The Asymmetry, in Math
Payoff functions, kinks, and the convexity that makes options different from stock
Two investors place differently-shaped bets on the same stock. The first buys 100 shares at $200 — a $20,000 position. The second buys one at-the-money call expiring in 30 days at the prevailing 30% implied volatility — a contract that costs about $7.
22 per share, or $722 for the 100-share contract. The stock rallies to $220 by expiration. The shareholder makes exactly $2,000 — twenty dollars per share, the linear payoff a long position always pays on a $20 move.
The call holder also collects $2,000 (the contract's intrinsic value at expiration is $20 per share × 100 shares), but they only put up $722 to be there. Their profit is $1,278 on $722 of capital — a 177% return. The shareholder's return on the same move is 10%.
Now run the experiment the other direction. The stock falls to $180.
That is the opening. Finishing a lesson is where it stops being interesting and starts being useful: the full lesson runs to 8 sections and ends with 6 practice questions. A free account is what opens the rest, and the other 255 lessons in the Academy with it. No card.
What this lesson covers
- 1Why the kink creates value above intrinsic
- 2Convexity is the technical name for the asymmetry
- 3Payoff Diagram Builder
- 4Payoff functions and their derivatives
- 5What an at-the-money call is worth at different volatilities
- 6GameStop, January 2021 — convexity in extremis
- 7Where to see this on the platform
- 8Summary