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This paper discusses the valuation of credit default swaps, where default isannounced when the reference asset price has gone below certain level from thelast record maximum, also known as the high-water mark or drawdown. We assumethat the protection buyer pays premium at fixed rate when the asset price isabove a pre-specified level and continuously pays whenever the price increases.This payment scheme is in favour of the buyer as she only pays the premium whenthe market is in good condition for the protection against financial downturn.Under this framework, we look at an embedded option which gives the issuer anopportunity to call back the contract to a new one with reduced premium paymentrate and slightly lower default coverage subject to paying a certain cost. Weassume that the buyer is risk neutral investor trying to maximize the expectedmonetary value of the option over a class of stopping time. We discuss optimalsolution to the stopping problem when the source of uncertainty of the assetprice is modelled by L\'evy process with only downward jumps. Using recentdevelopment in excursion theory of L\'evy process, the results are givenexplicitly in terms of scale function of the L\'evy process. Furthermore, thevalue function of the stopping problem is shown to satisfy continuous andsmooth pasting conditions regardless of regularity of the sample paths of theL\'evy process. Optimality and uniqueness of the solution are established usingmartingale approach for drawdown process and convexity of the scale functionunder Esscher transform of measure. Some numerical examples are discussed toillustrate the main results.

Optimal valuation of American callable credit default swaps under drawdown of L\'evy insurance risk process