Here, I will discuss the Call option, which grants the right to purchase a stock at a predetermined price in the future.
What Determines the Price of a Call Option?
- Upper Bound by Stock Price: The price of the option is capped by the price of the underlying stock. Naturally, an option’s price cannot exceed the stock’s price.
- Lower Bound by Exercise Profit: The option’s price is also bounded from below by the profit that can be made upon exercise. If exercising the option can immediately yield X$, then the option’s price cannot be less than X$.
- Zero Stock Price Equals Zero Option Price: When the stock price is zero, the option price is also zero. If there were any possibility of the stock price increasing, it would have some value today, but since it is zero, it indicates it will also be zero in the future (NPV=0).
- Stock Price Increase Raises Option Price: As the stock price rises, the option’s price increases, because the exercise price is fixed. The higher the stock price, the greater the potential profit from exercising the option, thus increasing its price.
- Convergence of Option Price to Exercise Profit: As the stock price rises, the option price approaches the profit obtainable from exercising it (exceeds it but approaches). More precisely, it converges to the stock price minus the NPV of the exercise price.
- As the stock price rises, the likelihood of exercising the option increases, because the probability of the stock price rising further decreases, though it still remains.
Importance of the NPV of the Exercise Price
Why do we consider the NPV of the exercise price? Suppose the option does not exist as a security you can sell. Instead, you simply have a contract granting you the right to buy the stock at a predetermined X$. When the time comes, you must have X$ in your account to exercise the option. Thus, you need to deposit the PV of X$ now so that with interest, you have X$ in the future.
From this, we derive two additional important conclusions:
- The higher the risk-free interest rate, the higher the option’s price.
- The longer the period until the option’s expiry, the higher the option’s price.
- Option Price Exceeds Immediate Exercise Profit: The minimum value of the option, or its lower bound, always exceeds the profit obtainable from immediate exercise because there is a probability that the stock price will rise further.
- Profit is not capped from above, while loss is capped by the exercise price. Thus, the probabilistic net effect is always positive.
From this, we derive the following key conclusion:
- Higher Stock Price Volatility Increases Option Value: The greater the stock price volatility, the higher the option’s value, as there are more opportunities for additional profit.
Source:
“Principles of Corporate Finance” by F. Allen, R. A. Brealey, & S. Myers