Figure 10-20. Value-at-risk for geometric Brownian motion and jump diffusion
Credit Value Adjustments
Other important risk measures are the credit value-at-risk (CVaR) and the credit value
adjustment (CVA), which is derived from the CVaR. Roughly speaking, CVaR is a
measure for the risk resulting from the possibility that a counterparty might not be able to
honor its obligations — for example, if the counterparty goes bankrupt. In such a case
there are two main assumptions to be made: probability of default and the (average) loss
level.
To make it specific, consider again the benchmark setup of Black-Scholes-Merton with the
following parameterization:
In [ 79 ]: S0 = 100.
r = 0.05
sigma = 0.2
T = 1.
I = 100000
ST = S0 * np.exp((r - 0.5 * sigma ** 2 ) * T
+ sigma * np.sqrt(T) * npr.standard_normal(I))
In the simplest case, one considers a fixed (average) loss level L and a fixed probability p
for default (per year) of a counterparty:
In [ 80 ]: L = 0.5
In [ 81 ]: p = 0.01
Using the Poisson distribution, default scenarios are generated as follows, taking into
account that a default can only occur once:
In [ 82 ]: D = npr.poisson(p * T, I)
D = np.where(D > 1 , 1 , D)
Without default, the risk-neutral value of the future index level should be equal to the
current value of the asset today (up to differences resulting from numerical errors):
In [ 83 ]: np.exp(-r * T) * 1 / I * np.sum(ST)
Out[83]: 99.981825216842921
The CVaR under our assumptions is calculated as follows:
In [ 84 ]: CVaR = np.exp(-r * T) * 1 / I * np.sum(L * D * ST)
CVaR
Out[84]: 0.5152011134161355
Analogously, the present value of the asset, adjusted for the credit risk, is given as follows:
