Vanna-Volga methods applied to FX derivatives : from theory to market practice

We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black-Scholes price through the so-called `probability of survival' and the `expected first exit time'. Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data

Analytically tractable stochastic volatility models in asset and option pricing

Questa tesi si compone di quattro saggi sui modelli stocastici di volatilità in asset e option pricing. Più precisamente, questa tesi si concentra sul tasso di interesse stocastico e sui modelli di volatilità stocastica multiscala, con applicazioni in vari prodotti finanziari. Nel primo saggio viene presentato un modello ibrido Heston-CIR (HCIR) con un tasso di interesse stocastico. In questo saggio, sono state dedotte formule elementari esplicite per i momenti relativi alle distribuzioni dei prezzi delle azioni, nonché formule efficaci per approssimare i prezzi delle opzioni. Utilizzando i prezzi di call e put options europei sul l'indice S&P 500 americano, questo studio empirico dimostra che il modello HCIR è migliore del modello di Heston nell'interpretazione e nella previsione dei prezzi di call e put options. Il secondo saggio è un ulteriore estensione del modello HCIR con due diverse applicazioni. La prima applicazione utilizza il modello HCIR per interpretare la struttura a termine dei rendimenti e per prevedere la loro tendenza al rialzo /ribasso. La seconda analisi è basata sui valori della “long-term health endowment policy”. L'analisi empirica mostra che il tasso di interesse stocastico gioca un ruolo cruciale come fattore di volatilità e fornisce un modello multi-fattore che supera il modello di Heston nel predire prezzo della polizza assicurativa sanitaria. Nel terzo saggio è stato proposto un modello ibrido di tipo Heston Hull-White (HHW) per descrivere le dinamiche di un prezzo di asset con volatilità e tasso di interesse stocastici che tiene conto di valori negativi. Nel lavoro sono state dedotte sia formule elementari esplicite per la funzione di densità di probabilità di transizione della variabile prezzi degli asset che formule in forma chiusa per stimare il prezzo delle opzioni. Nella prima analisi empirica è stato calibrato il modello HHW utilizzando la cosiddetta implied volatility. La seconda analisi empirica si concentra sui prezzi di futures Eurodollar e i corrispondenti prezzi delle opzioni europee con una generalizzazione del modello di Heston con tassi di interesse stocastici. I risultati relativi all’approssimazione e alla previsione sono molto interessanti. Ciò conferma l'efficienza del modello di HHW e la necessità di considerare possibili valori negativi di tasso di interesse. Il quarto saggio descrive un modello di Heston ibrido e multiscala per il tasso di cambio a pronti FX per poter considerare tassi di interesse stocastici. Il trattamento analitico del modello è descritto in dettaglio sia considerando delle misure legate alla fisica che misura di neutralità al rischio. In particolare, è stata derivata una formula per la funzione di densità di transizione di probabilità usando un integrale a una dimensione di una funzione integrale elementare che viene utilizzato per il prezzaggio delle opzioni call e put European Vanilla.This dissertation consists of four related essays on stochastic volatility models in asset and option pricing. More precisely, this dissertation focuses on stochastic interest rate and multiscale stochastic volatility models, with applications in various financial products. In first essay, a hybrid Heston-CIR (HCIR) model with a stochastic interest rate process is presented. In this essay, explicit elementary formulas for the moments of the asset price variables as well as efficient formulas to approximate the option prices are deduced. Using European call and put option prices on U.S. S&P 500 index, empirical study shows that the HCIR model outperforms Heston model in interpreting and predicting both call and put option prices. The second essay is a further extension of the HCIR model with two different applications. The first application is using HCIR model to interpret bond yield term structure and to forecast their upward/downward trend. The second analysis is based on the values of the long-term health endowment policy. The empirical analysis shows that the stochastic interest rate plays a crucial role as a volatility factor and provides a multi-factor model that outperforms the Heston model in predicting health endowment policy price. In the third essay, a hybrid Heston Hull-White (HHW) model is designed to describe the dynamics of an asset price under stochastic volatility and interest rate that allows negative values. Explicit elementary formulas for the transition probability density function of the asset price variable and closed-form formulas to approximate the option prices are deduced. In first empirical analysis, the HHW model is calibrated by using implied volatility. The second empirical analysis focuses on the Eurodollar futures prices and the corresponding European options prices with a generalization of the Heston model in the stochastic interest rate framework. Both the results are impressive for approximation and prediction. This confirms the efficiency of HHW model and the necessary to allow for negative values of interest rate. The fourth essay describes a multiscale hybrid Heston model of the spot FX rate which is an extension of the model De Col, Gnoatto and Grasselli 2013 in order to allow stochastic interest rate. The analytical treatment of the model is described in detail both under physical measure and risk neutral measure. In particular, a formula for the transition probability density function is derived as a one dimensional integral of an elementary integral function which is used to price European Vanilla call and put options

Prudent Valuation & Model Risk Quantification

This paper is a master's thesis by Erik Kivilo and Carl Olofsson at the Faculty of Engineering (LTH) at Lund University. The research and writing took place during the fall of 2014 under the supervision of Magnus Wiktorsson at the Division of Mathematical Statistics at Lund University, and Per Thastrom at EY, Copenhagen. The thesis concerns prudent valuation of fair-valued nancial instruments under current regulation on capital requirements for credit institutions and investment rms within the EU. The aim is to explain the concept of prudent valuation, and develop statistical methods for the calculation of additional valuation adjustments (AVAs) required by the regulations. As there has been little focus on model risk in previous regulations, the main objective is to quantify model risk AVA in a way that is compliant, using current research on prudent valuation and model risk. The method suggested in this paper captures the instantaneous valuation uncertainty related to model risk as dened in the regulation. The rst step of the method is to dene a group of plausible models and calibration approaches for the instrument type. Each combination of model and calibration is then assigned a probability weight based on the number of parameters in the model, and a measure of t that includes all available market data. When prices for the instrument have been calculated for all the dierent models and calibrations, the probability weights are used to form a cumulative price probability distribution for the instrument. The method is to our understanding in line with the current regulation, and should according to us hold some advantages compared to other proposed methods

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

Mathematics of Quantitative Finance

The workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants

An Empirical Analysis of European Credit Default Swap Spread Dynamics

I analyze the dynamics of European credit default swap spreads by estimating CDS spreads via an extension of the structural credit risk models by Black and Cox (1976) as well as Leland (1994), the so called CreditGrades model proposed by Finger et al. (2002). Using two different procedures in approximating the asset volatility surface of obligors, the models are calibrated by means of historical equity volatility and volatility extracted out of at-the-money options. I discover that model performance strongly depends on the distribution of input parameters clustered by economical sectors. Model spreads exhibit significant correlation with market spreads and seem to predict market spreads contingent on sectors and model calibration techniques. The gap between model and market spreads, derived model spreads and empirical market spreads are analyzed by running panel regressions in fashion of Collin-Dufresn et al. (2001) and Bedendo et al. (2011). These show that times of disconnectedness between credit and equity markets, model inherent misspecifications as well as possible market inefficiencies can contribute to the inability to estimate spreads reliably. Robustness checks show that determinants of gap, model and market spreads are sector specific, time varying and tenor dependent. Keywords: Credit Risk; Credit Risk Modelling; Structural Models; Credit Risk Management; Quantitative Finance.I analyze the dynamics of European credit default swap spreads by estimating CDS spreads via an extension of the structural credit risk models by Black and Cox (1976) as well as Leland (1994), the so called CreditGrades model proposed by Finger et al. (2002). Using two different procedures in approximating the asset volatility surface of obligors, the models are calibrated by means of historical equity volatility and volatility extracted out of at-the-money options. I discover that model performance strongly depends on the distribution of input parameters clustered by economical sectors. Model spreads exhibit significant correlation with market spreads and seem to predict market spreads contingent on sectors and model calibration techniques. The gap between model and market spreads, derived model spreads and empirical market spreads are analyzed by running panel regressions in fashion of Collin-Dufresn et al. (2001) and Bedendo et al. (2011). These show that times of disconnectedness between credit and equity markets, model inherent misspecifications as well as possible market inefficiencies can contribute to the inability to estimate spreads reliably. Robustness checks show that determinants of gap, model and market spreads are sector specific, time varying and tenor dependent. Keywords: Credit Risk; Credit Risk Modelling; Structural Models; Credit Risk Management; Quantitative Finance

Valuation of Convertible Bonds

Convertible bonds are hybrid financial instruments with complex features. They have characteristics of both debts and equities, and usually several options are embedded in this kind of contracts. The optimality of the conversion decision depends on equity price, interest rate and default probability of the issuer. The decision making can be further complicated by the fact that most convertible bonds have call provisions allowing the bond issuer to call back the bond at a predetermined call price. In this thesis, we first adopt a structural approach where the Vasicek-model is applied to incorporate interest rate risk into the firm's value process which follows a geometric Brownian motion. Default is triggered when the firm's value hits a lower boundary. The complex nature of the firm's capital structure and information asymmetry may make it hard to model the firm's value and the capital structure. In this case, reduced-form models are applied for the study of convertible bonds. We adopt a parsimonious, intensity-based default model, in which the default intensity is modeled as a function of the pre-default stock price. We first analyze the contract features of the convertible bonds and show that callable and convertible bonds can be decomposed into a straight bond and a game option component. Then the no-arbitrage prices of the European- and American-style callable and convertible bonds are derived. In American-style contracts, the focus is on the analysis of the strategic optimal behavior. The bondholder and issuer choose their stopping times to maximize or minimize the expected payoff respectively. For the bondholder it is optimal to select the stopping time which maximizes the expected payoff given the minimizing strategy of the issuer, while the issuer will choose the stopping time that minimizes the expected payoff given the maximizing strategy of the bondholder. The no-arbitrage price can be approximated numerically by means of backward induction. In the structural model, the recursion is carried out alongside a recombining binomial tree. Whereas in the reduced-form approach, the optimization problem is formulated and solved with the help of the theory of doubly reflected backward stochastic differential equations. In practice, it is often a difficult problem to calibrate a given model to the available data. Determining the volatility of the firm's value process or stock price process is not a trivial problem. We therefore assume that the volatility of the firm's value process/stock price process lies between two extreme values, and combine it with the results on game option in incomplete market to derive certain pricing bounds for callable and convertible bonds. The maximizing strategy of the bondholder and the choice of the most pessimistic pricing measure from his perspective determine the lower bound of the no-arbitrage price. Whereas the minimizing strategy of the issuer and the most pessimistic expectation from his aspect construct the upper bound of the no-arbitrage price. Numerically, to make the computation tractable a constant interest rate is assumed. The pricing bounds can be calculated with recursion alongside a recombining trinomial tree or with the finite-difference method

Statistical hedging with neural networks

This thesis investigates the problem of statistical hedging with artificial neural networks (ANNs). The statistical hedging is a data-driven approach that derives hedging strategy from data and hence does not rely on making assumptions of the underlying asset. Consider an investor who sells an option and wishes to hedge it with some amount of underlying asset. ANNs can be used to determine this number by minimising the discrete hedging error. In the first chapter, we provide a comprehensive literature review of papers on the topic of using ANNs for option pricing and hedging, as well as other related ones. Based on our research experience and summary of papers, we provide several advices that we believe are critical in using ANNs for option pricing and hedging problem. In particular, we point out an existing information leakage issue in the literature when preparing data. This review is invaluable for future researchers who are wish to work in this topic. In the second chapter, we consider the hedging problem in the single period case. The ANN is designed to output a hedging ratio directly, instead of first learning to prices. The experiments are taken on simulated Black-Scholes (BS), Heston, end-of-day S&P 500, and tick Euro Stoxx 50 datasets. The results show the ANN can significantly outperform the BS benchmark, but is only comparable to linear regressions on sensitivities. Hence, we illustrate that the edge of the two statistical hedging methods arises mainly from the existence of the leverage effect. Moreover, the information leakage found in the literature is reproduced. It’s shown that a wrong in- and out-of-sample split can overestimate the performance of statistical hedging methods. This leakage can be further exploited by tagging independent variables. Building on the previous chapter, sensitivity analysis are given in the third chapter. They concern data cleaning on the two historical datasets, different simulation parameters of the two simulated datasets, and data preparations. In particular, we show that the statistical hedging methods can also exploit drift and convexity apart from the leverage effect. In the last chapter, our model is extended to multiple periods on the Black-Scholes data. The replicating portfolio is rebalanced with a fixed frequency over the option’s life. We show again the ANN and linear regression methods outperform the BS benchmark, and their performance are comparable

A High-Low Model of Daily Stock Price Ranges

We observe that daily highs and lows of stock prices do not diverge over time and, hence, adopt the cointegration concept and the related vector error correction model (VECM) to model the daily high, the daily low, and the associated daily range data. The in-sample results attest the importance of incorporating high-low interactions in modeling the range variable. In evaluating the out-of-sample forecast performance using both mean-squared forecast error and direction of change criteria, it is found that the VECM-based low and high forecasts offer some advantages over some alternative forecasts. The VECM-based range forecasts, on the other hand, do not always dominate –the forecast rankings depend on the choice of evaluation criterion and the variables being forecasted.daily high, daily low, VECM model, forecast performance, implied volatility