Can you help to decide which model to use to price bondoptions ? Advantages and disadvantages. Black Scholes offers a Put-Call parity while the Binomial model covers the smile effect, but does not have a put-call parity. Any suggestions ?regardsSiggi

I'm not sure exactly what you mean.The classic Black-Scholes formula is not suitable for bond options. It requires assuming interest rates are constant, in which case the bond will not change in price. However, with a little tinkering, you can get a model that is reasonable for undemanding uses. It has the advantage that it can be computed easily and quickly, in a simple Excel formula, and all derivatives are analytic.A simple binomial model is not going to help much. What is going to evolve at each node? A spot interest rate? Then you need a formula to relate the long-term rate to the possible spot rate paths. Still, you could put together a simple binomial model that might be more accurate than the Black-Scholes, at the cost of greater computation time and giving up analytic formula.You need at least two parameters to get any sort of bond option pricing model that is suitable for serious analysis. At the least you need to let a spot rate and long-term rate vary. Changes in interest rate volatility are also important, and they are highly correlated to changes in rates.You don't mention whether you are considering risky bonds or convertible bonds, then you will need an equity parameter as well.

The key point about pricing bond options is not whether you can include the smile or not (with most non-trivial models that's easy) but the relative maturities of the option and the underlying bond. Call them TO and TB. Obviously TO < TB, but how close are they? Is pull to par important? Can you assume that the underlying bond behaves like a stock (usually yes, if TO << TB)? Depending on these two timescales you could have different models. P

I used to be market maker in bond options for many years. What Paul is saying is the main topic here. In most bond options markets the time to maturity on the options (TO) are very short compared to the time to maturity on the underlying bonds (TB). If that is the case the pull-to par effect is not that important. The "market standard" in the most liquid bond options is Black-76. Yes I and others had access to a lot of arbitrage free yield curve models (and reasonable good understanding..) , but then (and still?) I think most bond option traders prefered the simple Black-76. For bond options with time to maturity close bond maturity (especially American style) there is a different story, here arbitrage free yield curve model is very important. Concerning the smile, yes there is some type of smile/skew, but for European: T0 small compared with TB one can use Black-76 and different vol for every maturity and strike just as in the equity market.Rule of tumb if T0/TB <1/6 then Black-76 "okay" else you have to use some type of arbitrage-free yield based or put a lot of effort in adjusting the volatility for Black-76.

>The classic Black-Scholes formula is not suitable for bond options. It requires assuming interest rates are constant, in which case the bond >will not change in price. However, with a little tinkering, you can get a model that is reasonable for undemanding uses. It has the advantage >that it can be computed easily and quickly, in a simple Excel formula, and all derivatives are analytic.In Black-76 one assume the underlying is the bond price. In that sense one indirectly assume stochastic yield. The only rate one assume is "constant" is the discount rate in the Black-76 model (to discount the option value only). For options where TO<<TB the effect of change in the discount rate has minimal effect compared with other factors....and one can also take this into account in various ways... Not a completely consistent theory, but most yield based models are neither perfect.For some prop trading relative value strategy it could naturally be valuable to spend time on some more sophisticated models even for bond options where TO<<TB.

It's not the logical inconsistency that bothers me, it's the error. I agree that with a long duration bond and a short option expiry, treating the bond price as a brownian random walk and the short-term interest rate as constant might give reasonable valuations.But consider, for example, a three month call on a five-year note. The option price volatility will depend on the five year interest rate six to eight times as much as the three month interest rate (the three month rate is much more volatile than the five year rate, but it operates on the value over a shorter time interval). For most serious purposes, ignoring that fraction of volatility is unacceptable. For a three month call on a typical stock, the volatility due to changes in underlying stock price is more than 100 times the volatility due to changes in the three month interest rates. The reason is that stocks are more volatile than bonds, not that bond prices depend explicitly on interest rates.But if the option term is shorter than three months or the underlying bond is longer than five years, my objection loses it's force.The other advantage of a consistent bond option pricing model, such as a two parameter tree, is that it can be calibrated with market instruments. Using straight Black-Scholes requires estimating the volatility parameter from historical data. That is often unsatisfactory as both the level and structure of interest rate volatility changes much more than stock market volatility.

Aron I am not sure if I understand you right but:>For most serious purposes, ignoring that fraction of volatility is unacceptable.We are actually not ignoring this fraction of volatility, or are we?okay let's assume 3 month option on 5 year note. Using the Black-76 approach the model is actually not assuming that even the short (3 month rate is constant). The 3 month rate is naturally affecting the volatility of the bond (and the volatility of the yield of the bond) so by assuming the underlying is the bond we actually take this into account indirectly. Only for the discounting of the option premium we assume the 3 month rate is constant.Very simplified this is what you would do (assuming constant vol) to price the bond option. Look at historical vol for a 5 year over a 3 month, then the effect of bond maturity getting shorter and shorter is "taken into account" in some way, not the best way but still. The highly stochastic short rate is taken into account in this approach. Not for the discounting but for the most important part.In practice naturally all types of adjustments depending on who you are, how often you hedge, how long do you plan to keep the option (are you trading vega or gamma?).Conclusion: Black-76 approach- Assume even short rate is stochastic with respect to bond price volatility.- Assume constant short rate for discounting.Not consitent, but good enough to take millions from the 4 factor guys....

Where would we all be without the 4-factor guys paying the mortgage?!I tend to think in terms of interest rates as having two roles. 1) for present valuing final payoff and 2) for the growth rate (risk-neutral) when the quantity modelled is also traded. Since the five-year note in this example is traded you don't need to worry about the five-year interest rate, only the 'spot rate' for the next three months. So it's not really any different from an option on a lognormal asset, as long as the volatility of the bond price return (not the yield) can be modelled. P

Not consistent, but good enough to take millions from the 4 factor guys.... >>I'm afraid I don't get the point. What precisely is wrong with 4 factor models? Are the people who use them not practical enough? Are the models too complicated to implement and too slow to use in real time? If someone as experienced as Collector would use 4 factor models, all things being equal, would he do any less than with Black 76? Or is it that, given the bid/ask spread, commissions, stale data, and all other practical nuisances, the improvements that they offer are simply not worth their hassle?

Hi Omar 4-factor guy.Well nothing wrong with 4-factor models, I just tried to "indicate" that more complex models not necessary are better, and that they in some circumstances actually can be worse. I just think the Black-76 model (used for interest rate options) got bad reputation by a lot of academics. Academics trying to guerilla marketing their yield factor models, and telling that Black-76 is useless because of constant interest rate, no pull to par effect.Or is the Black-76 model actually a one factor model. Some years ago I meet Professor Longstaff on a derivatives conference in Denmark. He first spent a lot of years in academia, then he moved to Wall Street for a few years (head of some interest rate derivatives research: Salmon Brothers I think?)., before he moved back to academia again, UCLA. It was very inspiring to hear what Longstaff had to say about practica. On wall street he had figured out they where considering almost every part of the yield curve as a "separate" instrument, using different input for every bond maturity, option maturity, strike etc. If I got him right (probably not) he considered this as a type of string models.By using a 1, 2, 3, or 4 factor yield curve model the whole idea is to value the whole yield curve with one "complex" model using reasonable few input parameters, to get everything consistent. Most people on Wall Street (at least how it used to be) use the same model, but with different input for every deal. So to compare a 1, 2, 3, or 4 factor model with Black-76 in practice you have to compare with what you can do with a whole string of Black-76 models. Using a string of Black-76 models will however require much more input estimates than a 4 factor model. So to use the string approach you have to be on the top of the market, knowing the liquidity of each bond, doing a lot of adjustments in vol for moneyness, option exp…..You can naturally use a whole string of 4 factor models, but isn’t the whole point to calibrate it "once" and use it for "all" options?. A whole string of 4 factor models would also be quite slow?? Also limited work published on how to calibrate yield factor models to the vol smile, and even less practical experience?I think Longstaff got inspired by his years at wall street and tried to extend this string approach to some new type of more consistent string/multi factor models. Unfortunately I haven’t got much time to study his recent papers….guess can be downloaded from the UCLA site?>If someone as experienced as Collector would use 4 factor models, all >things being equal, would he do any less than with Black 76?Defiantly YES! (Speaking for myself only)Personally I just got to the point where I think I understand 90% of how to use the Black-Scholes model in practice. I also believed the same 10 years ago, and every year I learn something new about using this great model. A 4-factor model is way to advanced for me to even think about. Or is 4-factors actually all to little for me, I am frankly not sure.There are so many trying to guerilla marketing yield curve models, so here are some sales points for Black-76 type models:Different versions of the Black-76 model is still the market standard for quoting vol’s for 90% of the interest rate option market:- Caps and floor, - swaptions (European style 90% of the swaptions market) - as well as bond options where T0<<TB.- Most people also use this for money market options (eurodollar options etc...)But what is "using a model"? At least this is the model for communicating with the market, one can naturally have one or more models for other purposes, risk management etc....A) There is more than 30 years experience with this model (at least from equity).B) A lot of research have been published about it’s weaknesses and strength. (well not so much about its strength).C) Black-76 is the standard way of communicate vol in the interest rate market.D) A whole string of Black-Scholes type models is far from a single factor model.E) The model is super fast.F) Cheap to implement, instead you can pay the traders. JBut as always you should have a whole toolbox of models. It never hurts to have access to 1, 1.5, 2, 3, 4….as well as multi-multi factor models. It all depends on your trading style (market maker, pure prop trading, your boss, your budget, your skills). If you can get risk management to use a 4-factor model to do market to market on your positions you could probably at least do some internal arbitrage. Life is to serious to be serious.Omar I am not against 4-factor models, but I still I think I can compete with most 4-factor guys with a simple Black-Scholes model, but the option model is naturally only a small part of it.

hi Collector, I also think a simpler model is better because it is easier to use for traders, risk-managers … but sometimes we simply cannot create a simple formula such as Balck-Scholes ... hence the 4-factor models. I guess CMOs amd prepayments are a good example, From what I’ve heard these models need frequent calibration to different market sectors … it’s a real pain, but for the time being we simply don’t have a better choice. Now maybe in ten years a brilliant person will come with a simple unifying theory that will make these four factor calibrating models obsolete.

hi Collector, I also think a simpler model is better because it is easier to use for traders, risk-managers … but sometimes we simply cannot create a simple formula such as Balck-Scholes ... hence the 4-factor models. I guess CMOs amd prepayments are a good example, From what I’ve heard these models need frequent calibration to different market sectors … it’s a real pain, but for the time being we simply don’t have a better choice. Now maybe in ten years a brilliant person will come with a simple unifying theory that will make these four factor calibrating models obsolete. >>Actually, mortgage backed models tend to be parsimonious in parameters because they need to include the history of those parameters rather than just the current value. That is, the value of a security does not just depend on the current level of prepayments and interest rates, but the history of those parameters since the pool was originated. Or, looking forward, if you simulate a tree of possible parameter changes, the value of the security depends not just on what node you are looking at but on the path taken to reach that node.This results in complicated models with simple cross-sections. Using a 1-year and 10-year interest rate is usually sufficient to model the term structure of all the yield curves, and a simple markov transition probability model with three or four states is good enough for prepayments.