A Conditional Valuation Approach for Path-Dependent Instruments

Credit exposure is the amount a bank can potentially lose in the event that one of its counterparties default. The measurement of exposure on derivative contracts is very important because it is used not only to set up the trading limits but also as an essential input to the bank's economic and regulatory capital calculation. The new Basel guideline has recommanded for a more forward-looking approach (in contrast to the tradtional loan-based method) to the treatment of counterparty exposure to the bank's trading book. We present a value-at-future methodology for calculating the exposures of derivative instruments across the future times. The standard valuation models used to price the instruments for mark-to-market are not applicable for calculating exposure on path-dependent instruments whose value at the future time may depend on either some event at an ealier time or in some cases on the entire path leading to the future date. For such path-dependent instruments, we propose in this paper the notion of conditional valuation, which is based on probabilistic conditional expectation techniques. Using the properties of the Brownian bridges, we show that analytic solutions are readily availabe for the exposure or value-at-future on a number of path-dependent instruments such as barrier options, average options, variance swaps and swap-settled swaptions