205 research outputs found

    Pricing swing options and other electricity derivatives

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    The deregulation of regional electricity markets has led to more competitive prices but also higher uncertainty in the future electricity price development. Most markets exhibit high volatilities and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. A good understanding of the stochastic price dynamics is required for the purposes of risk management and pricing derivatives. In this thesis we examine a simple spot price model which is the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of this spot price process at maturity T. With some restrictions on the set of possible martingale measures we show that the risk neutral dynamics remains within the class of considered models and hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulas for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process. Further contributions of this thesis include a short discussion of interpolation methods to generate a continuous forward curve based on the forward contracts with delivery periods observed in the market, and an investigation into optimal martingale measures in incomplete markets. In particular we present known results of q-optimal martingale measures in the setting of a stochastic volatility model and give a first indication of how to determine the q-optimal measure for q=0 in an exponential Ornstein-Uhlenbeck model consistent with a given forward curve

    Stock Price Dynamics and Option Valuations under Volatility Feedback Effect

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    According to the volatility feedback effect, an unexpected increase in squared volatility leads to an immediate decline in the price-dividend ratio. In this paper, we consider the properties of stock price dynamics and option valuations under the volatility feedback effect by modeling the joint dynamics of stock price, dividends, and volatility in continuous time. Most importantly, our model predicts the negative effect of an increase in squared return volatility on the value of deep-in-the-money call options and, furthermore, attempts to explain the volatility puzzle. We theoretically demonstrate a mechanism by which the market price of diffusion return risk, or an equity risk-premium, affects option prices and empirically illustrate how to identify that mechanism using forward-looking information on option contracts. Our theoretical and empirical results support the relevance of the volatility feedback effect. Overall, the results indicate that the prevailing practice of ignoring the time-varying dividend yield in option pricing can lead to oversimplification of the stock market dynamics.Comment: 23 pages, 7 figures, 2 table

    Parametric and Nonparametric Volatility Measurement

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    Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. We define three different volatility concepts: (i) the notional volatility corresponding to the ex-post sample-path return variability over a fixed time interval, (ii) the ex-ante expected volatility over a fixed time interval, and (iii) the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete- and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently-popularized realized volatility measures for (non-trivial) fixed-length time intervals.

    Stochastic modelling in volatility and its applications in derivatives

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    This thesis consists of three articles concentrating on modelling stochastic volatility in commodity as well as equity and applying stochastic volatility models to evaluate financial derivatives and real options. Firstly, we introduce the general background and the incentive of considering stochastic volatility models. In Chapter 2 we derive tractable analytic solutions for futures and options prices for a linear-quadratic jump-diffusion model with seasonal adjustments in stochastic volatility and convenience yield. We then calibrate our model to data from the fish pool futures market, using the extended Kalman filter and a quasi-maximum likelihood estimator and alternatively using an implied-state quasi-maximum likelihood estimator. We find no statistical evidence of jumps. However, we do find evidence for the positive correlation between salmon spot prices and volatility, seasonality in volatility and convenience yield. In addition we observe a positive relationship between seasonal risk premium and uncertainty within the EU salmon demand. We further show that our model produces option prices that are conform with the observation of implied volatility smiles and skews. In Chapter 3, we introduce a linear quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply general method of moments (GMM) and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear quadratic volatility process and a Lévy-driven Ornstein-Uhlenbeck process. We compare the quality of fit for several mod- els, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normal distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. A Monte Carlo experiments is employed to confirm this. Finally, in Chapter 4, we study the optimal stopping problems in the context of American options with stochastic volatility models and the optimal fish harvesting decision with stochastic convenience yield models, in the presence of drift ambiguity. From the perspective of an ambiguity averse agent, we transfer the problem to the solution of a reflected backward stochastic differential equation (RBSDE) and prove the uniform Lipschitz continuity of the generator. We then propose a numerical algorithm with the theory of RBSDEs and a general stratification technique, and an alternative algorithm without using the theory of RBSDEs. We test the accuracy and convergence of the numerical schemes. By comparing to the one dimensional case, we highlight the importance of the dynamic structure of the agent’s worst case belief. Results also show that the numerical RBSDE algorithm with stratification is more efficient when the optimal generator is attainable

    Parametric and Nonparametric Volatility Measurement

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    Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. We define three different volatility concepts: (i) the notional volatility corresponding to the ex-post sample-path return variability over a fixed time interval, (ii) the ex-ante expected volatility over a fixed time interval, and (iii) the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete- and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently-popularized realized volatility measures for (non-trivial) fixed-length time intervals.

    Topics in volatility models

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    In this thesis I will present my PhD research work, focusing mainly on financial modelling of asset’s volatility and the pricing of contingent claims (financial derivatives), which consists of four topics: 1. Several changing volatility models are introduced and the pricing of European options is derived under these models; 2. A general local stochastic volatility model with stochastic interest rates (IR) is studied in the modelling of foreign exchange (FX) rates. The pricing of FX options under this model is examined through the use of an asymptotic expansion method, based on Watanabe-Yoshida theory. The perfect/partial hedging issues of FX options in the presence of local stochastic volatility and stochastic IRs are also considered. Finally, the impact of stochastic volatility on the pricing of FX-IR structured products (PRDCs) is examined; 3. A new method of non-biased Monte Carlo simulation for a stochastic volatility model (Heston Model) is proposed; 4. The LIBOR/swap market model with stochastic volatility and jump processes is studied, as well as the pricing of interest rate options under that model. In conclusion, some future research topics are suggested. Key words: Changing Volatility Models, Stochastic Volatility Models, Local Stochastic Volatility Models, Hedging Greeks, Jump Diffusion Models, Implied Volatility, Fourier Transform, Asymptotic Expansion, LIBOR Market Model, Monte Carlo Simulation, Saddle Point Approximation

    Pricing Inflation and Interest Rates Derivatives with Macroeconomic Foundations

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    I develop a model to price inflation and interest rates derivatives using continuous-time dynamics linked to monetary macroeconomic models: in this approach the reaction function of the central bank, the bond market liquidity, and expectations play an important role. The model explains the effects of non-standard monetary policies (like quantitative easing or its tapering) on derivatives pricing. A first adaptation of the discrete-time macroeconomic DSGE model is proposed, and some changes are made to use it for pricing: this is respectful of the original model, but it soon becomes clear that moving to continuous time brings significant benefits. The continuous-time model is built with no-arbitrage assumptions and economic hypotheses that are inspired by the DSGE model. Interestingly, in the proposed model the short rates dynamics follow a time-varying Hull-White model, which simplifies the calibration. This result is significant from a theoretical perspective as it links the new theory proposed to a well-established model. Further, I obtain closed forms for zero-coupon and year-on-year inflation payoffs. The calibration process is fully separable, which means that it is carried out in many simple steps that do not require intensive computation. The advantages of this approach become apparent when doing risk analysis on inflation derivatives: because the model explicitly takes into account economic variables, a trader can assess the impact of a change in central bank policy on a complex book of fixed income instruments, which is not straightforward when using standard models. The analytical tractability of the model makes it a candidate to tackle more complex problems, like inflation skew and counterparty/funding valuation adjustments (known by practitioners as XVA): both problems are interesting from a theoretical and an applied point of view, and, given their computational complexity, benefit from a tractable model. In both cases the results are promising.Open Acces

    Eksoottisten valuuttaoptioiden hinnoittelu

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    The popularity of exotic foreign exchange rate options has grown rapidly during the past decade. High profit margins and rapid market growth have made the market particularly lucrative for the banks. On the other hand, the correct pricing of exotic options requires more sophisticated models than the traditional Black-Scholes. The objective of this thesis is to build, implement, and validate a pricing model for the exotic foreign exchange rate options. Based on previous research, this thesis models the stochastic behavior of the foreign exchange rates as a stochastic volatility – jump-diffusion process with piecewise constant model parameters. The process is defined in both continuous and discrete times. The continuous time process is used for pricing European options in a semi-closed form, which enables an efficient model calibration. The discrete time model is used for pricing exotic options with Monte Carlo. The model is calibrated using a method customized specifically for the purposes of this thesis. The model is validated by analyzing its performance with real market data from the beginning of July to the end of August 2007. The convergence of the closed-form and Monte Carlo solution option prices shows that the model is internally consistent. The comparison of the model implied and market implied option prices indicate that the model is market consistent. The analysis of the robustness suggests that the model and its calibration are mathematically meaningful.Eksoottisten valuuttaoptioiden suosio on kasvanut voimakkaasti viimeisen vuosikymmenen aikana. Korkeiden tuottomarginaalien ja nopean kasvun vuoksi markkina on pankeille erittäin houkutteleva. Toisaalta eksoottisen optioiden oikea hinnoittelu vaatii perinteistä Black-Scholes mallia monimutkaisempien hinnoittelumallien käyttöä. Tämän työn tavoitteena on kehittää, implementoida ja validoida hinnoittelumalli eksoottisille valuuttaoptioille. Aiempiin tutkimustuloksiin nojaten, tämä tutkimus mallintaa valuuttakurssien käyttäytymistä stokastista volatiliteettia ja hyppydiffuusiota kuvaavalla yhdistelmämallilla, jonka parametrit ovat paloittain vakioita. Malli määritellään sekä jatkuvassa että diskreetissä ajassa. Jatkuvan ajan mallia käytetään eurooppalaisten optioiden hinnoitteluun puolisuljetussa muodossa, jota tarvitaan mallin tehokasta kalibrointia varten. Diskreetin ajan mallia käytetään eksoottisten optioiden hinnoitteluun Monte Carlo simuloinnin avulla. Malli kalibroidaan tätä työtä varten räätälöidyllä menetelmällä. Mallin toiminta validoidaan testaamalla mallia todellisella markkinadatalla heinäkuun alusta elokuun loppuun 2007 ulottuvalla ajanjaksolla. Puolisuljetun muodon ja Monte Carlo ratkaisujen optiohintojen yhtäpitävyys osoittaa mallin olevan sisäisesti konsistentti. Mallin tuottamien hintojen ja markkinahintojen yhtäpitävyys validoi mallin markkinakonsistenttiuden. Parametrien käyttäytyminen osoittaa, että malli ja sen kalibrointi ovat matemaattisesti mielekkäitä
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