Suppose that a bank buys an option from a client. The option...

Suppose that a bank buys an option from a client. The option is uncollateralized and there are no other transactions outstanding with the client. The expected values of the option at the midpoint of years 1,2 , and 3 are 6,5, and 4 . The probability of the counterparty defaulting in each of the three years is $3 \%$. The probability of the bank defaulting in each of the three years is $2 \%$. Estimate the bank's CVA and DVA for the transaction. Assume no recovery in the event of a default and zero interest rates.

Key Concepts

Credit Valuation Adjustment (CVA)

CVA is the adjustment to the value of a derivative that accounts for the risk of a counterparty defaulting on its obligations. This concept involves estimating the potential loss expected from such a default, using the expected exposure at various future time points and the probability of default over those periods. It quantifies the risk inherent in uncollateralized transactions where there is no protection if the counterparty fails to meet its obligations.

Debit Valuation Adjustment (DVA)

DVA is the mirror concept to CVA and represents the adjustment to the value of a derivative resulting from the bank’s own default risk. It reflects the potential gain from the bank’s obligation to pay, which may not be fully realized if the bank itself defaults. DVA calculations involve the expected exposure and the likelihood of the bank defaulting, emphasizing how a party’s own credit risk can alter the derivative's fair value.

Expected Exposure

Expected exposure refers to the anticipated future positive value of a derivative position at specific points in time. In risk management, this measure is used to estimate how much risk there is on the balance sheet and is particularly useful in computing CVA and DVA. It takes into account the underlying market dynamics that affect the derivative’s value over time, serving as a crucial input in credit risk adjustments.

Default Probability

Default probability is a key element in assessing credit risk, indicating the likelihood that a counterparty (or the bank itself) will default within a certain time period. It is used to weight the expected exposure at different future intervals when calculating CVA and DVA. Accurate estimation of default probabilities is essential since even small changes in these probabilities can significantly affect the valuation adjustments for over-the-counter derivatives.

Uncollateralized Transactions and Recovery Rates

Uncollateralized transactions are those in which no collateral is posted to mitigate credit risk, thereby making them more sensitive to counterparty credit issues. When combined with an assumption of zero recovery in the event of default, the risk adjustments (CVA and DVA) are directly based on full loss upon default. This scenario emphasizes the importance of incorporating default probabilities and expected exposures without the offsetting effects that collateral or recovery rates might provide.

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