Chapter 29Quanto, Timing, and Convexity Adjustments1期权期货及其衍生品第29弹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 relation 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 yield2期权期货及其衍生品第29弹Relationship Between Bond Yields and Prices (Figure 29.1, page 669) BondPriceYieldY3B 1Y1Y2B 3B 23期权期货及其衍生品第29弹Convexity Adjustment for Bond Yields (Eqn 29.1, p. 670)Suppose a derivative provides a payoff at time T dependent on a bond yield, yT observed at time T. Define: G(yT) : price 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 4期权期货及其衍生品第29弹Convexity Adjustment for Swap RateThe 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 equal to the forward swap rate5期权期货及其衍生品第29弹Example 29.1 (page 671)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 compounding)The convexity adjustment is 10.9 bps so that the value of the instrument is 101.09/1.13 = 75.956期权期货及其衍生品第29弹Example 29.2 (Page 671-672)An instrument provides a payoff in 3 years = to the 3-year swap rate multiplied by $100Payments are made annually on the swapVolatility is 22%Yield curve is flat at 12% (with annual compounding)The convexity adjustment is 36 bps so that the value of the instrument is 12.36/1.123 = 8.807期权期货及其衍生品第29弹Timing Adjustments (Equation 29.4, page 673)The expected value of a variable, V, in a world that is FRN wrt P(0,T*) is the expected value of the variable in a world that is FRN wrt P(0,T) multiplied by where R is the forward interest rate between T and T* expressed with a compounding frequency of m, sR is the volatility of R, R0 is the value of R today, sV is the volatility of F, and r is the correlation between R and V 8期权期货及其衍生品第29弹Example 29.3 (page 673)A derivative provides a payoff 6 years equal to the value of a stock index in 5 years. The interest rate is 8% with annual compounding1200 is the 5-year forward value of the stock indexThis is the expected value in a world that is FRN wrt P(0,5)To get the value in a world that is FRN wrt P(0,6) we multiply by 1.00535The value of the derivative is 12001.00535/(1.086) or 760.26 9期权期货及其衍生品第29弹Quantos(Section 29.3, page 674)Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currencyExample: contract providing a payoff of ST K dollars ($) where S is the Nikkei stock index (a yen number)10期权期货及其衍生品第29弹Diff SwapDiff swaps are a type of quantoA floating rate is observed in one currency and applied to a principal in another currency11期权期货及其衍生品第29弹Quanto Adjustment (page 675)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. 12期权期货及其衍生品第29弹Example 29.4 (page 675)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 exchange 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 15,260.2313期权期货及其衍生品第29弹Quantos continuedWhen 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 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 sS14期权期货及其衍生品第29弹Siegels Paradox15期权期货及其衍生品第29弹When is a Convexity, Timing, or Quanto Adjustment NecessaryA 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 option16期权期货及其衍生品第29弹