Option Implied Jump Risk And Risk Premium

Starting directly from modeling the dynamic process of option implied volatility surface which is observable from markets,and allowing jumps present in the dynamic process of the underlying asset price,we deduces the equation of implied volatility surface satisfying the dynamic no-arbitrage condition,thus extending the model framework of Carr&Wu(2016)to the case that underlying asset price is allowed to jump.Our results show that the effect of the jumping term on the implied volatility surface is simply reflected by the change of the instantaneous total volatility(the sum of the diffusion volatility and the jump volatility),so that the jump term does not cause equation of implied volatility surface getting more involved.Meantime,Our results keep all of characteristics of Carr&Wu(2016),especially that the no-arbitrage implied volatility surface equation still has a simple form of the second-order algebra polynomial,which greatly enhance the efficiency of the calibration model as avoiding the numerical integration which the traditional stochastic volatility models rely on.Furthermore,based on the new volatility surface model,we develope a new method that extracting the jump risk premium implicit in options through the implied volatility surface information,which has two advantages:First,compared with the existing literature which the risk premium is extracted,our method based on implied volatility surface avoids the latent state variable which may cause difficulty to model estimation process.Also,using the whole implied volatility surface obviously contains more information.Second,compared with the existing literature which using the stochastic volatility jump diffusion model,our method benifits from the property of quadratic algebra polynomial of the volatility surface equation which greatly improve the calculation efficiency.To this end,we first present a new form of the state price density function with jumps,which sets the market prices for jump risk,diffusion risk and volatility risk.On this basis,we induce the relationship between the volatility surface under risk neutral measure and under the realistic measure,and the expressions of the jump risk premium,the volatility risk premium,the diffusion risk premium and are given.Worth to mention,under the new form of state-price density function,the distribution of the jumping magnitudes keep the same form under the realistic measure and the risk neutral measure,and more intuitively reflects the risk premium of the random jump size.Finally,using the TXO data,we analyze the pricing performance of the model and the dynamic behavior of the volatility surface as well as the risk premiums.we also test the predictability of the jump risk premiums on the future excess return of the TAIEX index and on the future tail risk.The following conclusions are drawn:(1)Compared with the model without the jump term given by Carr&Wu’s(2016),incorporating of the jump into model has improved the pricing performance of the short-term option;(2)The state variables including instantaneous volatility,volatility of instantaneous volatility,drift rate of volatility,correlation coefficient of the underlying volatility surface,jumping intensity and jumping amplitude whichportray the whole volatility surface show a significant time-varying characteristic,changing as market situation changing;(3)The jump risk premium shows obvious time-varying characteristics,significantly higher and more volatile during the period before and after the financial crisis in 2007-2009 than those in other periods,the implied volatility risk premium of the index options also has obvious time-varying characteristics,the mean value during the sample period is negative,in line with most literatures’ results;(4)The extracted jump risk premium significantly predicts future excess return of TAIEX.When the traditional forecasting factors such as price-earnings ratio and price-dividend ratio are added to the forecasting process,it is found that the implied jump risk premium and the traditional forecasting factor are enhanced at the same time,and R square of the forecast regression is also increased.So the option implied jump risk premium is an effective supplement to the prediction information contained in the traditional forecasting factor;(5)The forecasting power of the volatility risk premium extracted by our model is superior to the volatility risk premium calculated by using the VIX index,which reflects the advantage of extracting the implicit risk premium from the whole volatility surface information;(6)The implied jump risk premium of the option has certain predictive power to the probability of the future tail risk.But it is not possible to accurately predict the state of the tail risk in the future market.The reason may be that the implied jump risk premium in the option contains a large proportion of the irrational expectation affected by investor’s emotions.