8.6 Example: Options
Measure value-at-risk as 1-day 95% USDvalue-at-risk. Count basis days as actual days. Assume cash valuation. A trader holds NYMEX Henry Hub natural gas futures and options with expirations out to a year. We shall specify a procedure for constructing primary mapping of the form
Futures are for 10,000 MMBtu. Puts and calls are for a single future and expire on that future’s last trading day. Out to a year, there are monthly expirations of futures, puts, and calls. A challenge in modeling exchange-traded options is the proliferation of strikes and expirations. We consider asset vector
and other components are similar. When implemented, the value-at-risk measure will need to be flexible to handle the actual strikes trading at any given time.
We determine holdings ω, and define
We shall employ futures prices, implied volatilities, and Libor rates as key factors. To avoid a proliferation of such factors, it is natural to model just a handful and interpolate to obtain the rest. Because such interpolation represents an approximation, we defer it for a subsequent remapping. For now, we directly model implied volatilities corresponding to each option as well as monthly Libor discount factors.
Specify risk factors
and other implied volatility components are specified similarly. Invoking put-call parity, we do not distinguish between put and call implied volatilities for a given strike and expiration.
The Libor component 1QLibor comprises interpolated cash-valuation Libor rates for maturities corresponding to the expiration dates of the NYMEX options. We refer to these as the 1st through 12th nearby Libor rates:
Define the mapping 1S = φ(1Q) as follows. Since futures margin daily, current values 0sFutures are zero and accumulated values 1SFutures are the margin payments for time 1:
Options components are defined with Black’s (1976) options pricing formula, which we denote BCall( ) or BPut( ), as appropriate. For example,
where y3 is years until expiration, as of time 1, for third nearby options.
We could almost use 1Q as a key vector. Indeed, we almost will. The only reason we choose not to do so is the contrived nature of our Libor rates 1QLibor, which are not quoted in the markets. USD Libor rates published by the BBA are for 2nd-day valuation and specific maturities. We specify key vector
All component vectors of 1R are identical to corresponding component vectors of 1Q except the last, which is
We define the mapping 1Q = (1R) by simply mapping each component vector of 1R to its counterpart in 1Q. The only exception is the component mapping 1Q Libor = Libor (1R BBALibor), which converts from 2nd-day valuation to cash valuation and interpolates to adjust maturities. It parallels techniques covered in Section 8.2, so we don’t provide details. Note that the interpolation is not an approximation—it is not a remapping. Since component of 1Q Libor are not observable in the market, the interpolation 1Q Libor = Libor (1R BBALibor) actually defines 1Q Libor.