Estimation of option implied surfaces that are consistent with observed market prices and stable over time is a fundamental problem in finance. This paper develops a general optimization based framework for estimation of the option implied risk-neutral density (RND) surface, while satisfying no-arbitrage constraints. Our developed framework considers all types of realistic surfaces and is hence not constrained to a certain function class. When solving the problem the RND is discretized, which leads to an optimization model where it is possible to formulate the constraints as linear constraints, making the resulting large-scale optimization problem convex and the solution a global optimum. This is a major advantage of our method compared to most estimation algorithms described in the literature, which are typically cast as non-convex optimization problems with multiple local optima. We show that our method produces smooth local volatility surfaces that can be used for pricing and hedging of exotic derivatives. The stability of our method is demonstrated through a time series study based on historical prices of S&P 500 index options.
Hedging of an option book in an incomplete market with transaction costs is an important problem in finance that many banks have to solve on a daily basis. In this paper we develop a stochastic programming (SP) model for the hedging problem in a realistic setting, where all transactions take place at observed bid and ask prices. The SP model relies on a realistic modelling of the important risk factors for the application, the price of the underlying security and the volatility surface. The volatility surface is unobservable and must be estimated from a cross-section of observed option quotes that contain noise and possibly arbitrage. In order to produce arbitrage-free volatility surfaces with high quality as input to the SP model a novel non-parametric estimation method is used. The dimension of the volatility surface is infinite and in order to be able solve the problem numerically we use discretization and PCA to reduce the dimensions of the problem. Testing the model out-of-sample for options on the Swedish OMXS30 index, we show that the SP model is able to produce a hedge that has both a lower realized risk and cost compared with dynamic delta and delta-vega hedging strategies.
Hedging of an option book in an incomplete market with transaction costs is an important problem in finance that many banks have to solve on a daily basis. In this paper, we develop a stochastic programming (SP) model for the hedging problem in a realistic setting, where all transactions take place at observed bid and ask prices. The SP model relies on a realistic modeling of the important risk factors for the application, the price of the underlying security and the volatility surface. The volatility surface is unobservable and must be estimated from a cross section of observed option quotes that contain noise and possibly arbitrage. In order to produce arbitrage-free volatility surfaces of high quality as input to the SP model, a novel non-parametric estimation method is used. The dimension of the volatility surface is infinite and in order to be able solve the problem numerically, we use discretization and principal component analysis to reduce the dimensions of the problem. Testing the model out-of-sample for options on the Swedish OMXS30 index, we show that the SP model is able to produce a hedge that has both a lower realized risk and cost compared with dynamic delta and delta-vega hedging strategies.
In this paper we develop a general optimization based framework for estimation of the option implied local variance surface. Given a specific level of consistency with observed market prices there exist an infinite number of possible surfaces. Instead of assuming shape constraints for the surface, as in many traditional methods, we seek the solution in the subset of realistic surfaces. We select local volatilities as variables in the optimization problem since it makes it easy to ensure absence of arbitrage, and realistic local volatilities imply realistic risk-neutral density- (RND), implied volatility- and price surfaces. The objective function combines a measure of consistency with market prices, and a weighted integral of the squared second derivatives of local volatility in the strike and the time-to-maturity direction. Derivatives prices in the optimization model are calculated efficiently with a finite difference scheme on a non-uniform grid. The framework has previously been successfully applied to the estimation of RND surfaces. Compared to when modeling the RND, it is for local volatility much easier to choose the parameters in the model. Modeling the RND produces a convex optimization problem which is not the case when modeling local volatility, but empirical tests indicate that the solution does not get stuck in local optima. We show that our method produces local volatility surfaces with very high quality and which are consistent with observed option quotes. Thus, unlike many methods described in the literature, our method does not produce a local volatility surface with irregular shape and many spikes or a non-smooth and multimodal RND for input data with a lot of noise.
In this paper we develop a general optimization based framework for estimation of the option implied local volatility surface. We show that our method produces local volatility surfaces with very high quality and which are consistent with observed S&P 500 index option quotes. Thus, unlike many methods described in the literature, our method does not produce a local volatility surface with irregular shape and many spikes for input data which contains a lot of noise. Through a time series study we show that our optimization based framework produces squared local volatility surfaces that are stable over time. Given a specic level of consistency with observed market prices there exist an innite number of possible surfaces. Instead of assuming shape constraints for the surface, as in many traditional methods, we seek the solution in the subset of realistic surfaces. We select squared local volatilities as variables in the optimization problem since it makes it easy to ensure absence of arbitrage, and realistic local volatilities imply realistic risk-neutral density- , implied volatility- and price surfaces. The objective function combines a measure of consistency with market prices, and a weighted integral of the squared second derivatives of local volatility in the strike and the time-to-maturity direction. Derivatives prices in the optimization model are calculated efficiently with a finite difference scheme on a non-uniform grid. The resulting optimization problem is non-convex, but extensive empirical tests indicate that the solution does not get stuck in local optima.
Accurate pricing of exotic or illiquid derivatives which is consistent with noisy market prices presents a major challenge. The pricing accuracy will crucially depend on using arbitrage free inputs to the pricing engine. This paper develops a general optimization based framework for estimation of the option implied risk-neutral density (RND), while satisfying no-arbitrage constraints. Our developed framework is a generalization of the RNDs implied by existing parametric models such as the Heston model. Thus, the method considers all types of realistic surfaces and is hence not constrained to a certain function class. When solving the problem the RND is discretized making it possible to use general purpose optimization algorithms. The approach leads to an optimization model where it is possible to formulate the constraints as linear constraints making the resulting optimization problem convex. We show that our method produces smooth local volatility surfaces that can be used for pricing and hedging of exotic derivatives. By perturbing input data with random errors we demonstrate that our method gives better results than the Heston model in terms of yielding stable RNDs.
In this paper we evaluate the density forecasts obtained from a cross-section of S&P 500 index option prices. The option implied density forecasts rely on a result derived by Heath and Platen (2006), which under certain assumptions allows us to transform risk-neutral densities into real-world densities. In order to remove liquidity premia from the real-world densities we use a transformation into densities implied by the Minimal Market Model. The accuracy of the estimated real-world density forecasts relies on using a recently developed method for estimation of risk-neutral densities of high quality. We find that our recovered real-world densities explains the realized return distribution for S&P 500 better than historical GARCH densities for a forecasting horizon of two days. This can be contrasted to the findings in two recent papers in the literature, who find that historical densities estimated from intra-day data performs as least as well as option implied densities for a forecasting horizon of one day.
In this paper we develop a methodology for simultaneous recovery of the real-world probability density and liquidity premia from observed S&P500 index option prices. Assuming the existence of a numeraire portfolio for the US equity market, fair prices of derivatives under the benchmark approach can be obtained directly under the real-world measure. Under this modeling framework there exists a direct link between observed call option prices on the index and the real-world density for the underlying index. We use a novel method for estimation of option implied volatility surfaces of high quality which enables the subsequent analysis. We show that the real-world density that we recover is consistent with the observed realized dynamics of the underlying index. This admits the identication of liquidity premia embedded in option price data. We identify and estimate two separate liquidity premia embedded in S&P500 index options that are consistent with previous findings in the literature.
In [Euro. J. Operat. Res. 143 (2002) 452, Opt. Meth. Software 17 (2002) 383] a Riccati-based primal interior point method for multistage stochastic programmes was developed. This algorithm has several interesting features. It can solve problems with a nonlinear node-separable convex objective, local linear constraints and global linear constraints. This paper demonstrates that the algorithm can be efficiently parallelized. The solution procedure in the algorithm allows for a simple but efficient method to distribute the computations. The parallel algorithm has been implemented on a low-budget parallel computer, where we experience almost perfect linear speedup and very good scalability of the algorithm. © 2003 Elsevier Science B.V. All rights reserved.
We develop a stochastic programming framework for hedging currency and interest rate risk, with market traded currency forward contracts and interest rate swaps, in an environment with uncertain cash flows. The framework captures the skewness and kurtosis in exchange rates, transaction costs, the systematic risks in interest rates, and most importantly, the term premia which determine the expected cost of different hedging instruments. Given three commonly used objective functions: variance, expected shortfall, and mean log profits, we study properties of the optimal hedge. We find that the choice of objective function can have a substantial effect on the resulting hedge in terms of the portfolio composition, the resulting risk and the hedging cost. Further, we find that unless the objective is indifferent to hedging costs, term premia in the different markets, along with transaction costs, are fundamental determinants of the optimal hedge. Our results also show that to reduce risk properly and to keep hedging costs low, a rich-enough universe of hedging instruments is critical. Through out-of-sample testing we validate the findings of the in-sample analysis, and importantly, we show that the model is robust enough to be used on real market data. The proposed framework offers great flexibility regarding the distributional assumptions of the underlying risk factors and the types of hedging instruments which can be included in the optimization model.
We show that traditional data sets for the U.S. Treasury yield curves contain large amounts of noise, in e.g. the Fama-Bliss discount file already the second factor loading for innovations in forward rates is a consequence of noise. We implement the quadratic and cubic McCulloch splines, Nelson-Siegel and Svensson models and compare these traditional models with a recently developed generalized optimization framework using daily CRSP data from 1961 to 2011. In out-of-sample tests, it is shown that the generalized optimization framework produces smaller pricing errors compared to the traditional methods. Factor loadings from the generalized optimization framework show that the short and long end of the forward rate curve move independently, where principal component 1-3 explain the long end, and subsequent principal components explain the short end. This is consistent with the behavior of the market where short rates are governed by central bank while long rates are dependent on e.g. the expectation of future inflation.
We propose a new method for certain multistage stochastic programs with linear or nonlinear objective function, combining a primal interior point approach with a linear-quadratic control problem over the scenario tree. The latter problem, which is the direction finding problem for the barrier subproblem is solved through dynamic programming using Riccati equations. In this way we combine the low iteration count of interior point methods with an efficient solver for the subproblems. The computational results are promising. We have solved a financial problem with 1,000,000 scenarios, 15,777,740 variables and 16,888,850 constraints in 20 hours on a moderate computer. © 2002 Elsevier Science B.V. All rights reserved.
We show that a Riccati-based Multistage Stochastic Programming solver for problems with separable convex linear/nonlinear objective developed in previous papers can be extended to solve more general Stochastic Programming problems. With a Lagrangean relaxation approach, also local and global equality constraints can be handled by the Riccati-based primal interior point solver. The efficiency of the approach is demonstrated on a 10 staged stochastic programming problem containing both local and global equality constraints. The problem has 1.9 million scenarios, 67 million variables and 119 million constraints, and was solved in 97 min on a 32 node PC cluster.
We build an investment model based on Stochastic Programming. In the model we buy at the ask price and sell at the bid price. We apply the model to a case where we can invest in a Swedish stock index, call options on the index and the risk-free asset. By reoptimizing the portfolio on a daily basis over a ten-year period, it is shown that options can be used to create a portfolio that outperforms the index. With ex post analysis, it is furthermore shown that we can create a portfolio that dominates the index in terms of mean and variance, i.e. at given level of risk we could have achieved a higher return using options.
In this paper we present a nonlinear dynamic programming algorithm for the computation of forward rates within the maximum smoothness framework. The algorithm implements the forward rate positivity constraint for a one-parametric family of smoothness measures and it handles price spreads in the constraining data set. We investigate the outcome of the algorithm using the Swedish Bond market showing examples where the absence of the positive constraint leads to negative interest rates. Furthermore we investigate the predictive accuracy of the algorithm as we move along the family of smoothness measures. Among other things we observe that the inclusion of spreads not only improves the smoothness of forward curves but also significantly reduces the predictive error.
Traditional methods for estimating yield curves are special cases of a generalized optimization framework. For pricing out-of-sample in both the Swedish and U.S. interest rate swap (IRS) markets, it is shown that the framework dominates or is close to dominating the traditional methods in the comparison by first order stochastic dominance. When measuring the perceived variance for each traditional method, it is shown that, for the same level of market consistency, the framework produces lower variance. For these new yield curves, PCA of innovations in forward rates shows that the first three loadings (shift, twist and butterfly) do not explain movements in the short end, and that the subsequent loadings explain uncorrelated movements in the short end.
After the credit crunch which started in 2007, significant basis spreads for exchanging floating payments of different tenors appeared. To deal with the problem, multiple yield curves estimation methods have been suggested. In this paper, a generalized optimization framework is extended to a multiple yield curve framework. As has been observed by practitioners, extending traditional cubic splines to multiple yield curves, though consistent with the market prices, does not provide smooth and realistic yield curves. When the parameters in the generalized optimization framework are selected to exactly match market prices, the yield curves are much more realistic, but small waves still remain due to noise in the input data. To avoid having a rough yield curve, we also study the least squares parameter setting in the generalized optimization framework. This method gives much smoother and more realistic yield curves with adjustments to market prices that are less than 0.2 basis points. When exact traditional methods are extended to estimate multiple yield curves, then even tiny pricing errors can cause a situation where the shape constraints prevent the method from finding realistic yield curves.
The vast size of real world stochastic programming instances requires sampling to make them practically solvable. In this paper we extend the understanding of how sampling affects the solution quality of multistage stochastic programming problems. We present a new heuristic for determining good feasible solutions for a multistage decision problem. For power and log-utility functions we address the question of how tree structures, number of stages, number of outcomes and number of assets affect the solution quality. We also present a new method for evaluating the quality of first stage decisions.
We study methods to simulate term structures in order to measure interest rate risk more accurately. We use principal component analysis of term structure innovations to identify risk factors and we model their univariate distribution using GARCH-models with Student’s t-distributions in order to handle heteroscedasticity and fat tails. We find that the Student’s t-copula is most suitable to model co-dependence of these univariate risk factors. We aim to develop a model that provides low ex-ante risk measures, while having accurate representations of the ex-post realized risk. By utilizing a more accurate term structure estimation method, our proposed model is less sensitive to measurement noise compared to traditional models. We perform an out-of-sample test for the U.S. market between 2002 and 2017 by valuing a portfolio consisting of interest rate derivatives. We find that ex-ante Value at Risk measurements can be substantially reduced for all confidence levels above 95%, compared to the traditional models. We find that that the realized portfolio tail losses accurately conform to the ex-ante measurement for daily returns, while traditional methods overestimate, or in some cases even underestimate the risk ex-post. Due to noise inherent in the term structure measurements, we find that all models overestimate the risk for 10-day and quarterly returns, but that our proposed model provides the by far lowest Value at Risk measures.
A good estimation of expected returns is imperative when optimal investments are determined with Stochastic Programming model. However, existing Stochastic Programming models do not include a model for the time-varying term premium. In this paper Duffee’s essentially affine model is used to capture the randomness in interest rates and the time-varying term premium. To determine optimal investments, a two-stage Stochastic Programming model without recourse is proposed which models borrowing, shorting and proportional transaction costs. The proposed model is evaluated over the period 1961-2011 and the Sharpe ratio is better than the one’s that corresponds to the market index, and Jensen’s alpha is positive.