1 / 32

Financial Models 15

Financial Models 15. Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html. Bonds and Interest Rates. Zero coupon bond = pure discount bond T-bond, denote its price by p(t,T). principal = face value,

rusti
Download Presentation

Financial Models 15

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Financial Models 15 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html T.Bjork, Arbitrage Theory in Continuous Time

  2. Bonds and Interest Rates Zero coupon bond = pure discount bond T-bond, denote its price byp(t,T). principal = face value, coupon bond - equidistant payments as a % of the face value, fixed and floating coupons. FinModels - 15

  3. Assumptions • There exists a frictionless market for T-bonds for every T > 0 • p(t, t) =1 for every t • for every t the price p(t, T) is differentiable with respect to T. FinModels - 15

  4. Interest Rates Let t < S < T, what is IR for [S, T]? • at time t sell one S-bond, get p(t, S) • buy p(t, S)/p(t,T) units of T-bond • cashflow at t is 0 • cashflow at S is -$1 • cashflow at T is p(t, S)/p(t,T) the forward rate can be calculated ... FinModels - 15

  5. The simple forward rate LIBOR - L is the solution of: The continuously compounded forward rate R is the solution of: FinModels - 15

  6. Definition 15.2 The simple forward rate for [S,T] contracted at t (LIBOR forward rate) is The simple spot rate for [S,T] LIBOR spot rate is FinModels - 15

  7. Definition 15.2 The continuously compoundedforward rate for [S,T] contracted at t is The continuously compoundedspot rate for [S,T] is FinModels - 15

  8. Definition 15.2 The instantaneous forward rate with maturity T contracted at t is The instantaneous short rate at time t is FinModels - 15

  9. Definition 15.3 The money market account process is Note that here t means some time moment in the future. This means FinModels - 15

  10. Lemma 15.4 For t  s  T we have And in particular FinModels - 15

  11. Models of Bond Market • Specify the dynamic of short rate • Specify the dynamic of bond prices • Specify the dynamic of forward rates FinModels - 15

  12. Important Relations Short rate dynamics dr(t)= a(t)dt + b(t)dW(t) (15.1) Bond Price dynamics (15.2) dp(t,T)=p(t,T)m(t,T)dt+p(t,T)v(t,T)dW(t) Forward rate dynamics df(t,T)= (t,T)dt + (t,T)dW(t) (15.3) W is vector valued FinModels - 15

  13. Proposition 15.5 We do NOT assume that there is no arbitrage! If p(t,T) satisfies (15.2), then for the forward rate dynamics FinModels - 15

  14. Proposition 15.5 We do NOT assume that there is no arbitrage! If f(t,T) satisfies (15.3), then the short rate dynamics FinModels - 15

  15. Proposition 15.5 If f(t,T) satisfies (15.3), then the bond price dynamics FinModels - 15

  16. Proof of Proposition 15.5 FinModels - 15

  17. Fixed Coupon Bonds FinModels - 15

  18. Floating Rate Bonds L(Ti-1,Ti) is known at Ti-1 but the coupon is delivered at time Ti. Assume that K =1 and payment dates are equally spaced. FinModels - 15

  19. This coupon will be paid at Ti. The value of -1 at time t is -p(t, Ti). The value of the first term is p(t, Ti-1). FinModels - 15

  20. T0 T1 Tn-1 Tn Forward Swap Settled in Arrears K - principal, R - swap rate, rates are set at dates T0, T1, … Tn-1 and paid at dates T1, … Tn. FinModels - 15

  21. Forward Swap Settled in Arrears If you swap a fixed rate for a floating rate (LIBOR), then at time Ti, you will receive where ci is a coupon of a floater. And at Ti you will pay the amount Net cashflow FinModels - 15

  22. Forward Swap Settled in Arrears At t < T0 the value of this payment is The total value of the swap at time t is then FinModels - 15

  23. Proposition 15.7 At time t=0, the swap rate is given by FinModels - 15

  24. Zero Coupon Yield The continuously compounded zero coupon yield y(t,T) is given by For a fixed t the function y(t,T) is called the zero coupon yield curve. FinModels - 15

  25. The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by FinModels - 15

  26. Macaulay Duration Definition of duration, assuming t=0. FinModels - 15

  27. Macaulay Duration What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon. FinModels - 15

  28. $ r Meaning of Duration FinModels - 15

  29. Proposition 15.12 TS of IR With a term structure of IR (note yi), the duration can be expressed as: FinModels - 15

  30. $ r Convexity FinModels - 15

  31. FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S: -K at time T: KeR*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate FinModels - 15

  32. Exercise 15.7 Consider a consol bond, i.e. a bond which will forever pay one unit of cash at t=1,2,… Suppose that the market yield is y - flat. Calculate the price of consol. Find its duration. Find an analytical formula for duration. Compute the convexity of the consol. FinModels - 15

More Related