Set out the factors that drive an option's value
The idea
Six inputs determine an option's value: the underlying price, the exercise price, time to expiry, the risk-free rate, volatility of the underlying, and any cash flows on the underlying such as dividends. A call gains value as the underlying rises, as volatility rises, and generally as time to expiry lengthens; a higher risk-free rate helps a call and hurts a put because the exercise price is discounted more heavily. Dividends work the other way, reducing call values and raising put values, since the holder of the option does not receive them.
A practice question
What does a rise in volatility do to a call's value?
- Lowers itWhy not: More dispersion widens the range of favourable outcomes while the downside stays floored at zero.
- Raises it - correct
- Leaves it unchangedWhy not: Volatility is a direct input to option value, and a higher figure raises it, which is why options are used to take views on volatility.
Volatility measures how much a price swings. A call's payoff is Max(0, Price at expiry - Strike). The loss is limited to the premium paid, but the gain has no upper limit. Higher volatility makes big upward moves more likely without making the worst case worse, so the call is worth more. For example, a stock that could end at $80 or $120 makes a $100-strike call worth more than one that could end at $95 or $105. It raises a put's value too.
A harder one
Why does a higher risk-free rate help a call and hurt a put?
- Because calls are more sensitive to interest rates in generalWhy not: Restates the observation instead of explaining it; both are sensitive, in opposite directions.
- Because puts are written more often than callsWhy not: Relative supply does not determine how an input moves a theoretical value.
- Because the exercise price is discounted more - correct
The risk-free rate is the return on a safe investment. Present value of strike = Strike / (1 + r)^T. A higher rate makes this smaller, because money paid or received later is worth less today. The call holder pays the strike later, which is better for them when the rate is higher. The put holder receives the strike later, which is worse. So a higher rate helps the call and hurts the put because the exercise price is discounted more.
The hardest tier
Laura Kim covers a listed company for an options desk. A European call and a European put on the company's stock have the same strike and expiry. During a review of option sensitivities she asks which single change, with everything else unchanged, would raise the put's value while lowering the call's value. Which change should she identify?
- A rise in the risk-free interest rateWhy not: This works the other way round. A higher rate lowers the present value of the strike, which helps a call and hurts a put.
- A rise in the stock's expected volatilityWhy not: Higher volatility raises the value of both options, because both gain from a wider range of outcomes.
- A higher dividend forecast for the period - correct
Look at how each change affects the call and the put. A higher dividend forecast lowers the expected share price at expiry, because option holders do not receive dividends. A lower price hurts a call and helps a put, so the two move in opposite directions. A higher risk-free rate does the reverse: it helps the call and hurts the put, because the strike is discounted more heavily. Higher volatility raises the value of both options, since the loss is limited to the premium while a bigger move raises the payoff. Only the dividend forecast fits.
In the app: 12 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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