Quote...But if that is your theta, all your theta would be zero.Yes, the example was a bit sloppy in that there will not be exact offsetting, it was intended to make clear why the time change in a bond price on its own was not so relevant in practice (there is always something that largely offsets it). That was your original question I think.The martingale condition was something that expressed the example in more concrete terms -- I just alluded to a fairly standard no-arbitrage condition that you would have as a result of constructing a consistent theory. I didn't really intend to invent a "theta". QuoteConsider the Black-Scholes equation for a vanilla call [$]C[$] for a deterministic interest rate [$]r[$]\[(A-r)C=0,\]If you replace A with the generator of the diffusion (i.e., without the d/dt term) then I think you have theta. :)Quote...judgements mind you --- to the proposers of those statements. None has answered straightforwardly and almost all have been evasive! Don't you think it is strange, given what you have said? So your second warning does not apply to my attitude, does it? But of course, if you have a way other than asking questions to "figure out what they actually mean", but rather, say, stop pestering people to make their statements precise and making them more vague and perhaps even poetically ambiguous, or say practicing telepathy, do tell. I am all ears. I don't disagree with the basic approach, just its execution. If you really want information to flow, I would suggest that a) you reduce the volume of your posts (so that people will be more inclined to read them), and b) you remove the "Call me dumb" statements, which are essentially implying that the person responding is dumb. Up to you.

QuoteOriginally posted by: list1I am wondering which rate is more risk-free: 1) risk stochastic one that you call risk free r ( t ) or 2) rate which is defined by long-short bond portfolio and which differential does not have dw term?In the Vasicek paper these are shown (well, given no-arb assumptions) to be the same thing -- starting after Eq (10) -- no?

QuoteOriginally posted by: amikeQuoteOriginally posted by: list1I am wondering which rate is more risk-free: 1) risk stochastic one that you call risk free r ( t ) or 2) rate which is defined by long-short bond portfolio and which differential does not have dw term?In the Vasicek paper these are shown (well, given no-arb assumptions) to be the same thing -- starting after Eq (10) -- no?Let us briefly recall BS pricing idea. We construct hedged portfolio finding delta. The option price still undefined, it is only known of the class function eligible for call option price, ie deterministic smooth functions. Now they assumed that risk free is a known constant and it is also good to assume that it is deterministic continuous function. They put that change in value of their portfolio is the known risk free rate. That looks perfect. In the Vasicek paper he followed the same steps but he calls stochastic rate risk free. His hedged portfolio at initial moment is fully deterministic and he think that deterministic change in the value is equal to stochastic 'risk free' rate only because he called it risk free. Deterministic change in the value to put be equal to stochastic rate even if we call it risk free looks very confusion. It is clear that saying risk free stochastic rate implies only no default feature. Vasicek portfolio is defined at t and we can calculate the change of the value on[ t , t + dt) interval without risk free stochastic or deterministic rate r ( t ). The r is an excess information.

QuoteOriginally posted by: amikeThe martingale condition was something that expressed the example in more concrete terms -- I just alluded to a fairly standard no-arbitrage condition that you would have as a result of constructing a consistent theory. I didn't really intend to invent a "theta". That is confusing. I have already written down the zero coupon bond PDE in my second post. and I have been citing Vasicek's paper which lays out the derivation of the exact same bond PDE numerous times. That is not the point of our conversation, is it? I thought you would like to construct a particular theta I called [$]\Theta_{\mathrm{amike}}[$] and show it is zero. It contradicts what you said earlier "I think a mathematically precise version of "theta=0 for a zero coupon bond" would be something like: the price of a ZCB relative to the (risk-free) moneymarket numeraire: M_t=P_t(T)/B_t, is a martingale: E[M_s|t]=M_t, s>=t, and therefore theta could be the time derivative of this expectation, which must therefore be zero". QuoteOriginally posted by: amikeQuote... QuoteConsider the Black-Scholes equation for a vanilla call [$]C[$] for a deterministic interest rate [$]r[$]\[(A-r)C=0,\]If you replace A with the generator of the diffusion (i.e., without the d/dt term) then I think you have theta. :)Wowo, hold a minute. Are you saying your theta is the following (A without the [$]\frac{\partial}{\partial t}[$] term) ?\[\Theta_{\mathrm{amike}}=\Big(r\frac{\partial}{\partial r}+\frac{\sigma^2}2\frac{\partial^2 }{\partial r^2}-r\Big)C\]So by the Black-Scholes equation or the martingale argument, we have\[\Theta_{\mathrm{amike}}=-\frac{\partial C}{\partial t}=-\Theta.\]just the negative of the traditional delta. Presumably you would want to apply the same to the bond PDE, right? After all that exercise, we come back to square one [$]\frac{\partial C}{\partial t}[$]? This just negates the whole point of your example, i.e., the point of [$]d\mathbf E[M_t][$]. On top of that, [$]\frac{\partial C}{\partial t}[$] and [$]\frac{\partial P}{\partial t}[$] are DEFINITELY NOT zero, contradicting again your attempt to construct "a mathematically precise version of 'theta=0 for a zero coupon bond' ".Maybe it is time for you to write down YOUR theta in a clear and definitely manner once and for all, instead of me trying so hard to divine.

@amike:Do you agree?

QuoteOriginally posted by: bearishAt the risk of saying something stupid again (yes, it has happened before), the passage of time is not a risk. The only reason theta is on the list of things you keep track of as an option trader (and no, bond traders do not keep track of their theta -- don't try to ask them, they will object to any Greek stuff) is that it is a useful proxy for your gamma, with a bit of carry mixed in. To an option trader, theta is the risk of nothing happening, whereas gamma is the risk/benefit (up to sign) of too much happening. Bond traders, on the other hand, care about things like yield, carry, roll down, key rate durations, principal component exposures (if advanced), various kinds of spread exposures (e.g. swaps vs government bond yields), etc.@bearish and @ThinkDifferent:I have many questions.Suppose we are under the setting of the bond PDE as specified by the Vasicek paper, which you accepted as one of the best in finance. That means we have1) What is theta for bond?As amike has by his last post in effect, in logic --- though I am still awaiting his explicit response --- acknowledged the "new theta" is still [$]\frac{\partial P}{\partial t}[$]. So that is certainly not zero as he and others had claimed originally. Do you agree with this definition? If you -- or anyone else -- do not agree, please write down your own definition of theta and let's discuss.2) If the theta is $[$]\frac{\partial P}{\partial t}[$]$, is theta for bond useful?I do not know. From the theory of hedging, I would think it is as useful (or not useful) as theta in the option. Sure we know that all the Greeks are constrained linearly as the consequence of the PDE of an option or a bond. You say theta is a proxy of gamma. It is so only when the interest rate and the market risk premium is small. Let us assume the interest rate is zero, so your statement theta is a proxy of gamma is exactly true and they are essentially the same. Now how does your statement "to an option trader, theta is the risk of nothing happening, whereas gamma is the risk/benefit (up to sign) of too much happening" reconcile with the previous statement? 3) Are the greeks as useful for bonds as options?You say they are not. I would like to ask why. I thought bond traders would like to immunize their bond portfolios. Is immunization not just hedging? If it is, since the bond PDE look exactly like the option PDE, why is there a difference between bond hedging and option hedging as big as you say?4) What is the definition of yield and carry (even though I have said at the very top, but I have to emphasize again that we are under the stochastic short rate setting as specified by the Vasicek paper)?5) I think I can explain "roll down" in our setting. I will ask more questions after I formulate it clearly.6) Do "key rate durations, principal component exposures" basically cover the same thing, as the dimension of the Brownian motion in the short rate SPDE?7) I think "various kinds of spread exposures" can be treated by adding the factor of credit risk. So we should for now ignore this for the sake clarity. Do you agree?Why do we want to complicate the discussion as we have not settled on the previous questions yet. There are myriad of other complication you can add, counter party risk, collateral risk, etc. Why do we want to confuse ourselves?

lovenatalya, your questions make it reasonably you're still ignoring what everyone's been telling you all along... Let me try it again.1) There is no theta for a bond. Period. You defining a particular partial as "theta" doesn't magically change the language and the underlying framework used by practitioners. Given that your original post was a question, it's rather strange to see that you keep denying this and insisting on your interpretation.2) No, it's not useful.3) No, the Greeks are not useful for a bond portfolio, since there's a native, arguably more appropriate framework available.We can leave the other questions for another time...

QuoteOriginally posted by: Martinghoullovenatalya, your questions make it reasonably you're still ignoring what everyone's been telling you all along... Let me try it again.1) There is no theta for a bond. Period. You defining a particular partial as "theta" doesn't magically change the language and the underlying framework used by practitioners. Given that your original post was a question, it's rather strange to see that you keep denying this and insisting on your interpretation.2) No, it's not useful.3) No, the Greeks are not useful for a bond portfolio, since there's a native, arguably more appropriate framework available.We can leave the other questions for another time...I know people are repeating their mantra "there is no theta". They give no detailed reasoning and explanation of what they actually mean. Rather than I ignoring other people's statements, on the contrary, people are ignoring my questions all along. Aside from amike, almost all evade my direct questions.Let us deal with the first question first. Could you please answer my following questions one by one directly for once?1) I do not understand what you mean by "there is no theta". I have defined what I meant by theta (I don't want to repeat the millionth time the exact mathematical definition). Do you want a different definition for it? If so, what is it? Otherwise, you acknowledge my definition, is that correct? If so, by "no theta" do you mean the theta is zero?

QuoteIf it is, since the bond PDE look exactly like the option PDE, why is there a difference between bond hedging and option hedging as big as you sayYou love those PDEs, don't you...? Let me try to understand the logic here. You are saying that if two products are valued with the same PDE then the difference between their treatments (hedging, etc.) should be small? You do realize that, generally, the pricing PDE is the same for all products and it is boundary/terminal/earlyexercise conditions that make a difference?

QuoteOriginally posted by: ThinkDifferentQuoteIf it is, since the bond PDE look exactly like the option PDE, why is there a difference between bond hedging and option hedging as big as you sayYou love those PDEs, don't you...? Let me try to understand the logic here. You are saying that if two products are valued with the same PDE then the difference between their treatments (hedging, etc.) should be small? You do realize that, generally, the pricing PDE is the same for all products and it is boundary/terminal/earlyexercise conditions that make a difference?You see, the reason I am using the PDE's is only trying to pin down what we are discussing on a common simple platform. I am fully aware the PDE's are an approximation to what transpires in reality. But the scientific method is to start from a simplified setting, which is what is called a model. PDE's are but one such model. If you think another model is even better, I am all ears. We can always add more complexities afterwards. Do you agree? Look, we are still arguing about even the basics after all these conversation.I am aware when I wrote that sentence, there is a chance it will be questioned regarding its exact meaning. Yes, of course it goes without saying that the solution/hedging depends on the boundary (the early exercise is a particular form, the free boundary condition, of boundary condition) and initial condition. What I am saying is that there is a general statement (we can formulate it as a theorem) about the dynamic hedging, encapsulated by the notion of delta hedging the size of which depending on the bondary/initial condition, that is true for a size infinity set of boundary/initial conditions. The boundary/initial conditions of the European options and (zero coupon) bonds are but two simple elements of this infinite sized set. Other people on this topic seem to deny this, saying dynamic hedging is not needed at all for bonds --- not just the delta hedging sizes are different for bonds and European options. I do not understand and am trying to enquire why. If you do understand, please do tell.

QuoteOriginally posted by: lovenatalyaQuoteOriginally posted by: ThinkDifferentQuoteIf it is, since the bond PDE look exactly like the option PDE, why is there a difference between bond hedging and option hedging as big as you sayYou love those PDEs, don't you...? Let me try to understand the logic here. You are saying that if two products are valued with the same PDE then the difference between their treatments (hedging, etc.) should be small? You do realize that, generally, the pricing PDE is the same for all products and it is boundary/terminal/earlyexercise conditions that make a difference?You see, the reason I am using the PDE's is only trying to pin down what we are discussing on a common simple platform. I am fully aware the PDE's are an approximation to what transpires in reality. But the scientific method is to start from a simplified setting, which is what is called a model. PDE's are but one such model. If you think another model is even better, I am all ears. We can always add more complexities afterwards. Do you agree? Look, we are still arguing about even the basics after all these conversation.I am aware when I wrote that sentence, there is a chance it will be questioned regarding its exact meaning. Yes, of course it goes without saying that the solution/hedging depends on the boundary (the early exercise is a particular form, the free boundary condition, of boundary condition) and initial condition. What I am saying is that there is a general statement (we can formulate it as a theorem) about the dynamic hedging, encapsulated by the notion of delta hedging the size of which depending on the bondary/initial condition, that is true for a size infinity set of boundary/initial conditions. The boundary/initial conditions of the European options and (zero coupon) bonds are but two simple elements of this infinite sized set. Other people on this topic seem to deny this, saying dynamic hedging is not needed at all for bonds --- not just the delta hedging sizes are different for bonds and European options. I do not understand and am trying to enquire why. If you do understand, please do tell.In your view, what is the theta of a stock?

There is no subject to discuss theta of the bond. lovenatalya you use a formula which you call definition of the theta. Of course this is your right. Other point is that before yours theta no one used theta for the bond. This is a fact which can be accepted or rejected. In order to reject opposite point of view it is sufficient to introduce example where someone used term theta similar to yours. If you could not find an example then the best to try use other letter or you can say I wish to introduce my own notation theta and you do not care whether somebody used or not used theta in similar situation.

QuoteOriginally posted by: bearishIn your view, what is the theta of a stock?Theta of a stock is not defined. In term of the SPDE of stock [$]S(t)[$], [$]\frac{dS}{dt}[$] does not exist, because Brownian path is nowhere differentiable almost surely.I am looking forward to your answers to my previous questions. I seem to sense a reluctance in answering my questions.

QuoteOriginally posted by: list1There is no subject to discuss theta of the bond. lovenatalya you use a formula which you call definition of the theta. Of course this is your right. Other point is that before yours theta no one used theta for the bond. This is a fact which can be accepted or rejected. In order to reject opposite point of view it is sufficient to introduce example where someone used term theta similar to yours. If you could not find an example then the best to try use other letter or you can say I wish to introduce my own notation theta and you do not care whether somebody used or not used theta in similar situation.I have defined the notion "theta OF a BOND". Why is that a problem? Nobody except perhaps you objected to the definition. Are you saying I can not define a new name if nobody has before? Exactly because nobody has defined this new name before, there is no problem whatsoever but all the freedom in the world for me to define and use it. Problem only arises, when somebody has already used the same name before but in a different way. Only then I have to clarify or use a different name if my definition and the existing one contradict.A definition is not a view (proposition).Do you not agree with my logic?

QuoteOriginally posted by: lovenatalyaQuoteOriginally posted by: bearishIn your view, what is the theta of a stock?Theta of a stock is not defined. In term of the SPDE of stock [$]S(t)[$], [$]\frac{dS}{dt}[$] does not exist, because Brownian path is nowhere differentiable almost surely.I am looking forward to your answers to my previous questions. I seem to sense a reluctance in answering my questions.OK, I hate to do this, but the price of any finite maturity bond in the standard model equally well has a Brownian path. As does, incidentally, the price of a call option in the Black Scholes model. So that's not it.