Proceedings of the 25 Winter Simulation Conference. E. Kuhl,.. Steiger, F. B. Armstrong, and J. A. Joines, eds. THE DELIVERY OPTIO I ORTGAGE BACKED SECURITY VALUATIO SIULATIOS Scott Gregory Chastain Jian Chen Fannie ae 39 Wisconsin Ave Washington, DC 216, U.S.A. ABSTRACT A delivery otion exists in mortgage-backed security market, which has not been considered in existing mortgage ricing simulation literature. We exlain the delivery otion in the To Be Announced trade. We discuss how the resence of the delivery otion effects the use of the standard ricing simulation technique. This technique uses a risk neutral interest rate simulation with a reayment otion model to recover a rice which is an exectation over the ossible rate outcomes. The simulation technique uses onte Carlo integration with a suitable selected seudo or quasi-random sequence. To recover market rices a sread term called the Otion Adjusted Sread is required. We see that multile simulations are required to exlore the full structure of the delivery otion but suggest how to use one simulation to aroximate ricing even when the delivery otion is resent. 1 ITRODUCTIO We will use the abbreviation BS for mortgage-backed security. We will always be referring to fixed rate agency BS in this article. The existing literature on mortgagebacked security valuation has focused on the reayment otion (Richard and Roll 1989, Sahr and Sunderman 1992), which is considered the hardest asect of the roblem. To a lesser extent the roblem of efficient numerical simulation, in articular the use of low-discreancy sequences has received some attention (Paskov 1996, Åkesson and Lohoczky 2). Practitioners of mortgage-backed security valuation are aware of the need for a sread term to relicate market rices. As is the case in general for fixed income investing, the relative value to other markets is the redominate way of thinking about valuation. In the case of mortgage-backed securities the relative value to the swa market, the treasury market, and the agency bond market all receive attention. Thus the sread will the quoted with resect to some market; for examle the LIBOR sread or the treasury sread. An asect of the mortgage-backed security market which the ractitioners deal with but which has not been described in the context of valuation simulation in the literature is the resence of the delivery otion in the agency mortgage-backed security market. This delivery otion is commonly refereed to as the TBA trade. 2 THE TO BE AOUCED ARKET In the excellent descrition of the mechanics of the Agency BS market (Tierney 1997) in The Handbook of ortgage Backed Securities, the To Be Announced trade is described as being a trade of a generic ool. The seller announces the exact ool to be delivered two days before settlement. The BS commitment thus contains an otion held by the seller. The buyer secifies only the originating agency, the ass through rate, and the original maturity (usually either 3 years or 15 years). The couon aid by the mortgage holders is reduced to the ass through rate aid to investors by removal of fees for servicing, administration and credit guaranties. The investor ass through rates are commonly in half integer increments. All other ool characteristics are not secified at the time of the commitment. The seller does not need to own an aroriate BS at the time of commitment. Since the seller sometimes does not obtain an aroriate BS for delivery in time for the two day announcement the market has standard rocedures for handling these failed deliveries. 3 OPTIO ADJUSTED SPREAD We first describe the valuation method in the context of ricing a articular BS with known attributes. As described in (Chen 24), the valuation of a BS is defined as the exected value of the resent value of future cashflows: P E[ V ] E PV ( E c(, t t 1821
where P is the rice of the BS, V is the value of the BS, which is a random variable, deendent on the realization of the economic scenario, PV( is the resent value for cash flow at time t, is the discounting factor at time t, c( is the cash flow at time t, is the maturity of the BS. The exectation is taken with resect to a risk neutral measure. An interest rate model is needed for the calculation of the function, and a reayment model is requirement for the calculation of the ath deendent cashflows, c(. The cashflow function and in articular the reayment model deend on knowledge of the BS attributes. The exectation is aroximated using onte Carlo integration: P 1 d ( c ( (1) 1 t where a subscrit is added to indicate that the discount function and the cashflow function are ath deendent. When the interest rate model is calibrated to swa or treasury rate levels and otion values and the reayment model is calibrated to observed reayment atterns the rice returned by equation (1) will be found to not equal the observed market rice. A single sread term is added to the equation in the discounted function. This sread, s, is referred to as the otion adjusted sread. The discount function becomes ex{ [ ex{ [ ex{ [ e i 1, ex( r( Δ st. r( ] Δt}, r( + s] Δt} r( + s] Δt}ex{ st} Then s is solved for such that Pmarket E c(. t With onte Carlo integration we can rearrange terms to simlify solving for the sread E t 1 c( 1 1 t 1 t e st 1 d ( c 1 t d ( e ( st d ( c c ( (. We will use the abbreviation OAS for the otion adjusted sread. This is the standard resentation on the OAS aroach to BS valuation. A somewhat different aroach is resented in (Kalatay 25), where the reayment model is calibrated to market rices instead of being based on observed reayment atterns. 4 EXAIIG THE COPOETS OF OAS ost eole view OAS of BS as the excess return over a comarable risk-free fixed income investment, generally the LIBOR market, after adjustment of the otionality associated with underlying mortgage loans. Whether LIBOR rates are risk-free rates is still arguable, but it is viewed as the funding cost for most large financial institutions, so OAS can be a measurement of rofitability of BS investment, if financed via the LIBOR market. Theoretically OAS can be decomosed of several comonents. 4.1 Credit Sread on-agency BS have significant credit risk associated with them, so they are generally structured in such a way that different degrees of credit risk are ut on different layers of the structured BS, from B-rated u to AAA-rated. For those non-agency BS, credit sread generally consists a big ortion of the nominal sread, as well as the OAS. In senior/sub structure, the credit risk comes from the mortgage borrower, and the CO structure. For a back-end credit enhanced deal, there is additional credit risk coming from the insurance rovider. For Agency BS, the credit comonent is negligible. 4.2 Liquidity Sread Agency TBA market is the most liquid BS market, and generally carry little liquidity remium. However, for very seasoned Agency BS, and most on-agency BS, liquidity could be an imortant factor affecting the ricing and OAS numbers. This comonent is largely determined by market demand and suly. When market anics, the 1822
liquidity could lay an very imortant role in determining the rice, which is very well demonstrated in the 1998 Russian default event and the subsequent Long Term Caital anagement debacle. 4.3 odel Uncertainty Risk Premium Preayment model is of vital imortance to any BSrelated analysis, and OAS calculation is no excetion. Like any econometric forecasting model, reayment model has forecasting errors, no matter how well it is calibrated. First, there are lots of economic variables, which will affect reayment behavior, cannot enter the reayment model, such as local unemloyment rate, borrower education level, household income imrovement, etc. Second, human behavior is always affected by some non-rational factors, and cannot be fully deicted by any econometric model. And last but not least, eole change, so a good model last year might not be a good model this year. Thus OAS could be interreted as a risk remium to comensate for the model uncertainty. A very good examle for this henomenon is that for very well understood BS roduct, like Agency 3-year fixed rate mortgage, generally has low OAS. For a new roduct with exotic features, e.g., a hybrid adjustable rate mortgage with 4 year amortization term, and 3 year interest only eriod, and 5 year fixed interest rate eriod, generally will have a higher OAS. 4.4 Delivery Otion In the TBA market, at the time of commitment, the buyer and seller only need to agree on the issuing agency, note rate, and the seller need to notify the buyer of the exact ool to be delivered two days before the settlement date. So the seller has a very wide range of BS ools eligible for delivery, and she would naturally deliver the cheaest available BS ools. So the OAS calculated from TBA rice might not reresent the TBA universe, but the worst of the universe, and difference of the OAS numbers should be viewed as the otion cost for this cheaest-to-deliver otion. 4.5 egative OAS As we have mentioned above, OAS is generally viewed as the extra return over risk-free rate, so how is it ossible be negative? Actually, for recent months, the OAS for revailing TBA trades have been consistently negative. There could be several reasons for this observation: 1) OAS is the excess return over the life of the BS. If BS yield is relatively high, and volatility is low, which has been the case for recent eriod, eole are not very concerned with the BS otionality in the short term, and as long as they do not hold the BS till maturity, then can still make rofit on short-term trading; 2) There has been large urchase from foreign central banks, and as long as their funding cost remains lower than LIBOR, which is generally the case, and BS yields higher than alternative investment, which is treasury bond for most of them, buying BS is still a good investment. 5 VALUATIO OF THE TBA TRADE 5.1 Relative Value of the TBA Trade All descritions of the OAS method of BS valuation that we have found in the literature assume that the ool characteristics are known, that is there are defined inuts for the cashflow function. In (Tierney 1997) it is said that the TBA trade can be thought of as a urchase of the average of all similar ools. Putting this statement together with the definition of OAS ricing imlies that for the TBA trade one should base the cashflow function on the average characteristics of all deliverable ools. This aroach maintains the structure of the OAS ricing formulas. There are several other ossibilities for maintaining the OAS ricing formulas. All that is required is assuming one set of ool characteristics is sufficient for the ricing method. Besides the average of ossible ools, one might use the cheaest to deliver characteristics or the most likely to be delivered characteristics. The cheaest to deliver characteristics is based on the assumtion the seller has sufficient variety of ools available to deliver that they can select the worst ossible ool from the urchasers oint of view. On the other hand if the urchasers is actively buying a high volume of securities they may be able to detect delivery atterns and identify the most likely to be delivered. However, besides using the characteristics for average available ool or the worst available ools in the OAS analysis, we could also calculate the OAS for all reresentative ools, and assign them a factor loading, and calculate the weighted average OAS as the OAS for TBA trade. So actually there are three different set of OAS measures could be used: OAS for average ool(s), noted as OAS ( ool) ; OAS for cheaest ool(s), noted as OAS ( oolw ) ; Average OAS for all available ools OAS ( ool). The most flexible and influential characteristic in a TBA trade is the WALA (weighted average loan age), and we would like to examine how different WALA would affect the OAS in a TBA trade. Let s examine the following three TBA trades: Discount ools, which have rice lower than the rincial value, because they generally have mortgage rates lower than current rate, either because the mortgage bor- 1823
rower aid oints to get a lower rate, or the borrower get the mortgage in a revious eriod when the mortgage rate is lower then the current mortgage rate. Since the mortgage always ays the rincial value at reayment, the BS investor benefits from reayment because she aid a lower than ar rice, and get the ar value back. So in such a trade, seasoned ools should have a higher yield than new ools, and hence a higher OAS. Current ools, which have rice equal to or very close to the rincial value. ost likely these mortgage borrowers get the loan in recent eriod, or from revious eriods when mortgage rate are close to today s level. BS investor should be indifferent to reayment, as long as the ool kee current. Thus in a flat yield curve environment, the OAS curve with resect to WALA should be retty flat. And in a uward sloed yield curve, it should be slighted downward sloed, because reaid rincial are discounted lower. Premium ools, which have rice higher than the rincial value, because they have mortgage rates higher than current rate, either because the mortgage borrower could only get a high rate mortgage due to imerfect credit, or the borrower get the mortgage in a revious eriod when the mortgage rate is higher then the current mortgage rate, and she has not refinanced yet. To the oosite of discount ools, the OAS curve should be downward sloed, because BS investor is hurt by faster reayment. 5.2 Pricing Secified Pools with TBA OAS As we have mentioned before, in a TBA trade, the buyer and seller do not need to agree on the secific characteristics of the underlying BS ools at the time of transaction. However, in some case, the buyer is willing to ay above the TBA rice for secified ools, i.e., the current TBA rice for 5.5% couon BS is $1, and the buyer want to buy some ools originated in 22, and there is no market rice for this trade. Generally eole would use the OAS derived from the TBA market, and rice these ools accordingly. A ay-u will be calculated as the difference between the rices of the secified ools and the TBA market. Then there will be some buyer adjustment to make a final offer rice. Generally the buyer will not offer the full ay-u for the secified ools, because of the following reasons: 1) They are not sure about using the TBA OAS to rice seasoned ools; 2) They require higher OAS to comensate the uncertainty associated with seasoned ools; 3) Seasoned ools have better quality, thus higher OAS. 6 UERICAL EXAPLE We use the simle reayment model in Chen(24) to carry out the onte Carlo simulation of BS ricing, in order to acquire the OAS curve for different seasoned ools, given the TBA rice. The following table gives the hyothetical TBA trade rices: Table 1. TBA Secifications and Prices TBA TBA Price 7.5% WAC 1 8.% WAC 12 8.5% WAC 14 9.% WAC 16 9.5% WAC 17 We use a simle secant method to calculate the OAS, given the TBA rice: OAS OAS OAS OAS error k k 1 k+ 1 k k errork errork 1 The algorithm roves be very efficient, and can find the otimal solution at error level of 1e-6 within five iterations. And the following chart gives the OAS curve with resect to WALA for different TBA trades: OAS (b) 15 95 85 75 65 55 45 35 25 OAS Curve 1 2 3 4 5 6 WALA (year) 7.5% WAC 8.% WAC 8.5% WAC 9.% WAC 9.5% WAC Figure 1. OAS for Different Seasoned BS As exected, we see that the OAS curve for discount ools (7.5% WAC) are increasing along with the WALA. As the WAC increases to current couon (8.% WAC), the OAS curve is retty flat with slight downward sloe in the beginning. For the remium mortgage ools (9.% WAC), the OAS curve is obviously decreasing. One interesting henomenon is for the cusy couon mortgage (couon rate slightly higher than the current couon, in our examle, 8.5% WAC), the OAS actually demonstrated U - shaed curve, which can be demonstrated in the following grah. 1824
OAS (b) 75 74 73 72 71 7 69 68 67 66 65 OAS Curve 1 2 3 4 5 6 Figure 2. WALA (year) OAS Curve for 8.5% WAC BS This can be exlained by two factors in the reayment model: refinance seasoning and burnout effect. The refinance seasoning maxes out at month 3, and after that the burnout effect will slow down the reayment, so not surrisingly, we see the OAS has the lowest value at month 3, i.e., year 2.5. 7 SIULATIO COVERGECE Our simulation was imlemented in the ATLAB 7 rogramming environment. For the OAS analysis resented above we used the built in ATLAB 7 seudo-random number generator. ATLAB 7 documentation indicates that this generator is based on the work of Florida State University rofessor George arsaglia. In all cases we used uniform random numbers and converted to a normal distribution using the ATLAB language function norminv(). This may be less recise than using the built in ATLAB 7 normal random number generator but it allowed for us to switch uniform number generators without much code change. We used a fixed seed and fixed number of aths across the exeriments. The OAS calculation requires reeated calculation of rice; that is, reeated onte Carlo integration. By using a fixed seed and number of aths we believe that bias inherent in the simulation is consistent and we are able to make qualitative observations about the OAS curve. To understand the convergence roerties of our roblem we erformed exeriments on the rice function with a fixed OAS in which we varied the seed, the number of aths, and the number generator. We tried the ersenne Twister (atsumoto and ishimura 1998) seudo-random number generator in lace of the built in ATLAB 7 generator. We used a ATLAB language imlementation of the ersenne Twister written by Peter Perkins who is associated with the maker of ATLAB, The athworks. His imlementation is derived from the C code ublished by atsumoto and ishimura. For our seudo-random number tests we used the statistical roerties of onte Carlo integration to find confidence intervals for various choices of number of aths. The confidence intervals obtained with the ersenne Twister were similar to those obtained with the built in ATLAB 7 generator. The confidence interval is quoted around a single observation so there is more information in the width of the interval than the absolute location of the interval. We use the variance of the ath-wise observations in the simulation to estimate the simulations variance at the secified number of aths. This is the variance of reeated trials at that number of aths for true random numbers which can be considered as aroximated by reeated trials with a seudo-random number generator with varying seed. Table 2. Confidence Intervals with ersenne Twister Interval umber of Paths [13.9889,15.9426] 1 [14.1827,14.8183] 1, [14.5146,14.7167] 1, [14.5965,14.667] 1, We also considered the convergence with quasi onti Carlo. We slit the roblem into two sets of random dimensions (X,Y) in the time basis, which is the natural concet of dimension in this roblem. Let X reresent the interest rate shocks for the first 4 months on each ath and Y reresent the remaining months. We used a ATLAB language imlementation of the Sobol sequence made available by John Burkardt from Florida State University which roduces Sobol vectors in u to 4 dimensions. Thus in our decomosition (X,Y) we use Sobol for the X dimensions and ersenne Twister for the Y dimensions. The use of a low discreancy sequence means that we can no longer reort confidence intervals. The reorted results are for a fixed number of aths, 1, with a varying seed and the same BS arameters as used for the ersenne Twister exeriment. It is observed that the stability of the rice in the 1 ath Sobol exeriment exceeds that observed in the confidence interval for 1 aths for seudo-random generators. We have found the build in seudo-random generator in ATLAB 7 adequate for demonstrating the qualitative asects of OAS in the resence of the delivery otions. Pseudo-random sequences have the advantage of allowing us to work with confidence intervals. Table 3. 4 Dim Sobol Convergence Price Seed 14.585 2 14.5488 1, 14.5675 2, 1825
ACKOWLEDGETS The author would like to thank ichael C. Fu for his invitation to resent this aer at the Winter Simulation Conference. Any oinions exressed in this article are those of the authors and do not reflect the views of Fannie ae. REFERECES Åkesson, F. and J. P. Lehoczky. 2. Path generation for Quasi-onte Carlo simulation of mortgage backed securities, anagement Science, 46:(9). Chen, J. 24. Simulation-based ricing of mortgage backed securities. In Proceedings of the 24 Winter Simulation Conference, ed. R. G. Ingalls,. D. Rossetti, J. S. Smith, and B. A. Peters, 1589-1595. Piscataway, J: Institute of Electrical and Electronics Engineers. Kalotay, A., D. Yang, and F. J. Fabozzi. 25. An otiontheoretic reayment model for mortgages and mortgage-backed securities. Prerint to aear in International Journal of Theoretical and Alied Finance. atsumoto,. and T. ishimura. 1998. ersenne Twister: A 623-dimensionally equidistributed uniform seudo-random number generator, AC Transactions on odeling and Comuter Simulation, 8:(1), 3-3. Paskov, S. H. 1996. ew methodologies for valuing derivatives. In athematics of Derivative Securities, ed. S. Pliska and. Demster, 545-582, Cambridge University Press. Richard, S. F. and R. Roll. 1998. Preayments on fixedrate mortgage backed securities, Journal of Portfolio anagement, 15:(3), 73-82. Sahr, R. W. and. A. Sunderman. 1992. The effect of reayment modeling in ricing mortgage-backed securities, Journal of Housing Research, 3:(2), 381-4. Tierney, J. F.. 1997. Trading, settlement, and clearing rocedures for agency BS. In The Handbook of ortgage Backed Securities, ed. F. J. Fabozzi, 4th Edition, 81-9. ew York, Y: cgraw-hill. AUTHOR BIOGRAPHIES SCOTT GREGORY CHASTAI is currently an economist in Fannie ae. He received his Ph.D. in mathematics at the University of Florida. JIA CHE is currently a financial engineer in Fannie ae. He received his Ph.D. in comutational finance in the Robert H. Smith School of Business, at the University of aryland. His research interests include simulation and comutational finance, articularly with the ricing and hedging for interest rate derivatives and credit derivatives. 1826