Risk measurement, management and option pricing via a new log-normal sum approximation method

Abstract

Bu tezde temel olarak KoŞullu Riske Maruz De?ger (CVaR)'in risk y¨onetiminde kullanımıile geometrik ortalama sepet ve Asya tipi opsiyonların log-normal da?gılımların toplamınayeni bir yaklaŞım metodu ile fiyatlanması ¨uzerine odaklandık. ¨Oncelikli olarak, Rockafellerve Uryasev tarafından ortaya atılan CVaR'ın do?grusallaŞtırılması y¨ontemi ¨uzerinde c¸alıŞtık.Amac¸ fonksiyonu beklenen getiriyi maksimize etmek olan ve CVaR kısıtına sahip bir optimizasyonproblemi kurduk. Olası ara ¨odemelerden dolayı, orjinal hali tek d¨onem olan problemic¸ok d¨onemli probleme d¨on¨uŞt¨uren bir yeniden yatırım problemi ile uğra

In this thesis we mainly focused on the usage of the Conditional Value-at-Risk (CVaR) inrisk management and on the pricing of the arithmetic average basket and Asian options inthe Black-Scholes framework via a new log-normal sum approximation method. Firstly, weworked on the linearization procedure of the CVaR proposed by Rockafellar and Uryasev. Weconstructed an optimization problem with the objective of maximizing the expected returnunder a CVaR constraint. Due to possible intermediate payments we assumed, we had to dealwith a re-investment problem which turned the originally one-period problem into a multiperiodone. For solving this multi-period problem, we used the linearization procedure ofCVaR and developed an iterative scheme based on linear optimization. Our numerical resultsobtained from the solution of this problem uncovered some surprising weaknesses of the useof Value-at-Risk (VaR) and CVaR as a risk measure.In the next step, we extended the problem by including the liabilities and the quantile hedgingto obtain a reasonable problem construction for managing the liquidity risk. In this problemconstruction the objective of the investor was assumed to be the maximization of the probaivbility of liquid assets minus liabilities bigger than a threshold level, which is a type of quantilehedging. Since the quantile hedging is not a perfect hedge, a non-zero probability of havinga liability value higher than the asset value exists. To control the amount of the probable deficientamount we used a CVaR constraint. In the Black-Scholes framework, the solution ofthis problem necessitates to deal with the sum of the log-normal distributions. It is known thatsum of the log-normal distributions has no closed-form representation. We introduced a new,simple and highly efficient method to approximate the sum of the log-normal distributions usingshifted log-normal distributions. The method is based on a limiting approximation of thearithmeticmean by the geometric mean. Using our new approximation method we reduced the quantile hedgingproblem to a simpler optimization problem.Our new log-normal sum approximation method could also be used to price some options inthe Black-Scholes model. With the help of our approximation method we derived closed-formapproximation formulas for the prices of the basket and Asian options based on the arithmeticaverages. Using our approximation methodology combined with the new analytical pricingformulas for the arithmetic average options, we obtained a very efficient performance forMonte Carlo pricing in a control variate setting. Our numerical results show that our controlvariate method outperforms the well-known methods from the literature in some cases.Keywords: Risk measures, linearization of conditional value-at-risk, quntile hedging, pricingoptions based on arithmetic averages, variance reduction with control variates