# How to interpret the (expected) exposure and CVA of an option or a single share

I have a quick (hopefully simple) question regarding the interpretation of the expected exposure of a call option and a single share. I've done some computations on the formula for the expected exposure and this yielded that the expected exposure of both the option and the share, are equal to their initial value, i.e. $$EE(t)^{option}=V(t_0)$$ and $$EE(t)^{stock}=S(t_0)$$. I arrived at these results by using that both the discounted option value and the discounted stock value are martingales under the risk neutral measure. However, I'm reading mixed definitions on what just the term exposure actually is. Some say it is what you could lose on an investment, which would go well with my results, but others say it is what you could lose if things go bad, i.e. if you own a share worth $$100$$ euros/dollars, then this is your exposure no matter what you purchased it for.

Could anyone help me in clarifying what the concept of exposure/expected exposure means for these two objects? The concept is slightly easier to grasp for swaps, but for products as 'basic' as these, it seems to be harder to understand. The same holds for how to think about the CVA of a single share, which I also have a hard time wrapping my head around.

Thanks in advance!

## 1 Answer

For a very nice reference on this matter, I recommend Pykhtin and Zhu’s Guide to Modelling Counterparty Credit Exposure, a short paper that thoroughly defines these concepts.

Expected Exposure $$EE(t)$$ (also known as Expected Positive Exposure) for a trade with value $$V(t)$$ is given by: $$EE(t)=\mathbb{E}[\max(0,V(t))]$$ It is effectively “what you could lose on an investment” (you can also define discounted EE in which case you rightly find that for an option or a share the value at initial time $$t_0$$ of discounted EE is the value of the option or share). EE gives the expected value to you of the deal at a future time $$t$$, hence it is the loss you can incur if “things go bad”. What does that mean?

Well for an option, which is really a contract between two parties, the risk is that your counterparty in that deal goes bankrupt for example and does not pay you: then you will lose the value $$V(t)$$ if it is positive to you. CVA is then defined as the value of your potential loss throughout the life of the deal, ie. from $$t_0$$ to $$T$$, weighted by the probability of default $$\mathbb{P}$$: $$CVA(t_0)=\int_{t_0}^TEE(t)\text{d}\mathbb{P}(t)$$

However a stock is different because it is a property title on a company, and therefore its value should already capture the risk of default; I have never heard of EE or CVA being computed for stocks, only in the case of share lending (i.e. deals between banks and HFs which allow HFs to short shares).

• Thanks alot for the clarifications. I am in a setting where I have both an option and a stock in a portfolio, and am trying to compute the $EE(t)$ of the whole portfolio. How would you deal with the share in that situation? Just omit it completely in computing the EE or CVA or do you account for it in some other way? Because I feel like when talking about purchasing a whole portfolio like that there has to be accounted for the stock also. – Charlie Shuffler Jun 20 '19 at 22:07
• As far as I know, you would not take into acccount the share. In practice, a bank trades an option with a counterparty with which in has a netting agreement, namely a set of deals with common rules about collateral and what protocol to follow in case one of the two defaults. A share would not be included in such a netting set because it is not a deal between two counparties, rather a property title of one party over another. Thus that situation would not occur often. If it did, my opinion is CVA should not be calculated for a share because the share price already captures default risk. – Daneel Olivaw Jun 20 '19 at 22:17
• Alright that makes sense. Thank you very much for the detailed explanation! – Charlie Shuffler Jun 20 '19 at 22:20
• You’re welcome. – Daneel Olivaw Jun 20 '19 at 22:22
• To add to Daniel's answer, Counterpart Credit Risk is very specific. It is not the same thing as default risk for a bond or a stock. When you own a stock, you have a share of company, there is no counterparty here. The risk you have in this case is that the company's value drops (or that it eventually goes bankrupt and ceases to exist). For an OTC option, you have a contract with a counterparty that could default on the payoff payment when the time comes. This CCR is specific to the counterparty, and has nothing to do with the option value. – byouness Jun 21 '19 at 8:52