Abstracts
Thursday June 5, 14:00-14:30 | session 8.5 | Options, Futures | room G
The main contribution of this paper is to propose a novel analytical approach to price discretely sampled generalized variance swaps of the underlying based on the Heston's (1993) two-factor stochastic volatility model with simultaneous jumps in the asset price and variance processes. The new approach is adopted to produce the closed-form pricing formulae for variance swaps, gamma swaps, corridor variance swaps, and conditional variance swaps. Unlike Zheng and Kwok’s (2013) approach, the pricing formulae obtained in this paper are in much simpler forms than those presented by Zheng and Kwok (2013); the requirement of the parameter functions being twice differentiable in Zheng and Kwok (2013) have now been completely avoided. Furthermore, we show that the proposed methodology can be extended to price various types of generalized variance swaps such as self-quantoed variance swaps, entropy swaps, and proportional variance swaps introduced by Crosby (2013). The solution procedure presented in this paper will thus enable researchers to view this type of problems from a different angle.
Thursday June 5, 14:30-15:00 | session 8.5 | Options, Futures | room G
We consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the growth of the slope for infinite activity exponential Levy models. As auxiliary results, we obtain the limiting values of short maturity digital call options. Finally, we discuss when the at-the-money slope is consistent with the steepness of the smile wings, as given by Lee's moment formula.
Thursday June 5, 15:00-15:30 | session 8.5 | Options, Futures | room G
With the existence of active markets for volatility derivatives and options on the underlying instrument, the need for models that are able to price these markets consistently has increased. Although pricing formulas for VIX and vanilla options are now available for commonly employed models exhibiting stochastic volatility and/or jumps, it remains to be shown whether these are able to price both markets consistently. This paper aims to fill this vacuum. In particular, the Heston model, the Heston model with jumps in returns, and the Heston model with simultaneous jumps in returns and variance are studied. In all these models the characteristic function of log-returns is known in an analytically closed form, and options on the underlying index can be priced via Fourier transform techniques. Likewise, the three model specifications allow for tractable pricing of VIX options as demonstrated in [1]. Compared to the VIX option pricing formula in Lian and Zhu (2013), we derive a numerically simpler formula in the case of the Heston model with jumps in returns (but not variance).
We find that the full flexibility of having jumps in both returns and volatility added to a stochastic volatility model is essential. The jumps in returns allow for an improved fit to SPX options, while jumps in volatility are important to match the upward sloping implied volatility skew observed on VIX options. Moreover, we find that the SVJJ model with the Feller condition imposed and calibrated jointly to SPX and VIX options fails to fit both markets satisfactory with average relative pricing errors for the dates considered around 16-20%. Relaxing the Feller condition in the calibration improves the fit considerably and errors are down to around 8-11%. Still, the fit is not satisfactory for trading purposes and we conclude that one needs more flexibility in the model in order to jointly fit both options markets.
[1] Lian, G. and Zhu, S. (2013). Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance, 36 (1): 71-88.
Thursday June 5, 15:30-16:00 | session 8.5 | Options, Futures | room G
We study functionals of Brownian excursion, including first hitting time, last passage time, maximum and the time it is achieved. Our method is based on the analogy of Brownian excursion, Brownian motion and Bessel process, which can be related using conditional martingales. We derive the first hitting time of Brownian excursion in a closed form and provide three proofs using elementary arguments from probability theory emphasizing the nature of excursions. Relying on Pitman's Bessel bridge representation we deduce time reversibility and derive the last passage time of Brownian excursion. From the law of hitting time we implicate the law of the maximum and conclude with our main result, studying the joint probability of maximum and time it is achieved. These results are applied to address problems in option pricing. Since Madan, Roynette and Yor discovered that European option prices can be written in terms of last passage times where they allow great flexibility to the local martingale modeling the stock price, they came into focus of financial mathematics. We also discuss the pricing of options depending on the running maximum and the time it is achieved and being triggered when the drawdown of the underlying price exceeds a certain level within a prespecified time period.