Z_0 == discount factor

Background – in mathematical finance, DF is among the most basic yet practical concepts. Forward contracts (including equity fwd used in option pricing, FX fwd, FRA…) all rely directly on DF. DF is part of most arbitrage discussions including interview questions.

When we talk about a Discount Factor value there are always a few things implicit in the context

* a valuation date, which precedes

* a cash flow date,

* a currency

* a financial system (banking, riskfree bond…) providing liquidity, which provides

* a single, consistent DF value, rather than multiple competing values.

* [1] There's no uncertainty in this DF value, as there is about most financial contracts

– almost always the DF value is below 1.0

– it's common to chain up 2 DF periods

An easily observable security price that matches a DF value is the market price of a riskless zero-coupon bond. Usually written as Z_0. Now we can explain [1] above. Once I buy the bond at this price today (valuation date), the payout is guaranteed, not subject to some market movement.

In a math context, any DF value can be represented by a Z_0 or Z(0,T) value. This is the time-0 price of some physical security. Therefore, the physical security “Z” is a concrete representation of the abstract _concept_ of discount factor.

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