution for the Rough Heston model, with S.Gerhold and B.Smith
• Rough Heston with jumps - joint calibration to SPX/VIX level and skew as T→0, and issues with the quadratic rough Heston model, with B.Smith
• Optimality properties of Dupire and Brunick-Shreve diffusion-type processes - extremal prices of vol swaps and VIX futures given full marginals

Code

• Python code to fit GARCH(1,1) model with (skewed) t-distributed residuals to daily returns data - compute MLEs for model parameters, p-values for 8 goodness-of-fit tests and qq-plot (works well for major FX rates/US tech stocks/Indices/Crypto, + simulates synthetic path(s) with MLE params and MLEs for the synthetic path and plots stationary distribution for V and R, and MLEs using in-built Python functions). SPX Intraday prices typically better fit with GARCH model with power law kernel for e.g. 1-1.5hrs with 5 second intervals to ensure most price changes are non-zero
• Python code to fit GARCH(1,1) model using additional intraday data to reduce sample variance of MLEs, for SPX 3rdJan2022-16thJul2024 and EURUSD 3rdJan2022-23rdJune2023
• MATLAB to compute SPX and VIX smiles and confidence intervals for their associated OTM (out-of-the-money) option prices for a Quadratic Rough Heston Model (calibrations updated 21/06/24); Python version used for calibrations not included here as it requires various sublibraries and runs fast on GPU using tensorflow, MATLAB version here gives the same answers but is slow), additional files required: RbarSeries.m, gamma_inc.m
• Python code to maximize p-values for the RFSV model using realized variance data for SPX, EURUSD and AAPL, p-vals very low (easily modified for other log Gaussian models, e.g. Bacry-Muzy-Li or Abi-Jaber+Li). MATLAB version for model where fBM is replaced with a stationary fractional OU process) using normality tests: fOUmodelpvals.m, fOUMLEfunction.m (note estimating instantaneous variance with realized variance even using 1min bins is problematic since 6.5hr trading day corresponds to M=390 bins, but even if just estimating volatility of standard Brownian motion, from the CLT the standard error is 1.65*sqrt(2/M)=11.8% (factor of sqrt(2) appears because Var(Z^2)=2 if Z is N(0,1)) SPXRealizedVariance1minIntervals03Jan22to15Jul24.m normalitytest.m (in MATLAB as it appears difficult to compute incomplete Gamma function with negative 2nd argument in Python)
• Python code which shows GARCH(1,1) goodness-of-fit tests "passing" (i.e. failing to reject) for returns series generated by a single synthetic qRHeston stock price path with 50 intraday time steps (using qRHeston code above), so in this sense the GARCH(1,1) model is a fake qRHeston model here

Publications

• Markovian stochastic volatility with stochastic correlation - joint calibration and consistency of SPX/VIX short-maturity smiles, with B.Smith, Int. J. Theor. Appl. Finance, 26, 2-3, 2023
• The Riemann-Liouville field and its GMC as H→0, and skew flattening for the rough Bergomi model, with M.Fukasawa, S.Gerhold and B.Smith, Stat. Prob. Lett., Volume 181, February 2022
• Optimal trade execution for Gaussian signals with power-law resilience, with L.Sánchez-Betancourt and B.Smith, Quantitative Finance, 22(3), 585-596, 2022
• Small-time, large-time and H→0 asymptotics for the rough Heston model, with S.Gerhold and B.Smith, Mathematical Finance, 31(1), 203-241, 2021
• Rough volatility and CGMY jumps with a finite history and the Rough Heston model - small-time asymptotics in the k√t regime, with B.Smith and L.Viitasaari, Quantitative Finance, 21(4), 541-563, 2021
• The conditional law of the Bacry-Muzy and Riemann-Liouville log-correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces, with B.Smith, Stat. Prob. Lett., Volume 161, June 2020
• Pathwise superhedging for time-dependent barrier options on cadlag paths - finite or infinite tradeable European, One-Touch, lookback or forward starting options, Stoch. Proc. Appl, 129(3), 799-821, 2019
  
• Asymptotics for rough stochastic volatility models, with H.Zhang, SIAM J. Finan. Math., 8(1), 114-145, 2017
  
• Large-time option pricing using the Donsker-Varadhan LDP - correlated stochastic volatility with stochastic interest rates and jumps, with R.Kumar, Annals of Applied Probability, 26(6), 3699–3726, 2016
  
• Small-time asymptotics under local-stochastic volatility with a jump-to-default: curvature and the heat kernel expansion, with J.Armstrong, M.Lorig and H.Zhang, SIAM J. Finan. Math., 8(1), 82-113, 2017
  
• Small-time asymptotics for basket options - the bi-variate SABR model and the hyperbolic heat kernel on H³, with H.Zhang, SIAM J. Finan. Math, 7(1), 1-551, 2016
  
• Large deviations for the boundary local time of doubly reflected Brownian motion, with R.Kumar and H.Zhang, Stat. Prob. Lett., 96, 262-268, 2015
  
• The Large-maturity smile for the Stein-Stein model, Stat. Prob. Lett., 91, 145-152, 2014
  
• On the Markovian projection in the Brunick-Shreve mimicking result, Stat. Prob. Lett., 85, 98-105, 2014
  
• The Large-maturity smile for the SABR and CEV-Heston models, with A.Pogudin, Int. J. Theor. Appl. Finance, 16 (8), 2013
  
• Hitting times, occupation times, tri-variate laws and the forward Kolmogorov equation for a one-dimensional diffusion with memory, with A.Pogudin and H.Zhang, Adv. Appl. Probab., 45(3), 860-875, 2013
  
• Correction note for ‘The large-maturity smile for the Heston model’, with C.Bernard, Z.Cui, A.Jacquier, D.McLeish, A.Mijatović, Finance and Stochastics, 17 (1), 223-224, 2013
  
• The small-time smile and term structure of implied volatility under the Heston model, with A.Jacquier and R.Lee, SIAM J. Finan. Math., 3(1), 690-708, 2012
  
• The small-maturity smile for exponential Lévy models, with J.E.Figueroa-López, SIAM J. Finan. Math., 3(1), 33-65, 2012
  
• A diffusion-type process with a given joint law for the terminal level and supremum at an independent exponential time, Stoch. Proc. Appl., 121(12), 2802-2817, 2011
  
• Small-time asymptotics for an uncorrelated Local-Stochastic volatility model, with A.Jacquier, Appl. Math. Finance, 18(6), 517-535, 2011
  
• Large-time asymptotics for an uncorrelated stochastic volatility model, Stat. Prob. Lett., 81, 1230-1232, 2011
  
• A note on essential smoothness in the Heston model, with A.Jacquier and A.Mijatovic, Finance and Stochastics, 15(4), 781-784, 2011
  
• Exact pricing and large-time asymptotics for the modified SABR model and the Brownian exponential functional, Int. J. Theor. Appl. Finance, 14(4), 1–19, 2011
  
• Asymptotic formulae for implied volatility in the Heston model, with A.Jacquier and A.Mijatovic, Proc. R. Soc. A, 466(2124), 3593-3620, 2010
  
• The Large-maturity smile for the Heston model, with A.Jacquier, Finance and Stochastics, 15(4), 755-780, 2011
  
• Robust approximations for pricing Asian options and volatility swaps under stochastic volatility, with A.Jacquier, Appl. Math. Finance, 17(3), 241-259, 2010
• Short maturity asymptotics for a fast mean-reverting Heston stochastic volatility model, with J.P.Fouque and J.Feng, SIAM J. Finan. Math., 1(1), 126-141, 2010
  
• Small-time asymptotics for implied volatility under the Heston model, with A.Jacquier, Int. J. Theor. Appl. Finance, 12(6), 861-876, 2009

Older preprints

• Duality for three robust hedging problems - finite strikes and maturities with tradeable exotics, or given joint/marginal laws for the minimum and the terminal price, with R.Kumar (2015)
  
• Portfolio optimization with transient linear and non-linear price impact under exponential utility, with M.Weber and H.Zhang (2015)
  
• Sharp tail estimates for the correlated SABR model, with H.Zhang (2015)
  
• The large-time smile and skew for exponential Lévy models, with J.E.Figueroa-Lopez and A.Jacquier (2011)
  

Teaching

• FM14 Advanced Volatility Models and Path-dependent options
• FM04 Stochastic Analysis
• FM50 Bloomberg mini-course
• FM02 Risk Neutral Valuation
• CM338 Mathematical Finance II: Continuous Time
• FM01 Applied Probability and Stochastics
• Former PhD student: Benjamin Smith