Interest Rate Modeling Background Data Regression Time series - PowerPoint PPT Presentation

Interest Rate Modeling Jaime Frade Outline Objective Purpose Interest Rate Modeling Background Data Regression Time series Jaime Frade ARIMA CIR European Call option April 18, 2008 Jaime Frade Interest Rate Modeling

Interest Rate 1 Objective Modeling 2 Purpose Jaime Frade Outline 3 Background Objective 4 Data Purpose 3 month LIBOR Background 3 month T-rate Data 30 year T-rate Regression Time series 5 Regression Time series ARIMA CIR Forcasted value European Call 6 ARIMA option Forcasted value 7 CIR Forcasted value 8 European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective The main focus of this project is to create a model for 3 month Purpose Background LIBOR interest rates. This will be modeled three different ways Data Regression Time Series Regression Time series ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective The main focus of this project is to create a model for 3 month Purpose Background LIBOR interest rates. This will be modeled three different ways Data Regression Time Series Regression Time series ARIMA ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective The main focus of this project is to create a model for 3 month Purpose Background LIBOR interest rates. This will be modeled three different ways Data Regression Time Series Regression Time series ARIMA ARIMA Cox, Ingersoll, Ross (CIR) stochastic model. CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective The three models will be accessed and a short rate will be forecasted Purpose with some error. The forecasted rate will be used produce value of a Background European call option, with S , spot rate and , K , strike price. The Data payoff for this call option is Regression Time series ARIMA max[( S − K ] , 0) CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose The process for the short rate models a rate in a risk-neutral world. Background Interest rate behavior is similar to the behavior of stocks; however, Data rates appear to be pulled back to some long-run average level over Regression Time series time. This mean reversion is taken into consideration in only the CIR ARIMA model. CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background Interest rates were collected on a daily basis for the past three years. Data 3 month LIBOR Data was obtained from the Federal Reserve website. 3 month T-rate 30 year T-rate Regression Time series ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background Data 3 month LIBOR 3 month T-rate 30 year T-rate Regression Time series ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background Data 3 month LIBOR 3 month T-rate 30 year T-rate Regression Time series ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background Data 3 month LIBOR 3 month T-rate 30 year T-rate Regression Time series ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose To avoid multicollinearity, I regressed several models to predict either Background the current 3 month LIBOR rate or changes LIBOR rate. I decided to Data use a 1 step lag in the 3 month T-rate to predict Libor. Regression Time Model: series Forcasted value Y = β 0 + β 1 ∗ ( X t − 1 ) ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background Using a value of 1 . 5%, I predicted the 3 month LIBOR rate to be Data 4 . 55%, with a 95% confidence level (4 . 518720% , 4 . 585424%) Regression Time series Forcasted value ARIMA CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose ARIMA processes are just integrated ARMA processes. In other Background words, a process is ARIMA of order d if its d-th derivative is ARMA. Data The model can be written Regression Time series φ ( B ) (1 − B ) d X t = θ ( B ) Z t ARIMA Forcasted value CIR European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose A risk neutral process for r which contains a mean reverting drift as Background well as way to model non-negative rates. The change in the short rate in a short period of time is proportional to √ r . Data Regression Time series dr = a ( b − r ) dt + σ √ r dz ARIMA CIR Forcasted value European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background Using Monte Carlo simulations and other assumptions about the Data parameters of the model, the following value for the LIBOR rate was Regression Time forcasted. 4 . 5831179% series ARIMA CIR Forcasted value European Call option Jaime Frade Interest Rate Modeling

Interest Rate Modeling Jaime Frade Outline Objective Purpose Background RTS model: Cost of 0.045600363 Data CIR model: Cost of option is 0.045892182 Regression Time series ARIMA CIR European Call option Jaime Frade Interest Rate Modeling