Create Presentation
Download Presentation

Download Presentation
## Interest Rate Derivatives: The Standard Market Models

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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 • 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**Approaches to PricingInterest Rate Options**• Use a variant of Black’s model • Use a no-arbitrage (yield curve based) model**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**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**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**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**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)**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**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**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**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**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**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**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**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**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**Quanto, Timing, and Convexity Adjustments**Chapter 29**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**Bond**Price B3 B2 B1 Yield Y3 Y2 Y1 Relationship Between Bond Yields and Prices (Figure 29.1, page 668)**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**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**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**Example 29.2 (Page 670-671)**• An instrument provides a payoff in 3 years = to the 3-year swap rate multiplied by $100 • Payments are made annually on the swap • Volatility is 22% • Yield curve is flat at 12% (with annual compounding) • The convexity adjustment is 36 bps so that the value of the instrument is 12.36/1.123 = 8.80**Timing Adjustments (Equation 29.4, page 672)**The expected value of a variable, V, in a world that is FRN wrt P(0,T*) is the expected value of the variable in a world that is FRN wrt P(0,T) multiplied by where R is the forward interest rate between T and T*expressed with a compounding frequency of m, sR is the volatility of R, R0 is the value of R today, sV is the volatility of F, andris the correlation betweenRandV**Example 29.3 (page 672)**• A derivative provides a payoff 6 years equal to the value of a stock index in 5 years. The interest rate is 8% with annual compounding • 1200 is the 5-year forward value of the stock index • This is the expected value in a world that is FRN wrt P(0,5) • To get the value in a world that is FRN wrt P(0,6) we multiply by 1.00535 • The value of the derivative is 1200×1.00535/(1.086) or 760.26**Quantos(Section 29.3, page 673)**• Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currency • Example: contract providing a payoff of ST – K dollars ($) where S is the Nikkei stock index (a yen number)**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**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.**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**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**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**Interest Rate Derivatives: Model of the Short Rate**Chapter 30**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**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**Interest**rate HIGH interest rate has negative trend Reversion Level LOW interest rate has positive trend Mean Reversion (Figure 30.1, page 683)**Zero Rate**Zero Rate Maturity Maturity Zero Rate Maturity Alternative Term Structuresin Vasicek & CIR (Figure 30.2, page 684)**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**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**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**Diagrammatic Representation of Ho-Lee (Figure 30.3, page**687) r Short Rate r r r Time**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**Diagrammatic Representation of Hull and White (Figure 30.4,**page 688) r Short Rate r Forward Rate Curve r r Time**Black-Karasinski Model (equation 30.18)**• Future value of r is lognormal • Very little analytic tractability**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