SERVING THE QUANTITATIVE FINANCE COMMUNITY

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lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### 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?

bearish
Posts: 5903
Joined: February 3rd, 2011, 2:19 pm

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

Since bonds are linear instruments, they have no theta. You are confused by convexity, which looks deceptively gamma-like, but unlike a proper gamma, it is the (scaled) second derivative with respect to a non-price. This is actually important, and many people are very confused on the matter.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

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

@bearish:I do not understand what you mean by bond having no Theta. As time passes the value of a bond increases, holding interest rate fixed. Theta is the time derivative of the price. For the simplest case, the zero coupon bond with 1 dollar face value and constant interest rate $r$, the price is $P(t,T)=e^{-r(T-t)}$. $\Theta=\frac{\partial P}{\partial t}=re^{-r(T-t)}$. It is defined and positive. Maybe I have misunderstood what you have said. Please clarify.
Last edited by lovenatalya on April 27th, 2016, 10:00 pm, edited 1 time in total.

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

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

you have just defined duration again. it is not a real time decay in the sense of an option position that has convexity where the "theta" is supposed to be balanced by the gamma. for a zero coupon bond that is at a discount to its par value the yield on the bond less the cost of borrowing money to buy the bond is your carry. all you have done in your equation is to define one element of the carry.
knowledge comes, wisdom lingers

Martinghoul
Posts: 3256
Joined: July 18th, 2006, 5:49 am

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

I agree w/dave. If I understood the question correctly, carry is the relevant concept.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

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

QuoteOriginally posted by: daveangelyou have just defined duration again. it is not a real time decay in the sense of an option position that has convexity where the "theta" is supposed to be balanced by the gamma. for a zero coupon bond that is at a discount to its par value the yield on the bond less the cost of borrowing money to buy the bond is your carry. all you have done in your equation is to define one element of the carry.How did I "define duration again"? Duration is $\frac{\partial P}{\partial r}$ whereas I defined theta as $\frac{\partial P}{\partial t}$. They are not the same. How does this relate to carry? Could you please elucidate your point with some mathematics?

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

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

QuoteOriginally posted by: lovenatalyaQuoteOriginally posted by: daveangelyou have just defined duration again. it is not a real time decay in the sense of an option position that has convexity where the "theta" is supposed to be balanced by the gamma. for a zero coupon bond that is at a discount to its par value the yield on the bond less the cost of borrowing money to buy the bond is your carry. all you have done in your equation is to define one element of the carry.How did I "define duration again"? Duration is $\frac{\partial P}{\partial r}$ whereas I defined theta as $\frac{\partial P}{\partial t}$. They are not the same. How does this relate to carry? Could you please elucidate your point with some mathematics?if we have to use maths for this then we don't understand it.if this is were a coupon paying bond, the change in the price from one day to the next will be the accrued interest (all else being the same). for a zero coupon bond this is just the pull to par - ie the change in the price from one day to the next will be the yield for one day. it will also cost you to fund the position for one day. the difference is your carry cost.
knowledge comes, wisdom lingers

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

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

@daveangel:So you agree I was not "defining duration again". I think I have resolved the issue as described in the following derivation. As for the carry cost issue, you mentioned yield. Do you define yield as $P(r,t,T)=P(r,T,T)e^{-y(r,t,T)(T-t)}$? Then the yield is $y(r,t,T)=\frac{\partial P}{P\partial t}=\Theta$. However, this is just definition and does not help answer my question for how one should immunize (hedge/replicate) the bond (portfolio). In my analysis below, there is a relationship between $\Theta$ or yield and duration, convexity and the short rate. The relationship is akin to the Greeks between the Black-Scholes model as I was expecting in my original post. With regard to the issue of carry, in terms of what I have derived, would it be the term involving duration and convexity in the equation below for $\Theta$ or yield, duration, convexity and the short rate?The theta for bond acts exactly like the theta in option.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 $P(r,t,T)$ 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 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 a short hedging time period, the portfolio can be represented by one bond the duration of which matches that of the bond portfolio and a cash account. Now for slightly longer hedging time period, we require $\frac{\partial^2 P}{\partial r^2}=0$ so that at the next time step we would need to adjust little to annihilate the duration, and the bond PDE becomes $\displaystyle\frac{\partial P}{\partial t}=rP$. To that end, we need one more bond for the proxy portfolio.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

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

lovenatalya, let talk about bond which price at t is P ( r , t , T ). It seems P ( r , T , T ) is a known constant which does not depend on r.When you say " Using argument similar to that deriving the Black-Scholes equation, we derive the PDE for the bond price" it is not clear what do you mean. In BSE we have an option that you probably interpret as P given equation for dP but dr does not a price of underlying. If we follow BSE derivation we should define a $contract$ written on r and construct a portfolio of long P and short portion of $contract$ on r and consider change in the value of this portfolio to verify that risk term with dB is eliminated. If that is done we might arrive at a version of BSE. It might be you have also another way to derive a version of BSE? r itself could not be used to eliminate risk we need a contract

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

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

@list1:Good question! You are right that $r$ is not a traded asset and we need a traded asset that depends on $r$. The underlying in the BSE is replaced by another bond with a different maturity. The portfolio of the hedge portfolio of these two bonds has value$$\Pi=P_1-\Delta P_2$$and $$\Delta=\frac{\frac{\partial P_1}{\partial r}}{\frac{\partial P_2}{\partial r}}$$annihilates the stochastic term of $dB$ just as in the derivation of BSE. Now$$d\Pi=r\Pi dt$$Collect all $P_1$ terms on one side and all $P_2$ terms on the other side of the equation. We will see the two sides can not depend on the maturity $T$. If we write this $T$ independent function as $\mu-\lambda(r,t)\sigma$, we obtain the diffusion PDE for the bond similar to the BSE.
Last edited by lovenatalya on April 28th, 2016, 10:00 pm, edited 1 time in total.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

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

QuoteOriginally posted by: lovenatalya@list1:Good question! You are right that $r$ is not a traded asset and we need a traded asset that depends on $r$. The underlying in the BSE is replaced by another bond with a different maturity. The portfolio of the hedge portfolio of these two bonds has value$$\Pi=P_1-\Delta P_2$$and $$\Delta=\frac{\frac{\partial P_1}{\partial r}}{\frac{\partial P_2}{\partial r}}$$annihilates the stochastic term of $dB$ just as in the derivation of BSE. Now$$d\Pi=r\Pi dt$$Collect all $P_1$ terms on one side and all $P_2$ terms on the other side of the equation. We will see the two sides can not depend on the maturity $T$. If we write this $T$ independent function as $\mu-\lambda(r,t)\sigma$, we obtain the diffusion PDE for the bond similar to the BSE.if one follows BS' portfolio construction then $P_2$ should be underlying of the $P_1$ . In our case in order to eliminate risk term we seem need to assume that sigma in both bonds is the same. Intuitively this might be wrong but who knows everything can be theoretically assumed. On the other hand then we should assume that sigma of the bond for any maturity is the same fixed constant. That is not obvious fact too.
Last edited by list1 on April 28th, 2016, 10:00 pm, edited 1 time in total.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

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

"If one follows BS' portfolio construction then $P_2$ should be underlying of the $P_1$." Essentially this is the case and I thought of saying exactly that in my last post. "In our case in order to eliminate risk term we seem need to assume that sigma in both bonds is the same." I assume by sigma you mean the volatilities of the two bonds. Why should that be the case? The ratio of the volatility of the two assets is $\Delta$ which is not necessarily $1$, just as in BS. See the $\Delta$ express in my last post. However, all the bonds do depend on the single short rate process with volatility $\sigma$. This should be intuitively obvious. What would have been intuitively wrong is to assume the contrary that different bonds depend on different short rate processes. Do you not agree?
Last edited by lovenatalya on April 29th, 2016, 10:00 pm, edited 1 time in total.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

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

$\Delta$ is fixed at t while P$_2( u )$ is a variable on [ t , t + dt). You can look athttp://www.slideshare.net/list2do/bs-consept-d ... g-57300457 for details.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

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

@list1:Does your argument carry through for the derivation of Black-Scholes equation?
Last edited by lovenatalya on April 29th, 2016, 10:00 pm, edited 1 time in total.

bearish
Posts: 5903
Joined: February 3rd, 2011, 2:19 pm

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

QuoteOriginally posted by: lovenatalya@list1:Does your argument carry through for the derivation of Black-Scholes equation?You are going down the rabbit hole...