This thesis mainly focuses on analyzing and pricing European swaption via Crank{Nicolson Finite Dierence method. This paper begins with some rather common instruments, denitions and valuations are also provided. MATLAB is the main computer language used throughout this paper, for the numerical examples, the MATLAB codes are also provide in the appendix in order for reader to reproduce the result. Also, the paper extends to price cancellable swap in the end.

Abstract The valuation of equity swap under the condition of risk neutral is similar to the forward interest rate swap with the same period. Therefore, its valuation formula is consistent to interest rate swap model in the traditional methods. But it is not the same as in pricing the cross-currency equity swap. The dymanic prices of foreign stock index and exchange rate, and the correlation coefficients between exchange rates and foreign assets also affect the swap rate. In this paper, we extend Chance and Rich(1998)¡¦s valuation formula of equity swaps, and apply Amin (1991)¡BAmin and Bodurtha(1995)¡BLin(1997)¡¦s dymanic prices of assets in discrete time period. To derive the risk neutral valuation formula of equity swap, it uses the method of transfer probability measure. This study finds the expected return of foreign stock index in the no arbitrage condition, in addition equal to foreign forward interest rate with the same period , must be add a correction term to reflect the exchange rate risk and the transfer of forward martingale measure. This paper also derives the pricing formula of equity swaptions¡Bcaps¡Bfloors¡Bvariable notional principal and blended equity swap. Finally, we find the volatility of foreign forward interest rate is the most important factor of pricing the swap rate from numerical simulation. And if the correlation of the volatility of exchange rate and foreign stock index¡Bthe correlation of the volatility of exchange rate and foreign forward interest rate are negative, the swap rate will be higher.

Unique and proprietary data of the illiquid, one-year non cancelable for three month Bermudan swaps (1Y NC 3M swaps) and one-year non callable for three months Bermudan CDs (1Y NC 3M CDs), provides evidence of market efficiency. The 1Y NC 3M swap and 1Y NC 3M CD markets efficiently reflected unexpected economic information. The 1Y NC 3M swaption premiums also followed the European one-year into three-month (1Y into 3M) swaption volatilities. Swaption premiums were computed by pricing non-optional instruments using the quoted 1Y NC 3M swap rates and the par value swap rates and taking the difference between them. Swaption premiums ranged from a slight negative premium to a 0.21 percent premium. The average swaption premium during the study period was 0.02 percent to 0.04 percent. The initial swaption premiums were over 0.20 percent while the final swaption premiums were 0.02 percent to 0.04 percent. Premiums peaked and waned throughout the study period depending on market uncertainty as reflected in major national economic announcements, Federal Reserve testimonies and foreign currency devaluations. Negative swaption premiums were not necessarily irrational or quoting errors. Frequently, traders obligated to provide market quotes to customers do not have an interest and relay that lack of interest to the customer through a nonaggressive quote. The short-dated 1Y NC 3M swaption premiums closely followed 3M into 1Y swaption volatilities, indicating the 3M into 1Y swaption market closely follows the 1Y NC 3M swaption market and that similar market factors affect both markets or both markets efficiently share information. Movements in 1Y NC 3M swaption premiums and in 3M into 1Y swaption volatilities reflected a rational response by market participants to unexpected economic information. As market uncertainty decreased in the market place, risk measured both by swaption premiums and swaption volatilities decreased; vice verse when economic factors showed increases in economic uncertainty.

Starting from basic financial mathematics, we cover the mathematics of pricing swaptions, options on interest rate swaps. We then continue to the topic of obtaining an approximately pure volatility exposure. This exposure to volatility, which in practice enables us to trade volatility according to our perceptions of the market, is obtained by buying or selling swaptions and appropriate amounts of the underlying interest rate swap contract. Taking offsetting positions in the underlying contract is called hedging and is covered in depth. We note that hedging can primarily be done in two ways, and discuss the advantages and disadvantages of each of them. After deriving the value formulas for such a swaption strategy aimed at isolating volatility exposure we end with a discussion on the transition from theory to practice.We find that this way of trading volatility is conceptually simple, but that pre-trade profitability analysis is difficult due to the sometimes poor availability of the sophisticated data needed to simulate such a swaption strategy. Despite the possible limitations in the data necessary to translate this theory into an experimental setup, this thesis serves as a good basis for further research on the profitability of a volatility trading strategy using interest rate swaptions.

Problem: The standard Black-Scholes framework cannot incorporate the volatility smiles usually observed in the markets. Instead, one must consider alternative stochastic volatility models such as the SABR. Little research about the suitability of the SABR model for Swedish market (swaption) data has been found. Purpose: The purpose of this paper is to account for and to calibrate the SABR model for swaptions trading on the Swedish market. We intend to alter the calibration techniques and parameter values to examine which method is the most consistent with the market. Method: In MATLAB, we investigate the model using two different minimization techniques to estimate the model’s parameters. For both techniques, we also implement refinements of the original SABR model. Results and Conclusion: The quality of the fit relies heavily on the underlying data. For the data used, we find superior fit for many different swaption smiles. In addition, little discrepancy in the quality of the fit between methods employed is found. We conclude that estimating the α parameter from at-the-money volatility produces slightly smaller errors than using minimization techniques to estimate all parameters. Using refinement techniques marginally increase the quality of the fit.

Fixed income markets are vast markets, involving a large number of actors including financial institutions, state actors, asset managers and corporations. An import part of these markets are contracts written on the xIBOR rates. This report is concerned with the trying to provide prices for options written on these rates, in particular for swaptions that are not at-the-money (atm) utilizing prices in the cap market. Different methods have been suggested in the literature for solving this problem. In particular we study the method suggested by Hagan et al where one calibrates a SABR model to the caplet surface with the same expiry as the swaption. One then assumes that the swaption contract with the same expiry follows the same SABR dynamics as the caplet, but with a recalibrated initial volatility to fit the atm point. We also study the approach suggested by Rebonato and Jäckel. They derive a model for swaption prices based on the individual volatilities of the forward rates that the underlying interest rate swap consists of, as well as the correlation between the forward rates.  Both of these approaches are studied empirically for the STIBOR market. The data set span between 2016 and 2021 and consists of the yield curve, flat cap volatilities and swaption volatilities. We use the 1Y1Y and 5Y5Y swaption surfaces, where the prices are not only quoted atm, to verify our model. We conclude that despite the SABR model being able to fit the caplet prices well, the method suggested by Hagan does not capture the swaption smile. The Rebonato and Jäckel approach also falls short of capturing the smile and produces similar results as the Hagan et al method. This is suggested to be due to the Hagan method capturing the caplet smile well, and the constant correlation assumption made in this thesis.

This document is a Master Thesis report in financial mathematics for KTH. This Master thesis is the product of an internship conducted at Nexialog Consulting, in Paris. This document is about the innovative use of Constrained Gaussian process regression in order to build an arbitrage free swaption cube. The methodology introduced in the document is used on a data set of European Swaptions Out of the Money. / Det här dokumentet är en magisteruppsats i finansiel matematik på KTH. Detta examensarbete är resultatet av en praktik som ufördes på Nexialog Consulting i Paris.Detta dokument handlar om den innovativa användningen av regression för gaussiska processer med bivillkor för att bygga en arbitragefri swaption kub. Den metodik som introduceras i dokumentet används på en datamängd av europeiska swaptions som är "Out of the Money".

摘要 許多公司在發行可贖回公司債時(Callable Bond)，為了規避利率變動的風險因此簽訂利率交換(IRS)契約，此外，考慮到提前贖回的可能性，更進一步承做利率交換選擇權(Swaption)，在利率交換選擇權的部分，一般又會配合特定贖回時點而設計，因此可以視為百慕達式的利率交換選擇權(Bermudan Swaption)。大致而言，百慕達式利率交換選擇權(Bermudan Swaption)可以分為兩類，一類是不論履約時點為何均固定交換期間長度的選擇權，又可稱為Constant Maturity Bermudan Swaption，另一類則是固定商品到期日，即選擇權到期期間與利率交換期間相加為固定常數，換言之，越晚做提前履約的動作，則利率交換的期間也相對便短。 至於在評價部分，百慕達式或美式這些具有提前履約特性的選擇權其封閉解並不存在，因此需要利用到其他的近似解或是數值方法來幫助我們評價。由於本文採用BGM(1997)的市場利率模型(Libor Market Model)，在其高維度的特性下，樹狀方法以及有限差分法並不適用，因此本文選擇使用蒙地卡羅法來幫助我們評價，同時採用Longstaff and Schwartz (2001)的最小平方蒙地卡羅法(Least Squares Monte Carlo Method)來解決傳統蒙地卡羅法無法處理提前履約的困擾。 最後，本文將利用BGM(1997)的利率模型配合Longstaff and Schwartz (2001)的方法實際評價三種商品，包含了上述兩種不同類型的百慕達式利率交換選擇權(Bermudan Swaption)，再加上由中信金所發行的利率交換選擇權(Swaption)，並探討歐式與百慕達式商品價格之差異。

許多公司在發行公司債的時候，會給此公司債一個可提前贖回的特性，此種公司債稱為可贖回公司債(Callable Bond)，用來規避利率變動風險的金融商品也與我們熟知的利率交換不同，稱為可取消利率交換(Cancelable Swap)。其實可取消利率交換可以拆解成百慕達利率交換選擇權(Bermudan Swaption)加上利率交換，由於利率交換之評價較簡單也有市場一致的評價方法，因此百慕達利率交換選擇權便成為評價的重點。 評價的部分，由於百慕達式的商品有提前履約的特性，造成其封閉解不存在，因此需要利用其他的近似解或是數值方法來求它的價格。由於本文採用BGM(1997)的市場利率模型(Libor Market Model)，其高維度的性質導致數狀方法與有限差分法使用起來較無效率，因此本文選擇使用蒙地卡羅法做為評價的方法，同時利用Longstaff and Schwartz(2001)的最小平方蒙地卡羅法(Least Squares Monte Carlo Method)來解決提前履約的問題。 最後，本文將採用2種利率波動度假設與2種不同利率間相關係數的假設，共4種組合，在歐式利率交換選擇權的市場波動度下進行校準，使用校準出來的參數進行評價來得到4種價格。再進行商品的敏感度分析(Sensitivity Analysis)和風險值(Value at Risk)的計算。