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# Introduction to XVA

The motivation is to give people the basics of XVA, and feel that they are nonsense, so they found a note and finally found this.

thank

This series is modeled in a biased manner and is not directly related to the content of “predictive”, “trading strategy”, “option strategy” and other close transactions. Those who are not interested can have the upper right corner.

First, the entry point: an opponent credit risk modeling framework

Although there are many value adjustments, as an entry point, we choose the credit risk of the opponent that is the easiest to explain. The so-called counterparty credit risk refers to not the general corporate credit risk, but only when there is an exposure to the enterprise (he owes me money), the risk of loss of the asset value due to the breach of contract is called the adversary risk. .

Because this special credit risk is only for the counterparty, not only to measure the default rate, but also to measure the time of default, so the credit risk related to the general "credit rating" is very different, so a special framework is needed. We claim that because of this risk, the opponent's credit value is adjusted to CVA, and our credit value to the opponent is DVA (we owe them money adjustment), so CVA is a two-way concept. This paper will then directly become the CVA of assets or liabilities without distinguishing between CVA and DVA.

We consider the default of both parties in a time interval [t, t+dt]:

1. We and our opponents have lived to t+dt;

2. We lived to t+dt, but the opponent died in dt;

3. We died in dt, but the opponent is still alive;

4. We and our opponents both died in dt.

First, if you want to consider the default rate of t to t+dt, the condition is that both parties live to time t. So we must first model the survival rate. Here, a property is used: the event of the Poisson distribution obeyed by the occurrence number, and the distribution of the event for the first time of the event is just an exponential distribution. We can make the distribution of the default event and the distribution of the time of the event a common hazard rate (default probability per unit time). Therefore, the probability that a party will live from 0 to t is:

Here is a point of modeling. Because we know that the discount factor DF (form is generally: ) is the existence of the multiplier. The survival rate is also used as a multiplier, so the hazard rate is formally used as an interest rate. In fact, in the highly circulated credit product market, the default rate is used as “interest rate”. The short-term rate corresponds to the hazard rate, and ytm corresponds to the credit spread (intermediate difference LGD).

We use our and cpty to represent us and our counterparts, so that we have four occurrence probabilities and cash flows under the above four default conditions:

1. probability of occurrence ; cash flow , ( representing the cash flow generated by derivatives in dt

2. probability of occurrence ; cash flow

(RR indicates the recovery rate at the time of default, so 1-RR is the default loss rate, and P^+ indicates that it is open.

3. probability of occurrence ; cash flow (P^- indicates negative exposure

4. probability of occurrence ; cash flow

We know that CVA can be expressed as:

Give the conditional probability that both parties live to time t: we will get a general model of the price of a derivative considering the credit risk of the opponent within a given time period. For the sake of the simplicity of the results here are some provisions.

1. The agreement "risk discount rate" is (here is the form of hazard as interest rate)

2. All default probabilities are not related to exposure (no risk of misdirection)

3. Defining the expectation of open exposure is the desired open EPE (corresponding to the negative exposure expected to be ENE)

4. The probability of simultaneous default by both parties in a short period of time is small enough to be ignored

Then the price of the derivative is:

The two-way CVA_bi is the sum of the asset CVA and the liability CVA (that is, DVA, here negative):

They can be expressed as:

We define credit overflow difference is:

This method is a general adversary credit risk adjustment method. It is noted that 2 and 4 are stronger assumptions in the above regulations, so the more mature models will correct these two points.

Second, the replicable counterparty credit risk and capital cost

In order to make a natural transition to other value adjustments, we need to improve the concept of copying (hedging) in the credit risk of the opponent in order to better understand which is the cost of capital we need to spend.

First we naturally define the total EPE and total ENE as:

Then the above assets and liabilities CVA can be expressed as:

If we and the counterparty want to hedge the credit risk of each other's opponents, the best way is to take the tripartite organization to buy the other party's CDS. (The above s - credit spread is actually the credit spread in the quotation form of CDS) So in theory, we can use the cash deducted from the CVA assets to buy the equivalent counterparty CDS to completely replicate this part of the risk. The hedged position is then:

Which is open and has a poor credit overflow offer good liquidity of counterparties company cds.

And if we want to hedge our own risk of default, we can't issue our own CDS (violation operation), so we can only borrow funds with the current value equal to ENE. The borrowed funds are in addition to putting them in a risk-free market, because the liquidity premium generated by borrowing this part becomes the extra cost of our borrowing. Therefore, there is one more value in the risk of the opponent:

We call this one the LVA for liquidity value adjustment. In fact, this part is not part of the credit value adjustment and belongs to the fund value adjustment (FVA). Here we introduce FVA, which we can define as the adjustment of all the cost of capital due to the exposure of the counterparty, which will be described in detail below.

to be continued