Pricing Constant Maturity Floaters with Embedded Options Using Monte Carlo Simulation

A popular interest rate security used by Austrian banks are so called secondary market yield oaters with caps or oors or other types of path dependent em bedded options Since there are no analytical pricing formulas for these constant maturity instruments numerical techniques have to be employed In this paper we present Monte Carlo Simulation techniques to price these products Because these techniques are computationally intensive emphasis is put on a parallel implementation of these products We nd that a parallel implementation en hances the performance of the numerical analysis considerably and makes an accurate pricing of complex interest rate sensitive products possible Introduction and Motivation The pricing of interest rate dependent nancial instruments is one of the most important areas in asset pricing theory In case of simple instruments with deterministic nondefaultable cash ows pricing is based on an arbitrage ar gument on the basis of pure discount bond prices Hence if we have given a term structure of spot rates it is easy to derive the price of a xed coupon bond This research was a part of the Special Research Program SFB F AURORA supported by the Austrian Fonds zur Foerderung der wissenschaftlichen Forschung The theoretical price then simply is the present value of future cash ows Ma ny interest rate dependent products are not based on a xed interest rate In case of oating rate securities we have to distinguish two di erent classes of instruments depending on whether the maturity of the reference interest rate is smaller or equal to the period until the next interest rate adjustment takes place In case the maturity of the reference interest rate is smaller than the adjustment period it can be shown that the price of the oater at the time of the next adjustment is equal to its face value so that the pricing of the oater is identical to that of a xed income deterministic cash ow instrument In case the maturity of the reference interest rate exceeds the period of adjustment we are faced with a so called constant maturity instrument and the pricing becomes much more involved In particular one needs an interest rate model that allows for projections of future interest rates that can be used to forecast uncertain interest payments Hull and White see developed a single factor model that we will make use of in this paper to price constant maturity instruments The use of constant maturity instruments is very popular among Austrian banks see In particular instruments based on the so called secondary market yield SMY are frequently used Examples include loans credits as well as bonds that use the SMY as a reference interest rate The SMY is an index of Austrian government bond yields currently traded in the secondary market and can be interpreted as an average yield to maturity Since the maturity spectrum of Austrian government bonds ranges from less than one year to thirty years the time bucket to which the SMY corresponds is in the range of ve to seven years Hence all instruments that make use of the SMY t into the class of constant maturity oaters The popularity of the SMY stems from its properties given a normal shaped yield curve If bank deposits liabilities have on average a short maturity and loans with the SMY as interest rate assets have a longer time to maturity the normal shape of the yield curve guarantees a remarkable pro t for the bank Things are however quite di erent during a period of an inverse yield curve Despite these arguments the pricing of constant maturity instruments is an interesting theoretical issue and becomes even more involved if the interest rate product is characterized by option features In this paper we take up the issue of pricing CMI s with embedded options on the basis of numerical techniques As far as the embedded options are concerned we look at the following products As mentioned above the use of SMY oating instruments are very popular in Austria Moreover some of these products have the follwing type of interest rate rate adjustment There is an initial reference rate for that applies as long as the variable interest rate does not hit a lower or an upper bound If the variable rate the SMY passes the bounds an interest rate adjustment is made that depends on a factor of adjustment as well as the previous adjustment levels Hence these products are characterized by caps and oors but the cap oor rate depends on past interest rate realizations Therefore the products are path dependent To capture these characteristics we make use of the Hull and White interest rate tree model and use Monte Carlo techniques to price the instruments Using the Hull and White trinomial tree implies that we generate an entire term structure on the basis of a single factor risk factor which is the short term interest rate The exibility of the Hull and White model however guarantees that the term structures generated by the tree are consistent with todays observed one Moreover the model can be calibrated to t the current volatility structure as well Making use of Monte Carlo simulation techniques together with the Hull and White interest rate tree gives us enough exibility to price a very large spectrum of interest rate products There is one disadvantage to this approach however A Monte Carlo simulation is computationally very intensive Therefore we pre sent two possible implementations of our model One is a sequential version and the other one makes use of data parallel structures We present performance evaluations based on this two implementations and sensitivity analysis as the accuracy of the pricing tool is concerned The pricing module presented in this paper is part of a larger model that has been developed within the AURORA research program This model ts into the category of a nancial planning tool that uses stochastic optimization techniques to derive optimal nancial alloca tion decisions see Our paper is organized as follows In the next section we present the Hull and White interest rate tree model and discuss its applications to the pricing of our products In section we present a description of the SMY oaters with caps and oors Section discusses the numerical implementation as well as the numerical results and section concludes the paper The Hull and White Interest Rate Model As outlined in the introduction our aim is to derive a pricing tool for constant maturity instruments based on the interest rate model developed by Hull and White see The Hull and White model ts into the class of one factor arbitrage free models and is based on the following dynamic speci cation for the instantaneous short rate