Martin Forde, Maths Dept.
Articles
Code
 Python code to fit GARCH(1,1) model with (skewed) tdistributed residuals to daily returns data 
compute MLEs for model parameters, pvalues for 8 goodnessoffit tests and qqplot (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 inbuilt Python functions).
SPX Intraday prices typically better fit with GARCH model with power law kernel for e.g. 11.5hrs with 5 second intervals to ensure most price changes are nonzero
 Python code to fit GARCH(1,1) model using additional intraday data to reduce sample variance
of MLEs, for SPX 3rdJan202216thJul2024 and EURUSD 3rdJan202223rdJune2023
 MATLAB to compute SPX and VIX smiles and confidence intervals for their associated OTM (outofthemoney) 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 pvalues for the RFSV model
using realized variance data for SPX, EURUSD and AAPL, pvals very low (easily modified for
other log Gaussian models, e.g. BacryMuzyLi or AbiJaber+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) goodnessoffit 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 shortmaturity smiles, with B.Smith, Int. J. Theor. Appl. Finance,
26, 23, 2023
 The RiemannLiouville 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 powerlaw resilience, with L.SánchezBetancourt and B.Smith, Quantitative Finance, 22(3), 585596, 2022
 Smalltime, largetime and H→0 asymptotics for the rough Heston model, with S.Gerhold and B.Smith, Mathematical Finance, 31(1), 203241, 2021
 Rough volatility and CGMY jumps with a finite history
and the Rough Heston model  smalltime asymptotics in the k√t regime, with B.Smith and L.Viitasaari, Quantitative Finance, 21(4), 541563, 2021
 The conditional law of the BacryMuzy and
RiemannLiouville logcorrelated 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 timedependent barrier options on cadlag paths  finite or infinite tradeable European, OneTouch, lookback or forward starting options, Stoch. Proc. Appl,
129(3), 799821, 2019
 Asymptotics for rough stochastic volatility models, with H.Zhang, SIAM J. Finan. Math., 8(1), 114145, 2017
 Largetime option pricing using the DonskerVaradhan LDP  correlated stochastic volatility with stochastic interest rates and jumps, with R.Kumar, Annals of Applied Probability, 26(6), 3699–3726, 2016
 Smalltime asymptotics under localstochastic
volatility with a jumptodefault: curvature and the heat kernel expansion, with J.Armstrong, M.Lorig and H.Zhang, SIAM J. Finan. Math.,
8(1), 82113, 2017
 Smalltime asymptotics for basket options  the bivariate SABR model and the hyperbolic heat kernel on H^{3}, with H.Zhang, SIAM J. Finan. Math, 7(1), 1551, 2016
 Large deviations for the boundary local time of doubly reflected Brownian motion, with R.Kumar and H.Zhang, Stat. Prob. Lett., 96, 262268, 2015
 The Largematurity smile for the SteinStein model, Stat. Prob. Lett., 91, 145152, 2014
 On the Markovian projection in the BrunickShreve mimicking result, Stat. Prob. Lett., 85, 98105, 2014
 The Largematurity smile for the SABR and CEVHeston models, with A.Pogudin, Int. J. Theor. Appl. Finance, 16 (8), 2013
 Hitting times, occupation times, trivariate laws and the forward Kolmogorov equation for a onedimensional diffusion with memory, with A.Pogudin and H.Zhang, Adv. Appl. Probab., 45(3), 860875, 2013
 Correction note for ‘The largematurity smile for the Heston model’, with C.Bernard, Z.Cui, A.Jacquier, D.McLeish, A.Mijatović, Finance and Stochastics, 17 (1), 223224, 2013
 The smalltime smile and term structure of implied volatility under the Heston model, with A.Jacquier and R.Lee, SIAM J. Finan. Math., 3(1), 690708, 2012
 The smallmaturity smile for exponential Lévy models, with J.E.FigueroaLópez, SIAM J. Finan. Math., 3(1), 3365, 2012
 A diffusiontype process with a given joint law for the terminal level and supremum
at an independent exponential time, Stoch. Proc. Appl., 121(12), 28022817, 2011
 Smalltime asymptotics for an uncorrelated LocalStochastic volatility model, with A.Jacquier, Appl. Math. Finance, 18(6), 517535, 2011
 Largetime asymptotics for an uncorrelated stochastic volatility model, Stat. Prob. Lett., 81, 12301232, 2011
 A note on essential smoothness in the Heston model, with A.Jacquier and A.Mijatovic, Finance and Stochastics, 15(4), 781784, 2011
 Exact pricing and largetime 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), 35933620, 2010
 The Largematurity smile for the Heston model, with A.Jacquier, Finance and Stochastics, 15(4), 755780, 2011
 Robust approximations for pricing Asian options and volatility swaps under stochastic volatility, with A.Jacquier, Appl. Math. Finance, 17(3), 241259, 2010
 Short maturity asymptotics for a fast meanreverting Heston stochastic volatility model, with J.P.Fouque and J.Feng,
SIAM J. Finan. Math., 1(1), 126141, 2010
 Smalltime asymptotics for implied volatility under the Heston model, with A.Jacquier, Int. J. Theor. Appl. Finance, 12(6), 861876, 2009
Older preprints
Teaching
 FM14 Advanced Volatility Models and Pathdependent options
 FM04 Stochastic Analysis
 FM50 Bloomberg minicourse
 FM02 Risk Neutral Valuation
 CM338 Mathematical Finance II: Continuous Time
 FM01 Applied Probability and Stochastics
 Former PhD student: Benjamin Smith

