Credit default swaptions (CDS options) are credit derivatives that are widely used by finan-cial institutions such as banks and hedging companies to manage their credit risk. These options are usually priced using Black-Scholes model, but the assumptions underlying this model do not always hold especially when solving complex financial problems. The proposed solution is to use numerical methods such as finite difference method (FDM) to approximate the solution of the Black-Scholes PDE in cases where closed form solutions cannot be obtained. The pricing of swaptions are important in financial markets, hence we specifically discuss the pricing of interest rate swaptions, CDS options, commodity swaptions and energy swap-tions using Black-Scholes model. Simple parabolic PDE known as heat equation given at (Higham, 2004) forms a foundations to understand the application of FDM when solving a PDE. Since, Black-Scholes PDE is also a parabolic equation it is transformed to a form of a heat equation (diffusion equation) by applying change of variables technique. FDM, specifically Crank-Nicolson method can be applied to the heat equation but in this dissertation it is applied directly to the Black-Scholes PDE to approximate its solution. Therefore, it is preferable to use Crank-Nicolson method because it is known to be second- order accurate, unconditionally stable, very flexible, suitable and can accommodate varia- tions in financial problems, (Duffy, 2008). The stability of this method is investigated using a matrix approach because it accommodates the effect of boundary conditions. To test the convergence of Crank-Nicolson method, it is compared with the Black-Scholes method used in (Tucker and Wei, 2005) to price CDS options. Conclusively the results obtained by Crank-Nicolson method to price CDS options are similar to those obtained using Black-Scholes method. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Credit default swaptions (CDS options) are credit derivatives that are widely used by finan-cial institutions such as banks and hedging companies to manage their credit risk. These options are usually priced using Black-Scholes model, but the assumptions underlying this model do not always hold especially when solving complex financial problems. The proposed solution is to use numerical methods such as finite difference method (FDM) to approximate the solution of the Black-Scholes PDE in cases where closed form solutions cannot be obtained. The pricing of swaptions are important in financial markets, hence we specifically discuss the pricing of interest rate swaptions, CDS options, commodity swaptions and energy swap-tions using Black-Scholes model. Simple parabolic PDE known as heat equation given at (Higham, 2004) forms a foundations to understand the application of FDM when solving a PDE. Since, Black-Scholes PDE is also a parabolic equation it is transformed to a form of a heat equation (diffusion equation) by applying change of variables technique. FDM, specifically Crank-Nicolson method can be applied to the heat equation but in this dissertation it is applied directly to the Black-Scholes PDE to approximate its solution. Therefore, it is preferable to use Crank-Nicolson method because it is known to be second- order accurate, unconditionally stable, very flexible, suitable and can accommodate varia- tions in financial problems, (Duffy, 2008). The stability of this method is investigated using a matrix approach because it accommodates the effect of boundary conditions. To test the convergence of Crank-Nicolson method, it is compared with the Black-Scholes method used in (Tucker and Wei, 2005) to price CDS options. Conclusively the results obtained by Crank-Nicolson method to price CDS options are similar to those obtained using Black-Scholes method. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Due to the interesting financial moment we are living, my motivations to write this Master thesis has mostly been the behavior of interest rates and models that can be used predict them. Thus, in this dissertation I have presented theHull-White model and the way to calibrate it against market data so it can be used to price interest rate derivatives. The reader can find both theoretical and practical presentations and examples along with the code to program them byhim/herself.

With the increasing popularity of interest rate swaps the need to understandswaptions, an option of an interest rate swap, is of great importance. A swap-tion can be used in both speculative purposes and to hedge against changesin interest rates. The most important thing to understand is the pricing for-mula. By starting at the basic rate instrument, bonds, we will work our waytowards the pricing formula for a swaption, the Black76 model. The Black76model is a variant of the Nobel prize winning formula Black-Scholes-Merton.With the pricing model we can start looking at the main scope of this thesis,a hedging strategy against swaptions from a clearinghouse perspective.Clearinghouses are central to the modern financial market. They act asa middleman in order to clear trades from clearing members and have anoversight of the financial market. In case a clearing member defaults, theclearinghouse will gain control over the defaulted portfolio. The clearing-house will host an auction of the portfolio which they strive to hold after5 to 15 days. When they hold the portfolio, they are exposed to the risksand therefore it’s of great importance to be able to hedge the assets in theportfolio. In this thesis a strategy and algorithm have been developed todelta-hedge swaptions in order to be delta-neutral under stable market con-ditions.In the thesis we will consider two cases. The first case is when the clear-inghouse receives the portfolio long before the swaptions maturity. In thiscase forward swaps are used to hedge and in order to reduce the number offorward swaps obtained, compression is used. The second case is when theswaption maturity will be reached within the period the clearinghouse holdsthe portfolio. For the days before maturity is reached, forward swaps andcompression is used. After maturity is reached interest rate swaps is used tohedge.For both cases the result is very close to achieving delta-neutrality. Withnormalized deltas with respect to the notional amount the mean delta ex-posure is of the magnitude 10−4 for the first case and 10−6 for the second.However, one thing to keep in mind is that everything is based on simu-lated values under some simplifying assumptions. This thesis should be asolid ground for future studies where more extreme scenarios are considered.With more extreme scenarios one could investigate the possibility to hedgewith Gamma or another Greek such as Vega. / Med den ökande användningen av ränteswappar är det av stor vikt att förståswaptioner, vilket är en option på en ränteswapp. En swaption kan användasbåde för spekulativa syften och för att hedgea mot risker i ränteförändringar.Det viktigaste att förstå är hur man prissätter en swaption. Eftersom swap-tioner baseras på underliggande tillgångar så kommer vi börja med det mestgrundläggande, obligationer, och arbeta oss fram till modellen vi kommer an-vända, Black76. Black76-modellen är en variant av den nobelprisvinnandemodellen Black-Scholes-Merton. Med denna modell kan vi börja undersökadet huvudsakliga syftet med avhandlingen, en hedgningsstrategi för swap-tioner från perspektivet av ett clearingshus.Clearinghus är en central del av den moderna finansmarknaden. De agerarsom en mellanhand för att hantera affärer mellan clearingmedlemmar ochhar en översikt över marknaden. Ifall en clearingmedlem går i konkurs,kommer clearinghuset att ta över portföljen med tillgångar. Clearinghusetkommer att hålla en auktion för att sälja av portföljen. De strävar efteratt hålla auktionen så snabbt som möjligt och det sker generellt efter 5 till15 dagar. Medan de har portföljen så är de exponerade mot riskerna i till-gångarna och därför är det av största vikt att kunna hedga tillgångarna. Iden här avhandlingen har en strategi och en algorithm tagits fram för attanvända delta-hedging för att uppnå delta-neutralitet under normala mark-nadsrörelser.Vi kommer att undersöka två olika fall. Det första fallet är när portföljen tasöver när det är lång tid kvar till swaptionens förfallodatum. Då kommer viatt använda forward swaps för att hedgea och för att minska antalet swapparkommer vi att använda kompression. Det andra fallet är när förfallodatumetuppnås under tiden som clearinghuset håller i portföljen. Dagarna innan för-fallodatumet kommer vi hedgea med forward swaps med kompression. Närdatumet är nått så kommer vi istället att använda ränteswappar.I båda fallen är resultaten nära att uppnå delta-neutralitet. Med normalis-erade deltan med avsenende på det nominella beloppet är medelvärdet avdelta-exponeringen av magnituden 10−4 för det första fallet och 10−6 för detandra. Men, det är värt att komma ihåg att allting är baserat på simuler-ade värden under förenklade antaganden. Denna avhandling bör utgöra enbra grund för vidare studier där man kan undersöka mer extrema mark-nadsrörelser. Med extremare rörelser skulle man kunna undersöka hedgn-ingsstrategier med andra Greker som till exempel Gamma och Vega.

This article presents a new approach for calculating sensitivities of European swaptions. The sensitivities are found by applying an adjoint method to a stochastic volatility model, namely the SABR/LIBOR market model. This market model predicts the volatility smile and follows the market fluctuations more accurately than earlier used deterministic volatility market models for complex derivatives. The new adjoint method involves not only sensitivity calculations, it also presents a way of estimating the time discretization error using an a posteriori approach. The error calculation is described in this document but not investigated further. The first step in order to calculate the sensitivities is to calibrate the SABR/LIBOR market model to some market data. In our calculations we used data from June 15 2011 with 6 month intervals between the maturity times. When this calibration is complete all of the parameters in the SABR/LIBOR market model are specified and we can continue with the sensitivity calculations using the new adjoint method. The results from these calculations show that the method is a good choice for estimating sensitivities if we consider a complex financial derivative like the European swaption. The method is quite computational so we recommend that it is only used on a small number of securities with respect to a large number of parameters. The method provides more market-driven price and sensitivity estimations than earlier used methods and can benefit hedging of portfolios.

Inflation-indexed instruments provide a real return and protect investors from the erosion of the purchasing power of money. Hence, inflation-indexed markets grow very fast day by day. In this thesis, we focus on pricing of the inflation-indexed swaps and swaptions which are the most liquid derivative products traded in the inflation-indexed markets. Firstly, we review the Hull-White extended Vasicek model in the HJM framework. Then, we use this model to price inflation-indexed swaps. Also, pricing of inflation-indexed swaptions is given using Black&rsquo / s market model.

<p>Diss. Stockholm : Handelshögskolan, 2007 viii, s. [1]-4: sammanfattning, s. [7]-104: 3 uppsatser</p>

This thesis consists of extensions of results on a perpetual American swaption problem. Companies routinely plan to swap uncertain benefits with uncertain costs in the future for their own benefits. Our work explores the choice of timing policies associated with the swap in the form of an optimal stopping problem. In this thesis, we have shown that Hu, Oksendal's (1998) condition given in their paper to guarantee that the optimal stopping time is a.s. finite is in fact both a necessary and sufficient condition. We have extended the solution to the problem from a region in the parameter space where optimal stopping times are a.s. finite to a region where optimal stopping times are non-a.s. finite, and have successfully calculated the probability of never stopping in this latter region. We have identified the joint distribution for stopping times and stopping locations in both the a.s. and non-a.s. finite stopping cases. We have also come up with an integral formula for the inner product of a generalized hyperbolic distribution with the Cauchy distribution. Also, we have applied our results to a back-end forestry harvesting model where stochastic costs are assumed to exponentiate upwards to infinity through time. / Graduation date: 2013

本文探討可違約互換率之匯率連動選擇權的評價，外國以及本國違約交 換率的動態是建立在LIBOR 市場模型的框架。為了簡化推導過程，我們將 原本本國以及外國交換率的雙動態轉為單一動態, 因此違約以及履約價將轉 換為一個固定的常數比率來評價可違約互換率之匯率連動選擇權。由於商品 本身是考量違約的情況，因此使用遠期的存活測度來評價可違約互換率之匯 率連動選擇權。最後在數值分析的部分我們使用蒙地卡羅來模擬可違約互換 率之匯率連動選擇權，理論值與模擬值的結果接近。 / This study prices quanto options on defaultable swap rates (QODSR) in which domestic and foreign defaultable swap rates are considered in the LIBOR market model. We use two fixed ratios to price the QODSR with the default and strike rate property. The forward default-swap measure provides a simple method for valuing the QODSR. Numerical analysis is performed and compared with the Monte Carlo method to investigate the effects of volatility and default on the QODSR.