1、Chapter 30Convexity, Timing, and Timing, and Quanto Adjustments,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,1,Forward Yields and Forward Prices,We define the forward yield on a bond as the yield calculated from the forward bond priceThere is a non-linear relatio
2、n between bond yields and bond pricesIt follows that when the forward bond price equals the expected future bond price, the forward yield does not necessarily equal the expected future yield,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,2,Relationship Between Bond
3、 Yields and Prices (Figure 30.1, page 694),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,3,Convexity Adjustment for Bond Yields (Eqn 30.1, p. 695),Suppose a derivative provides a payoff at time T dependent on a bond yield, yT observed at time T. Define: G(yT) : pr
4、ice of the bond as a function of its yield y0 : forward bond yield at time zerosy : forward yield volatilityThe expected bond price in a world that is FRN wrt P(0,T) is the forward bond priceThe expected bond yield in a world that is FRN wrt P(0,T) is,Options, Futures, and Other Derivatives, 9th Edi
5、tion, Copyright John C. Hull 2014,4,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 (approximately)where G(y) defines the relationship between price and yield for a bond lasting between T and T+t that pays a coupon e
6、qual to the forward swap rate,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,5,Example 30.1 (page 696),An instrument provides a payoff in 3 years equal to the 1-year zero-coupon rate multiplied by $1000Volatility is 20%Yield curve is flat at 10% (with annual compou
7、nding)The convexity adjustment is 10.9 bps so that the value of the instrument is 101.09/1.13 = 75.95,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,6,Example 30.2 (Page 696-697),An instrument provides a payoff in 3 years = to the 3-year swap rate multiplied by $10
8、0Payments are made annually on the swapVolatility is 22%Yield curve is flat at 12% (with annual compounding)The convexity adjustment is 36 bps so that the value of the instrument is 12.36/1.123 = 8.80,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,7,Timing Adjustme
9、nts (Equation 30.4, page 698),The expected value of a variable, V, in a world that is FRN wrt P(0,T*) is the expected value of the variable in a world that is FRN wrt P(0,T) multiplied by where R is the forward interest rate between T and T* expressed with a compounding frequency of m, sR is the vol
10、atility of R, R0 is the value of R today, sV is the volatility of F, and r is the correlation between R and V,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,8,Example 30.3 (page 698),A derivative provides a payoff 6 years equal to the value of a stock index in 5 ye
11、ars. The interest rate is 8% with annual compounding1200 is the 5-year forward value of the stock indexThis is the expected value in a world that is FRN wrt P(0,5)To get the value in a world that is FRN wrt P(0,6) we multiply by 1.00535The value of the derivative is 12001.00535/(1.086) or 760.26,Opt
12、ions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,9,Quantos(Section 30.3, page 699-702),Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currencyExample: contract providing a payoff of ST K dollars ($) where
13、 S is the Nikkei stock index (a yen number),Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,10,Diff Swap,Diff swaps are a type of quantoA floating rate is observed in one currency and applied to a principal in another currency,Options, Futures, and Other Derivatives
14、, 9th Edition, Copyright John C. Hull 2014,11,Quanto Adjustment (page 700),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
15、the correlation between V and W.,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,12,Example 30.4 (page 700),Current value of Nikkei index is 15,000This gives one-year forward as 15,150.75Suppose the volatility of the Nikkei is 20%, the volatility of the dollar-yen e
16、xchange rate is 12% and the correlation between the two is 0.3The one-year forward value of the Nikkei for a contract settled in dollars is 15,150.75e0.3 0.20.121 or 15,260.23,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,13,Quantos continued,When we move from the
17、 traditional risk neutral world in currency Y to the tradional risk neutral world in currency X, the growth rate of a variable V increases byrsV sSwhere sV is the volatility of V, sS is the volatility of the exchange rate (units of Y per unit of X) and r is the correlation between the two rsV sS,Opt
18、ions, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,14,Siegels Paradox,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,15,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 payoffNo adjustment is necessary for a vanilla swap, a cap, or a swap option,Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull 2014,16,
