Suppose we wanted to price a 3yr equity call option that terminates worthless if the 5yr swap rate hits 6% at any point in the next 3 years. Also suppose we choose GBM for the stock price and 1F HW for the interest rate.Let's say for simplicity we use weekly time steps. Does this general valuation methodology work?1. Generate series of 1 week short rates for the next 3 years2. Based on correlation, generate series of stock prices weekly for the next 3 years assuming that the stock price expected return in a given week equals the randomly simulated rate from step 1.3. Calculate the equty payoff and discount back at the pathwise sequence of short rates4. For each week calculate the 5y swap rate (we have the HW formula for discount bond prices so this can be done analytically)5. If any of these is greater than 6%, multiply the payoff from 3 by zero6. Repeat many times and averageThis should correspond to a rolling CD as the numeraire for the stock price, and no convexity adjustment for the 5y rate ... and I wanted to make sure this is correct. Any thoughts would be appreciated.

Is this for a real trade?This thing will have a killer cross-gamma exposure as it gets close to knocking out, esp if the equity call is way in the money.Does the client really need to have an equity call option knocked out by the 5yr CMS?I would try to modify the structure so that it was path independent. You could estimate how high the 3yr into 5yr swap would likely be, conditional on it having hit 6% at some point. Say this is 6.5% for argument's sake. Then you could have the underlying equity option final payoff decrease smoothly as the forward swap goes from maybe 6.25% to 6.75%. It might be worth capping the equity option so that the magnitude of the cross gamma is limited.The whole thing could be expressed as the product of an equity call spread and a swaption spread payoff, whilst expressing roughly the same economic view, so that you would have much more manageable cross gammas and vastly less model risk. Then again if you are one in a crowd of guys making a price and you have committed to putting up a price for "anything", you could make a bad price for this but suggest the other structure, with a good price.I will let a fixed income expert comment on the suitability of the 1F HW model, but wouldn't you need to include skew and volatiity term structure effects for the equity option in your original structure? In my modified version the full skew of the stock option market and the swaption market could be incorporated without great difficulty.

Erstwhile,Thanks very much for your informative reply! Would you mind just lending some advice on this: of course I realize that GBM is not a good model for the stock price, and that 1FHW is not a good model for rates. But let's just take all of that as a given ... are the technical details of my proposed method correct? i.e. do I have any issues with numeraires or convexity adjustments ... hybrids are new for me.As for your other comments, I agree, the option is difficult to hedge under the situation you mention. I will definitely consider the proposal and examine some alternative structures ... I appreciate the time you took to write it up.Rgds

Glad it was helpful.To be honest, I was an IR quant until about the end of 1990, and the most complex thing I was involved with in IR derivatives was arbitraging swaptions against cap/floors.So take what I am saying with a grain of salt, and I would appreciate it if an IR derivs person would chime in and correct me.First I would say that if your one factor in the HW model is the one week short rate, you will need a convexity adjustment if you want to simulate the 5yr CMS rate. This rate is not equal to the forward swap rate. I should fill in this hole in my knowledge by looking up Clopinette's postings on CMS stuff in these forums!I think the adjustments comes from a combination of several things, such as the correlation between the forward swap rate and both the short discount rate and the DV01 (they call this measure LVL I seem to recall, and you get a quanto-like effect when you change to the LVL measure). So I don't know how to do it, but yes you need a convexity adjustment. Could someone help us out here?In terms of your general approach in simulating both the short rate and also the equity, which in turn is trying to have the "right drift", this sounds reasonable to me, but I would worry that something might be amiss in the details. Again, advice from a monte carlo expert, or anyone else who has simulated dynamic yield curves and dynamic equity at the same time, would be appreciated - I bet many of us could learn something interesting here!

The mechanics sound right. Whatever measure you're simulating under you will have to solve for the short rate drift. If you choose the spot measure, the drift of the equity will indeed be the (state-specific) short rate. There is no "convexity" required - the 5-year swap rate is a function of the short rate at any given (stochastic) value of the latter. The suitability of using a one-factor model for valuing barrier options is another issue though.

Thanks Erstwhile and ggt for your replies.ggt, I am glad I have the honour of being the subject of your 1st post!So ggt, based upon what you are saying, let's say I choose my time step to be weekly and my numeraire to be the "rolling 1-week CD." Then at each point in time I can calculate the 5y swap rate directly without convexity adjustments. My terminology may be weak ... I am not sure but I think this is what is meant by the "spot" measure.As for the stock price, I think you are saying that in this context I would not need a drift to the stock if I set my stock drift equal to the stochastically generated 1-week rate. Not sure ... but anyway I suppose I can check this by calculating the risk-neutral expected present value of the stock price for all future times and making sure it equals to day's price. This seems to make sense, i.e. if I have expected returns of r1, r2, ..., rn and then I discount back at that sequence, then on each path the expected present value of S_t should be S_0 for any t. As an aside, is it typical to calculate the drift adjustments analytically or numerically in situations where one is required?Many thanks to both of you for your replies.Rgds

The honor's mine, especially if you found my comments of use!The way to look at this might be as follows: Imagine you're simulating the stock price in a BS world, where the interest rates are deterministic. Then, under the "spot" martingale measure, the drift of the stock price is the (deterministic) short interest rate and the numeraire at every time t is exp(\Int_{0}^{t}r(u)du) - which is path-independent. If you now move to a world where the interest rate as well as the stock price are stochastic and keep the same numeraire (which becomes path-dependent as it now depends on a stochastic quantity) the only remaining unknown quantity is the no-arbitrage drift of the short interest rate (as well as the vol structure but that's another story). This you find by revaluing the zero-coupon bonds. After that you can value any (normalised by the numeraire) payoff. The calculation of the drifts can be either analytical or numerical - it really depends on the context. Even in cases where the drift is some continuous function of the state variables of the model you may want to numerically solve for the drift so that there is no arbitrage in your discretised version of the continuous model (and, I think, HW is one such case).I hope this helps

Definitely useful and interesting comments - don't hold back on posting in future!

Thank you Erstwhile. I am humbled to hear that from one of the senior members of this forum.

Thanks again to all. I agree with Erstwhile, this has been an educational discussion! Thanks ggt, helpful explanation.Now to close the issue in my own mind I would like to repeat back what you are saying about short rate drift in order to make sure I understand it. When you say "This you find by revaluing the zero-coupon bonds. After that you can value any (normalised by the numeraire) payoff," I take that to mean the following:1. For each path, based upon my stochastically generated sequence of 1-week rates, calculate discount factors successively at each time step in the MC path.2. For each time step compute the average discount factor across all MC scenarios.3. If the average discount factors match the time-0 discount factors obtained from the swap curve, my drift adjustment is correct. Otherwise, numerically compute an adjustment at each time step that will force this to be the case.Vol I assume is calibrated from caps or swaptions of your choice.Now I have a model which I can use to simulate the stock price, and can calculate the discounted value of the derivative on every path using the pathwise discount factors (i.e. the payoffs normalized by the value of the numeraire). How does this sound? I think I have it this time.Rgds

Sounds right to me.

Paper on Hybrids by L.C.G. Rogershttp://www.statslab.cam.ac.uk/~chris/papers/Po ... ybrid3.pdf