Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

Abstract

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.

Appendix: Additional proofs

Lemma A.1

Let $V$ be as in (1.8), with $\mu(Y_{t})$ and $\sigma(Y_{t})$ replaced by $\bar{\mu}_{t}$ and $\bar{\sigma}_{t}$ defined in (4.8). Also let $\bar{\sigma}'$, $\bar{\sigma}''$, $\bar{\alpha}$ and $\bar{\gamma}$ be the stopped processes in (4.20) and $\bar{\sigma}_{t}^{*}:=\sqrt{\frac{1}{t}\int_{0}^{t}\bar{\sigma}_{s}^{2}ds}$. Then the following relations hold for any $p\geq1$:

• (i) ${\mathbb {E}}[|\bar{\mu}_{t}-\mu_{0}|^{p}] = O(t^{\frac{p}{2}}),\quad t\to0$.

• (ii) ${\mathbb {E}}[|\bar{\sigma}_{t}-\sigma_{0}|^{p}] = O(t^{\frac{p}{2}}),\quad t\to0$.

• (iii) ${\mathbb {E}}[|\bar{\sigma}^{*}_{t}-\sigma_{0}|^{p}] = O(t^{\frac {p}{2}}),\quad t\to0$.

• (iv) ${\mathbb {E}}[\bar{\sigma}_{t}^{*}-\sigma_{0}] = O(t), \quad t\to0$.

• (v) For $\xi_{t}^{1},\xi_{t}^{2}$ and $\xi_{t}^{1,0}$ defined as in (4.20) and (4.21), we have ${\mathbb {E}}[|\xi_{t}^{1}|]=O(t)$ and ${\mathbb {E}}[|\xi _{t}^{2}|]+{\mathbb {E}}[|\xi_{t}^{1}-\xi_{t}^{1,0}|] = O(t^{\frac{3}{2}})$, $t\to0$.

• (vi) ${\mathbb {E}}[|\bar{\sigma}_{t}^{*}-\sigma_{0}-\sigma_{0}'\gamma_{0}\frac {1}{t}\int_{0}^{t}W_{s}^{1}ds|]=O(t),\quad t\to0$.

Proof

Let $L$ be a common Lipschitz constant for $\mu$, $\sigma$ and $\gamma$.

(i) By the Lipschitz-continuity of $\mu$ and the Burkholder–Davis–Gundy (BDG) inequality, we can find a constant $C_{p}$ such that

$$\begin{aligned} {\mathbb {E}}[|\bar{\mu}_{t}-\mu_{0}|^{p}] & \leq L^{p}{\mathbb {E}}[|Y_{t\wedge \tau}-y_{0}|^{p}]\\ &\leq L^{p}C_{p}\bigg({\mathbb {E}}\bigg[\int_{0}^{t}\bar{\alpha}_{s}ds\bigg]^{p}+{\mathbb {E}}\bigg[\int_{0}^{t}\bar{\gamma}^{2}_{s}ds\bigg]^{\frac {p}{2}}\bigg) =O(t^{\frac{p}{2}}),\quad t\to0, \end{aligned}$$

since $\alpha$ and $\gamma$ are bounded.

(ii) is proved in a similar way, and for (iii) we use the boundedness of $\sigma$, Jensen’s inequality and (ii) to write

$$\begin{aligned} {\mathbb {E}}[|\bar{\sigma}^{*}_{t}-\sigma_{0}|^{p}] &\leq\frac{1}{(2m)^{p}}{\mathbb {E}}\bigg[\frac{1}{t}\int_{0}^{t}(\bar{\sigma}^{2}_{s}-\sigma^{2}_{0})ds\bigg]^{p}\\ &\leq\bigg(\frac{M}{m}\bigg)^{p}\frac{1}{t}\int_{0}^{t}{\mathbb {E}}[\bar{\sigma}_{s}-\sigma_{0}]^{p}ds =O(t^{\frac{p}{2}}),\quad t\to0. \end{aligned}$$

(iv) We can write

$${\mathbb {E}}[\bar{\sigma}_{t}^{*}-\sigma_{0}] =\frac{1}{2\sigma_{0}}{\mathbb {E}}[(\bar{\sigma}_{t}^{*})^{2}-\sigma_{0}^{2}] +{\mathbb {E}}\bigg[\big((\bar{\sigma}_{t}^{*})^{2}-\sigma_{0}^{2}\big)\bigg(\frac{1}{(\bar{\sigma}_{t}^{*}+\sigma_{0})}-\frac{1}{(2\sigma_{0})}\bigg)\bigg], $$

where the second term is of order $O(t)$ by (iii), while for the first term we have by Itô’s lemma

$$\begin{aligned} & {\mathbb {E}}[(\bar{\sigma}_{t}^{*})^{2}-\sigma_{0}^{2}] \\ &= {\mathbb {E}}\bigg[\frac{1}{t}\int_{0}^{t}\bigg(\int_{0}^{s}2\bar{\sigma}_{u}\bar{\sigma}'_{u}\bar{\gamma}_{u}dW_{u}^{1}+\int_{0}^{s}\big(2\bar{\sigma}_{u}\bar{\sigma}'_{u}\bar{\alpha}_{u}+(\bar{\sigma}'_{u})^{2}+\bar{\sigma}_{u}\bar{\sigma}''_{u}\big)du\bigg)ds\bigg]\\ &=O(t),\quad t\to0, \end{aligned}$$

since the expected value of the stochastic integral is zero.

(v) By Cauchy’s inequality and Itô’s isometry, we have

$$\begin{aligned} {\mathbb {E}}[|\xi_{t}^{2}|] \leq\sqrt{\int_{0}^{t}{\mathbb {E}}\bigg[\int _{0}^{s}\bigg(\bar{\sigma}'_{u}\bar{\alpha}_{u}+\frac{1}{2}\bar{\sigma}''_{u} \bar{\gamma}^{2}_{u}\bigg)du\bigg]^{2} ds}=O(t^{\frac{3}{2}}),\quad t\to0. \end{aligned}$$

Similarly,

$$\begin{aligned} {\mathbb {E}}[|\xi_{t}^{1}-\xi_{t}^{1,0}|] \leq\sqrt{\int_{0}^{t}\int _{0}^{s}{\mathbb {E}}[\bar{\sigma}'_{u}\bar{\gamma}_{u}-\sigma'_{0} \gamma _{0}]^{2}duds}=O(t^{\frac{3}{2}}),\quad t\to0, \end{aligned}$$

because the boundedness of $\sigma'$ and $\gamma$ allows us to find a constant $K$ such that

$$\begin{aligned} {\mathbb {E}}[\bar{\sigma}'_{u}\bar{\gamma}_{u}-\sigma'_{0}\gamma_{0}]^{2} &\leq K{\mathbb {E}}[\bar{\gamma}_{u}-\gamma_{0}]^{2} + K{\mathbb {E}}[\bar{\sigma}'_{u}-\sigma'_{0}]^{2}\\ &\leq2LK{{\mathbb {E}}[Y_{u\wedge{}\tau}-y_{0}]^{2}} = O(u),\quad u\to0, \end{aligned}$$

where in the last step we again used the BDG inequality. Similarly, Cauchy’s inequality and Itô’s isometry yield ${\mathbb {E}}[|\xi _{t}^{1}|]=O(t)$, as $t\to0$.

(vi) follows from the triangle inequality and the following three identities. First, by (iii) above, we have

$$\begin{aligned} {\mathbb {E}}\bigg[\bigg|\bar{\sigma}_{t}^{*}-\sigma_{0}-\frac{(\bar{\sigma}_{t}^{*})^{2}-\sigma_{0}^{2}}{2\sigma_{0}}\bigg|\bigg] =\frac{1}{2\sigma_{0}}{\mathbb {E}}[\bar{\sigma}_{t}^{*}-\sigma _{0}]^{2}=O(t),\quad t\to0. \end{aligned}$$

Second, by Itô’s lemma,

$$\begin{aligned} & {\mathbb {E}}\bigg[\bigg|(\bar{\sigma}_{t}^{*})^{2}-\sigma_{0}^{2}-\frac {1}{t}\int_{0}^{t}\int_{0}^{s}2\bar{\sigma}_{u}\bar{\sigma}'_{u}\bar{\gamma}_{u}dW_{u}^{1}ds\bigg|\bigg]\\ &\leq\frac{1}{t} \int_{0}^{t}\int_{0}^{s}{\mathbb {E}}\bigg[\bigg|2\bar{\sigma}_{u}\bar{\sigma}'_{u}\bar{\alpha}_{u}+\frac{1}{2}\big((\bar{\sigma}'_{u})^{2}+\bar{\sigma}_{u}\bar{\sigma}''_{u}\big)\bar{\gamma}^{2}_{u}\bigg|\bigg]duds\\ &=O(t),\quad t\to0, \end{aligned}$$

since the integrand in the last integral is bounded. Third, Cauchy’s inequality and Itô’s isometry can be used to show

$$\begin{aligned} {\mathbb {E}}\bigg[\bigg|\frac{1}{t}\int_{0}^{t}\int_{0}^{s}(\bar{\sigma}_{u}\bar{\sigma}'_{u}\bar{\gamma}_{u}-\sigma_{0}\sigma'_{0}\gamma _{0})dW_{u}^{1}ds\bigg|\bigg] =O(t),\quad t\to0, \end{aligned}$$

by following similar steps as in the proof of (v). □