Calculate DVA in Example 24.6. Assume that default can happen in the middle of each month. The default probability of the bank is 0.001 per month for the two years and the recovery rate in the event of a bank default is 30%.

Data From Example 24.6:

A bank has entered into a forward contract to buy 1 million ounces of gold from a mining company in 2 years for \(\$ 1,500\) per ounce. The current 2 -year forward price is \(\$ 1,600\) per ounce. We suppose that only two intervals each 1 -year long are considered in the calculation of CVA. The probability of the company defaulting during the first year is \(2 \%\) and the probability that it will default during the second year is 3\%. The risk-free rate is 5\% per annum. A \(30 \%\) recovery in the event of default is anticipated. The volatility of the forward price of gold is \(20 \%\).

In this case, \(q_{1}=0.02, q_{2}=0.03, F_{0}=1,600, K=1,500, \sigma=0.2, r=0.05\), \(R=0.3, t_{1}=0.5\), and \(t_{2}=1.5\).

\[\begin{gathered}d_{1}\left(t_{1}\right)=\frac{\ln (1600 / 1500)+0.2^{2} \times 0.5 / 2}{0.2 \sqrt{0.5}}=0.5271 \\d_{2}\left(t_{1}\right)=d_{1}-0.2 \sqrt{0.5}=0.3856\end{gathered}\]

so that

\[v_{1}=e^{-0.05 \times 2.0} \times(1-0.3)[1600 N(0.5271)-1500 N(0.3856)]=92.67\]

Similarly \(v_{2}=130.65\).

The expected cost of defaults is

\[q_{1} v_{1}+q_{2} v_{2}=0.02 \times 92.67+0.03 \times 130.65=5.77\]