Mapping derivatives

At this point, we have seen how to map primary assets. In particular, in the previous
section we have seen how to map asset flows at arbitrary maturity points to maturity
points contained within fundamental asset set. In this section, we shall see how to
map assets that are derivatives of primary assets contained within the set of risk
factors. We do this by approximating the derivative by primary asset flows.
Primary asset values
The key to our approach is expressing the value of derivatives in terms of the values
of primary asset flows. We will do this by developing expressions for the value of
derivatives in terms of basic variables that represent the value of primary asset flows.
We shall term these variables primary asset values (PAVs). An example of such a
variable would be the value at time t of a US dollar to be delivered at (absolute) time
T by the US government. We use the notation USDTreasury
T (t) for this variable. We
interpret these variables as equivalent ways of expressing value. Thus, just as one
might express length equivalently in centimeters, inches, or cubits, one can express
value equivalently in units of spot USD or forward GBP for delivery at time T by a
counterparty of LIBOR credit quality.
Just as we can assign a numerical value to the relative size of two units of length,
we can compare any two of our units of value. For example, the spot FX rate GBP/
USD at time t, XGBP/USD(t), can be expressed as
XGBP/USD(t)óGBPt (t)/USDt (t)2
Since both USDt (t) and GBPt (t) are units of value, the ratio XGBP/USD(t) is a pure
unitless number. The physical analogy is
inch/cmó2.54
which makes sense since both inches and centimeters are units of the same physical
quantity, length. The main conceptual difference between our units of value and the
usual units of length is that the relative sizes of our units of value change with time.
The zero-coupon discount factor at time t for a USD cashflow with maturity qóTñt,
DUSD
q (t), can be expressed as
DUSD
q (t)óUSDT(t)/USDt (t)
(We use upper case D here just to distinguish it typographically from other uses of d
below.)
Thus, the key idea is to consider different currencies as completely fungible units
for expressing value rather than as incommensurable units. While it is meaningful
to assign a number to the relative value of two currency units, an expression like 196