Broad Philosophical Remarks Financial Securities Mortgages Mortgage Backed Securities Traditional

Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach An Option-Theoretic Model for the Valuation of Mortgage-Backed Securities Deane Yang Polytechnic University Brooklyn, New York USA 2007 March 1 Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Outline Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach What is Finance? Economics is the social science that studies the production, distribution, and consumption of goods and services. Finance is the study of how individuals, businesses, and organizations manage their tangible assets and liabilities. Positive versus normative. Positive finance is the study and analysis of what is done. Normative finance addresses what should be done. Mathematical finance. Primarily normative finance. Determine optimal strategies for managing financial assets and liabilities. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach What is Mathematical Finance? The goal is to create new knowledge and insights in finance. Specifically, to identify and analyze situations where the desired outcome is dominated by logical consequences of rigorously defined financial principles and sufficient data is available for the analysis. Mathematics is the key tool New mathematical ideas and techniques are developed as needed. Mathematical finance is not the attempt to predict future behavior through the statistical analysis of observed financial data. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Fundamental Principles of Mathematical Finance Caveats There are no universal principles or laws. Everything is only an approximation that works some of the time. It is useful to distinguish between man-made conventions and “universal” mathematical or financial principles. All are true but for very different reasons. Principles Everybody is trying to make as much money as possible with minimal risk. (Law of One Price) If two financial assets generate the same future cash flows under the same conditions and have identical risks, then their fair values are the same. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Most Common Use of a Mathematical Model Determining the fair value of a complicated financial asset that is made up of simpler financial assets for which market prices are directly available. What is fair value? It is the price at which under “normal circumstances”, willing buyers and sellers are willing to transact. What is the fair value used for? Accounting requirements Legal and regulatory requirements Internal reporting requirements Risk management Transaction negotiations Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Who Uses Mathematical Models? Example: A derivatives trading desk in an investment bank Analogous to a combination of a chop shop and a car factory. The desk buys certain financial assets, chops them up into simpler parts, and reassembles the parts into new financial assets, which get sold. The desk makes money if revenue from sales exceeds expenses, including cost of purchases made. The value of the parts is determined directly from market prices. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Why Are Mathematical Model Needed? The derivatives desk uses mathematical models for two purposes. Hedging. Figuring out the best way to chop up the purchased assets and assemble into assets to be sold. Valuation. Determining the value of each whole asset, given the value of the parts. Convexity. The value of the whole may be greater or less than the sum of the parts! Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach What is a Financial Security? Partial ownership of a large financial asset. A stock is partial ownership of a business. A bond is partial ownership of a loan. An asset-backed security is partial ownership of a collection of smaller assets (loans, mortgages, credit card receivables, etc.) Designed to facilite transactions Legal assurances and protections are provided to the buyer of a security, guaranteeing the security is really what it claims to be. Legally structured so that ownership can be transferred from one person or institution to another with minimum cost and paperwork. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Stocks A share of stock is part ownership in the company. If the stock is listed on an exchange, it is easy to buy or sell at any instant during business hours. If a company chooses to, it can pay dividends to the shareholders. If the company goes bankrupt, you are last in line among all the people demanding their money back from the company. The fair value of a stock depends solely on what price buyers are willing to pay and sellers are willing to receive for the stock. Mathematical finance has very little to say about valuation of stocks, but it plays a fundamental role in explaining how the fair value of an option on the stock is related to the stock value. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Bonds A bond is a loan. The bond investors are collectively the lender, and the company issuing the bond is the borrower. The company pays a fixed interest rate during the life of the bond and returns the original amount borrowed (i.e. invested) when the bond matures. If the company goes bankrupt, bond investors are near the beginning of the line of people demanding their money back. The value of a bond is equal to the sum of the values of its cash flows, plus the value of the put option if there is one, and minus the value of the call option if there is one. Mathematical finance plays a fundamental role in this determination. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach What is a Mortgage? A loan for purchasing a house. The house always serves as collateral. Most common structure is a 30 year fixed interest rate amortizing loan (fixed monthly payment). Homeowner has right to repay any or all of the principal at any time. Most common reason is sale of home. In particular, if new mortgage rates are low enough, homeowner has the right to take out a new mortgage and repay the old one. This is called “refinancing”. Until relatively recently, most mortgages were issued by local banks who funded them through deposits. To increase funding available for mortgages, mortgage-backed securities were created. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Financial Analysis of a Mortgage A mortgage is equivalent to a callable amortizing bond. There is a well-established theory and practice, requiring an suitably calibrated arbitrage-free stochastic interest rate model, for determining the best time to refinance the mortgage (i.e., calling the bond). When you refinance, you are buying back the remaining mortgage (bond) “at par”. The fair value of the remaining mortgage is equal to the value of the future cash flows (owned by the lender) minus the value of the refinancing option (owned by you). You should refinance if you can buy back the mortgage for less than what the remaining mortgage is worth. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach How Should a Homeowner Decide When to Refinance? Currently, banks and mortgage brokers offer homeowners rules of thumb. Years ago, they would recommend refinancing only if the new rate is more than 1% below the old rate. Today, with lower refinancing costs, they recommend acting when the new rate is only 0.50% below the old rate. But do the banks and mortgage brokers have the best interests of the homeowner in mind? Also, the rules of thumb do not account for opportunity lost for future refinancings or the cost of refinancing too often. A more sophisticated solution can be found at http://www.kalotay.com/calculators This uses the same arbitrage-free stochastic interest rate model that Wall Street traders use to tell you whether you should refinance or not. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach What is a Mortgage-Backed Security? It is a financial security, usually issued by Fannie Mae or Freddie Mac, backed by a pool of mortgages owned by the issuer. The security pays its owners the following cash flows: All principal payments made by the homeowners. A prespecified portion of all interest payments. The portion held back from the investors is used for two purposes: Pay a mortgage servicing company (often, the original bank that issued the mortgages) for the servicing. Pay Fannie Mae, Freddie Mac, or Ginnie Mae a guarantee fee. If a homeowner defaults, then the guarantor pays the investors the remaining principal and recovers whatever it can from the sale of the home. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Why Mortgage-Backed Securities? It dramatically increases the amount of money available for mortgages. Agency-backed MBS are perceived as being almost as safe as Treasuries but with a higher yield. Demand for the MBS allows the fixed mortgage rate to be lower. Lower mortgage rates increase housing sales. Rising housing prices increase perception of safety of MBS. Home mortgages in the U.S: $10 trillion outstanding. $2.5 trillion originated in 2006. Mortgage-backed securities: $6 trillion outstanding. $4 trillion issued by Fannie Mae, Freddie Mac, or Ginnie Mae. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Challenges of Valuing a Mortgage-Backed Security The cash flows are uncertain. Need to model principal payments (“prepayments”): Sale of home. Refinancing. Default. Curtailment (voluntary overpayment to shorten the life of the mortgage). The MBS investor is short the refinancing option. If the option were held by an option trader, then standard arbitrage-free models (Black-Scholes et al) can be used. It is more difficult when the option is held by ordinary homeowners, who do not necessarily exercise the option optimally. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Traditional MBS Modeling All prepayments except for refinancing are relatively stable in time under different economic and interest rate conditions. Model them using a deterministic prepayment speeds based on history. Refinancings, however, depend on interest rates and require a stochastic interest rate model. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Traditional Modeling of Refinancings Use a stochastic interest rate model calibrated to either the Treasury or swap curve Assume that a future mortgage rate is determined (using a fixed ad hoc formula) by, say, the 2 year and 10 year Treasury yields Assume that the amount of refinancings at any given time in the future is given by a fixed ad hoc function of the difference between the outstanding mortgage rate and the modeled future mortgage rate Adjust the formula and function to fit historical refinancing behavior Do a Monte Carlo simulation of prepayments and discount the corresponding MBS cash flows to obtain the expected value of the MBS Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach What’s Wrong With This? A mortgage is a callable bond rate must account for the value of the refinancing option It is unrealistic to model the mortgage rate, which is a callable interest rate, as a deterministic function of two non-callable interest rates (the 2 year and 10 year Treasury yields). A mortgages rate must account for the value of the refinancing option. The refinancing speed is modeled using ad hoc functions fit to history and not on fundamental financial principles that drive the refinancing. This statistical or econometric approach to financial modeling has never worked well for any length of time. Market participants long ago abandoned this approach to valuing all types of financial securities, except for MBS. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Traditional Modeling of Burnout “Burnout” refers to the observed phenomena that when there are two consecutive periods of falling interest rates, the rate of refinancings in a given mortgage pool is significantly lower during the second period than the first The reason is no mystery. People eager to refinance do so during the first period, leaving the pool more weighted towards people who are slow to refinance. Yet traditional MBS models do not model this naturally, so burnout has to be added to the model in an ad hoc manner. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Current State of MBS Valuation Models For years MBS valuation models have broken down virtually every year, as new interest rate environments appear and new refinancing behavior is observed, forcing MBS model builders to reconfigure their models. MBS traders do not trust their own quants’ models and use their own ad hoc models (heavily using linear regression in Excel) to price MBS. Although MBS quants defend their models, we have yet to find a single satisfied customer. Is there a better way? Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Some Perspectives Market participants have abandoned historical or econometric modeling for valuing all types of financial securities, except for MBS. For everything but MBS, people use arbitrage-free forward-looking models that are calibrated to market prices and implied volatility (never historical volatility). There have been earlier attempts to develop arbitrage-free option-theoretic valuation models for MBS by both academics and practitioners, but all have been flawed. Traditional MBS quants deride option-theoretic models, because homeowners don’t use a model to make refinancing decisions Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach A Better Approach A new robust option-based model is proposed in A. Kalotay, D. Yang, F. Fabozzi, An Option-Theoretic Prepayment Model for Mortgages and Mortgage-Backed Securities, International Journal of Theoretical and Applied Finance, December 2004. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Key features Two stochastic interest rate models are used in parallel, one used for valuing the mortgage from the perspective of the homeowner and one used for discounting the MBS cash flows. The mortgage interest rate model is calibrated to current mortgage rates and allows modeling of future mortgages rates as callable interest rates. The MBS interest rate model is calibrated to MBS prices and reflects the perceived credit risk of an MBS. Homeowners are assumed to have a range of refinancing behavior, ranging from those who refinance optimally to those who wait too long. The distribution of refinancing behavior is calibrated against current MBS prices (not historical behavior!). Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Burnout As an MBS ages and refinancings occur, the distribution of refinancing behavior is adjusted by assuming that the most eager refinancers leave the pool first. This causes the refinancings to slow down as an MBS ages, making burnout a natural consequence of the model. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach Does It Work Better? We have been testing the model periodically over the last 5 years. During that period, mortgage rates dropped dramatically, leading to new highs in refinancings, followed recently by a mild increase, slowing refinancings significantly. Traditional models have continued to need regular major readjustments throughout this period. Our model, when tested against market MBS prices, has produced impressively good results using relatively stable model parameter values. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe Broad Philosophical Remarks Financial Securities Mortgages Mortgage-Backed Securities Traditional Modeling of Mortgage-Backed Securities A Better Approach One More Advantage An option-theoretic model can be implemented using a lattice instead of Monte Carlo simulation. Consequences: Much more accurate results. Much faster computation times. 10,000 MBS can be valued in one minute on an ordinary PC. In contrast, the best competing models will value about 100 per minute. Deane Yang An Option-Theoretic Model for the Valuation of Mortgage-Backe