I have recently read with great interest the article by Hull and White "The valuation of credit default swap options", available for free download. However, I do not see the practical usefulness of the CDS options. First of all, they would be options on options, which as we know have already gained relatively little popularity in the equity derivatives markets (more of a noise than a signal). Secodly, I cannot find any concrete applications where a client should compelling decide to use these instruments versus others. Third, there is the problem of pricing an ultra complex product and cannot imagine a dealer that would bear that model risk if not at horrendous prices (that would limit the attractivness of the product).Admittedly, this is a question for creative people, that I am not. Thanks!

According to me, we will see in the near future an increasing interest for multi-callable credit-sensitive structures.At begining it would be on single-name, after maybe on multi-name ones. Nevertherless, there is still a huge amount of work to be done.

Thanks for the paper, but the other paper I have cited mentions the fact that these options on CDS have been gaining more and more importance in the marketplace. Because the article was published in January, hence possibly written by December/October, it sounds like the authors have been a little bit too optimistic about the options (perhaps to add importance to their work). Do you really think that Hull and White need to playaround with thesew things? Also, the question remains open: why should an investor ever use the option on CDS??Thanks!!!

QuoteOriginally posted by: CreditGuyI have recently read with great interest the article by Hull and White "The valuation of credit default swap options", available for free download. However, I do not see the practical usefulness of the CDS options. First of all, they would be options on options, which as we know have already gained relatively little popularity in the equity derivatives markets (more of a noise than a signal). Secodly, I cannot find any concrete applications where a client should compelling decide to use these instruments versus others. Third, there is the problem of pricing an ultra complex product and cannot imagine a dealer that would bear that model risk if not at horrendous prices (that would limit the attractivness of the product).Admittedly, this is a question for creative people, that I am not. Thanks!To the extent that most of the term structure models map to default intensities models easily, I would expect the valuation approach to be very similar...

for pricing and hedging see also the thread on the hedging of CDS and default swaptionsI think, there are numerous applications for options on CDS, just think of all the embedded options (prepayment, cancellation) that lurk around in loan contracts, or committed lines of credit... And in some CDO hedging strategies you also have some volatility risk that wants to be managed.Apparently, JPMorgan are already posting prices for CDSOptions regularly. Quote originally posted by CreditGuyDo you really think that Hull and White need to playaround with thesew things? Hull and White have a consultancy contract with GFInet (a large credit derivatives broker)... Besides, if you do research, it is always good to be early. I wrote my paper with the CDS option pricing formula in 1999

with all due respect, thinking that you're going to pick apart the embedded options in loan contracts with this kind of thing sounds a little far-fetched to mein the beginning, the CDS contract was supposed to connect bonds and loans, but in fact it does not do this to first-order approximationproblem being that there isn't a simple 1-1 map between bond spreads and loan spreads, the driving force being that investor constituencies in the two different contracts are substantially different, as are the qualities of the instruments in different timesas practical example, in the old old days of loan trading one would do spread trades - long loans short bonds at a ratio, or the other way aroundas another, I would have a strong preference for junk loans over junk bonds now because the loans are more likely to remain current through bankruptcy, and I need at least a minimum of liquidity to keep the ship from sinking... to do the same with distressed bonds, one needs the ability to remain underwater for a very long time with no oxygen at all. Somebody is probably going to suggest a strategy that is short govie zeros and long govie coupons, but it's not that easy.

Here's how I'd do, it...write a european default swaption payoff in the same way you would IRS swaption...max[B(fee-feestrike),0]where B=sum of terms looking like DFti*Pi*deltaTi, where Pi is prob of surival to i. I claim that B is a valid numeraire. This is because it can be written as a portfolio of risk free and credit risky discount bonds.thus, modeling fee as a stochastic process, we'd see that fee is a martingale relative the numeraire B. then, valuation is just like vanilla IRS swaptions. namely, defaultswaption value = B*BlackCommodity term in fee. Voila!

HiI have been trying to implement a Bermudan option on an underlying CDS. I have read Schonbucher'spaper "A Tree Implementation of a Credit Spread Model for Credit Derivatives". However, I mustcode the problem on a symbolic manipulation package for finance similar to Mathematica. Therefore,I had to come up with a more abstract scheme to price the product, rather than just code the tree modelas specified by Schonbucher.The underlying process I am taking is the Hull-White process for the hazard ratedlambda=(k(t)-a*lambda)dt +sigma*dW. Note that I consider interest rates asdeterministic (although with term structure).Following Schonbucher, we know that for given \tau>0, the density of the time of thefirst default as seen from time zero is for T>0Pd(0,t)=Expect Value[\lambda(t) e^{-int_0^t lambda(\tau) d\tau}]so this defines, via the Feynman-Kac theorem a backwards Kolmogorov equation problemfor Pd(0,t) which is\frac{\partial Pd(t,\lambda)}{\partial t}+1/2\sigma^2\frac{\partial^2 Pd(t,\lambda)}{\partial \lambda^2}+[k(t)-a\lambda]{\partial Pd(t,\lambda)}{\partial \lambda}=\lambda Pd(t,lambda)with the terminal data Pd(T,\lambda)=\lambda. Similarly for the survival density,Ps(0,t)=Expect Value[e^{-int_0^t lambda(\tau) d\tau}]So the PDE is the same but the terminal data Ps(T,\lambda)=1.This is similar to the derivation of the Black-Scholes formula, however, I am not totally sure one can apply the Feynman-Kac since the discount term e^{-int_0^t lambda(\tau) d\tau} contains lambda as well, while for Black-Scholes the discount term does not depende on the stock price.So my first question to anyone is, is this right, can I use these two PDE’s to backwards integrate and solve the Pd(0,t) and Ps(0,t) probabilities? If so that makes my life easier, as the symbolic manipulation package for finance I am using allows me to simple write the PDE in the form above and solves it without me having to write any code. Second question is, in order to value the option on a cds, I exercise if the ratio between the sum default payments = (1-Recovery) \sum_{i=1}^{N Periods} DiscountFactor_i * Pd(0,t) and the sum coupon payments = \sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(0,t), where K is the fixed rate coupon, is bigger than K. If the ration is bigger than K, exercise,otherwise do not.Third and last question, how to calibrate k(t), in the Hull-White process for the hazard ratedlambda=(k(t)-a*lambda)dt +sigma*dW? I think “a” and sigma we will get from historical data andgood guesses, but k(t) should be such that the Pd(0,t) at coupon payment dates matches the probability of default coming from stripping cds spread rates in the market, right?Hope someone will be able to clarify these three questions for me. If so, thanks in advance!Eurico

Eurico, first of all a remark: You can calculate Ps(0,t) and Pd(0,t) in closed-form in your setup. The formula for Ps(0,t) is given in the paper, and by taking minus the derivative with respect to t of that, you get Pd(0,t).That should also make the calibration very easy.Now your questions:1.The Feynman-Kac formula does hold here. So you can write (I write x for lambda)f(x,t) = E[ exp{-\int_t^T x(s) ds } F(x(T)) | x(t) = lambda]and then f satisfies: E[df] = x f. In p.d.e. this is f_t + (k-ax) f_x + + sigma^2/2 f_xx - x f = 0 as you wrote yourself,with appropriate boundary and final conditions. (f(x,T) = F(x))2.1A slight correction on your exercise condition:The sum of the default payments is: (1-Recovery) \sum_{i=1}^{N Periods} DiscountFactor_i * Pd(0,t_i) delta_iYou are approximating an integral over all possible default times, so you must have the delta_i in it.2.2 This is not the exercise condition, but (part of) the early exercise payoff (maybe I am being a bit pedantic, but just making sure everything is right).The early exercise payoff is all you need to price this thing.It is the value of entering a CDS at K at time T*. Imagine you exercise the option at rate K and sell the CDS directly in the market for rate s, then you would earn s-K until default or maturity of the CDS. (Assuming the option is to enter the CDS as protection buyer.)The value of that is (s-K) \sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(T*,t_i)and that is what you should base the early exercise decision on.2.3Then you can go and do backwards induction. You will get an additional interest-rate drift in the Feynman-Kac pde, but as you assumed interest-rates to be constant, that should not be a problem. Early exercise decisions are then made by the usual free-boundary methods. 3.You would want to calibrate k to CDS. If you have the closed-form solutions for Ps, calibration of k(t) should be easy. With the vol and mean-reversion you are on your own (but everybody is guessing).

Philipp,Thanks for your quick reply.After reading your message, I now have a working version of a Bermudan option on an underlying CDSworking via Monte Carlo simulation using the Longstaff-Schwarz method for handling the exercise part.I am also working at the same time on the PDE version of the code using the PDE derived from Feynman Kac.I have correct the exercise condition as you recomended. The sum of the default payments is indeed: (1-Recovery) \sum_{i=1}^{N Periods} DiscountFactor_i * Pd(0,t_i) delta_iHowever, I noticed that in our contracts, the underlying CDS pays (1-Recovery)*Notional at default time, not at the end of the period. Therefore, this requires the total of default payments to be calculated as an integral.I have now taken this into account.After re-reading your paper I agree that the value of the option is (s-K) \sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(T*,t_i) delta_i(you forgot the delta_i there, didn't you?) where s is given by s=TotalDefaultPayments/TotalFixedRatePayments. Is it correct to say that (s-K) isthe price within the survival measure and \sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(T*,t_i) delta_iis the correction to take it into risk neutral measure?Just one more question, the fact that we have a Bermudan option to price, rather than an european optiondoes not affect the form of this correction term, does it?Thanks again, back to work now,Eurico

However, I noticed that in our contracts, the underlying CDS pays (1-Recovery)*Notional at default time, not at the end of the period. Therefore, this requires the total of default payments to be calculated as an integral.I have now taken this into account.usually (i.e. with quarterly payments) that should not make too much of a difference...After re-reading your paper I agree that the value of the option is (s-K) \sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(T*,t_i) delta_i(you forgot the delta_i there, didn't you?)oops, yes, the delta_i should be in it. where s is given by s=TotalDefaultPayments/TotalFixedRatePayments. Is it correct to say that (s-K) isthe price within the survival measure and \sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(T*,t_i) delta_iis the correction to take it into risk neutral measure?yes, you can see it that way. (s-K) is the price within the survival measure and taking the CDS fee stream as numeraire.\sum_{i=1}^{N Periods} K DiscountFactor_i * Ps(T*,t_i) delta_i is then the correction to take it back to the cash-numeraire and the risk-neutral value.Just one more question, the fact that we have a Bermudan option to price, rather than an european optiondoes not affect the form of this correction term, does it?It does not affect the correction. You need an early exercise strategy for a Bermudan option, and the resulting strategy that you will get using MC simulation and LSLS-early exercise will be the early exercise strategy conditional on survival (like everything in the survival measure).So (for formal completeness) you only have to complete the early exercise strategy with a specification of the exercise strategy in default. That should be trivial: Either you call at the first opportunity after default (in the case of a call on protection) or never exercise the option after default (in the case of a put on protection).

Thanks Philipp,I have now a working model, but I am still confused with the calibration. Ithink that, in analogy with the calibration for interest rates, I need to calibratek(t)~a \phi + \partial phi / \partial twhere phi=-\partial Ps(t,T)/\partial t and Ps(t,T) is the probability of survival from t to T.But then, is I do a Monte Carlo simulation, how do I choose the hazard ratespot rate=\lambda(0)? Do I use the \lambda(0) inferred from prices of CDS in the market?ThanksEurico