Jesper’s research interest include: term structure modeling, volatility smiles, and numerical. Together with Leif B.G. Andersen, Vladimir V. Piterbarg is the author of the authoritative, 1,200 page long, three. Approach to the term structure of interest rates, complete models of stochastic volatility, portfolio turnpike. Andersen and Vladimir V. Piterbarg: Interest Rate Modeling 235. With suggestions for efficient implementation. The last chapter of Volume 2 covers some practical issues in LMM implementation, such as, for example, interpolation, and introduces the swap.
. Part of the book series (MANAGEMENT SC.) Abstract This chapter is intended to provide an overview of state-of-the-art term structure models used in pricing and risk managing interest rate dependent financial products as well as forecasting interest rates in economic scenario generators for market and counterparty credit risk management purposes.
After introducing a general overview of the post-crisis markets environment we will provide insight into post-crisis modelling of term structures via short rate models and Libor Market models for multiple curves and show how these models are applied in economic scenario generators used for risk management and pricing purposes alike.
Introduction Prepayment modeling is crucial to the analysis of mortgage-backed securities (MBS). Prepayments by individual mortgage holders affect both the amount and timing of cash flows - and for collateralized mortgage obligations (for example, interest-only securities), prepayment can greatly affect the value of the securities.
PSA Model The most basic prepayment model is the Public Securities Association (PSA) model, which assumes a ramp-up phase and then a constant conditional prepayment rate (CPR). The PSA model can be generated in MATLAB using the Financial Instruments Toolbox function. % Parameters for MBS passthrough to be priced Settle = datenum( '15-Dec-2007'); Maturity = datenum( '15-Dec-2020'); IssueDate = datenum( '15-Dec-2000'); GrossRate =.0475; CouponRate =.045; Delay = 14; Period = 12; Basis = 4;% Generate cash flows and dates for baseline case using 100 PSA CFlowAmounts, CFlowDates = mbscfamounts(Settle,Maturity, IssueDate. GrossRate, CouponRate, Delay,100); CFlowTimes = yearfrac(Settle,CFlowDates); NumCouponsRemaining = cpncount(Settle, Maturity, Period,Basis, 1, IssueDate); Richard and Roll Model While prepayment modeling often involves complex and sophisticated modeling, often at the loan level, this example uses a slightly modified approach based on the model proposed by Richard and Roll in 6. The Richard and Roll prepayment model involves the following factors.
% Zero Curve - this data is hardcoded for now, but could be bootstrapped% using the bootstrap method of IRDataCurve. % G2 model from Brigo and Mercurio with time homogeneous volatility% parameters G2PP = LinearGaussian2F(irdc,a,b,sigma,eta,rho); LIBOR Market Model Implementation After the volatility and correlation have been calibrated, Monte Carlo simulation is used to evolve the rates forward in time. The object is used to simulate the forward rates. While factor reduction is often used with the LMM to reduce computational complexity, there is no factor reduction in this example. 6M LIBOR rates are chosen to be evolved in this simulation. Since a monthly prepayment vector must be computed, interpolation is used to generate the intermediate rates. Simple linear interpolation is used.
NPeriods = NumCouponsRemaining; nTrials = 100; DeltaTime = 1/12;% Generate factors and short rates Tenor = 1/12 1 2 3 4 5 7 10 15 20 30; G2PPSimZeroRates = G2PP.simTermStructs(nPeriods, 'NTRIALS',nTrials. 'Tenor',Tenor, 'DeltaTime',DeltaTime); SimDates = daysadd(Settle,360.DeltaTime.(0:nPeriods),1);% Tenors that will be recovered for each simulation date.
The stepsize is% included here to facilitate computing a discount factor for each% simulation path.% Remove any paths that go negative NegIdx = squeeze(any(any(G2PPSimZeroRates. % Compute the baseline zero rate at each cash flow time CFlowZero = interp1(ZeroTimes,ZeroRates,CFlowTimes, 'linear', 'extrap');% Compute DF for each cash flow time CFlowDFZero = zero2disc(CFlowZero,CFlowDates,Settle);% Compute the price of the MBS using the zero curve PriceZero = CFlowAmounts.CFlowDFZero';% Generate the cash flows for each IR Path G2PPCFlowAmounts = mbscfamounts(Settle. Repmat(Maturity,1,nTrials), IssueDate, GrossRate, CouponRate, Delay, , G2PPSMM(2:end,:));% Compute the DF for each IR path G2PPCFlowDFSim = cumprod(exp(squeeze(-G2PPSimZeroRates(:,1,:).DeltaTime)));% Present value the cash flows for each MBS G2PPPriceInd = sum(G2PPCFlowAmounts.G2PPCFlowDFSim',2); G2PPPrice = mean(G2PPPriceInd);% Repeat for LMM LMMCFlowAmounts = mbscfamounts(Settle. Repmat(Maturity,1,LMMNTRIALS), IssueDate, GrossRate, CouponRate, Delay, , LMMSMM(2:end,:));% Present value the cash flows for each MBS LMMPriceInd = sum(LMMCFlowAmounts.LMMMonthlyDF',2); LMMPrice = mean(LMMPriceInd); The results from the different approaches can be compared.
The number of trials for the G2 model will typically be less than 100 due to the filtering out of any paths that produce negative interest rates. Additionally, while the number of trials for the G2 model in this example is set to be 100, it is often the case that a larger number of simulations need to be run to produce an accurate valuation. # of Monte Carlo Trials: 73 # of Time Periods/Trial: 156 MBS Price with PSA 100: 1.0187 MBS Price with Custom G2PP Prepayment Model: 0.9871 MBS Price with Custom LMM Prepayment Model: 0.9993 Conclusion This example shows how to calibrate and simulate a G2 interest-rate model and how to use the generated interest-rate paths in a prepayment model loosely based on the Richard and Roll model. This example also provides a useful starting point to using the G2 and LMM interest-rate models in other financial applications. Bibliography This example is based on the following books, papers and journal articles. Andersen, L.
Piterbarg (2010). Interest Rate Modeling, Atlantic Financial Press. Mercurio (2001). Interest Rate Models - Theory and Practice with Smile, Inflation and Credit (2nd ed. Springer Verlag. ISBN 978-3-540-22149-4. Hayre, L, ed., Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities.
New York: John Wiley & Sons, 2001b. Karpishpan, Y., O. Turel, and A. Hasha, Introducing the Citi LMM Term Structure Model for Mortgages, The Journal of Fixed Income, Volume 20 (2010) 44-58. Rebonato, R., K. McKay, and R. White (2010).
The Sabr/Libor Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives. John Wiley & Sons. Richard, S. Roll, 1989, 'Prepayments on Fixed Rate Mortgage-Backed Securities',Journal of Portfolio Management. Office of Thrift Supervision, 'Net Portfolio Value Model Manual', March 2000.
J., Belikoff, A. L., Levin, K. And Tian, X., Analysis of Mortgage Backed Securities: Before and after the Credit Crisis (January 5, 2007).
Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity; Bielecki, Tomasz,; Damiano Brigo and Frederic Patras, eds., February 2011. Available at SSRN: See Also Related Examples. More About.