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this has the most intuitive explanation on this topic I have read. I liked the example when he generates a series with negative correlation but positive cointegration. Discuss this paper
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by Rama Cont, Valdo Durrleman, Jose Da Fonseca Abstract: We propose a market-based approach to the modelling of implied volatility, in which the implied volatility surface is directly used as the state variable to describe the joint evolution of market prices of options and their underlying asset. We model the evolution of an implied volatility surface by representing it as a randomly fluctuating surface driven by a finite number of orthogonal random factors. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data on SP500 and DAX options.
We illustrate how this approach extends and improves the accuracy of the well-known 'sticky moneyness' rule used by option traders for updating implied volatilities. Our approach gives a justification for the use of 'Vegas' for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors.
Keywords: Implied volatility, volatility risk, risk management, portfolios of options Discuss this paper
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by CAROL ALEXANDER and LEONARDO M. NOGUEIRA ABSTRACT There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is modelspecific. The condition is very easy to check in local volatility models having only a few stochastic parameters.
Investors in equity options experience two problems that compound each other. In contrast to fixed-income and currency markets, there are thousands of underlyers and tens of thousands of options, and each underlyer can have a potentially large volatility skew. How can an options investor gauge which option provides the best relative value? In this paper, we make use of a method for estimating the fair volatility smile of any equity underlyer from information embedded in the time series of that underlyer Discuss this paper
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by Vassilis Galiotos The purpose of this project is to explain to some extent the importance of stochastic volatility models and implied volatility. The model that is studied is the Heston model (1993). Our findings confirm the common belief that the implied volatility smile slopes downwards at the money if the correlation between the spot returns and the volatility is positive. Similarly, if the correlation is negative the implied volatility slopes upwards.
Abstract This project discusses two methods for obtaining static replicating portfolios for barrier options. The first method discussed is the method of (Carr & Chou 1997a) and the second is the method of (Derman, Ergener & Kani 1995). The methods are tackled from both a theoretical point of view as well as from a practical implementation point of view. Hence, Matlab code has also been provided implementing these methods. The inputs and outputs of this code is also discussed. The type of barrier options dealt with in this project are vanilla barrier options. That is, your basic up- and-(out/in) and down-and-(out/in), constant-barrier, standard European put and call options. Discuss this paper
In the original Vasicek model interest rates are calculated assuming that volatility remains constant over the period of analysis. In this study, we constructed a stochastic volatility model for interest rates. In our model we assumed not only that interest rate process but also the volatility process for interest rates follows the mean-reverting Vasicek model. We derived the density function for the stochastic element of the interest rate process and reduced this density function to a series form. The parameters of our model were estimated by using the method of moments. Finally, we tested the performance of our model using the data of interest rates in Turkey.
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Presentation by Yoann BOURGEOIS and Marc MINKO IInnttrroodduuccttiioonn Single stocks in the Equity Market generally are not stationary. But, their yields, in many cases are. From the econometrical point of view, they are generally told to be Integrated of order 1. Cointegration is a mathematical theory that helps to handle the problem generated by non-stationary data. With the help of this theory, we propose to build linear combinations of these single stocks that are stationary. Such combinations can be traded and are called synthetic assets. Eventually, these stationary assets have the mean reversion property and we will use this property in order to set up arbitrage strategies Discuss this paper
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by Kristina Andersson In the original Black-Scholes model, the risk is quanti ed by a constant volatility parameter. It has been proposed by many authors that the volatilities should be modeled by a stochastic process to obtain a more realistic model. The volatility that corresponds to actual market data for option prices in Black-Scholes model is called the implied volatility. This volatility is in general dependent on the strike price, in contrast to the underlying assumption of Black-Scholes model. As a function of strike it forms a curve called "volatility smile". To explain this smile it has been proposed to study models allowing for a volatility driven by a stochastic process. In the present paper a review of stochastic volatility is presented and three stochastic volatility models are studied in some detail. We study the volatility smile of these models and show that in some cases we can reproduce a smile similar to the curves occuring in reality. We also study a corrected Black-Scholes pricing formula
Outline 1 General Ideas 2 Bond market terminologies 3 Dynamic term structure models 4 Model design and estimation 5 Statistical arbitrage trading Discuss this paper
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Fabio Mercurio and Andrea Pallavicini We test both the SABR model [4] and the shifted-lognormal mixture model [2] as far as the joint calibration to swaption smiles and CMS swap spreads is concerned. Such a joint calibration is essential to consistently recover implied volatilities for non-quoted strikes and CMS adjustments for any expiry-tenor pair. Discuss this paper
This article introduces the concept of a statistical arbitrage opportunity (SAO). In a finite-horizon economy, a SAO is a zero-cost trading strategy for which (i) the expected payoff is positive, and (ii) the conditional expected payoff in each final state of the economy is nonnegative. Unlike a pure arbitrage opportunity, a SAO can have negative payoffs provided that the average payoff in each final state is nonnegative. If the pricing kernel in the economy is path independent, then no SAOs can exist. Furthermore, ruling out SAOs imposes a novel martingale-type restriction on the dynamics of securities prices. The important properties of the restriction are that it (1) is model-free, in the sense that it requires no parametric assumptions about the true equilibrium model, (2) can be tested in samples affected by selection biases, such as the peso problem, and (3) continues to hold when investors' beliefs are mistaken. The article argues that one can use the new restriction to empirically resolve the joint hyothesis problem present in the traditional tests of the efficient market hypothesis. Discuss this paper
Pricing American options requires solving an optimal stopping problem and therefore presents a challenge for simulation. This article investigates connections between a weighted Monte Carlo technique and regression-based methods for this problem. The weighted Monte Carlo technique is shown to be equivalent to a least-squares method in which option values are regressed at a later time than in other regression-based methods. This Discuss this paper
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by Ionut¸ Florescu1,3 and Frederi G. Viens Abstract We treat the problem of option pricing under the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be mean- reverting. Assuming that only discrete past stock information is available, we adapt an interacting particle stochastic filtering algorithm due to Del Moral, Jacod and Protter (Del Moral et al., 2001) to estimate the SV, and construct a quadrinomial tree which samples volatilities from the SV filter’s empirical measure approximation at time 0. Proofs of convergence of the tree to continuous-time SV models are provided. Classical arbitrage-free option pricing is performed on the tree, and provides answers that are close to market prices of options on the SP500 or on blue-chip stocks. We compare our results to non-random volatility models, and to models which continue to estimate volatility after time 0. We show precisely how to calibrate our incomplete market, choosing a specific martingale measure, by using a b nchmark option. Key words and phrases: incomplete markets, Monte-Carlo method, options market, option pricing, particle method, random tree, stochastic filtering, stochastic volatility. Discuss this paper
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Dietmar P.J. Leisen Abstract This paper constructs a sequence of discrete{time models, that converge to stochas- tic volatility models. It generalizes the well{known binomial models from the Black{ Scholes setup to bivariate di usions, applying back{and{forth transformations in the style of Nelson and Ramaswamy (1990). Our guideline in the construction is a general convergence theorem for the weak convergence of processes; this imposes restrictions on the discretization into a grid and the successors of grid points. We discuss the implementation for the Hull and White (1987) model and calculate prices for European{ and American{style put options. Convergence is smooth and fairly accurate with re nements of 20 time steps.
Abstract: Current Monte Carlo pricing engines may face computational challenge for the Greeks, because of not only their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fourni Discuss this paper
Quoted- Using a two-factor approach to modelling spread options reveals the paradox of negative vegas. Mark Garman shows how this can help users gain a better understanding of these complex instruments
Abstract We investigate a method for pricing the generic spread option beyond the classical two-factor Black-Scholes framework by extending the fast Fourier Transform technique introduced by Carr & Madan (1999) to a multi-factor setting. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. This enables us to incorporate stochasticity in the volatility and correlation structure { a focus of concern for energy option traders { by introducing additional factors within an ane jump-di usion framework. Furthermore, computational time does not increase signi cantly as additional random factors are introduced, since the fast Fourier Transform remains two dimensional in terms of the two prices de ning the spread. This yields considerable advantage over Monte Carlo and PDE methods and numerical results are presented to this e ect.
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Browse our more than 2,500 Math pages filled with short and easy-to-understand explanations. Click on one of the following subject areas: Algebra, Trigonometry, Calculus, Differential Equations, Complex Variables, Matrix Algebra, or Mathematical Tables.
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Outline The aim of these lectures is to introduce some of the techniques from stochastic analysis that are employed in mathematical finance. This is a huge area, so we can certainly do no more than scratch the surface, but we will see that mathematics has been of fundamental importance in the revolution that has taken place in the financial markets over the last twenty-five years.
Although we use financial examples for motivation, Brownian motion and stochastic calculus play an important r Discuss this paper
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Rolf Poulsen Static hedging of barrier options is more sensitive to model risk than -hedging. Still, under realistic conditions wrong static hedges may very well outperform correct -hedges. Especially after some natural adjustments. Discuss this paper
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Emanuel Derman Deniz Ergener Iraj Kani SUMMARY This paper presents a method for replicating or hedging a target stock option with a portfolio of other options. It shows how to construct a replicating portfolio of standard options with varying strikes and maturities and fixed portfolio weights. Once constructed, this portfolio will replicate the value of the target option for a wide range of stock prices and times before expiration, without requiring further weight adjustments. We call this method static replication. It makes no assumptions beyond those of standard options theory. You can use the technique to construct static hedges for exotic options, thereby minimizing dynamic hedging risk and costs. You can use it to structure exotic payoffs from standard options. Finally, you can use it as an aid in valuing exotic options, since it lets you approximately decompose the exotic option into a portfolio of standard options whose market prices and bid-ask spreads may be better known. Replicating an Exotic Option with a Portfolio of Standard Options.
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Morten Nalholm Rolf Poulsen Abstract We investigate how sensitive di erent dynamic and static hedge strategies for barrier options are to model risk. We nd that using plain vanilla options to hedge o ers considerable improvements over usual -hedges. Further, we show that the hedge portfolios involving options are relatively more sensitive to model risk, but that the degree of misspeci cation sensitivity is robust across commonly used models.
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Nikesh Agarwal 91164 Vikash Madhogaria 91166 Supreena Narayanan 80293 Introduction In this report, we explain and analyze a trading strategy, popularly known as Pairs trading. We begin by explaining what a pair trading strategy entails. Since there are various ways of implementing the strategy, we describe the methodology selected by us in section 3. Thereafter, we look at the returns from the strategy and benchmark it to the S&P 500 index in Section 4. In section 5, we examine the risks involved in pairs trading. Section 6 looks at some of the limitations we faced while trading and Section 7 points out some mistakes we made. Finally, we discuss some risk control measures in Section 8 and conclude in Section 9 with comments on whether we would implement the strategy in real life and if so, with what changes.
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This book was published in April 2002 by Cambridge University Press under the title "Stochastic Integration with Jumps" in the series Encyclopedia of Mathematics and its Applications.
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Anatoly Malyarenko Abstract This paper describes the simulation of interest rate in Vasicek Model and stochastic behavior of interest rate appeared in this applet. A brief description of the model is given, including the equation and parameters. Then the model is implemented throw a Java applet where the codes are examined in detail. Discuss this paper
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Alan Bain The following notes aim to provide a very informal introduction to Stochastic Calculus,and especially to the It Discuss this paper
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