Background

Delta-one is the colloquial term given to what can be understood as call-put parity, although it can arguably be extended to include a variety of derivative products. Consider Izabella Kaminska’s description from Financial Times:

"From a product point of view it’s simple. Delta one equals ETFs, sector swaps, dividend swaps and thematic baskets. It can include warrants and options too. Anything really, providing it involves returning a one-to-one performance to your client."

Call-put parity is interesting from both a finance and mathematical perspective. Here, I will share a mathematical proof to demonstrate why it is that call-put parity can occur.

The function we want to prove

$c(x)-p(x)=x-a$

The proof

  1. Definitions: $c(x)=\max\{x-a,0\}, p(x)=\max\{a-x,0\}$

  2. Re-write the functions:

$c(x)=\begin{cases} x-a & \mbox{if }x-a>0,\mbox{ i.e. }x>a,\\ 0 & \mbox{if }x\le a, \end{cases}$

$p(x)=\begin{cases} a-x & \mbox{if }a-x>0,\mbox{ i.e. }x<a,\\ 0 & \mbox{if }x\ge a. \end{cases}$

  1. We can solve the equation using two cases.

Case 1: $x\ge a$

if $x>a, c(x)=x-a, p(x)=0$
so $c(x)-p(x)=x=c(x)=x-a.$

Case 2: $ x<a$

If $x<a, c(x)=0, p(x)=a-x$ so $c(x)-p(x)=-p(x)=x-a.$

Conclusion

Call-put parity, or delta-one, is an interesting proposition from both a financial and mathematical perspective. Of course, it is in arbitrage that the real money is made (e.g. situations in real life where the one-to-one parity does not exist).