# 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

[8.67]

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

[8.68]

where

[8.69]

[8.70]

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

[8.71]

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

[8.72]

where

[8.73]

[8.74]

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 ^{1}*Q** ^{Libor}* comprises interpolated cash-valuation Libor rates for maturities corresponding to the expiration dates of the NYMEX options. We refer to these as the 1

^{st}through 12

^{th}nearby Libor rates:

[8.75]

Define the mapping ^{1}** S** = φ(

^{1}

**) as follows. Since futures margin daily, current values**

*Q*^{0}

*s**are zero and accumulated values*

^{Futures}^{1}

*S**are the margin payments for time 1:*

^{Futures}[8.76]

Options components are defined with Black’s (1976) options pricing formula, which we denote *B ^{Call}*( ) or

*B*( ), as appropriate. For example,

^{Put}[8.77]

where *y*_{3} is years until expiration, as of time 1, for third nearby options.

We could almost use ^{1}** Q** 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

^{1}

*Q**, which are not quoted in the markets. USD Libor rates published by the BBA are for 2*

^{Libor}^{nd}-day valuation and specific maturities. We specify key vector

[8.78]

All component vectors of ^{1}** R** are identical to corresponding component vectors of

^{1}

**except the last, which is**

*Q*[8.79]

We define the mapping ^{1}** Q** = (

^{1}

**) by simply mapping each component vector of**

*R*^{1}

**to its counterpart in**

*R*^{1}

**. The only exception is the component mapping**

*Q*^{1}

*Q*^{ Libor}=

^{Libor}(

^{1}

*R*^{ BBALibor}), which converts from 2

^{nd}-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

^{1}

*Q*^{ Libor}are not observable in the market, the interpolation

^{1}

*Q*^{ Libor}=

^{Libor}(

^{1}

*R*^{ BBALibor}) actually defines

^{1}

*Q*^{ Libor}.