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Your assignment must be submitted electronically (Microsoft Excel or Open Office workbook and
a text document — Microsoft Word or PDF) before midnight on the due date.
IThe text document must be typeset (using for exam-
ple BT E)( or Microsoft Word). Handwritten, scanned assignments will not be accepted. Please make sure your spreadsheets are sufficiently documented so that someone reasonably familiar with the underlying theory is able to use them — in addition to correctness, this is a criterion by which the spreadsheets will be marked.
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On all tasks, implement your calculations in a Microsoft Excel or Open Office workbook (you may use VBA if you wish) and in an accompanying document (PDF, RTF or Microsoft Word) supply a brief explanatory text, providing where appropriate a mathematical justification for your calculations and an interpretation of your results. Please note that not every question necessarily has a single -right- answer. Answers to such questions will be assessed as to whether they are well argued, and the arguments substantiated (for example by mathematical or analytical derivation, or references to the literature).
Any sources used in the preparation of the assignment must be properly cited. Please also refer to the -Statement on plagiarism- in the Subject Outline.
1. Consider the data in the Microsoft Excel workbook YieldCurveData. xlsx. It contains the United States Treasurys -Daily Yield Curve- data (sourced from www . treasury .gov) for the years 2007 and 2015. These rates are what the US Treasury calls -CMT rates- (see https : //www . treasury .gov/resource-center/faqs/Interest-Rates/Pages/f aq. aspx).
(a) (2 marks) Convert these rates into continuously compounded yields.
(b) (3 marks) Assuming a one-factor Gauss/Markov HJM model with mean reversion fixed at a = 0.1, use the data for the six-month yields to determine an appropriate choice of the volatility level a. Do this separately for 2007 and 2015.
(c) (5 marks) Assuming a one-factor Gauss/Markov HJM model, use the data for the one-month and thirty-year yields to determine an appropriate choice of the mean reversion parameter a and volatility level a. Do this separately for 2007 and 2015. Discuss the results in comparison with your results for (b) above.
(d) (10 marks) Suppose that at the beginning of January 2007 you have sold an at-the-money caplet covering a three-month accrual period beginning in three months time. If you hedge this caplet with daily rebalancing using the natural hedge instruments, what would be your profit/loss when this option matures? Assume that caplet price and hedges are calculated using a one-factor Gauss/Markov HJM model with the parameters determined in (c). Assume that any profit/loss is invested/borrowed by buying/selling the zero coupon bond maturing at the same time as the option. Repeat this for a caplet sold at the end of, respectively, March 2007, June 2007, September 2007, March 2015, June 2015, and September 2015. Discuss the possible causes of profit/loss.
(e) (4 marks) Repeat (d) for 2015 caplets, but using the 2007 model parameters. Discuss the result.
(f) (4 marks) Repeat (d) for all caplets, but using as hedge instruments two zero coupon bonds, one of which initially has a time to maturity of six months, and another which initially has a time to maturity of thirty years. Discuss the result.
2. Suppose that the dynamics of the default-free interest rate term structure are given by a lognormal LIBOR Market Model (i.e., assume that LIBOR is an approximately default-free interest rate) driven by a one-dimensional Brownian motion, where the relative volatility of forward LIBOR is given by the deterministic function
A(t,T) = 4e- *(T -t)
with = 0.15 and a = 0.05.
The current term structure is flat at 5% continuously compounded for all maturities. Suppose further that the CDS spreads (annual, in arrears, in basis points) for a corporate entity A are deterministic and given by:
Assume that default, recovery in default, and default-free interest rates are mutually independent. Expected recovery is 40%. You may ignore accrued interest in the case of default. Entity A approaches a bank of negligible default risk, with the wish of A to enter into a 10-year interest swap, where A pays floating and receives fixed annually, with simple compounding. Where necessary, you may use the -frozen coefficient- approximation discussed in class.
(a) (12 marks) In a manner analogous to the case of counterparty credit risk in a Gauss/Markov HJM model covered in the spreadsheet discussed in class, calculate the EFV, EE and PFE for this trade.
(b) (5 marks) Taking into account counterparty credit risk, what is the level of the fixed coupon on this swap, which results in a zero initial mark-to-market value of the swap? If some of the above assumptions are relaxed, in which case would the bank be exposed to -wrong-way risk- in this transaction?
(c) (5 marks) Suppose the bank has an existing swap with the same counterparty, where the bank pays floating and receives fixed annually, with simple compounding. This swap has exactly three years left to run, and the fixed leg was set at 4.75% (simple compounding). In this situation, what is the answer to (b) if there is no netting agreement in place? If there is a netting agreement in place?
Maturity
Spread
1 yr
180
2 yrs
190
3 yrs
220
5 yrs
250
10 yrs
270