Question

Suppose that the payoff from a derivative will occur in 10 years and will equal the 3-year
US dollar swap rate for a semiannual-pay swap observed at that time applied to a certain
principal. Assume that the swap yield curve (used for discounting) is flat at 8%
(semiannually compounded) per annum in dollars and 3% (semiannually compounded)
in yen. The forward swap rate volatility is 18%, the volatility of the 10-year ‘‘yen per
dollar’’ forward exchange rate is 12%, and the correlation between this exchange rate
and US dollar interest rates is 0.25. What is the value of the derivative if (a) the swap rate
is applied to a principal of $100 with a dollar payoff and (b) it is applied to 100 million
yen with a yen payoff?”

Answer 1

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(a) In this case we must make a convexity adjustment to the forward swap rate. Define 4 100 G(y) = + = (1+y/2) (1 + y/2) so that 6 2i 300 G'(y) = -) + 1-1 (1+ y/2)+1 (1 + y/2) + i(i+1) 1050 G"(y) = EI (1 + y/2) +2 (1 + y/2) G'(0.08) =-262.11 and G"(0.08) = 853.29 so that the convexity adjustment is 1 -x0.08? x0.18²x10x 853.29 -0.00338 262.11 The adjusted forward swap rate is 0.08 +0.00338 = 0.08338 and the value of the derivative in millions of dollars is 0.08338 x 100 -= 3.805 1.0420 (b) When the swap rate is applied to a yen principal we must make a quanto adjustment in addition to the convexity adjustment. From Section 30.3 this involves multiplying the forward swap rate by e -0.25x0.120.18x10 = 0.9474. (Note that the correlation is the correlation between the dollar per yen exchange rate and the swap rate. It is therefore -0.25 rather than +0.25.) The value of the derivative in millions of yen is 0.08338 x 0.9474x100 = 5.865 1.01520