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--- abstract: 'We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.' address: 'Department of Mathematics, Imperial College London' author: - Antoine Jacquier and Patrick Roome title: 'Large-Maturity Regimes of the Heston Forward Smile' --- [^1] Introduction ============ Consider an asset price process $\left({\mathrm{e}}^{X_t}\right)_{t\geq0}$ with $X_0=0$, paying no dividend, defined on a complete filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0},{\mathbb{P}})$ with a given risk-neutral measure ${\mathbb{P}}$, and assume that interest rates are zero. In the Black-Scholes-Merton (BSM) model, the dynamics of the logarithm of the asset price are given by ${\mathrm{d}}X_t=-\frac{1}{2}\sigma^2 {\mathrm{d}}t +\sigma {\mathrm{d}}W_t$, where $\sigma>0$ represents the instantaneous volatility and $W$ is a standard Brownian motion. The no-arbitrage price of the call option at time zero is then given by the famous BSM formula [@BS73; @M73]: $C_{\textrm{BS}}(\tau,k,\sigma) := {\mathbb{E}}\left({\mathrm{e}}^{X_\tau}-{\mathrm{e}}^k\right)_+ ={\mathcal{N}}\left(d_+\right)-{\mathrm{e}}^k{\mathcal{N}}\left(d_-\right)$, with $d_\pm:=-\frac{k}{\sigma\sqrt{\tau}}\pm\frac{1}{2}\sigma\sqrt{\tau}$, where ${\mathcal{N}}$ is the standard normal distribution function. For a given market price $C^{\textrm{obs}}(\tau,k)$ of the option at strike ${\mathrm{e}}^k$ and maturity $\tau$, the spot implied volatility $\sigma_{\tau}(k)$ is the unique solution to the equation $C^{\textrm{obs}}(\tau,k)=C_{\textrm{BS}}(\tau,k,\sigma_{\tau}(k))$. For any $t,\tau>0$ and $k\in{\mathbb{R}}$, we define as in [@B04; @VL] a forward-start option with forward-start date $t$, maturity $\tau$ and strike ${\mathrm{e}}^k$ as a European option with payoff $\left({\mathrm{e}}^{X_{\tau}^{(t)}}-{\mathrm{e}}^k\right)^+$ where $\label{eq:XtTauDef}X_{\tau}^{(t)}:=X_{t+\tau}-X_t$ pathwise. By the stationary increment property, its value is simply $C_{\textrm{BS}}(\tau,k,\sigma)$ in the BSM model. For a given market price $C^{\textrm{obs}}(t,\tau,k)$ of the option at strike ${\mathrm{e}}^k$, forward-start date $t$ and maturity $\tau$, the forward implied volatility smile $\sigma_{t,\tau}(k)$ is then defined (see also [@B04]) as the unique solution to $C^{\textrm{obs}}(t,\tau,k)=C_{\textrm{BS}}(\tau,k,\sigma_{t,\tau}(k))$. The forward smile is a generalisation of the spot implied volatility smile, and the two are equal when $t=0$. The literature on implied volatility asymptotics is extensive and has drawn upon a wide range of mathematical techniques. Small-maturity asymptotics have received wide attention using heat kernel expansion results [@Benarous]. More recently, they have been studied using PDE methods [@BBF; @Hagan; @Pascucci], large deviations [@DFJV; @FJSmall], saddlepoint methods [@FJL11], Malliavin calculus [@Gobet; @KT04] and differential geometry [@GHLOW; @PHL]. Roger Lee [@Lee] was the first to study extreme strike asymptotics, and further works on this have been carried out by Benaim and Friz [@BF1; @BF2] and in [@Guli; @GuliMass; @GS; @FGGS; @DFJV; @Marco]. Large-maturity asymptotics have only been studied in [@Tehr; @FJ09; @JMKR; @JM12; @FJM10] using large deviations and saddlepoint methods. Fouque et al. [@Fouque] have also successfully introduced perturbation techniques in order to study slow and fast mean-reverting stochastic volatility models. Models with jumps (including Lévy processes), studied in the above references