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Name the six factors that influence the price of an option.

FRM® Part I · Financial Markets and Products · Properties of Options · FRM1.FM.13.a

The idea

Six factors drive the price of a stock option: the current stock price, the strike price, the time to expiration, the volatility of the stock, the risk-free interest rate, and the dividends expected during the option's life. A higher stock price raises a call and lowers a put. A higher strike does the reverse. Higher volatility raises both, because the option holder keeps the upside and is protected on the downside. A higher risk-free rate raises calls and lowers puts, since the strike is paid or received later. Higher expected dividends lower calls and raise puts, because the stock drops on the ex-dividend date. Longer maturity usually raises value, but a European option can be an exception, especially a deep in-the-money put. Memorise the sign table, because exams test it with short scenarios.

A practice question

From the question bank, with the reason each wrong option is wrong.

A European call and a European put on the same non-dividend stock have the same strike and maturity. The stock price rises while all other factors are unchanged. What happens to the two option values?

  1. Both increase in valueWhy not: Treats the stock price as if it behaved like volatility; only volatility raises both a call and a put.
  2. The call increases in value and the put decreases in value - correct
  3. The call decreases in value and the put increases in valueWhy not: Reverses the signs; a higher stock price makes a call more likely to finish in the money and a put less likely.
  4. Neither changes, because the strike price is fixed and the payoffs depend only on itWhy not: Ignores that the payoffs depend on S - K for a call and K - S for a put, so the option value moves with the stock price.

A call pays max(S - K, 0), so a higher S helps it; a put pays max(K - S, 0), so a higher S hurts it. Only higher volatility raises both.

A harder one

Central banks raise short-term rates, lifting the continuously compounded risk-free rate. Holding the stock price, volatility, strike and maturity constant, what is the effect on European options on a non-dividend stock?

  1. Calls decrease in value and puts increase in value, because the stock must earn a higher returnWhy not: Confuses the discount effect on the strike with the required return on the stock; the stock price is held constant.
  2. Both calls and puts increase in value, because higher rates raise uncertaintyWhy not: Confuses interest rates with volatility; rates have opposite effects on calls and puts.
  3. Calls increase in value and puts decrease in value, because the present value of the strike falls - correct
  4. Both calls and puts decrease in value, because the present value of the payoff fallsWhy not: Discounts the payoff of both options but ignores that the strike is paid by a call holder and received by a put holder.

A higher rate lowers K x exp(-rT). The call holder pays the strike later, which is worth more; the put holder receives the strike later, which is worth less. Calls rise, puts fall.

The hardest tier

A risk manager holds a European call and a European put on the same non-dividend stock. Which single change, all else equal, increases the value of both options?

  1. A rise in the risk-free rate, which lowers the present value of the strikeWhy not: Calls gain and puts lose from a higher rate, since the present value of the strike falls.
  2. A rise in the volatility of the stock - correct
  3. A rise in the strike price, which makes the call harder to finish in the moneyWhy not: A higher strike lowers a call and raises a put, so the effects on the two options offset in sign.
  4. A rise in the expected dividends paid before expiry by the stockWhy not: Dividends lower the call and raise the put; the signs again differ between the two options.

Volatility is the only factor whose increase raises both: the holder keeps unlimited upside (call) or large downside gain (put) while losses are limited to the premium. Rates, strikes and dividends move calls and puts in opposite directions.

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