Credit Risk Exposure

[ftp2002a]

Hi, I got stuck with this question and I don't know how to arrive at this figure:

You have purchased a one‐year European 650‐strike call option on a stock index from your
counterparty. The index currently stands at 600, its volatility is 25%, and the risk‐free interest rate is 4%
per annum, with continuous compounding. Assuming that the rate of change of the index is normally
distributed, estimate your maximum potential credit exposure to your counterparty, using a 95%
confidence level. Your contract is for 1,000 units of the index.
Answer: $263,143.52

Can anybody explain this question to me? Thanks a lot!

 

[David Harper CFA FRM]

Hi Louis,

(This is an instructive FRM question, IMO). See Hull Ch 13.
1. Expected return under lognormal = 4% - 25%^2/2 = 0.8750% ("volatility erodes exp return")
2. This question expects NORMSINV(95%) = 1.64 so that 95% conf interval = 1.64 * 25% vol ~ 41% (basically a 95% credit VaR, from the counterparty's perspective)
3. So, the 95% CVaR = 0.8750% + 41.x% =~ 42%
4. So, grow 600 continuously = 600*EXP(42%) = $913; i.e., worst 95% expected continuous compound @ 42% gets the stock to $913
5. 913 - 650 = $263

David

 

[[email protected]]

if I understand your method correctly, you bascially use the equation for Geometric Brownian motion and get an expected stock price on the 95% tail, then take the differnece of the expected future stock price and the strike price.

However, when I looked at this problem, the first thought was to use the black-schole model, and the value of the option will be the credit exposure.

So, today's price = 600
95% Conf. move of today's price = 600*exp (1.64*25%/sqrt(252))=616
Strike = 650
Volatility = 25%
risk free rate = 4%
Using Black-Schole, the value of the call option is approx. $58

 

[RomanS]

Hi,
The call price doesn't help you in this question for at least 2 reasons. (1) The call price is a value at present, i.e. t=0 but you want to need something in t=T, right? (2) You want to estimate the counterparty exposure and for that purpose you need payouts in t=T, specifically the payout you can expecte to occur at a 95% confidence interval. Hope that was useful. Best Roman

 

[rahul.goyl]

Hi David,

Could you elaborate on Step 2 & 3, what is the intution behind below formula
Sigma*Vol (+-) R

Thanks

 

[David Harper CFA FRM]

Hi Rahul,

As the buyer (long) of the option, potential credit exposure to the counterparty is a gain (in the money); i.e., the counterparty risk is that the seller (short) will not pay future intrinsic gain. So, from today, the VaR-type exposure is:
1. You do expect the asset price to drift up (per the expected return); this is the "deterministic" component
2. But, also, volatility can create further upside or downside. This is typically (in simple parametric approach) given by: sigma*volatility; e.g., 5% of the time, if normal, return will be at least or above 1.65*volatility

so, as the long, the upside 5% scenario (ie.., 95% of the time gain will be less than) = drift + 1.65*volatility, or +return + 1.65 * volatility

To me, the confusion is that: VaR is a worst expected LOSS concept. But potential future exposure (or current exposure, or any of the counterparty metrics) are GAIN measures because you are exposed to this risk if you gain in the position.

Hope this helps, thanks, David

 

[rahul.goyl]

Hi David,

Thanks a lot for such strong intution, understood though PFE is positive (Gain) but still the whole amount is at risk in terms of CVaR context if counterparty defaults. (Correct me If am wrong)

Thanks
Rahul