Implied basket correlation dynamics

A Joint Analysis of the KOSPI 200 Option and ODAX Option Markets Dynamics

A semiparametric factor model for CDO surfaces dynamics

Return Dynamics With Lévy Jumps: Evidence from Stock and Option Prices

ABSTRACTDistributional Divergence and Statistical Experiments are used herein for a positive stochastic process. This framework provides, under mild assumptions, Risk Neutral Probability (-ies) for a stock price process which does not have necessarily either to satisfy a Stochastic Differential Equation or to follow a model, both non-realistic assumptions. The results contribute in understanding the relation between statistical contiguity and market's informational efficiency. -price of European option is obtained, confirming the universal quote of the Black–Scholes–Merton price for the class of calm stock prices that includes log-normal price. Other consequences are presented.

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.

A Moving Boundary Approach to American Option Pricing

Modelling the dynamics of credit derivatives is a challenging task in nance and economics. The recent crisis has shown that the standard market models fail to measure and forecast nancial risks and their characteristics. This work studies risk of collateralized debt obligations (CDOs) by investigating the evolution of tranche spread surfaces and base correlation surfaces using a dynamic semiparametric factor model (DSFM). The DSFM oers a combination of exible functional data analysis and dimension reduction methods, where the change in time is linear but the shape is nonparametric. The study provides an empirical analysis based on iTraxx Europe tranches and proposes an application to curve trading strategies. The DSFM allows us to describe the dynamics of all the tranches for all available maturities and series simultaneously which yields better understanding of the risk associated with trading

Implied volatility can be considered as a function of strike level and time to maturity. As it is calculated from the actual trading options, it contains dynamic, multi-dimensional information of options, modelling the implied volatility is an interesting task for researchers. Dynamic semiparametric factor models (DSFM) are used to model the implied volatility surface. It employs semiparametric factor functions and time variate loadings to reduce the dimensions of the data. This master thesis applies joint analysis with the time variate factor loadings resulted from DSFM, in order to discuss the relationship between index options and stock options. The data of DAX index option and its liquid components stock options will be applied in analysis. The result of the joint analysis shows, that the index option has long term relationship with its stock options. It is unlikely to disperse the risk by trading the stock options under the same index.

Leveraged ETF Options Implied Volatility Paradox: A Statistical Study

We provide a framework for learning risk-neutral measures (Martingale measures) for pricing options from high frequency financial data. In a simple geometric Brownian motion model, a price volatility, a fixed interest rate and a no-arbitrage condition suffice to determine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these assumptions to obtain a class of allowable risk-neutral measures. We then propose a framework for learning the appropriate risk-neural measure. Since the riskneutral measure prices all options simultaneously, we can use all the option contracts on a particular underlying stock for learning. We demonstrate the performance of these models on historical data. In particular, we show that both learning without a no-arbitrage condition and a no-arbitrage condition without learning are worse than our framework; however the combination of learning with a no-arbitrage condition has the best result. These results indicate the potential to learn Martingale measures with a no-arbitrage condition providing just the right constraint. We also compare our approach to standard Binomial models with volatility estimates (historical volatility and GARCH volatility predictors). Finally, we illustrate the power of such a framework by developing a real time trading system based upon these pricing methods.

Arbitrage-Free Prediction of the Implied Volatility Smile

The stock price that is observed in the market contains irregular fluctuations that cannot be explained by the theory. We assume this kind of fluctuation to be observation noise. By eliminating observation noise from the underlying stock price, we can evaluate the option that better reflects the entity price. We define the stock price that eliminates observation noise as the entity stock price. The most important problem in eliminating the observation noise is the choice of unknown constant parameters of the market model. We develop a parameter estimation method by using the relationship between stock price and its option price or future price. We obtain satisfactory results in testing the algorithm by some simulations. We apply the parameters obtained by this method to estimate the stock price by eliminating observation noise from the real market data. We then calculate the call option price based on the estimated entity stock price.

In the last decade, most studies on the interaction between the financial sector and economic growth has focused mainly on the banks and the stock market. Recently, growing attention has shifted to the interaction between the non-bank financial intermediaries (NBFIs) such as the insurance companies and economic growth. In this study, we examined the impact of insurance contributions on economic growth in Nigeria over a twenty seven year period, between 1981 and 2008 using a dynamic factor model. The proposed technique described a number of methods designed to analyze a functional relationship between the volume of insurance contribution and economic growth in terms of underlying but unobservable random quantities called factors. The factor loadings indicate which common trend is related to which set of time series. The result obtained through this approach shows that the functional relationship between the volume of insurance contribution and economic growth in Nigeria is a first order vector autoregressive model. Key words: Insurance contribution, dynamic factor model, economic growth.

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let $$V(S_t, t)$$ be the price of an option at time t when the share price of the underlying stock is $$S_t$$ . See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond $$B_t$$ and a risky asset stock $$S_t$$ . They evolve as where r is the interest rate, $$\mu $$ is the drift, $$\sigma $$ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price $$B_t^{-1}V(S_t,t)$$ . By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure . Also, under the risk-neutral measure $$\mu =r$$ . We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.

In a viable single-period model with one stock and k ≥ 2 scenarios the completeness of the market is equivalent to the uniqueness of the risk neutral probability; this equivalence allows to price every derivative security with a unique fair price. When the market is incomplete, the set of all possible risk neutral probabilities is not a singleton and for every non attainable derivative security we have a bid-ask interval of possible prices. In literature, different methods have been proposed in order to select a unique risk neutral probability starting with the real world probability p. Contrary to the complete case, in all these models \(\textbf{p}\) is really used for the option pricing and its elicitation is a crucial point for every criterion used to select a risk neutral probability. We propose a method for the valuation problem in incomplete markets which can be used when p is a partial conditional probability assessment as well as when we have different expert opinions expressed through conditional probability assessments. In fact, it is not always possible to elicit a probability distribution p over all the possible states of the world: the information that we have could be partial, conditional or even not coherent. Therefore we will select a risk neutral probability by minimizing a discrepancy measure introduced in [2] and analized in [3] between p and the set of all possible risk neutral probability, where p can be a partial conditional probability assessments or it can be given by the fusion of different expert opinions.KeywordsExpert OpinionCall OptionConditional EventMartingale MeasureStrike PriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.