A new look at short-term implied volatility in asset price models with jumps (Q2788693)

The paper under review is to study the behavior of the implied volatility smile for option close to expiry in the exponential Lévy class of asset price models with jumps. The central object in option market, the implied volatility, is not easy to obtain a closed form, therefore, the study of the asymptotic behavior in a variety of asymptotic regimes.NEWLINENEWLINEThe market data suggest a moving log-strike \(k_t =\theta \sqrt{t\ln (1/t)} \neq 0\) for \(t>0\) such that the corresponding implied volatility has a nontrivial limit \(\lim_{t\to 0^+} \hat{\sigma}(t, k_t) \neq 0\) and \(k_t\) converges to the ATM log-strike value \(\lim_{t\to 0^+} k_t =0\) if \(S_0=1\) after rescale.NEWLINENEWLINESection 2 sets the Theorem 2.1 and Proposition 2.3 for the exponential Lévy model of assets with certain assumptions. Theorem 2.1 describes how the Blumenthal-Getoor indices of the positive and negative jumps of the Lévy process influence the asymptotic behavior of option prices at short maturity in the case of the tails of the Lévy measure of \(X\) around zero have asymptotic power-like behavior. Proposition 2.3 implies that in the absence of a Brownian component, the call and put prices of options strike at \(e^{\pm k_t}\) tend to zero at the rate equal to \(t\) if the Lévy process \(X\) has paths of finite variation, the asymptotic behavior of short-dated option prices does not depend, up to order \(o(t)\), on the Blumenthal-Getoor indices \(BG(X^{\pm})=\alpha_{\pm}\).NEWLINENEWLINETheorem 3.1 in Section 3 shows that the implied volatility \(\sigma_t(\theta)= \hat{\sigma}(t, k_t)\), defined to be the unique solution for the \(C^BS(t, k_t, \sigma)= C(t, k_t)\) for \(k_t = \theta (t \ln (\frac{1}{t}))^{1/2}\), has the explicit expression with the asymptotic rate \(O(\frac{1}{\ln^2 \frac{1}{t}})\). For the Lévy process X with jumps of finite variation, \(\ln C(t, k_t) =\min\{1, \frac{1}{2}+\frac{\theta^2}{2\sigma^2}\} \log t + o (\log \frac{1}{t})\) as \(t\to 0^+\). The main asymptotic result of the implied volatility \(\sigma_t (\theta)\) is given in Corollary 3.3 for the log-strike \(k_t=\theta (t \ln (\frac{1}{t}))^{1/2}\) such that the tail around zero of the Lévy measure \(\nu\) of X behaves as \(v(\pm x, \pm \infty) \sim c_{\pm }x^{- \alpha_{\pm}}\), and the Lévy process X has positive jumps when \(\theta > 0\) and negative jumps when \(\theta < 0\). Section 4 assumes the Lévy process X follows the widely used CGMY (Carr, Geman, Madan and Yor model) [\textit{P. Carr} et al., ``The fine structure of asset returns: an empirical investigation'', J. Bus. 75, No. 2, 305--333 (2002; \url{doi:10.1086/338705})] with a specified Lévy density \(\nu (x)\) in (4.1) to illustrate some numerical computations in Section 3. Authors first compare their algorithm with \textit{I. Wang} et al. [``Robust numerical valuation of European and American options under the CGMY process'', J. Comput. Finance 10, No. 4, 31--69 (2007; \url{doi:10.21314/jcf.2007.169})] and plot the dependence of the normalized option price and the normalized ``Bachelier'' price in Figure 4.1, investigate numerically the convergence of the OTM option prices in Theorem 2.1 in Section 4.3, and Corollary 3.3 in Section 4.4. Section 5 gives the theoretical insights into the shape of the short-maturity smile with the actual market smiles and empirically assess the parameterizations of the implied volatility smiles in terms of the theta, delta, and strike. For two currency pairs (USDJPY and EURUSD), the plotted implied volatilities give the market prices for the options with the following strikes, ATM, 25-delta call and put, 10-delta call and put, and maturities ranging from 1 day to 2 months, respectively. A discussion on the stability of the algorithm and the implied volatility function of the parameter \(\theta\) is briefly stated. Section 6 and Appendix are technical proofs of Theorem 2.1, Proposition 2.3, Theorem 3.1 and Corollary 3.3. Theorem 2.1 is proved by first Lévy process with no diffusion part to have the asymptotic behavior with the jumps, and then for a spectrally negative Lévy process with zero mean and zero diffusion part. Proposition 2.3 is proved by Itô-Tanaka formula and results of \textit{K.-I. Sato} [Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press (1999; Zbl 0973.60001)]. Theorem 3.1 is proved for the \(\theta>0\) case first by the change of variable and the regularity of the coefficients, and for the case \(\theta <0\) by applying the result of the positive log-strike under the share measure with its Radon-Nikodym derivative. Corollary 3.3 is proved by using Theorem 2.1 and Theorem 3.1 as well as the put-call symmetry. Some of the equation labels are not properly referred, for instance, on page 173 in the proof of Proposition 2.3, ``the proof of (2.10).'' really means the proof of (2.7). It would be interesting to know if the implied volatility has similar asymptotic behavior near the maturity time under the same Lévy process assumptions.