This article focuses on calculating credit valuation adjustment (CVA) for commodity futures with wrong-way risk (WWR) and counterparty credit deterioration simultaneously. Kelin Pan presents an analytical expression that calculates CVA using integration of a commodity futures exposure and the conditional probability of a credit event under WWR and credit downgrades

The commodity futures price is one of the key market factors in valuing counterparty credit risk (CCR). Since a risky asset is priced by reducing the credit valuation adjustment (CVA) of a risk-free asset, the toughest aspect of CCR is CVA pricing. It increases the difficulty of pricing CVA when the correlation between counterparty exposure and a credit event – in this case, wrong-way risk (WWR) – is considered. According to Basel III (Bank for International Settlements 2010), the major credit losses during the credit crisis were due to counterparty credit downgrades or deterioration rather than actual default. Adopting this point of view, the CVA model should also incorporate credit deterioration.

Several research papers have studied CVA with WWR. Pykhtin & Rosen (2010) adopted a Gaussian copula model to calculate WWR and obtained an analytical expression using a normal credit exposure assumption. Hull & White (2012) postulated a linear model between exposure and default. Rosen & Saunders (2012) developed a simulation algorithm to capture the CVA for both general and specific WWR. Among commodities pricing models, Gabillon (1991) and Gibson-Schwartz (1990) showed two classical models for crude oil futures.

In this article, we present a CVA with WWR model for commodity futures contracts. The credit exposure for commodity futures is calculated based on the Gabillon (1991) two-factor model. The model parameters are calibrated to market data for crude oil futures. The credit deterioration indicator is calculated using the Standard & Poor’s (S&P) credit transition matrices (Vazza & Kraemer 2013). An analytical expression for commodity CVA with WWR is obtained. The numerical results show that CVA is a function of market-credit correlation, maturity time, risk-rating grade and credit transition period.

## Commodity pricing model

A popular model for commodity futures is the Gabillon (1991) model. The advantage of the Gabillon model is the analytical expression and easy estimation of model parameters it provides. Using the analogy of a cantilever can explain oil futures prices as well as commodity futures prices, provided the latter follow the cantilever assumption. The two state variables in the Gabillon model are the short- and long-term price of crude oil ($S$ and $L$, respectively), which are two stochastic processes:

 $\displaystyle\mathrm{d}S$ $\displaystyle=\mu_{S,t}\,\mathrm{d}t+\sigma_{S,t}\,\mathrm{d}W_{t1}$ $\displaystyle\mathrm{d}L$ $\displaystyle=\mu_{L,t}\,\mathrm{d}t+\sigma_{L,t}\,\mathrm{d}W_{t2}$

where $\mathrm{d}W_{t1}$ and $\mathrm{d}W_{t2}$ are two correlated Wiener processes with $\mathrm{d}W_{t1}\,\mathrm{d}W_{t2}=\rho\,\mathrm{d}t$, and $\rho$ is the correlation coefficient of the two Wiener processes. The short-term price is the nearest traded contract. For example, on March 19, 2018, the nearest contract was CLJ18, delivering in April 2018. The long-term price could be CLJ23, delivering in April 2023. The model on crude oil futures price is a function of $L$, $S$, $t$ and $T$ (maturity). Using Itô’s lemma, the crude oil futures price function yields:

 $\mathrm{d}F=\mu^{\mathrm{F}}\,\mathrm{d}t+\sigma^{\mathrm{F}}F\,\mathrm{d}W_{t}$

where $\mu^{\mathrm{F}}$ and $\sigma^{\mathrm{F}}$ are the drift (Gabillon 1991, equation (13)) and the volatility (Gabillon 1991, equation (23)) of futures price, respectively. The risk-neutral property of futures price implies a drift term of zero for the above equation, which leads to the following solution (Gabillon 1991, equations (19) and (24)):

 $F=A_{t,T}S^{B_{t,T}}L^{1-B_{t,T}}$

where:

 \left.\begin{aligned}\displaystyle B_{t,T}&\displaystyle=\exp\bigg\{{-}\int_{t% }^{T}\eta_{u}\,\mathrm{d}u\bigg\}\\ \displaystyle A_{t,T}&\displaystyle=\exp\bigg\{\int_{t}^{T}[(r-\delta_{u})B_{u% ,T}+\tfrac{1}{2}\nu_{u}B_{u,T}(B_{u,T}-1)]\,\mathrm{d}u\bigg\}\end{aligned}\right\} (1)

In addition, $r$ is the interest rate, while $\eta_{u}$, $\delta_{u}$ and $\nu_{u}$ are model parameters, calibrated using market data. The dynamic of futures price under the risk-neutral property is:

 $\displaystyle\mathrm{d}F_{t,T}=\sigma_{t,T}^{\mathrm{F}}F_{t,T}\,\mathrm{d}W_{t}$ (2) $\displaystyle\sigma_{t,T}^{\mathrm{F}}=[\sigma_{\mathrm{S}}^{2}B_{t,T}^{2}+% \sigma_{\mathrm{L}}^{2}(1-B_{t,T})^{2}+2\rho\sigma_{\mathrm{S}}\sigma_{\mathrm% {L}}B_{t,T}(1-B_{t,T})]^{1/2}$ (3)

where $\sigma_{\mathrm{S}}$ and $\sigma_{\mathrm{L}}$ are the volatilities of the short- and long-term price, respectively. Equation (2) implies the futures price is lognormally distributed. Using Itô’s lemma for the function $f(F)=\ln F$ yields:

 $F_{t,T}=F_{0,T}\exp\bigg\{{-}\hbox{\small\displaystyle\frac{1}{2}}\int_{0}^{% t}(\sigma_{u,T}^{\mathrm{F}})^{2}\,\mathrm{d}u+\int_{0}^{t}\sigma_{u,T}^{% \mathrm{F}}\,\mathrm{d}W_{u}\bigg\}$

This process was not discussed by Gabillon (1991). It was, however, discussed by Schwartz & Smith (2000, page 899, paragraph 1). In the latter paper, the Itô process is used for function $\smash{\mathrm{e}^{X}}$, where $X=\ln(F)$. We use function $\ln(F)$ directly. Therefore, the first term of the power function is negative. When $t=T$, we have $\smash{F_{T,T}=S_{T}}$. Thus, the spot price at future time $T$ is expressed as:

 $S_{T}=F_{0,T}\exp\bigg\{{-}\hbox{\small\displaystyle\frac{1}{2}}\int_{0}^{T}% (\sigma_{u,T}^{\mathrm{F}})^{2}\,\mathrm{d}u+\int_{0}^{T}\sigma_{u,T}^{\mathrm% {F}}\,\mathrm{d}W_{u}\bigg\}$ (4)

We further assume the volatility of the futures depends only on today’s information (calculation date $t$). This implies $\smash{\sigma_{u,T}^{\mathrm{F}}=\sigma_{t,T}^{\mathrm{F}}}$. Thus, (4) becomes:

 $S_{T}=F_{0,T}\exp\{-\tfrac{1}{2}(\sigma_{t,T}^{\mathrm{F}})^{2}T+\sigma_{t,T}^% {\mathrm{F}}\sqrt{T}Z\}$ (5)

where $Z$ is a standard normal random variable. Note $\smash{F_{0,T}}$ is the futures price at contract time $t=0$. According to (3) and (5), three parameters $\smash{\sigma_{\mathrm{S}}}$, $\smash{\sigma_{\mathrm{L}}}$ and $\smash{B_{0,T}}$ are required to be calibrated by market data. $\smash{\sigma_{\mathrm{S}}}$ and $\smash{\sigma_{\mathrm{L}}}$ are calibrated using market data directly, eg, by employing CLJ18 and CLJ23 contracts, as mentioned previously. In order to estimate $\smash{B_{0,T}}$ via (1), we assume $\eta_{t}$ follows a five-order polynomial function instead of the constant employed by Gabillon (1991):

 $\eta_{t}=\sum_{i=0}^{5}a_{i}t^{i}$

Substituting the above function into (1) yields:

 $B_{t,T}=\exp\bigg\{{-}\sum_{i=0}^{5}b_{i}(T^{i+1}-t^{i+1})\bigg\}$ (6)

where $T$ and $t$ are maturity and current time, respectively, and $\smash{b_{i}}=\smash{a_{i}/(i+1)}$ ($i=0,1,\dots,5$). In order to estimate $b_{i}$, the following six time series are created using crude oil futures contract data from December 20, 2017 to March 19, 2018: CLJ18, CLJ19, CLJ20, CLJ21, CLJ22 and CLJ23 for April-traded contracts.

## Conclusions

An analytical expression has been obtained for CVA with WWR for commodity futures. The mark-to-market value of commodity futures is lognormally distributed based on the Gabillon model. The credit deterioration indicator is introduced to consider credit downgrade events. A Gaussian copula model has been employed to specify the correlation between counterparty exposure and credit events. The impact of WWR on CVA is studied explicitly using the obtained formula. The CVA is a function of the credit deterioration indicator, correlation coefficient and asset class. The proposed model captures the impact of WWR on CVA and considers credit granularity by introducing the credit deterioration indicator.

The proposed model may be simple and straightforward, but it illustrates several characteristics that have not been studied before. The model is also compared with a published benchmark model. Further work using Monte Carlo simulation for comparison as well as considering the impact of the volatility assumption can be done.

Kelin Pan is an independent consultant. The author thanks Professor Steve Kopp of the Department of Statistical and Actuarial Science, University of Western Ontario, for his comments. The author also thanks the referee for helpful comments. Email: [email protected]

## References

• Bank for International Settlements, 2010
Basel III: a global regulatory framework for more resilient banks and banking systems
Report, available at http://www.bis.org/publ/bcbs189.pdf
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The term structures of oil futures prices
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Stochastic convenience yield and the pricing of oil contingent claims
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CVA and wrong way risk
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Pricing counterparty risk at the trade level and credit valuation adjustment allocations
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CVA the wrong way
Journal of Risk Management in Financial Institutions 5(3), pages 252–272
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Short-term variations and long-term dynamics in commodity prices
Management Science 46(7), pages 893–911
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