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lecture_slides-TermStructure_Binomial_#1

# lecture_slides-TermStructure_Binomial_#1 - Financial...

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Financial Engineering & Risk Management Introduction to Term Structure Lattice Models M. Haugh G. Iyengar Department of Industrial Engineering and Operations Research Columbia University

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Fixed Income Markets Fixed income markets are enormous and in fact bigger than equity markets. According to SIFMA , in Q3 2012 the total outstanding amount of US bonds was \$35 . 3 trillion : Government \$10.7 30.4% Municipal \$3.7 10.5% Mortgage \$8.2 23.3% Corporate \$8.6 24.3% Agency \$2.4 6.7% Asset-backed \$1.7 4.8% Total \$35.3 tr 100% – in comparison, size of US equity markets is approx \$26 trillion. Fixed income derivatives markets are also enormous – includes interest-rate and bond derivatives, credit derivatives, MBS and ABS – will focus here on interest-rate and bond derivatives – using binomial lattice models. (The slides and Excel spreadsheet should be suﬃcient but Chapter 14 of Luenberger is an excellent reference for the material in this section.) 2
Binomial Models for the Short Rate Will use binomial lattice models as our vehicle for introducing: 1. the mechanics of the most important ﬁxed income derivative securities - bond futures (and forwards) - caplets and caps, ﬂoorlets and ﬂoors - swaps and swaptions 2. the “philosophy” behind ﬁxed income derivatives pricing – more on this soon. Fixed-income models are inherently more complex than security models - need to model evolution of entire term-structure of interest rates . The short-rate , r t , is the variable of interest in many ﬁxed income models – including binomial lattice models r t is the risk-free rate that applies between periods t and t + 1 – it is a random process but r t is known by time t . 3

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The “Philosophy” of Fixed Income Derivatives Pricing We will simply specify risk-neutral probabilities for the short-rate, r t – without any reference to the true probabilities of the short-rate This is in contrast to the binomial model for stocks where we speciﬁed p and 1 - p – and then used replication arguments to obtain q and 1 - q . We will price securities in such a way that guarantees no-arbitrage Will match market prices of liquid securities via a calibration procedure – often the most challenging part. Will see that derivatives pricing in practice is really about extrapolating from liquid security prices to illiquid security prices. 4
Binomial Models for the Short-Rate ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± r 0 , 0 r 1 , 0 r 1 , 1 r 2 , 0 r 2 , 1 r 2 , 2 r 3 , 0 r 3 , 1 r 3 , 2 r 3 , 3 t = 0 t = 1 t = 2 t = 3 t = 4 We will take zero-coupon bond (zcb) prices to be our basic securities – will use Z k i , j for time i , state j price of a zcb that matures at time k Would like to specify binomial model by specifying all Z k i , j ’s at all nodes – possible but awkward if we want to ensure no-arbitrage .

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lecture_slides-TermStructure_Binomial_#1 - Financial...

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