(a) Construct two quantitative problem in the spirit of Q1 and Q2 in last year’s exam.
– Usually, Q1 is a tree problem with 3-5 sub-problems. This can relate to an existing derivative, or one you invent as long as the problems are solvable with the methods we have learnt.
– Usually, Q2 is a problem which relates to defaultable securities or credit risk shifting derivatives.
– In both Q1 and Q2, make sure that the sub-problems are of different complexity. Perhaps you want to start with easier ones and go towards the more complex. Each should be calibrated to approximately 40 minutes of working time.
(b) Construct an essay question in the spirit of Q3(b) in last year’s exam. This should relate to some conceptual issues and let the student to show that she/he understands the complexities of those issues. It should be calibrated to 25 minutes of working time.
(c) Construct two True, False, Uncertain, Imprecise problems in the spirit of Q3 (c) in last year’s exam. This should relate to some conceptual issues. The problem you write should sound slightly confusing for those who do not get the given concepts very clearly, but all the issues should be easy to spot for those who do. It should be calibrated to the 15 minutes of working time. Please submit a document with the problem and with a worked out solution similar to last year’s published version. (Please feel free to choose the format of the document at your convenience. For instance, Word, PDF or even scanned, hand-written pages are all fine as long as the problems and solutions are clear and legible.)
2. Monte Carlo Simulation [10%]
Build and calibrate a quantitative mortgage valuation model as described below by writing an appropriate code in Matlab or R or Excel (Please use only general toolboxes, without relying on ready-made financial routines, e.g. a routine calibrating Ho-Lee trees.) If you use Excel, please use T = 5 years and N = 1000 simulations at all points applicable below. If you are using a coding language, please use T = 10 and N = 100000. Otherwise, the choice of the software is yours. Please organize your assignment as follows:
There should be a detailed write up, which includes figures and procedures to get your answers, for example, some pseudocode. This should be self-contained and well-explained. In principal, your mark is based on this document. The other files are for us to replicate how you got your results.
We also expect a zip folder with the code and the data files or Excel sheets. Ideally, after expanding your folder, we can run your code/ see your Excel sheet generating all the figures and output on which the write-up is based on.
It is January 2019 and you are working in the back-office of the University Building Society (UBS) providing mortgages to university students and employees all over UK and issuing mortgage backed securities based on those mortgages. UBS has just decided to market a novel product, a T-year fixed rate mortgage. Your boss asks you to develop a model which helps to come up with the right interest rate for this product. If the interest rate is too high, no one will buy the product. If it is too low, UBS will lose on the mortgages. (Hint: These problems build on the example in 13.6 in Veronesi (2010) with longer maturity and a different yield curve. Perhaps a good way to start is to write the code using the original data in that example, so you can verify whether your code gives the same numbers. Then you can modify the input according to the problem below.)
(a) You decide to work with the binomial tree methodology. For this, you have to start with a binomial tree model of risk-neutral short rates. For robustness and simplicity, you decide to experiment both with the Ho-Lee model (HL) and the Black-Derman-Toy (BDT) model, with semi-annual (∆ = 0.5), continuously compounded short-rates. You estimate that the historical volatility of the level of short interest rates is σ = 1.73% while the volatility of the log of interest rates is σ = 21.42%. You should calibrate both type of trees to the current yield curve, which you can find as provided by the Bank of England in excel sheet yieldcurve.xls. (Treat the yields as continuously compounded.)
(b) As a benchmark, assume that the buyers of the mortgage will prepay optimally and will not default. Following our method of valuing mortgages in Chapter 4 of the lecture notes, find the fixed rates for which the value of the T-year mortgage is (approximately) par under the Ho-Lee model and under the Black-Derman-Toy model. Which one is higher? Do you have any intuition for that?
(c) Now you switch to the Monte Carlo methodology. Using N simulations plot a histogram of the simulated interest rates in year T under each of the short rate models. Can you comment on the difference? (Hint for Excel users: You might want to use the function INDEX() to help locate the simulated path on the value tree, so as to avoid repeating calculations. Look in the Help section of Excel for instructions. )
(d) Let us proceed with BDT from here. Use the corresponding mortgage rate you have found in problem (b). As a check on your calculations, you decide to revalue the mortgage contract using a Monte Carlo methodology. What is your point estimate and your confidence interval using N simulations? Is that consistent with your answer in (b)?
(e) You decide to incorporate additional assumptions on prepayment. (You continue with the BDT model and Monte Carlo). You realize that sometimes mortgages are prepaid even if it is not optimal. This typically happens when people move. You also realize that people tend to move more often during the summer months and tend to repay less often at the beginning of the contract. To have some guidance on the probability pi that a mortgage is prepaid in period i even when it is not optimal, you assume that pi is related to the so-called PSA measure of prepayment speed, discussed in Section 8.3.1 of Chapter 8 in Veronesi (2010). Under the PSA experience the conditional prepayment rate in period i (CP Ri) follows the rule of
CP Ri = min(12i × ∆ × 0.2, 6)(%)
where the first period is i = 1. (It starts at 0.2% in the first month, increases by 0.2% in each months then levels of at 6%.) In particular, you use the 50% of the PSA probability and assume that it doubles during the period including summer. (As we start in January, this is every even period.) To be more specific, you model pi as
pi = season index × (1 − (1 − 0.5 × CP Ri) ∆)
where ”season index” is 1 or 2, depending whether the period is odd or even. You also realize that for various reasons (e.g. inattention, financial or psychological cost of the refinancing decision) borrowers might not prepay, in periods when it would be optimal. They prepay with larger probability when interest rates are low, but this probability might not be 1. For instance, it makes sense to assume with that in each period when it would be optimal to prepay, they do so only with probability qi = a × e −b×ri where 0 < a ≤ 1 and b ≥ 0 are calibrated parameters. Smaller a implies less prepayment and larger b implies a larger sensitivity to interest rates. According to your calibration, a = 0.8, b = 20. (Do not let the typo on page 486 of Veronesi (2010) confuse you. The last expression in the one but last paragraph should be RAND() < qs i .) What is your point estimate and your confidence interval using N simulations on the value of the mortgage under these additional assumptions using the interest rate you have found in (b)? Comment on the effect of these assumptions. Under these additional assumptions, how would you change the mortgage rate you recommend to UBS?
(f) As the new product is very successful, UBS decides to issue a pass-through security, an interest rate only security and a principal only security backed by these mortgages. The maturity of these MBSs are the same. However, to recover the cost of securitization, UBS decides to offer them with an interest rate 50 bp lower than the mortgage rate you recommend in problem (e).(If your recommended mortgage rate is smaller than 50 bp, you can use 1% as the interest rate here.) Using the assumptions of problem (b) and a Monte Carlo with N simulation, what is your point estimate and confidence interval for value of the three MBSs?
