Lecture_13

Lecture_13 - 13 Swaps Caps Floors and Swaptions Swaps caps...

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1 13 Swaps, Caps, Floors, and Swaptions Swaps, caps, floors and swaptions are very useful interest rate securities. Imagine yourself the treasurer of a large corporation who has borrowed funds from a bank using a floating rate loan. A floating rate loan is a long-term debt instrument whose interest payments vary (float) with respect to the current rates for short-term borrowing. Suppose the loan was taken when interest rates were low, but now rates are high. Rates are projected to move even higher. The current interest payments on the loan are high and if they go higher, the company could face a cash flow crisis, perhaps even bankruptcy. The company’s board of directors is concerned.
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2 Is there a way you can change this floating rate loan into a fixed rate loan, without retiring the debt and incurring large transaction costs (and a loss on your balance sheet)? The solution is to enter into a fixed for floating rate swap or simultaneously purchase caps and floors with predetermined strikes. If you had thought about this earlier, you could have entered into a swaption at the time the loan was made to protect the company from such a crisis.
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3 A Fixed-Rate and Floating-Rate Loans In our simple discrete time model, the short-term rate of interest corresponds to the spot rate r(t) and each period in the model requires an interest payment. We define a floating rate loan for L dollars (the principal) with maturity date T to be a debt contract that obligates the borrower to pay the spot rate of interest times the principal L every period, up to and including the maturity date, time T. At time T, the principal of L dollars is also repaid.
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4 In our frictionless and default-free setting, this floating rate loan is equivalent to shorting L units of the money market account and distributing the gains (paying out the spot rate of interest times L dollars) every period. Paying out the interest as a cash flow maintains the value of the short position in the money market account at L dollars. At time T the short position is closed out.
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5 Table 13.1: Cash Flow from a Floating Rate Loan of a dollar (the Principal), with maturity date T. 01 2 time Borrow +1 T Pay interest –[r(0)–1] –[r(1)–1] –[r(T–1)–1] Pay principal –1
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6 As the floating-rate is market determined, it costs 0 dollars to enter into a floating-rate loan contract. Computing the present value of the cash flows paid on a floating-rate loan with a dollar principal and maturity date T makes this same point. Using the risk-neutral valuation procedure, the present value of the cash flows to the floating rate loan is: 1 ) t ( B ) T ( B 1 t E ~ ) t ( B ) 1 j ( B ] 1 ) j ( r [ 1 T t j t E ~ ) t ( r V = + + = = (13.1) Expression (14.1) shows that the value of the cash flows from the floating-rate loan at time t equals one dollar, which is the amount borrowed.
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7 We define a fixed rate loan with interest rate c for L dollars (the principal) and with maturity date T to be a debt contract that obligates the borrower
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This note was uploaded on 10/31/2010 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell.

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Lecture_13 - 13 Swaps Caps Floors and Swaptions Swaps caps...

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