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In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model, by developing efficient transform based pricing methods. This non-affine model gives prices of options on realized variance which allow upward sloping implied volatility of variance smiles. Heston's (1993) model, the benchmark affine stochastic volatility model, leads to downward sloping volatility of variance smiles - in disagreement with variance markets in practice. We show a robust method, using control variates, to express the Laplace transform of the variance call function in terms of the Laplace transform of realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm which gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps. Discuss this paper
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Options on Realized Variance by Transform Methods: A Non-Affine Stochastic Volatility Model
Gabriel G Drimus University of Copenhagen - Institute for Mathematical Sciences
September 19, 2009
Abstract: In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model, by developing efficient transform based pricing methods. This non-affine model gives prices of options on realized variance which allow upward sloping implied volatility of variance smiles. Heston's (1993) model, the benchmark affine stochastic volatility model, leads to downward sloping volatility of variance smiles - in disagreement with variance markets in practice. We show a robust method, using control variates, to express the Laplace transform of the variance call function in terms of the Laplace transform of realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm which gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps. Discuss this paper
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by RONGWEN WU and MICHAEL C. FU American-Asian options are average-price options that allow early exercise. In this paper, we derive structural properties for the optimal exercise policy, which are then used to develop an efficient numerical algorithm for pricing such options. In particular, we show that the optimal policy is a threshold policy: The option should be exercised as soon as the average asset price reaches a characterized threshold, which can be written as a function of the asset price at that time. By exploiting this and other structural properties, we are able to parameterize the exercise boundary, and derive gradient estimators for the option payoff with respect to the parameters of the model. These estimators are then incorporated into a simulation-based algorithm to price American-Asian options. Computational experiments carried out indicate that the algorithm is very competitive with other recently proposed numerical algorithms. Subject classifications: Finance, securities: option pricing. Simulation: perturbation analysis and stochastic approximation. Dynamic programming, models: structure of optimal policies Discuss this paper
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by YELIZ YOLCU The uncertainty attached to future movements of interest rates is an essential part of the Financial Decision Theory and requires an awareness of the stochastic movement of these rates. Several approaches have been proposed for modeling the one-factor short rate models where some lead to arbitrage-free term structures. However, no de nite consensus has been reached with regard to the best approach for interest rate modeling. In this work, we brie y examine the existing one-factor interest rate models and estimate the parameters of Vasicek and Hull-White (Extended Vasi Models by using Turkey's term structure. Moreover, a trinomial interest rate tree is constructed to represent the evolution of Turkey's zero coupon rates.
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documentation provided by StatSoft I like this website - it says elementary but find nice exmaplanation to advanced topics to Discuss this paper
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by Raquel Medeiros Gaspar This thesis consists of two papers that study forward price term structure models. Forward prices differ from futures prices in stochastic interest rate settings and become in their own right an interesting object of study. Forward prices with different maturities are martingales under different forward measures. This mathematical property makes the term structure of forward prices always connected with the term structure of bond prices, and this dependence makes forward price terms structure models relatively harder to handle. For finite dimensional factor models, the first paper (Chapter 1) studies general quadratic term structures. These term structures include as special cases the affine term structures and the Gaussian quadratic term structures, previously studied in the literature. We show, however, that there are other, non-Gaussian, quadratic term structures and derive sufficient conditions for the existence of these general quadratic term structures for bond, futures and forward prices. We exploit the connection with the term structure of bond prices and show that even in quadratic short rate settings we can have affine term structures for forward prices. Finally, we show how the study of futures prices is naturally embedded in a study of forward prices, that the difference between the two prices has to do with the correlation between bond prices and the price process of the underlying to the forward contract and that this difference may be deterministic in some (non-trivial) stochastic interest rate settings. In the second paper (Chapter 2) we study a fairly general Wiener driven model for the term structure of forward prices. The model, under a fixed martingale measure, Q, is described by using two infinite dimensional stochastic differential equations (SDEs). The first system is a standard HJM model for (forward) interest rates, driven by a multidimensional Wiener process W. The second system is an infinite SDE for the term structure of forward prices on some specified underlying asset driven by the same W. Since the zero coupon bond volatilities will enter into the drift part of the SDE for these forward prices, the interest rate system is needed as input into the forward price system. Given this setup we use the Lie algebra methodology of Bj Discuss this paper
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Riccardo Rebonato It is shown in this paper that it is not only possible, but indeed expedient and advisable, to perform a simultaneous calibration of a log-normal BGM interest-rate model to the percentage volatilities of the individual rates and to the correlation surface. One of the contributions of the paper it to show that the task can be accomplished in two separate and independent steps: the first part of the calibration (i.e. to cap volatilities) can always be accomplished exactly thanks to straightforward geometrical relationships; the fitting to the correlation surface, thanks to a simple theorem, can then be carried out in a numerically efficient way so that the calibration to the volatilities is not spoiled by the second part of the procedure. The ability to carry out the two tasks separately greatly simplifies the overall task. Actual calculations are shown for a 3- and 4-factor implementation of the approach, and the quality of the overall agreement between the target and model correlation surfaces is commented upon. Finally, the dangers of overparametrization, i.e. of forcing (near) exact fitting to certain portions of the correlation matrix, are analysed by looking at the cases of a trigger swap, a Bermudan swaption and a oneway floater (resettable cap).
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CAROL ALEXANDER EMESE LAZAR GARCH processes constitute the major area of time series variance analysis, hence the limit of these processes is of considerable interest for continuous time volatility modelling. The continuous time limit of the GARCH(1,1) model is fundamental for limits of other GARCH processes, yet it has been the point of much debate between econometricians. The seminal work of Nelson (1990) derived the GARCH(1,1) limit as a stochastic volatility process, uncorrelated with the price process. But then a subsequent paper of Corradi (2000) that derives the limit as a deterministic volatility process and several other contradictory papers followed. We reconsider this continuous limit, arguing that because the strong GARCH model is not aggregating in time it is incorrect to consider its limit. Instead it is legitimate to use the weak definition of GARCH that is aggregating in time. This model differs from strong GARCH by defining the discrete time process on the best linear predictor of the squared errors, rather than the conditional variance itself. We prove that its continuous limit is a stochastic volatility model with correlated Brownian motions in which both the variance diffusion coefficient and the price-volatility correlation are related to the skewness and kurtosis of the physical returns density. Under certain assumptions our limit model reduces to Nelson Discuss this paper
Option pricing is one of the most important areas in financial mathematics and incorporates many different disciplines in mathematics. One such example is the use of the Fourier transform. Peter Carr and Dilip Madan in 1998 [CM98] developed it to compute the option price numerically by using the fast Fourier transform. In fact, making a change to the option price function to enforce integrability, we can calculate its Fourier transform knowing the characteristic function of the underlying asset. The Black-Scholes model can be used to determine the analytic form of the option value as a function of the strike. Therefore we can obtain the option price by giving an analytic expression of the Fourier transform of the modified value and get the price by Fourier inversion. The essay aimed at computing the option value numerically following this method and using the fast Fourier transform algorithm which is popular due to its fast calculation
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Carl Chiarella Chih-Ying Hsiao Abstract This paper studies the impact of stochastic volatility (SV) on optimal investment decisions. We consider three different SV models: the generalized Stein/Stein model, the Heston Model and an extended Heston Model with a CEV (Constant Elasticity of Variance) volatility process and derive the the long-term optimal investment strategies under each of these processes. Since the volatility is not a directly observable variable, we need to deal with a partial information problem. In constructing the optimal investment strategies based on the estimation results for the ASX 200 Index, we provide the investment recommendations calculated on a range of parameter values within the standard error bands obtained from the estimation procedure. In order to obtain the investment strategies based on the CEV volatility model, we provide a computational solution since analytical solutions are no longer available. All three investment strategies based on the three SV models suggest a positive intertemporal hedging term in addition to the static mean-variance portfolio. However, in their details the three investment strategies differ from each other. We also find that the investment strategies are quite sensitive to the CEV parameter in the extended Heston model. Key words: Asset allocation under stochastic volatility, partial information problem, extended Kalman filter, the Heston model, CEV process.
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Bernard Mawah U.U.D.M. Project Report 2007:18
Contents 1. An Introduction to Arbitrage-free pricing 1.1 What is arbitrage? 1 1.2 Prerequisites of arbitrage 1 1.3 Price convergence 1 1.4 Fundamental theorem of arbitrage 1.5 Fundamental theorem of arbitrage in a finite market 2 1.6 Options 3 2. Stochastic Calculus 4 2.1 Stochastic Processes 4 2.2 Information 5 2.3 Martingales 8 2.4 It Discuss this paper
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Floyd B. Hanson and Guoqing Yan Outline 1. Introduction. 2. Stochastic-Volatility Jump-Diffusion Model. 3. European Option Prices. 4. Computing Fourier integrals and Inverses. 5. Numerical Results for Call and Put Options. 6. Conclusions. Discuss this paper
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Peter Carr and Dilip B. Madan This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically.
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Sergiy Butenko Alexander Golodnikov Stanislav Uryasevy Abstract. This paper develops trading strategies for liquidation of a financial security which maximize the expected return. The problem is formulated as a stochastic programming problem, which utilizes the scenario representation of possible returns. Two cases are considered, a case with no constraint on risk and a case when the risk of losses associated with trading strategy is constrained by Conditional Value-at-Risk (CVaR) measure. In the first case, two algorithms are proposed; one is based on linear programming techniques, and the other uses dynamic programming to solve the formulated stochastic program. The third proposed algorithm is obtained by adding the risk constraints to the linear program. The algorithms provide path-dependent strategies which sell some fractions of security depending upon price sample-path of security up to the current moment. The performance of the considered approaches is tested using a set of historical sample-paths of prices.
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GUR HUBERMAN WERNER STANZL Abstract: We study optimal liquidity trading in a framework where trade size has a price impact. A liquidity trader wishes to trade a fixed number of shares within a certain time horizon and to minimize the mean and variance of the costs of trading. Explicit formulas for the optimal trading strategies show that risk-averse liquidity traders reduce their order sizes over time and execute a higher fraction of their total trading volume in early periods when price volatility increases or price sensitivity decreases. In the presence of transaction fees, numerical simulations suggest that traders want to trade more frequently when price volatility or price sensitivity goes up. In the multi-asset case, price effects across assets have a substantial impact on trading behavior, as does continuous-time trading Discuss this paper
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Dimitris Bertsimasy, Paul Hummelz, and Andrew W. Lox
Abstract We derive dynamic optimal trading strategies that minimize the expected cost of trading blocks of securities over a xed time horizon. Given xed blocks si of shares of stock i to be traded within a nite number of periods T, i = 1; : : : ; n, and given price-impact functions that yield the execution price of an individual trade as a function of the shares of stock i traded and current market conditions, we obtain the optimal sequence of trades as a function of market conditions|closed-form expressions in some cases|that minimizes the expected cost of executing si within T periods. We also propose an approximation algorithm for incorporating constraints, a particularly important extension in practice. To illustrate the practical relevance of our methods, we apply them to a hypothetical portfolio of 25 stocks by estimating their price-impact functions using historical trade data from 1996 and deriving the optimal execution strategies for the portfolio, and by performing several Monte Carlo simulation experiments.
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Abstract: In this paper, we assume that log returns can be modelled by a Levy process. We give explicit formulae for option prices by means of the Fourier transform. We explain how to infer the characteristics of the Levy process from option prices. This enables us to generate an implicit volatility surface implied by market data. This model is of particular interest since it extends the seminal Black Scholes [1973] model consistently with volatility smile.
Keywords: Levy process, Fourier and Laplace transform, Smile
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Robert Almgreny and Neil Chriss Abstract We consider the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. For a simple linear cost model, we explicitly construct the efficient frontier in the space of time-dependent liquidation strategies, which have minimum expected cost for a given level of uncertainty. We may then select optimal strategies either by minimizing a quadratic utility function, or by minimizing Value at Risk. The latter choice leads to the concept of Liquidity-adjusted VAR, or L-VaR, that explicitly considers the best tradeoff between volatility risk and liquidation costs.
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Andreas Heigl Abstract This master thesis describes how to price options by means of Genetic Programming. The underlying model is the Generalized Autoregressive Conditional Heteroskedastic (GARCH) asset return process. The goal of this master thesis is to nd a closed-form solution for the price of European call options where the underlying securities follow a GARCH process. The data are simulated over a wide range to cover a lot of existing options in one single equation. Genetic Programming is used to generate the pricing function from the data. Genetic Programming is a method of producing programs just by de ning a problemdependent tness function. The resulting equation is found via a heuristic algorithm inspired by natural evolution. Three di erent methods of bloat control are used. Additionally Automatic De ned Functions (ADFs) and a hybrid approach are tested, too. To ensure that a good con guration setting is used, preliminary testing of many di erent settings has been done, suggesting that simpler con gurations are more successful in this environment. The resulting equation can be used to calculate the price of an option in the given range with minimal errors. This equation is well behaved and can be used in standard spread sheet programs. It o ers a wider range of utilization or a higher accuracy, respectively than other existing approaches.
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Christian Schmitt Abstract Various empirid studies have shown that the time-varying volatility of asset returns can be described by GARCH (generalized autoregressive conditional heteroskedasticity) models. The corresponding GARCH option pricing model of Duan (1995) is capable of depicting the "smile-effect" which often can be found in option prices. In some derivative markets, however, the slope of the smile is not symmetrical. In this paper an option pricing model in the context of the EGARCH (Exponential GARCH) process will be developed. Extensive numerical analyses suggest that the EGARCH option pricing model is able to explain the different slopes of the smile curve.
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Abstract There is extensive empirical evidence that index option prices systematically differ from Black- Scholes prices. Out-of-the-money put prices (and in-the-money call prices) are relatively high compared to the Black-Scholes price. Motivated by these empirical facts, we develop a new discretetime dynamic model of stock returns with Inverse Gaussian innovations. The model allows for conditional skewness as well as conditional heteroskedasticity and a leverage effect. We present an analytic option pricing formula consistent with this stock return dynamic. An extensive empirical test of the model using S&P500 index options shows that the new Inverse Gaussian GARCH model Discuss this paper
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Chun-Yang Liu Abstract The GARCH model has been successful in describing the volatility dynamics of asset return series. However, tree-based GARCH option pricing algorithms su Discuss this paper
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Abstract We derived a new analytic approximation formula based on the asymptotic expansion approach for pricing bond options in an HJM model. We also developed a variance reduction method for Monte Carlo simulations utilizing asymptotic expansion, and examined its accuracy and confirmed its validity in a realistic two-factor model.
Abstract The 1993 Laplace transform approach of Geman and Yor is a celebrated advance in valuing Asian options. Its insights are fundamental from both a mathematical and a financial perspective. In this paper, we discuss two observations regarding the financial relevance of its results. First, we show that the Geman and Yor Laplace transform is not that of an Asian option price, as reported in Geman and Yor and other papers. We nonetheless show how the Geman and Yor Laplace transform can be used to obtain the price of an Asian option. Second, we find that following Geman and Yor these Laplace transfoms are available only if the risk-neutral drift is not less than half the squared volatility. Using complex analytic techniques, we lift this restriction, thus extending the financial applicability of the Laplace transform approach Discuss this paper
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