Modeling, Computing Methods Of Implied Volatility And Its Application

Existing research on volatility mainly focuses on endogenous volatility evaluation and prediction. The model of endogenous volatility was established on the basis of market feature of financial asset returns. Endogenous volatility was always regarded as the forecasting of forward volatility subconsciously. Nevertheless, it had been proven by theory and practices that: though endogenous volatility describes some features of asset price forward fluctuation, its management of forward volatility risk is limited to last data and information. The mainly defects are endogenous volatility can not be observed and only depend on last information, endogenous volatility evaluations are influenced by measuring methods, measuring intervals and sample data; moreover, endogenous volatility can not be traded. As a result, in the existing pricing theories, the risk caused by stochastic volatility can not be hedged by asset trading. At certain degree, implied volatility could overcome these defects.According to the points of SchÇ’nbuche(r1999), standard options showed high liquidity and high trading volume, derivatives based on standard options were getting more and more; secondly, the strike price and expiration span of standard options were getting wider and wider, making the implied volatility tend to be continuous; thirdly, under Efficient Market Hypothesis, the difference between market price and theory price of options showed that implied volatility contain other extra information, which is not contained in underlying assets price; last, most traders price standard option by implied volatility, according to Stein(1989), the option market could be regarded as the market for volatility trading. So, the risk of implied volatility could be hedged by trading of standard options. Based on this point, according to the market feature of abroad and domestic implied volatility, we focus on research of implied volatility from two facets, local volatility and stochastic volatility. The existing local implied volatility model ignores the problem of term structure, so, we presented local implied volatility surface in this paper which overcomes defects of the existing implied volatility model; we gave a market-based stochastic squared-root implied volatility surface model on the basis of abroad and domestic implied volatility market feature and variety of implied volatility influence factors. Under absence of arbitrage, we concluded no-arbitrage drift restrictions of stochastic implied surface. We studied the existence of a solution given fixed strike price, any expiration. In application, numerical method is used for exotic derivative pricing.The paper is organized as follows: 1. Firstly, several different volatility concepts were presented, classified and compared. Secondly, volatility relevant literature was overviewed from the following aspects: (1)market features of volatility (including endogenous volatility and exogenous volatility)were discussed, wherein, exogenous volatility possesses such market features as strike structure, term structure, and volatility surface; this paper presented three properties of volatility on the basis of volatility market feature and forecasting performance. (2)Divided endogenous volatility models into three classes. They were deterministic volatility models, Time serial models, and stochastic models, features and relative merits of each class were discussed. (3) literature pertaining forecasting performance of volatility was overviewed and compared, this is the very theory and practice basis, on which market properties of volatility was brought forward.2. We examined local implied volatility model, on this basis, we brought forward the local implied volatility surface model. Firstly, we defined the concepts of instant implied volatility and average implied volatility; secondly, we discussed the methods of weighted average, Black-Scholes estimation, and the one based on at-the-money option, we also examined each method with numerical value and warrant data; according to the examination results, we found that implied volatility is not only function of strike price, but also influenced by expiration, as a result, we brought forward the local implied volatility surface model; the result of numerical study showed that the average forecasting of volatility can be increased by the local implied volatility surface model.3. We studied market features of domestic stock derivatives by experiences. underlying assets price decides derivative value, in return, derivatives influence underlying assets. We focused on two main research objects: the first one, started from the effect of listing of option and warrant introduction, studied the relation between derivative market and underlying asset market, studied the effect of warrants introduction on stoke price by test of experiences, we found that warrants introduction on causes the effect of price increase and trading volume increase. The second one, supply of share-reformed warrants and warrants with attached bond is extremely limited, under supply cause price increase necessarily, so, we thought of the limited supply of warrants as one of the reasons why implied volatility is higher than historical volatility. We studied mainland warrants and Taiwan call warrants comparatively by experiences, the results showed that the influence of warrant supply on mainland warrant implied volatility is apparent statistically4. We brought forward the stochastic square-root implied volatility surface model, and concluded the drift restrictions of implied volatility under risk-neutral condition. On hypothesis that implied volatility surface is function of expiration and strike price, according to standard option discounted price is martingale, we concluded the no-arbitrage condition restricts for implied volatility with squared-root volatility model. We discussed different appearances of drift conditions under different conditions; we also present the equation between implied volatility and forward volatility. Under real risk measure condition, we proved that the price of implied volatility risk existed for fixed strike and any expiration for call option.Novelties of this paper mainly focus on the following points.(1) We proposed algorithm of the static implied volatility surface based on the simple computation of implied volatility, finding it improved the valuation of implied volatility of deep in or out the money optons by numerical method.In simple computation of implied volatility methods, the calculation method based on at-the-money option was brought forward from strike structure of implied volatility. In practice, the model ignored the influence of expiration variation on option price. We found by numerical examination and experiences that implied volatility varied with expiration. Given a certain at-the-money option, we considered the difference between of at-the-money and other options is a function of strike price, expiration and volatility. By differentiating the price difference, we concluded the local stochastic implied volatility surface model.We proved by numerical calculation results that the evaluation of implied volatility can by improved with static implied volatility surface model on average. Especially, implied volatility of the options with deep in or out the money was estimated accurately compared with current methods. We also found the calculation error of implied volatility was not more than 5% for options with expiration more than six months.(2) We examined introduction effects of warrants and the factors which affected abnormity of implied volatility by expererience. We found the positive price and volume effect, also, smaller volatility of underlying assets; found that the implied volatility of domestic warrants was apparently influenced by net buying pressure.Generally, financial derivatives assets market interacted with underlying assets market, which derivatives assets could have the powers of completing market, dimilishing short-sall restrictions and improving information envirolment. We examined the listing effect of domestic warrants by experience, finding the effect of positive price and trade volume increase, but the effect was not significance statistically, also, above 70% smaller volatility of underlying assets. Warrant is a kind of low cost, high leverage, new, rare derivative; the successful listing of warrants gives good information to market; all these factors push more and more investors into stock market, and further cause stock price and trading value increase. For created warrants, because of the limitation of institution, created securities companies can only create by purchasing same number stocks for mortgaging call warrant, this also cause stock price and trading volume increase.Because of special background of domestic warrant listing and the features of new capital market, scarcity of warrant is one of the most important factors that distort warrant price. Existing research mainly focus on the skew effect of implied volatility from strike price. In this paper we tried to research the skew effect of implied volatility from another brand new viewpoint, supply and demand. Contrast by more mature Taiwan call warrant, we defined the buyer (seller)-motivated trade volume as net buying, defined the net demand regulated by market circulation as net buying pressure. Results showed that net buying pressure statistically significant influence the implied volatility of domestic warrants compared with call warrants of Taiwan. At the same time, created put warrants affected by net buying pressure were weaker than no-created put warrants and call warrants, which proving the creating behavior is more efficency for put warrants than call.(3) Brought forward the squared-root stochastic implied volatility surface concluded the no-arbitrage drift of implied volatility and the equation between the implied volatility and the actual volatility of stoke price; we proved the existence of a no-arbitrage condition solution for fixed strike price and any expiration pricing system.We found by experiences that implied volatility linear model constructed objectively can not describe the term structure of implied volatility very well, because it is not scientific enough. Whereas, square implied volatility surface model can not avoid negative volatility. In order to overcome the above mentioned defects, we selected squared-root model as the stochastic implied volatility surface model. We found by theoretically that the no-arbitrage condition restricts of implied volatility is not only simple formally but also correlated with expiration of squared-root. These two very features are not possessed by objective linear model or square model. The risk-neutral condition restricts demonstrate the interrelation between implied volatility and forward volatility. Under no-arbitrage condition constrain, we concluded the equation between implied volatility and forard volatility under limiting state.Given option price and other variables, the forward volatility can be derived and used to pricing other standard options. If derivative price is not known, exotic derivative pricing partial differential equations based on stock and stochastic volatility surface can be concluded under no-arbitrage drift conditions. The portfolio of stocks and standard options can replicate the value of exotic derivatives and hedge implied volatility risk; at last, we got the value of exotic derivatives by numerical methodology.