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?

  • $\begingroup$ The question is ill-posed. It is unclear what "bond time value risk" means and AFAIK it is not defined in any textbook. How can you "hedge" against the passage of time? What risk are we talking about? The question seems to mix concepts from option theory (theta, gamma) and from bond theory (duration, convexity) in a confusing way. $\endgroup$ – Alex C Dec 30 '16 at 4:55
  • $\begingroup$ @AlexC: The phrase "time value risk" may be ill-worded. I will examine it later. Your critique on "mix concepts from option theory..." is not an issue since the zero coupon bond is a special option on the short rate $r$ with constant $1$ as its terminal condition. It is shown in detail in my answer below. $\endgroup$ – Hans Dec 30 '16 at 5:17

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.

  • $\begingroup$ "there is no volatility input to calculate bond prices" is false. There is volatility in bond price. See my answer assuming diffusive process for the short rate. $\endgroup$ – Hans Dec 30 '16 at 17:07
  • $\begingroup$ In all practical applications, the vega of a bond is zero. There is no doubt about that. Ask any bond trader or risk manager. $\endgroup$ – dm63 Dec 30 '16 at 20:35
  • $\begingroup$ Are you saying the short rate is deterministic? What about the interest rate options? What is the use of the slew of stochastic interest rate models, Hull-White, CIR, Ho-Lee, Vasicek, LIBOR market? $\endgroup$ – Hans Dec 30 '16 at 20:49
  • $\begingroup$ I'm just saying that when vega of any instrument is calculated within these models, it is done while leaving forward interest rates unchanged. That is the convention. Yes, it contradicts the idea that a bond generated from a short rate model will have vega. $\endgroup$ – dm63 Dec 30 '16 at 21:44

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.

  • $\begingroup$ I don't understand how it follows from dP/dt=r P that "we need one more bond in the proxy portfolio". Also, what are the properties (duration, etc.) of this "other bond"? $\endgroup$ – noob2 May 2 '16 at 17:10
  • 1
    $\begingroup$ @noob2: Thank you for your question. The properties of the two hedging bonds are to make the whole portfolio duration zero and convexity zero. But I need to think twice about the derivation. I have edited the wording of my answer. I was mainly answering why theta of the bond price was not considered. The need for two bonds under discrete time hedging is admittedly not clear yet. I will think about it and get back on that. $\endgroup$ – Hans May 3 '16 at 0:35

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