Me trying to study and remember this for CFA. Tried to make it as reader friendly as possible.
Puts and calls are what you examples of derivatives. Derivatives are financial instruments and are called such because they derive value on an underlying asset (similar to maybe Christians whose value comes from God or the fact that Jesus saved them or that God created them).
Short intro, a put is an option to sell, meaning, if I have a stock I bought at Php100, and the Exercise Price of the put is Php 90, and when the price goes below Php 90, I can demand the seller of the put to buy back the shares at Php 90, limiting my losses to Php 10.
A call is an option to buy, meaning, the current price is Php 100, I don’t have money now, and the prices seem to go up, but I’ll fix my losses now, and buy a call with an exercise price of Php 105, so no matter how high the prices will go, I can buy at Php 105.
Put-call parity is this equation: c + X = p + S, with
c – price of call
X – bond purchased amounting to exercise price
p – price of put
S – price of underlying asset.
Meaning, the payoff of buying a call and a bond amounting to the exercise price is equal to the payoff of buying a put and the underlying asset. Why is this important? Because you can create synthetic financial instruments with it! You can create a synthetic call by transforming the equation:
c = p + S – X, meaning you buy a put and the underlying asset and you sell a bond (meaning you borrow money which you will use to buy the underlying asset). So having a put, an underlying asset, and a loan is all equal to owning a call. You can also create a synthetic put, bond or asset!
So why bother buying/selling 3 instruments when you can just buy a call? This exercise is not telling you to actually buy and sell these 3 instruments. What this is saying is the put-call parity should exist or else, there is an arbitrage opportunity. Arbitrage basically means getting more returns with zero risk (and we all know more returns should be accompanied with more risk). So in short, if c + X <> p + S, somebody screwed up the pricing and there is money to be made somewhere.
Proof that the put-call parity exists:
Assume that current price of share is 100, X is 150.
| Transaction | Current Value | Meaning | Payoff Formula | Value at Expiration |
| Buy Call | c | Price you paid for the option | max(0, S-X) | 0 (worthless, you’d rather buy the stock at current prices than at exercise price) |
| Buy Bond | X/(1+r)^T | The money you will use to buy at X, you invest it first in the Tbills | X | 150 Maturity Value |
| Total | 150 | |||
| Buy Put | p | Price you paid for the option | max(0, X-S) | 50 (Stock price is 100 but you can sell at 150) |
| Buy Underlying Asset | S (150) | Price you paid for the asset | S (100) | 100 (Current price) |
| Total | 150 |
So theeerrree… that’s the put-call parity.