Monthly Archives: December 2012

Potential Future Exposure for a Derivatives Position

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Potential Future Exposure (PFE) is a simple way to measure the riskiness of a derivatives portfolio, given some statistical assumptions about the distribution of future prices of the derivative portfolio’s underlying assets.  These assumptions are wrong, but not so wrong that the PFE will not provide some valuable information. The PFE uses the model of the underlying asset price process that the Black-Scholes option formula uses, and therefore shares the same assumptions, so the PFE can illustrate some of its limitations.

The PFE is easy to model in Excel.  We start with the asset price process,

ST = Ste(r – 1/2σ^2)(T-t) +/- σZ(T-t)^2

which is described here and here. For this example I have taken the WTI Crude Oil Cal13 prices as the underlying, meaning we have a portfolio of derivatives: futures, forwards, options, or a mixture of all of them with WTI as the underlying.

The r for each month will be the LIBOR rate of corresponding term, the sigma will be the implied volatility taken from the option on each contract month, and the Z will be a confidence level that we set.  Here we use 95%, or 1.65 (standard deviations).  In Excel, use NORMSINV(.95) to get 1.65.  If our assumptions were correct, which they aren’t, 95% of the time the portfolio’s worst loss would be inside the interval we create.

Apply the above formula to each contract month, adding the second term (with the Z factor) to get the upper interval and subtracting to get the lower, and using each month’s specific r, sigma, and time to expiration for each monthly contract (T-t), and we get this result:

It is a simple matter to get an exposure in dollar terms.  We take the portfolio’s exposure in barrels, net the longs and shorts for each month, and multiply each month’s underlying by each month’s corresponding price on the lower curve (since we would be losing if the underlying went down).  If the portfolio has options, treat them as being delta barrels long or short.  This works since the option delta is calculated using the same r, sigma, S, and (T-t) as our underlying process uses here.

Note that the upper bound is farther away from the current forward curve than the lower bound.  This is due to the lognormal distribution of asset prices.  The downside is floored at zero and the upside is infinite.

Costless Collars and the Volatility Skew

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Costless collars are one of the most-used hedging structures for commodity producers.  In this structure, the downside protection, the put option, is financed by selling a call.  In other words, the insurance against prices falling is paid for by giving up some of the upside in the case of a price rise.

The producer will set his downside protection target and then look at the call option market to see how much upside he will have to give up.  Say, for example, that the February WTI crude oil futures contract is currently at $89 and the at-the-money implied volatility is 27% annualized.  If the producer wants to limit his downside at $80, he will have to pay $.53/bbl for the $80 strike put option.  He can sell a $99.50 strike call option for $.53/bbl to finance his put (this is assuming no bid/ask spread for simplicity).

The option market, however, never has the same volatility for different strikes, despite the assumptions of the Black-Scholes and Black 76 (which is used in this example) option pricing models.  In fact, the February $80 put option is trading with a 29.5% implied volatility.  The $99.50 call has a 25.5% implied volatility.  This is because the market is putting a higher likelihood on a down move than on an up move, and the producer will have to pay more for that.

The $80 put is selling for $.69/bbl, while the $99.50 call is selling for $.43/bbl.  The structure will not be costless.  A costless structure will have to include a call with a $97.25 strike.  The producer will have to give up $2.25/bbl more of upside potential than if the volatility across all strikes was the same as the at-the-money volatility.  This is the effect of the volatility skew.  The payoff of the structure will look like this, with the producer’s price following the market between $80 and $97.25: