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We want to use the Duration to convert forward price volatility to yield volatility with following relation $$\dfrac{\Delta F_B}{F_B} = -D \Delta y_F.$$ But how to calculate the the yield of forward bond price? if we know the zero curve $Z(t^*,T).$ In the example it seems directly use the rate of zero curve. I only know the how to calculate the yield of a bond with following relation: $$\dfrac{\Delta B}{B} = -D \Delta y$$ here $B$ is the bond price.

Example: Consider a European put option on a 10-year bond with a principal of 100.

The coupon is 8% per year payable semiannually.

The life of the option is 2.25 years

The strike price of the option is 115.

The forward yield volatility is 20%.

The zero curve is flat at 5% with continuous compounding.

Here is some computation. What I don't understand is the yield with semiannual compounding $5.0630\%.$ I know how to calculate the yield of a coupon bond, but here how to obtain the yield of a forward $5.0630\%?$

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You need to convert from continuous compounding to simple compounding. With a continuous rate $r_c$, \$1 grows in one year to $exp(r_c)$. With semi-annual compounding at an annualized rate $r_s$ it grows to $(1+r_s/2)^2$. Equating the two you can convert one into an equivalent yield on the other.

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  • $\begingroup$ thanks, but why can we regard the The zero curve is flat at 5% as the yield of forward price? $\endgroup$ – A.Oreo Aug 15 '17 at 1:06
  • $\begingroup$ pls see my update. Usually how to calculate the yield of a bond forward? I only know for the bond price and its yield. $\endgroup$ – A.Oreo Aug 15 '17 at 1:16
  • $\begingroup$ If the zero curve is flat at 5%, then the yields of all bonds and indeed the forward yields of all bonds are 5%. $\endgroup$ – dm63 Aug 15 '17 at 3:40
  • $\begingroup$ if it's not flat, e.g $z(t^*,T),$ then how to calculate the yield? $\endgroup$ – A.Oreo Aug 15 '17 at 3:59
  • $\begingroup$ Given $z(t, T_1)$ and $z(t, T_2)$ it can be easily calculated. E.g. In a continuous setting the forward rate $r_F$ should satisfy (dropping the time t dependency): $\exp(r_{T_1}T_1)\exp(r_F(T_2-T_1) = \exp(r_{T_2}T_2)$. Here the $r_{T_i}$ are just the yields corresponding to the zeros. Your question and edit are a bit all over the place though. Maybe think a bit more about what it is exactly you don't understand? $\endgroup$ – Bram Aug 15 '17 at 21:01

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