Background Substraction: MOG2 y GMG

In a gas storage agreement the purchaser agrees to pay a fixed series of reservation charges, to the

0 5 10 15 20 25 −4 −2 0 2 4x 10 6

Futures Contract by Promptness

MMbtu

Minimum Notional Delta Neutral Hedge

0 5 10 15 20 25 −3 −2 −1 0 1 2 3x 10 6

Futures Contract By Promptess

MMbtu

Intrinsic Hedge

Figure 5 Top graph represents the minimum notional delta neutral hedge determined from the PDE solution. Bottom graph depicts the intrinsic hedge.

operating strategy subject to the limitations of the facility. Such a contract exposes each counter-

party in the trade to the credit risk of the other. If market conditions change so that the reservation

charges agreed to in the beginning are cheap relative to the floating cash flows generated by the

storage plant, then the purchaser of the storage contract has exposure to the seller’s credit risk.

If the seller defaults the purchaser losses out on their upside and has a legal claim against the

seller equal to the mark-to-market value of a long position in the remaining portion of the storage

contract at the time of the seller’s default minus the market value of any working gas in the facility

(which always remains the property of the counter-party who injected it).

If on the other hand the market conditions move the opposite way so that the remaining fixed

reservation charges are worth substantially more than the variable cash flows of the storage facility,

then the seller is exposed to the default risk of the purchaser and in the event of the purchaser’s

value of the short position in the storage contract (once adjusted for the value of the working gas

inventory) at the time of default.

To correctly price the storage contract one must adjust the valuation for the market value of each

of these credit exposure profiles. In addition, to avoid having too much credit exposure to any single

counter-party, most firms place limits on the size of the allowable credit exposures. These limits

typically are based on some tail measure of the maximum exposure profile. In order to monitor

these credit limits, potential future exposure (PFE) profiles that include all of the trades with a

given counter-party must be continually calculated and if exposure limits are ever exceeded further

trading with the counter-party must cease until the exposures can be brought back to internally

acceptable levels.

Using the RBF-PDE methodology as mentioned we determined that the value of the floating

side of the storage contract was $1.0041×107_{. We shall assume that the fixed side, total reservation} charges were determined so that the mark-to-market of the trade at inception was zero and that

the reservation charges were paid in equal monthly installments.

Once the RBF expansion coefficients were calculated we conducted 10000 scenarios and deter-

mined the value of the storage contract at each time-step and at each market state of the simulation.

At any market state and time step the credit exposure for the holder of the long position of the

trade is simply the maximum of the difference between the mark-to-market of the floating and

fixed side obligations (adjusted for working gas values) and zero, (if the fixed side is more valuable

than the floating side then the holder of the long position owes more than their counter-party and

hence has no credit exposure to that counter-party). In contrast, at any market state and time

step the credit exposure for the holder of the short position of the trade is simply the maximum of

the difference between the mark-to-market of the fixed side and floating side obligations (adjusted

for working gas values) and zero.

Figure 6 shows the expected exposure and 95th percentile exposure faced by the purchaser (long

position) in the storage contract, while Figure 7 depicts the same profiles but from the point

create the exposure profiles shown. On the one hand, as time increases so too does the size of

the uncertainties in market states and since credit exposure is only concerned with the positive

market movements from each perspective, future uncertainty increases expected exposure and 95th

percentile exposures. On the other hand as time increases the storage contract becomes shorter

thereby decreasing the values of the remaining portion of the trade. When uncertainty dominates

exposures increase and when decreasing time-to-maturity dominates exposures tend to decrease.

The futures curve is depicted in Figure 4 and shows the months in which the remaining compo-

nents of the trade are more or less valuable depending on whether one is interested in the long or

short perspective. The arch like shapes of the profiles in these Figures behave as would be expected.

For example in Figure 6 we see the exposure profiles for the holder of a long position in the storage

contract. The futures curve in Figure 4 shows that there is a peak price point around the 1 year

mark. Prior to this peak the holder of the long position on average has a large exposure to the

seller. If the seller defaults before this point a large amount of value on average would be lost.

However, after this point the exposure predictably drops significantly because these large expected

exposures are in the past.

To calculate the credit value adjustment from the expected exposure profiles one would have

to include the values of all other trades between these two counter-parties and include them in

the exposure calculations as well as any collateral payments, re-couponing agreements or any

other credit mitigation tools agreed to between the counter-parties. From the perspective of the

purchaser, the probability of default and recovery rate of the seller in each time interval as implied

by Credit-Default-Swap (CDS) spreads on the seller or a reasonable proxy must be determined.

Then the market value of the credit risk assumed by the purchaser of the storage contract is the

sum over each time step of the expected exposure value, times the probability of default in that

interval, times the recovery rate, times the appropriate discount factor. This is the amount by

which the purchaser would want the seller to pay up front in order to compensate for the credit

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6x 10

6 _{Expected Exposure and 95th Percentile Profiles (Long Position)}

Time (years from start of contract)

Exposure $

Expected Exposure Profile 95th Percentile Exposure Profile

Figure 6 Expected exposure and 95th percentile exposure for the holder of a long position in the gas storage contract assuming a reservation charge consistent with the value obtained from the PDE solution.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3x 10 6

Expected Exposure and 95th Percentile Profiles (Short Position)

Time (years from start of contract)

Exposure $

Expected Exposure Profile 95th Percentile Exposure Profile

Figure 7 Expected exposure and 95th percentile exposure for the holder of a short position in the gas storage contract assuming a reservation charge consistent with the value obtained from the PDE solution.

On the other hand the seller would do the same calculation but with the sides of the trades

reversed and the default probabilities and recovery rates of the purchaser used instead of those of

the seller, and from this the seller would calculate their own credit charge that they would require

in order to compensate them for the credit risk of the purchaser. The net value between these two

charges is the final CVA of the trade. For a more detailed of CVA see Zhu and Pykhtin (2007).

From the 95th percentile plots we see that the maximum credit exposure (PFE) from the long

position perspective is close to $6 million and from the short position perspective it is around $

2.5 million (the long position has unlimited up-side while the short position exposure is capped at

the discounted value of the remaining reservation charges). Therefore at inception the holder of

the long position (assuming no other trades or credit mitigation agreements exist with the seller)

would subtract $6 million from their credit limit with this counter party. If at any point in time

either through the addition of new trades or through market movements, the PFE exceeds the

internal limit no further trades with the seller would be allowed until the exposure was reduced.

Similar trading restrictions might also exist for the seller hence illustrating the need for proper

PFE monitoring.

In calculating the exposure profiles above, because of the path dependency, we needed to value

the storage contract and determine the required exercise strategy 7,300,000 times (10,000 scenarios

with 730 time-steps). Even if a simulation based method could be found with a computational cost

on the same order of magnitude as the RBF-PDE method used here, if such a method only produced

a value at one initial market state and inventory level it would take approximately 7,300,000 times

longer to calculate these CVA and PFE profiles. This illustrates the reason why Monte-Carlo based

methods are impractical for counter-party credit risk calculations.