Hi,I would like to know how to price a "memory cap", i.e. an instrument having the following payoff:1) at first payment date it pays: max(0; Libor(T_1) - K) where T_1 is the first fixing date and K is the strike2) at subsequent payment dates, it pays: max(0; Libor(T_i) - K) + previous couponIt should be straightforward to price it by Montecarlo methods, once one has chosen a given model for simulation (e.g. BGM), but I wonder if there is some sort of approximation that does not make use of any models, just price the memory cap in a "Black framework".Thanks for help!

Hi Baoh,This sounds like a sum of knock-in annuities with discrete knock-in conditions. If at time Tfix_i we have Ci = Libor(Tfix_i) – K > 0, then I think you say that at all subsequent payment times {Tpay_k with Tpay_k > Tfix_i} this same amount Ci forms part of the payment. This subsequent payment at each such Tpay_k is deterministic, conditional on Ci > 0 at Tfix_i. So it seems to me that the memory cap can be [edit: approximately] obtained from the ordinary Black cap by taking the Tfix_i caplet and substituting a sum of discount factors Sum(k) {DF(Tpay_k)} with Tpay_k > Tfix_i instead of the usual single DF(Tpay_i). Each discount factor can be regarded as discounting a payment in arrears with an extra DF from Tpay_k to Tpay_i, followed by the usual discounting to the current date from Tpay_i. Rearranging the sum, at each Tpay_k, you get Ck plus all previous non-zero coupons sum(i){Ci} with i<k. All such payments at Tpay_k are discounted by DF(Tpay_k). Since this sounds pretty simple, maybe I misunderstood what you meant.If it is what you meant, I would assume that the strike K would have to be set quite out of the money in order to compensate for the extra annuity payments.--------

This product is path-dependent: It will be very easy for you to see that there is no static hedge of this product with vanilla caplets because of the floor on the coupon. That is because as soon as a floor has been exercised, it is impossible to write the payoff simply. Therefore it is not possible to price it with Black formulas.In fact even with no floor on the coupon you would still need to apply some convexity adjustement to value the coupons PV.

JanDash,I'm not sure that would be correct...Indeed let us concentrate on the single caplet fixing at T_i; its price is:where L(T_i) is the Libor fixing at T_i and D(T_{i+1}) is the(stochastic) discount factor.Applying the tower property of the expected value operator and conditioningwith respect to the filtration at T_{i+1}, I get:where P(S,T) is the zero-coupon bond price at time S expiring at T.Up to my knowledge, there is no way of pricing this term, neither usingconvexity adjustments. Am I right? I basically agree with Clopinette, and I guess it takes a Montecarlo simulation to price it.

Yes, I agree with you and Clopinette that to do the problem “exactly”, in general you would have to run a numerical code. However you said originally: I wonder if there is some sort of approximation that does not make use of any models, just price the memory cap in a "Black framework".My previous response is the “Black framework” approximation for which you asked, and I think it will work reasonably well. The approximation is that the average values of the extra DFs are factored out of the expectation value. To be truthful, I did the calculation in my head in responding to your question and forgot it was an approximation. In any case, you can check how well it works when you get done with your code.There is a somewhat better approximation that I found useful in CMT calculations that might be useful to you, which I describe in my book. The numerical calculations for CMT products are highly numerically intensive. Most of the computer time is spent calculating the zero-coupon bonds at each node. We can substantially speed up the calculations. The basic idea is to use a fast analytic method based on a mean-reverting Gaussian process for discount factors, which is a numerically reasonable approximation. The actual values of the future short-term rates come from a separate code, for example a lognormal short-rate process (in that case, no negative rates actually appear).Also for what it’s worth, I think the calculation could be done exactly in a Gaussian model.-------

JanDash,actually I was previously thinking about some sort of convexity adjustment, but then I realized a gross approximation was perhaps needed...I had in mind to use the following:which I guess is the same as you were suggesting, if I got it right.Beg your pardon, what do you mean by CMT calculations? And, pray, what is your book, so that I might have a look therein?Thanks!

Yes, our approximations a la Black are the same. In your notation, Caplet(T(i),K) contains the discount factor P(0,T(i+1)) , right?The book is “Quantitative Finance and Risk Management, a Physicist’s Approach”, 800 pages, ISBN 9812387129, World Scientific (2004). You can get a 15% discount through 6/30/05. Go to http://www.worldscibooks.com/economics/5436.html and use the discount code WSPC5436. CMT (Constant-Maturity Treasury) rates are US rates obtained by fitting the US treasury curve, and are contained in a weekly Federal Reserve Bank “H15” report. A forward CMT rate is composite, and can be built up from short rates using an underlying stochastic short-rate model. It is equal to the coupon that is obtained from setting the calculated forward treasury coupon bond to par at each forward short-rate node at each time. Since many discount factors are involved both in CMT calculations and in your memory cap, I thought the approximation I used for CMT products would be useful for you.--------

JanDash, yes the Caplet(T(i),K) already contains the discount factor.I'll try have a look at your book. Thanks for help!