On the optimal exercise boundaries of swing put options

We use probabilistic methods to characterise time-dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon. Motivated by financial applications, we consider a payoff of immediate stopping of “put” type, and the underlying dynamics follows a geometric Brownian motion. The optimal stopping region relative to each optimal stopping time is described in terms of two boundaries, which are continuous, monotonic functions of time and uniquely solve a system of coupled integral equations of Volterra-type. Finally, we provide a formula for the value function of the problem

Optimal Redeeming Strategy of Stock Loans

A stock loan is a loan, secured by a stock, which gives the borrower the right to redeem the stock at any time before or on the loan maturity. The way of dividends distribution has a significant effect on the pricing of the stock loan and the optimal redeeming strategy adopted by the borrower. We present the pricing models sub ject to various ways of dividend distribution. Since closed-form price formulas are generally not available, we provide a thorough analysis to examine the optimal redeeming strategy. Numerical results are presented as well.Comment: 17 pages, 4 figure

Pricing swing options and other electricity derivatives

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

A class of recursive optimal stopping problems with applications to stock trading

In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. We then apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case.Comment: 35 pages, 2 figures. In this version, we provide a general analysis of a class of recursive optimal stopping problems with both finite-time and infinite-time horizon. We also discuss other application

Pricing swing options in the electricity market

The thesis deals with how to price swing options in the electricity market by using a least squares Monte Carlo method. This is a simulation method which uses a backwards moving algorithm where the optimal decision is calculated at every time step. Regression is used for the optimal decision and in this thesis both a polynomial regression and a cubic smoothing spline are used. They are both shown to be rather good estimators for the regression. Two variation of contracts are priced. For the rst only one exercise right that can be used when exercising and for the second one several exercise rights can be used when exercising. Volume restrictions are also used. The algorithm implemented in this thesis give similar results to the ones of previous authors and when we can not compare with other authors it give us results fairly close to our expectations. The thesis also examines the optimal exercise strategy for a swing option and the boundaries for when to use an exercise right are calculated.Prissattning av swing optioner inom elmarknaden Allmant om swing optioner inom elmarknaden I ett kontrakt for kop utav elektricitet ar det specicerat hur mycket som kommer att levereras och nar, och det gar oftast inte att andra pa. Om innehavaren av ett sadant kontrakt skulle vilja oka eller minska leveransen av elektricitet under perioden for kontraktet, behover den kopa eller salja elektricitet till priset pa marknaden. Det priset kan dock vara mycket ogynnsamt, speciellt om priset for narvarande uppvisar valdigt hoga eller laga priser. For att undvika denna prisrisk kan man anvanda en swing option. Den tillater innehavaren att kopa och/eller salja elektricitet ett visst antal ganger till forbestamda priser. Det nns manga olika varianter av swing optioner, t.ex. sa kan volymen som kan kopas eller saljas med en swing option vara reglerad. Ett kontrakt pa en swing option kan skrivas med fysisk leverans eller nansiellt utan att nagon faktisk leverans av elektricitet sker. Metod for att prissatta swing optioner For att prissatta en swing option kan man anvanda en minsta kvadrat Monte Carlo-metod. Det ar en simuleringsmetod dar man simulerar ett stort antal banor enligt en modell for priset av elektricitet. Banorna representerar mojliga utfall av priset vid framtida tidpunkter. Med en algoritm kan man sedan hitta de optimala tidpunkterna att anvanda optionen, d.v.s. dar man far storst utbetalning, for varje simulerad bana. For att bestamma de optimala tidpunkterna behover algoritmen vid varje tidpunkt fatta beslut om optionen ska anvandas eller inte. Det beslutet beror pa hur stor utbetalning man kan fa av att anvanda den vid den aktuella tidpunkten, kontra hur stor utbetalning man kan forvantas fa vid framtida tidpunkter. Forvantade utbetalningen kan uppskattas genom att anvanda en minsta kvadrat regression. Nar de optimala tidpunkterna och utbetalningarna fran dem ar hittade kan man rakna ut priset genom att ta genomsnittet av vardet pa utbetalningarna for alla banorna. Examensarbetets innehall I detta arbete har jag tittat pa hur den valda metoden fungerar och unders okt om algoritmen beter sig som forvantat. Det som jag har kommit fram till att minsta kvadrat regressionen ar en bra uppskattning av de forvantade framtida utbetalningarna och algoritmen ger darfor korrekta resultat. For innehavaren av en swing option ar det intressant att veta nar det ar optimalt att anvanda optionen for att fa ut sa stort varde av den som mojligt. For att fa en indikation om detta kan man vid varje tidpunkt titta pa vid vilka elektricitetspriser som algoritmen beslutade att anvanda optionen. Om det faktiska priset sedan ar samma eller ett pris som ger en annu battre utbetalning kan det da vara en antydan om att det ar lage att anvanda optionen. Jag har i examensarbetet undersokt hur priserna som ligger pa gransen mellan att anvanda optionen eller inte ser ut. Resultaten visar att ju langre tid som gar desto mindre gynnsamma priser kravs for att det ska vara optimalt att anvanda optionen. Det innebar att i slutet av optionens livslangd ar man villig att acceptera en mindre utbetalning an i borjan av optionens livslangd. En innehavare av ett kontrakt pa en swing option som skrivits med fysisk leverans har mindre nytta av att veta nar det ar optimalt att anvanda optionen. Detta beror pa att det ar innehavarens behov av elektricitet som styr nar optionen bor anvandas, oavsett om det ar en optimal tidpunkt eller inte

Optimal stopping problems in mathematical finance

This thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options, are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model, in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and its maximum drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising are shown to be the first times, at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations

Jump-diffusion models with two stochastic factors for pricing swing options in electricity markets with partial-integro differential equations

[Abstract] In this paper we consider the valuation of swing options with the possibility of incorporating spikes in the underlying electricity price. This kind of contracts are modelled as path dependent options with multiple exercise rights. From the mathematical point of view the valuation of these products is posed as a sequence of free boundary problems where two consecutive exercise rights are separated by a time period. Due to the presence of jumps, the complementarity problems are associated with a partial-integro differential operator. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank–Nicolson semi-Lagrangian method for the time discretization of the differential part of the operator, jointly with the explicit treatment of the integral term by using the Adams–Bashforth scheme and combined with biquadratic Lagrange finite elements for space discretization. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature. Moreover, we employ appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results in order to illustrate the proper behaviour of the numerical schemes.This work has been partially funded by MINECO of Spain (Project MTM2016-76497-R), Xunta de Galicia grants GRC2014/044 and ED431C 2018/33, including FEDER funds, and Bilateral German–Spanish DAAD Project No 57049700 HiPeCa: High Performance Calibration and Computation in Finance, Programme Acciones Conjuntas Hispano-Alemanas funded by German DAAD and the Fundación Universidad.Xunta de Galicia; GRC2014/044Xunta de Galicia; ED431C 2018/33Germany. German Academic Exchange Service; 5704970