138 research outputs found
Call and put implied volatilities and the derivation of option implied trees
Standard methodologies for the derivation of implied trees from option prices are based on the validity of the put-call parity. Muzzioli and Torricelli (2002) propose a methodology which accounts for PCP violations. Based on this latter approach the present paper advances in two main directions. First we propose a different methodology in order to imply the interval of artificial probabilities at each node of the tree. Secondly, we perform an empirical validation of the implied tree obtained, both in the sample and out of sample, by using DAX index options data set covering the period from January 4, 1999 to December 28, 2000. Numerical results are compared with one of the most used standard methodologies, i.e. Derman and Kaniâs. The results suggest that the estimation proposed, by taking into account the informational content of both call and put prices, highly improves both the in-the-sample fitting and the out-of-sample performance.Binomial Method; Put-Call Parity; Choquet Pricing; Interval Tree.
Multivariate Option Pricing with Copulas.
In this paper we suggest the adoption of copula functions in order to price multivariate contingent claims. Copulas enable us to imbed the marginal distributions extracted from vertical spreads in the options markets in a multivariate pricing kernel. We prove that such kernel is a copula function, and that its super -replication strategy is represented by the Fréchet bounds. As applications, we provide prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, we provide no-arbitrage pricing bounds, as well as the values consistent with independence of the underlying assets. As a final reference value, we use a copula function calibrated on historical data.option pricing; basket options; copula functions; non-normal returns
Numerics of Implied Binomial Trees
Market option prices in last 20 years conrmed deviations from the Black and Scholes (BS) models assumptions, especially on the BS implied volatility. Implied binomial trees (IBT) models capture the variations of the implied volatility known as \volatility smile". They provide a discrete approximation to the continuous risk neutral process for the underlying assets. In this paper, we describe the numerical construction of IBTs by Derman and Kani (DK) and an alternative method by Barle and Cakici (BC). After the formation of IBT we can estimate the implied local volatility and the state price density (SPD). We compare the SPD estimated by the IBT methods with a conditional density computed from a simulated diusion process. In addition, we apply the IBT to EUREX option prices and compare the estimated SPDs. Both IBT methods coincide well with the estimation from the simulated process, though the BC method shows smaller deviations in case of high interest rate, particularly.Implied Tree Models, Implied Volatility, Local Volatility, Option Pricing
Algorithmic optimization and its application in finance
The goal of this thesis is to examine different issues in the area of finance and application of financial and mathematical models under consideration of optimization methods.
Prior to the application of a model to its scope, the model results have to be adjusted according to the observed data. For this reason a target function is defined which is being minimized by using optimization algorithms. This allows finding the optimal model parameters. This procedure is called model calibration or model fitting and requires a suitable model for this application.
In this thesis we apply financial and mathematical models such as Heston, CIR, geometric Brownian motion, as well as inverse transform sampling, and Chi-square test. Moreover, we test the following optimization methods: Genetic algorithms, Particle-Swarm, Levenberg-Marquardt, and Simplex algorithm.
The first part of this thesis deals with the problem of finding a more accurate forecasting approach for market liquidity by using a calibrated Heston model for the simulation of the bid/ask paths instead of the standard Brownian motion and the inverse transformation method instead of compound Poisson process for the generation of the bid/ask volume distributions. We show that the simulated trading volumes converge to one single value which can be used as a liquidity estimator and we find that the calibrated Heston model as well as the inverse transform sampling are superior concerning the use of the standard Brownian motion, resp. compound Poisson process.
In the second part, we examine the price markup for hedging or liquidity costs, that customers have to pay when they buy structured products by replicating the payoff of ten different structured products and comparing their fair values with the prices actually traded. For this purpose we use parallel computing, a new technology that was not possible in the past. This allows us to use a calibrated Heston model to calculate the fair values of structured products over a longer period of time. Our results show that the markup that clients pay for these ten products ranges from 0.9%-2.9%. We can also observe that products with higher payoff levels, or better capital protection, require higher costs. We also identify market volatility as a statistically significant driver of the markup.
In the third part, we show that the tracking error of an passively managed ETF can be significantly reduced through the use of optimization methods if the correlation factor between Index and ETF is used as target function. By finding optimal weights of a self-constructed bond- and the DAX- index, the number of constituents can be reduced significantly, while keeping the tracking error small.
In the fourth part, we develop a hedging strategy based on fuel prices that can be applied primarily to the end users of petrol and diesel fuels. This enables the fuel consumer to buy fuel at a certain price for a certain period of time by purchasing a call option. To price the American call option we use a geometric Brownian motion combined with a binomial model
Modelling the implied probability of stock market movements
In this paper we study risk-neutral densities (RNDs) for the German stock market. The use of option prices allows us to quantify the risk-neutral probabilities of various levels of the DAX index. For the period from December 1995 to November 2001, we implement the mixture of log-normals model and a volatility-smoothing method. We discuss the time series behaviour of the implied PDFs and we examine the relations between the moments and observable factors such as macroeconomic variables, the US stock markets and credit risk. We find that the risk-neutral densities exhibit pronounced negative skewness. Our second main observation is a significant spillover of volatility, as the implied volatility and kurtosis of the DAX RND are mostly driven by the volatility of US stock prices. JEL Classification: C22, C51, G13, G15Option prices, risk-neutral density, spillover, Volatility
Corridor implied volatility and the variance risk premium in the Italian market
Corridor implied volatility introduced in Carr and Madan (1998) and recently implemented in Andersen and Bondarenko (2007) is obtained from model-free implied volatility by truncating the integration domain between two barriers. Corridor implied volatility is implicitly linked with the concept that the tails of the risk-neutral distribution are estimated with less precision than central values, due to the lack of liquid options for very high and very low strikes. However, there is no golden choice for the barriers levelsâ, which will probably change depending on the underlying asset risk neutral distribution. The latter feature renders its forecasting performance mainly an empirical question. The aim of the paper is twofold. First we investigate the forecasting performance of corridor implied volatility by choosing different corridors with symmetric and asymmetric cuts, and compare the results with the preliminary findings in Muzzioli (2010b). Second, we examine the nature of the variance risk premium and shed light on the information content of different parts of the risk neutral distribution of the stock price, by using a model-independent approach based on corridor measures. To this end we compute both realised and model-free variance measures which accounts for drops versus increases in the underlying asset price. The comparison is pursued by using intra-daily synchronous prices between the options and the underlying asset.corridor implied volatility, variance swap, corridor variance swap, variance risk
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