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Martin Forde

Publications and source records attributed to Martin Forde.

11 recordsLinked to original sources
arXiv2025

L\'evy processes as weak limits of rough Heston models

We show weak convergence of the time-$t$ marginals for the integrated variance in a re-scaled rough Heston model to an Inverse Gaussian L\'{e}vy process. This shows we can obtain such a limit without having to impose that the true Hurst exponent $H$ for the model is $\frac{1}{2}$ as in [Abi Jaber, & De Carvalho, 2024], or that $H\searrow -\frac{1}{2}$ as in [Abi Jaber, Attal, & Rosenbaum, 2025], so the result potentially has increased financial relevance. We later extend the analysis to the case where $V$ has jumps, showing weak convergence of the finite-dimensional distributions of the integrated variance to a deterministic time-change of the first-passage time process to lower barriers for a more general class of spectrally positive L\'evy processes. This convergence result is then strengthened to a functional setting, namely on the space of c\`adl\`ag functions on the non-negative half-line endowed with the $M_1$ topology.

math.PR↗
arXiv2021

Asymptotics for rough stochastic volatility models

Using the large deviation principle (LDP) for a re-scaled fractional Brownian motion $B^H_t$ where the rate function is defined via the reproducing kernel Hilbert space, we compute small-time asymptotics for a correlated fractional stochastic volatility model of the form $dS_t=S_tσ(Y_t) (\barρ dW_t +ρdB_t), \,dY_t=dB^H_t$ where $σ$ is $α$-Hölder continuous for some $α\in(0,1]$; in particular, we show that $t^{H-\frac{1}{2}} \log S_t $ satisfies the LDP as $t\to0$ and the model has a well-defined implied volatility smile as $t \to 0$, when the log-moneyness $k(t)=x t^{\frac{1}{2}-H}$. Thus the smile steepens to infinity or flattens to zero depending on whether $H\in(0,\frac{1}{2})$ or $H\in(\frac{1}{2},1)$. We also compute large-time asymptotics for a fractional local-stochastic volatility model of the form: $dS_t= S_t^β |Y_t|^p dW_t,dY_t=dB^H_t$, and we generalize two identities in Matsumoto&Yor05 to show that $\frac{1}{t^{2H}}\log \frac{1}{t}\int_0^t e^{2 B^H_s} ds$ and $\frac{1}{t^{2H}}(\log \int_0^t e^{2(μs+B^H_s)} ds-2 μt)$ converge in law to $ 2\mathrm{max}_{0 \le s \le 1} B^H_{s}$ and $2B_1$ respectively for $H \in (0,\frac{1}{2})$ and $μ>0$ as $t \to \infty$.

q-fin.PR↗
arXiv2020

Small-time, large-time and $H\to 0$ asymptotics for the Rough Heston model

We characterize the behaviour of the Rough Heston model introduced by Jaisson\&Rosenbaum \cite{JR16} in the small-time, large-time and $α\to 1/2$ (i.e. $H\to 0$) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model (cf.\ \cite{FZ17}, \cite{FGP18a} et al.), and the rate function is equal to the Fenchel-Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in \cite{ER19}, but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space-time scaling property which means we only need to solve this equation for the moment values of $p=1$ and $p=-1$ so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity. The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in \cite{FJ11}. Finally, using Lévy's convergence theorem, we show that the log stock price $X_t$ tends weakly to a non-symmetric random variable $X^{(1/2)}_t$ as $α\to 1/2$ (i.e. $H\to 0$) whose mgf is also the solution to the Rough Heston VIE with $α=1/2$, and we show that $X^{(1/2)}_t/\sqrt{t}$ tends weakly to a non-symmetric random variable as $t\to 0$, which leads to a non-flat non-symmetric asymptotic smile in the Edgeworth regime. We also show that the third moment of the log stock price tends to a finite constant as $H\to 0$ (in contrast to the Rough Bergomi model discussed in \cite{FFGS20} where the skew flattens or blows up) and the $V$ process converges on pathspace to a random tempered distribution.

q-fin.PR↗
arXiv2016

Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion

We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $ρ< 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $β=1, ρ\le 0$. \bl{For uncorrelated stochastic volatility models, our results also include a SABR-type model with $β=1$ and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model.} We later augment the model with a single jump-to-default with intensity $\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $ρ=0$, log-moneyness $x > 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \& Spector \cite{AS12} for the case $ρ=0$.

q-fin.PR↗
arXiv2016

Small-time asymptotics for basket options -- the bi-variate SABR model and the hyperbolic heat kernel on $\mathbb{H}^3$

We compute a sharp small-time estimate for the price of a basket call under a bi-variate SABR model with both $β$ parameters equal to $1$ and three correlation parameters, which extends the work of Bayer,Friz&Laurence [BFL14] for the multivariate Black-Scholes flat vol model. The result follows from the heat kernel on hyperbolic space for $n=3$ combined with the Bellaiche [Bel81] heat kernel expansion and Laplace's method, and we give numerical results which corroborate our asymptotic formulae. Similar to the Black-Scholes case, we find that there is a phase transition from one "most-likely" path to two most-likely paths beyond some critical $K^*$.

q-fin.PR↗
arXiv2014

Large deviations for the boundary local time of doubly reflected Brownian Motion

We compute a closed-form expression for the moment generating function $\hat{f}(x;λ,α)=\frac{1}λ\mathbb{E}_x(e^{αL_τ})$, where $L_t$ is the local time at zero for standard Brownian motion with reflecting barriers at $0$ and $b$, and $τ\sim \mathrm{Exp}(λ)$ is independent of $W$. By analyzing how and where $\hat{f}(x;\cdot,α)$ blows up in $λ$, a large-time large deviation principle (LDP) for $L_t/t$ is established using a Tauberian result and the Gärtner-Ellis Theorem.

math.PR↗
arXiv2011

The small-maturity smile for exponential Levy models

We derive a small-time expansion for out-of-the-money call options under an exponential Levy model, using the small-time expansion for the distribution function given in Figueroa-Lopez & Houdre (2009), combined with a change of numéraire via the Esscher transform. In particular, we quantify find that the effect of a non-zero volatility $σ$ of the Gaussian component of the driving Lévy process is to increase the call price by $1/2σ^2 t^2 e^{k}ν(k)(1+o(1))$ as $t \to 0$, where $ν$ is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility, which sharpens the first order estimate given in Tankov (2010). Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed Lévy models. We also consider a small-time, small log-moneyness regime for the CGMY model, and apply this approach to the small-time pricing of at-the-money call options.

q-fin.PR↗
arXiv2011

A note on essential smoothness in the Heston model

This note studies an issue relating to essential smoothness that can arise when the theory of large deviations is applied to a certain option pricing formula in the Heston model. The note identifies a gap, based on this issue, in the proof of Corollary 2.4 in \cite{FordeJacquier10} and describes how to circumvent it. This completes the proof of Corollary 2.4 in \cite{FordeJacquier10} and hence of the main result in \cite{FordeJacquier10}, which describes the limiting behaviour of the implied volatility smile in the Heston model far from maturity.

q-fin.PR↗
arXiv2010

Asymptotic formulae for implied volatility in the Heston model

In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.

q-fin.PR↗