Short-maturity asymptotics for VIX and European options in local-stochastic volatility models
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| Format: | Preprint |
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2024
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| author | Pirjol, Dan Wang, Xiaoyu Zhu, Lingjiong |
| author_facet | Pirjol, Dan Wang, Xiaoyu Zhu, Lingjiong |
| contents | We derive the short-maturity asymptotics for European and VIX option prices in local-stochastic volatility models where the volatility follows a continuous-path Markov process. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. Using large deviations theory methods, the asymptotics for the OTM options are expressed as a two-dimensional variational problem, which is reduced to an extremal problem for a function of two real variables. This extremal problem is solved explicitly in an expansion in log-moneyness. We derive series expansions for the implied volatility for European and VIX options which should be useful for model calibration. We give explicit results for two classes of local-stochastic volatility models relevant in practice, with Heston-type and SABR-type stochastic volatility. The leading-order asymptotics for at-the-money options are computed in closed-form. The asymptotic results reproduce known results in the literature for the Heston and SABR models and for the uncorrelated local-stochastic volatility model. The asymptotic results are tested against numerical simulations for a local-stochastic volatility model with bounded local volatility. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Short-maturity asymptotics for VIX and European options in local-stochastic volatility models Pirjol, Dan Wang, Xiaoyu Zhu, Lingjiong Pricing of Securities We derive the short-maturity asymptotics for European and VIX option prices in local-stochastic volatility models where the volatility follows a continuous-path Markov process. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. Using large deviations theory methods, the asymptotics for the OTM options are expressed as a two-dimensional variational problem, which is reduced to an extremal problem for a function of two real variables. This extremal problem is solved explicitly in an expansion in log-moneyness. We derive series expansions for the implied volatility for European and VIX options which should be useful for model calibration. We give explicit results for two classes of local-stochastic volatility models relevant in practice, with Heston-type and SABR-type stochastic volatility. The leading-order asymptotics for at-the-money options are computed in closed-form. The asymptotic results reproduce known results in the literature for the Heston and SABR models and for the uncorrelated local-stochastic volatility model. The asymptotic results are tested against numerical simulations for a local-stochastic volatility model with bounded local volatility. |
| title | Short-maturity asymptotics for VIX and European options in local-stochastic volatility models |
| topic | Pricing of Securities |
| url | https://arxiv.org/abs/2407.16813 |