Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

Posts Tagged ‘Payoff Diagrams

[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)

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)

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)

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)

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?).

Written by Vineet

October 28, 2016 at 7:07 pm