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Interest Rate Derivatives: The Standard Market Models

Interest Rate Derivatives: The Standard Market Models. Chapter 28. The Complications in Valuing Interest Rate Derivatives (page 647). We need a whole term structure to define the level of interest rates at any time

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Interest Rate Derivatives: The Standard Market Models

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  1. Interest Rate Derivatives: The Standard Market Models Chapter 28

  2. The Complications in Valuing Interest Rate Derivatives (page 647) • We need a whole term structure to define the level of interest rates at any time • The stochastic process for an interest rate is more complicated than that for a stock price • Volatilities of different points on the term structure are different • Interest rates are used for discounting the payoff as well as for defining the payoff

  3. Approaches to PricingInterest Rate Options • Use a variant of Black’s model • Use a no-arbitrage (yield curve based) model

  4. Black’s Model • Similar to the model proposed by Fischer Black for valuing options on futures • Assumes that the value of an interest rate, a bond price, or some other variable at a particular time T in the future has a lognormal distribution

  5. Black’s Model for European Bond Options (Equations 28.1 and 28.2, page 648) • Assume that the future bond price is lognormal • Both the bond price and the strike price should be cash prices not quoted prices

  6. Forward Bond and Forward Yield Approximate duration relation between forward bond price, FB, and forward bond yield, yF where D is the (modified) duration of the forward bond at option maturity

  7. Yield Volsvs Price Vols (Equation 28.4, page 651) • This relationship implies the following approximation where sy is the forward yield volatility, sBis the forward price volatility, and y0 is today’s forward yield • Often syis quoted with the understanding that this relationship will be used to calculate sB

  8. Theoretical Justification for Bond Option Model

  9. Caps and Floors • A cap is a portfolio of call options on LIBOR. It has the effect of guaranteeing that the interest rate in each of a number of future periods will not rise above a certain level • Payoff at time tk+1 is Ldk max(Rk-RK, 0) where L is the principal, dk=tk+1-tk, RK is the cap rate, and Rk is the rate at time tk for the period between tk and tk+1 • A floor is similarly a portfolio of put options on LIBOR. Payoff at time tk+1 is Ldk max(RK -Rk, 0)

  10. Caplets • A cap is a portfolio of “caplets” • Each caplet is a call option on a future LIBOR rate with the payoff occurring in arrears • When using Black’s model we assume that the interest rate underlying each caplet is lognormal

  11. Black’s Model for Caps(p. 657) • The value of a caplet, for period (tk, tk+1) is • L: principal • RK : cap rate • dk=tk+1-tk • Fk : forward interest rate • for (tk, tk+1) • sk: forward rate volatility

  12. When Applying Black’s ModelTo Caps We Must ... • EITHER • Use spot volatilities • Volatility different for each caplet • OR • Use flat volatilities • Volatility same for each caplet within a particular cap but varies according to life of cap

  13. Theoretical Justification for Cap Model

  14. Swaptions • A swaption or swap option gives the holder the right to enter into an interest rate swap in the future • Two kinds • The right to pay a specified fixed rate and receive LIBOR • The right to receive a specified fixed rate and pay LIBOR

  15. Black’s Model for European Swaptions • When valuing European swap options it is usual to assume that the swap rate is lognormal • Consider a swaption which gives the right to pay sK on an n -year swap starting at time T. The payoff on each swap payment date is where L is principal, m is payment frequency and sT is market swap rate at time T

  16. Black’s Model for European Swaptionscontinued (Equation 28.11, page 659) The value of the swaption is s0 is the forward swap rate; s is the forward swap rate volatility; ti is the time from today until the i th swap payment; and

  17. Theoretical Justification for Swap Option Model

  18. Relationship Between Swaptions and Bond Options • An interest rate swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond • A swaption or swap option is therefore an option to exchange a fixed-rate bond for a floating-rate bond

  19. Relationship Between Swaptions and Bond Options (continued) • At the start of the swap the floating-rate bond is worth par so that the swaption can be viewed as an option to exchange a fixed-rate bond for par • An option on a swap where fixed is paid and floating is received is a put option on the bond with a strike price of par • When floating is paid and fixed is received, it is a call option on the bond with a strike price of par

  20. Deltas of Interest Rate Derivatives Alternatives: • Calculate a DV01 (the impact of a 1bps parallel shift in the zero curve) • Calculate impact of small change in the quote for each instrument used to calculate the zero curve • Divide zero curve (or forward curve) into buckets and calculate the impact of a shift in each bucket • Carry out a principal components analysis for changes in the zero curve. Calculate delta with respect to each of the first two or three factors

  21. Quanto, Timing, and Convexity Adjustments Chapter 29

  22. Forward Yields and Forward Prices • We define the forward yield on a bond as the yield calculated from the forward bond price • There is a non-linear relation between bond yields and bond prices • It follows that when the forward bond price equals the expected future bond price, the forward yield does not necessarily equal the expected future yield

  23. Bond Price B3 B2 B1 Yield Y3 Y2 Y1 Relationship Between Bond Yields and Prices (Figure 29.1, page 668)

  24. Convexity Adjustment for Bond Yields (Eqn29.1, p. 668) • Suppose a derivative provides a payoff at time T dependent on a bond yield, yTobserved at time T. Define: • G(yT) : price of the bond as a function of its yield y0 : forward bond yield at time zero sy : forward yield volatility • The expected bond price in a world that is FRN wrt P(0,T) is the forward bond price • The expected bond yield in a world that is FRN wrt P(0,T) is

  25. Convexity Adjustment for Swap Rate The expected value of the swap rate for the period T to T+t in a world that is FRN wrt P(0,T) is where G(y) defines the relationship between price and yield for a bond lasting between T and T+tthat pays a coupon equal to the forward swap rate

  26. Example 29.1 (page 670) • An instrument provides a payoff in 3 years equal to the 1-year zero-coupon rate multiplied by $1000 • Volatility is 20% • Yield curve is flat at 10% (with annual compounding) • The convexity adjustment is 10.9 bps so that the value of the instrument is 101.09/1.13 = 75.95

  27. Example 29.2 (Page 670-671) • An instrument provides a payoff in 3 years = to the 3-year swap rate multiplied by $100 • Payments are made annually on the swap • Volatility is 22% • Yield curve is flat at 12% (with annual compounding) • The convexity adjustment is 36 bps so that the value of the instrument is 12.36/1.123 = 8.80

  28. Timing Adjustments (Equation 29.4, page 672) The expected value of a variable, V, in a world that is FRN wrt P(0,T*) is the expected value of the variable in a world that is FRN wrt P(0,T) multiplied by where R is the forward interest rate between T and T*expressed with a compounding frequency of m, sR is the volatility of R, R0 is the value of R today, sV is the volatility of F, andris the correlation betweenRandV

  29. Example 29.3 (page 672) • A derivative provides a payoff 6 years equal to the value of a stock index in 5 years. The interest rate is 8% with annual compounding • 1200 is the 5-year forward value of the stock index • This is the expected value in a world that is FRN wrt P(0,5) • To get the value in a world that is FRN wrt P(0,6) we multiply by 1.00535 • The value of the derivative is 1200×1.00535/(1.086) or 760.26

  30. Quantos(Section 29.3, page 673) • Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currency • Example: contract providing a payoff of ST – K dollars ($) where S is the Nikkei stock index (a yen number)

  31. Diff Swap • Diff swaps are a type of quanto • A floating rate is observed in one currency and applied to a principal in another currency

  32. Quanto Adjustment (page 674) • The expected value of a variable, V, in a world that is FRN wrt PX(0,T) is its expected value in a world that is FRN wrt PY(0,T) multiplied by exp(rVWsVsWT) • W is the forward exchange rate (units of Y per unit of X) and rVW is the correlation between V and W.

  33. Example 29.4 (page 674) • Current value of Nikkei index is 15,000 • This gives one-year forward as 15,150.75 • Suppose the volatility of the Nikkei is 20%, the volatility of the dollar-yen exchange rate is 12% and the correlation between the two is 0.3 • The one-year forward value of the Nikkei for a contract settled in dollars is 15,150.75e0.3 ×0.2×0.12×1 or 15,260.23

  34. Quantoscontinued When we move from the traditional risk neutral world in currency Y to the tradional risk neutral world in currency X, the growth rate of a variable V increases by rsV sS where sVis the volatility of V, sSis the volatility of the exchange rate (units of Y per unit of X) andr is the correlation between the two rsV sS

  35. Siegel’s Paradox

  36. When is a Convexity, Timing, or Quanto Adjustment Necessary • A convexity or timing adjustment is necessary when interest rates are used in a nonstandard way for the purposes of defining a payoff • No adjustment is necessary for a vanilla swap, a cap, or a swap option

  37. Interest Rate Derivatives: Model of the Short Rate Chapter 30

  38. Term Structure Models • Black’s model is concerned with describing the probability distribution of a single variable at a single point in time • A term structure model describes the evolution of the wholeyield curve

  39. The Zero Curve • The process for the instantaneous short rate,r, in the traditional risk-neutral world defines the process for the whole zero curve in this world • If P(t, T ) is the price at time t of a zero-coupon bond maturing at time T where is the average r between times t and T

  40. Equilibrium Models

  41. Interest rate HIGH interest rate has negative trend Reversion Level LOW interest rate has positive trend Mean Reversion (Figure 30.1, page 683)

  42. Zero Rate Zero Rate Maturity Maturity Zero Rate Maturity Alternative Term Structuresin Vasicek & CIR (Figure 30.2, page 684)

  43. Equilibrium vs No-Arbitrage Models • In an equilibrium model today’s term structure is an output • In a no-arbitrage model today’s term structure is an input

  44. Developing No-Arbitrage Model for r A model for r can be made to fit the initial term structure by including a function of time in the drift

  45. Ho-Lee Model dr = q(t)dt + sdz • Many analytic results for bond prices and option prices • Interest rates normally distributed • One volatility parameter, s • All forward rates have the same standard deviation

  46. Diagrammatic Representation of Ho-Lee (Figure 30.3, page 687) r Short Rate r r r Time

  47. Hull-White Model dr = [q(t ) – ar ]dt + sdz • Many analytic results for bond prices and option prices • Two volatility parameters, a and s • Interest rates normally distributed • Standard deviation of a forward rate is a declining function of its maturity

  48. Diagrammatic Representation of Hull and White (Figure 30.4, page 688) r Short Rate r Forward Rate Curve r r Time

  49. Black-Karasinski Model (equation 30.18) • Future value of r is lognormal • Very little analytic tractability

  50. Options on Zero-Coupon Bonds (equation 30.20, page 690) • In Vasicek and Hull-White model, price of call maturing at T on a bond lasting to s is LP(0,s)N(h)-KP(0,T)N(h-sP) • Price of put is KP(0,T)N(-h+sP)-LP(0,s)N(h) where

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