Optimal stopping problems with applications to mathematical finance

  • Yerkin Kitapbayev

Student thesis: Phd


The main contribution of the present thesis is a solution to finite horizon optimalstopping problems associated with pricing several exotic options, namely the Americanlookback option with fixed strike, the British lookback option with fixed strike,American swing put option and shout put option. We assume the geometric Brownianmotion model and under the Markovian setting we reduce the optimal stoppingproblems to free-boundary problems. The latter we solve by probabilistic argumentswith help of local time-space calculus on curves ([52]) and we characterise optimalexercise boundaries as the unique solution to certain integral equations. Then usingthese optimal stopping boundaries the option price can be obtained.The significance of Chapters 2 and 3 is a development of a method of scaling strikewhich helps to reduce three-dimensional optimal stopping problems, for lookbackoptions with fixed strike, including a maximum process to two-dimensional one withvarying parameter. In Chapter 3 we show a remarkable example where, for somevalues of the set parameters, the optimal exercise surface is discontinuous whichmeans that the three-dimensional problem could not be tackled straightforwardlyusing local time-space calculus on surfaces ([55]). This emphasises another advantageoffered by the reduction method.In Chapter 4 we study the multiple optimal stopping problems with a put payoffassociated to American swing option using local time-space calculus. To our knowledgethis is the first work where a) a sequence of integral equations has been obtainedfor consecutive optimal exercise boundaries and b) the early exercise premium representationhas been derived for swing option price. Chapter 5 deals with the shoutput option which allows the holder to lock the profit at some time τ and then attime T take the maximum between two payoffs at τ and T. The novelty of the workis that it provides a rigorous analysis of the free-boundary problem by probabilisticarguments and derives an integral equation for the optimal shouting boundary alongwith the shouting premium representation for the option price in some cases. Thisapproach can also be applied to other shout and reset options.In Chapter 6 we discuss a problem of the smooth-fit property for the American putoption in an exponential Levy model. In [2] the necessary and sufficient conditionwas obtained for the perpetual case. Recently Lamberton and Mikou [40] coveredalmost all cases for an exponential Levy model with dividends on finite horizon andwe study remaining cases. Firstly, we take the logarithm of the stock price as a Levyprocess of finite variation with zero drift and finitely many jumps, and prove thatone has the smooth-fit property without regularity unlike in the infinite horizon case.Secondly, we provide some analysis and calculations for another case uncovered in[40] where the drift is positive but for all maturities and removing the additionalcondition they used.The result of Chapter 1 is contained in the publication [33] and results of Chapters2-5 are exposed in preprints [34], [17] and [35] that are submitted for publication.
Date of Award1 Aug 2015
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorGoran Peskir (Supervisor) & John Moriarty (Supervisor)

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