Bond Immunization, Time Decay, and the Short-Rate Pricing PDE

# Why Is Bond Time Value Risk Not Considered in Bond Immunization?


# Why Is Bond Time Value Risk Not Considered in Bond Immunization?












I know bond portfolio immunization includes duration and (if the hedging period is longer) convexity matching. These are equivalent to taking the first and second partial derivatives of the bond portfolio price with respect to the short rate. I wonder whether we should also look at the time value increment of the bond price, which is the time partial derivative of the bond price, just as the theta in option price. For option Greeks, Theta, Delta and Gamma are related through the valuation or in the simple setting the Black-Scholes equation. However, there does not seem to be such a relation in place for bond. Or am I mistaken?

## Answer by Hans (score 0)

https://quant.stackexchange.com/a/25702

Suppose the short rate $r$ follows the diffusive process $$dr=\mu dt+\sigma dB$$ where $B$ is the standard Brownian motion. The price of a bond portfolio $P(r(t),t,T)$ at time $t$ maturing at time $T$ follows $$dP=\frac{\partial P}{\partial t} dt+\frac{\partial P}{\partial r}dr+\frac12\frac{\partial^2 P}{\partial r^2}dr^2=\Big(\frac{\partial P}{\partial t}+ \frac12\frac{\partial^2 P}{\partial r^2}+\mu\frac{\partial P}{\partial r}\Big)dt+\sigma\frac{\partial P}{\partial r}dB.$$ Note: It is not generally true that $\frac{\partial P}{\partial t}=rP$.

Using hedging argument similar to that deriving the Black-Scholes equation, we derive the PDE for the bond price $$\frac{\partial P}{\partial t}+\frac12\sigma^2\frac{\partial^2 P}{\partial r^2}+(\mu-\lambda\sigma)\frac{\partial P}{\partial r}-rP=0,$$ where $\lambda$ is the market risk premium. Thus, just like the Greeks for the equity option pricing $$\Theta+\sigma^2C-(\mu-\lambda\sigma)D-r=0$$ where $\Theta=\frac{\partial P}{P\partial t}$, $C$ is the duration and $D$ the convexity of the bond portfolio.

Substituting the bond PDE into that of $dP$, we have $$dP=\Big(rP+\lambda\sigma\frac{\partial P}{\partial r}\Big)dt+\sigma\frac{\partial P}{\partial r} dB.$$ So when the duration of the portfolio $\frac{\partial P}{\partial r}$ is made to vanish, the total derivative $dP=rPdt$, and the portfolio becomes a cash account. Therefore, in other words, for continuous time hedging, the portfolio can be represented by one bond the duration of which matches that of the bond portfolio and a cash account.

$$P(r(t),t,T)=\mathbf E\big[e^{-\int_t^T r}\big|r(t)\big]$$ is the solution of the PDE for the zero coupon bond. We will show that the above expression implies the bond PDE. $$P(r(t),t,T)=\mathbf E\big[e^{-\int_t^sr}\mathbf E[e^{-\int_s^Tr}|r(s)]\big|r(t)\big]=\mathbf E\big[e^{-\int_t^sr}P(r(s),s,T)\big|r(t)\big]$$ $u(r(s),s):=e^{-\int_t^sr}P(r(s),s,T)$ is a martingale. Apply Ito's Lemma to $u(r(s),s)$, we have

This is essentially the derivation of a special case of the Feynman-Kac formula.

Now for discrete time hedging, we will minimize the variance of the price difference of the whole portfolio (original bond portfolio together with the hedging portfolio), we require additionally $\frac{\partial^2 P}{\partial r^2}=0$ --- this statement needs reconsideration, later --- and the bond PDE becomes $\displaystyle\frac{\partial P}{\partial t}=rP$. To that end, we need one more bond for the proxy portfolio.

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/31728

In practice people do look at the time decay of bond portfolios, as follows: Often the "carry" is calculated , which means the profit or loss over the next day making the assumption that bond yields of all maturities stay the same. This assumption sonewhat conflicts with theory, since the more likely scenario is that bond yields move to their forward yields computed the day before, but nevertheless that is what people do.

There is no concept of true option style time decay in bonds, since there is no volatility input to calculate bond prices. That's a consequence of the fact that we assume bond prices (and forward rates) are a given , so they don't move when interest rate volatility moves. Arguably that assumption could be challenged.