Put-Call Parity: Understanding European Options

The No-Arbitrage Argument for Options

Just as we did with forward contracts, we will construct two portfolios that, despite being built from different components, have the exact same value at the options' expiration date, $T$. The no-arbitrage principle dictates that if their values are identical at time $T$, their costs at time 0 must also be equal.

Let's define two specific portfolios:

1. Portfolio A: The Fiduciary Call. This portfolio consists of one European call option and a risk-free investment of cash equal to the present value of the strike price.
2. Portfolio B: The Protective Put. This portfolio consists of one share of the underlying stock and one European put option.

The following video provides an excellent intuitive explanation of why these two portfolios must have identical payoffs at expiration.

As the video demonstrates, at expiration date $T$, the value of both portfolios is $\max (S_T,K)$.

• If $S_T>K$, the call is exercised (and the put expires worthless). The value of both portfolios is $S_T$.
• If $S_T\le K$, the put is exercised (and the call expires worthless). The value of both portfolios is $K$.

Since the payoffs are identical in all possible states of the world, the cost to establish these portfolios today must also be identical.

Stating and Applying the Put-Call Parity Formula

The insight from the previous section leads us directly to the put-call parity equation. Let's formally define the components:

• $C$: The price of a European call option at time 0.
• $P$: The price of a European put option at time 0.
• $S_0$: The price of the underlying stock at time 0.
• $K$: The strike price of both options.
• $r$: The continuously compounded risk-free interest rate.
• $T$: The time to expiration (in years).

The cost of the fiduciary call (Portfolio A) is $C+K{e}^{-rT}$.
The cost of the protective put (Portfolio B) is $P+S_0$.

Equating these costs gives the put-call parity formula:

This relationship holds for European options on a non-dividend-paying stock.

Application 1: Calculating an Unknown Option Price

The most direct application of the formula is to solve for the price of one of the assets, assuming the others are fairly priced. For example, if you know the call price, you can calculate the implied fair price of the corresponding put.

Example:
A stock is currently trading at $60. A 6-month European call option with a strike price of $58 is priced at $5.50. The risk-free rate is 4% per annum. What is the fair price of a 6-month European put with the same strike price?

• $S_0=60$
• $K=58$
• $C=5.50$
• $r=0.04$
• $T=0.5$

We can rearrange the parity formula to solve for $P$:

Plugging in the values:

The fair price for the put option is $2.35.

Application 2: Identifying Arbitrage Opportunities

Your background in probability will help you appreciate the power of this application. When the parity relationship does not hold, we have found two portfolios with identical, certain payoff structures but different initial costs. This is a clear arbitrage opportunity. The strategy is simple: buy the cheaper portfolio and sell the more expensive one.

The following video and text provide excellent walkthroughs of how to construct the arbitrage trade.

Let's formalize the strategy with a numerical example from a more academic source.

To summarize the arbitrage strategy:

• If $P+S_0>C+K{e}^{-rT}$: The protective put is overpriced.

  • Action: Sell the protective put (sell the put, short-sell the stock) and buy the fiduciary call (buy the call, invest $K{e}^{-rT}$ in a risk-free asset).
  • Result: You receive a guaranteed risk-free profit today equal to $(P+S_0)-(C+K{e}^{-rT})$. Your obligations at expiration are perfectly covered by your assets.

• If $P+S_0<C+K{e}^{-rT}$: The fiduciary call is overpriced.

  • Action: Sell the fiduciary call (sell the call, borrow $K{e}^{-rT}$) and buy the protective put (buy the put, buy the stock).
  • Result: You receive a guaranteed risk-free profit today equal to $(C+K{e}^{-rT})-(P+S_0)$.

Adjusting for Dividends

The basic formula assumes the underlying stock pays no dividends. If it does, the relationship must be adjusted. A person holding the stock (in the protective put portfolio) will receive the dividends, while the person holding the synthetic stock (in the fiduciary call portfolio) will not.

To restore parity, the spot price in the equation must be reduced by the present value of the dividends ($D$) expected to be paid during the life of the options.

The adjusted put-call parity formula for a stock paying discrete dividends is:

where $D$ is the present value of the expected dividends.

For a stock index that pays a continuous dividend yield $q$, the formula is:

This is directly analogous to the forward pricing formula with a known yield that we saw in the last lesson.

Conclusion

In this lesson, we have derived and applied one of the most fundamental relationships in financial engineering: put-call parity. Building on the no-arbitrage principle from our previous lesson, we showed that the prices of European puts and calls are not independent but are tightly bound together.

Key Takeaways:

• Put-Call Parity is based on constructing two portfolios (a protective put and a fiduciary call) that have identical payoffs at expiration and thus must have the same initial cost.
• The Formula (No Dividends): $C+K{e}^{-rT}=P+S_0$. This applies to European options with the same strike, expiration, and underlying.
• The Formula (With Dividends): $C+K{e}^{-rT}=P+S_0-D$, where $D$ is the present value of dividends.
• Applications: The formula can be used to calculate the implied fair price of an option or to identify and execute risk-free arbitrage trades if the market is mispriced.

We have now established a static relationship between option prices at a single point in time. In our next lesson, we will begin to explore the dynamics of option pricing, examining the various factors that cause option prices to change over time and with movements in the market.