# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Payoff diagrams

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Payoff diagrams are simply diagrammatic representation of payoffs at termination/expiration of a contract w.r.t value of the underlying $S_T$.

For example, for a baker’s forward contract (long forward) it is a plot of its payoff ($S_T - K$) at expiration ($T$) w.r.t the wheat’s spot price $S_T$. Let’s recap the payoffs for some of the contract we have already talked about, and then some:

1. Long Forward: $F = S_T - K$
2. Short Forward: $-F = K - S_T$
3. Long Call: $C = \mbox{max}(S_T - K, 0)$
4. Short Call: $-C = \mbox{- max}(S_T - K, 0)$
5. Long Put: $P = \mbox{max}(K - S_T, 0)$
6. Short Put: $-P = \mbox{- max}(K - S_T, 0)$

With this idea in place, we can also talk about the payoff diagram of an underlying itself w.r.t itself. This is of course trivial, because a stock is a stock is a stock, so if we buy a stock, that stock’s value is just the value of the stock.

Another easy one is that for a zero-coupon bond. A zero-coupon bond always return the face value at ‘expiration’ (maturity). It is often convenient in the context of options to talk about about a zero-coupon bond with face value $K$ and expiration $T$, i.e coinciding with that implicit in the forward contracts/options.

Let’s draw some pictures now. Since they are all basically plots of very simple functions as $\mbox{Payoff} = f(S_T)$, I take it that you recall enough of drawing functions of the kind $y = f(x)$ from your school days to not spend time explaining how you would draw these. Here we go.

Long Forward and Short Forward

Payoff Diagram – Long Forward (Baker) and Short Forward (Farmer)

Long Call and Short Call

Payoff Diagram – Long Call (Baker’s choice) and Short Call (Baker’s obligation)

Long Put and Short Put

Payoff Diagram – Long Put (Farmer’s choice) and Short Put (Farmer’s obligation)

Stock (underlying) and Zero-coupon Bond

Payoff Diagram – Stock (or Wheat or any underlying) and Zero-coupon bond (face value K with maturity T)

Note that all upward sloping lines have a slope of +1 and downward sloping lines have a slope of -1 (why?).