You use the Black-Scholes formula to find the theoretical price of a European call option (call) on a stock that pays no dividends. Which of the following is not an input to the formula?
Quiz: Option Valuation
Last reviewed: October 15, 2026
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6 questions to check your understanding of the chapter “Option Valuation”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.
Question 1
Answer and explanation
Answer: C
The formula has five inputs: the price of the underlying asset, the strike price, the time to expiry, the risk-free rate (a rate with no credit risk or similar risks), and the volatility. The expected return of the underlying (how fast investors expect it to rise in value) is not one of them. This is because the formula is derived from the idea of combining the underlying with risk-free investing and borrowing, and continuously rebalancing the mix so that its value moves in the same way as the call (replication). Setting the expected return equal to the risk-free rate gives the same formula, so this way of pricing is called risk-neutral valuation.
Of the five inputs, volatility is the one that cannot be observed directly in the market, but it is still an input to the formula. If you think a call on a stock that is expected to rise should be worth more, the expected return may seem to belong among the inputs as well, but it does not.
Learn more: The Black-Scholes Model
Question 2
As a fictional example, you have bought one unit of a European call with an underlying price of 100, a strike price of 100, a time to expiry of 1 year, a risk-free rate of 1% a year (continuously compounded), no dividends, and a volatility of 20%. Found with the Black-Scholes formula, the call's delta is 0.5596 and its gamma is 0.0197. If the underlying price rises from 100 to 105, what is the change in the call price estimated with delta and gamma?
Answer and explanation
Answer: B
Estimated with delta and gamma, the call price rises by about 3.04. The delta part is delta × the change in the underlying price: 0.5596 × 5 = 2.798. The gamma part is gamma × the change in the underlying price × the change in the underlying price ÷ 2: 0.0197 × 5 × 5 ÷ 2 = 0.24625. Adding the two gives 2.798 + 0.24625 = 3.04425, or about 3.04.
2.80 is the estimate from the delta part alone. Delta also grows while the underlying price moves from 100 to 105, so delta alone underestimates the rise. 3.29 comes from forgetting to divide the gamma part by 2 (2.798 + 0.0197 × 5 × 5). 2.55 comes from subtracting the gamma part (2.798 − 0.24625).
If the underlying price falls from 100 to 95, the delta part is −2.798 and the gamma part is the same 0.24625, so the estimate is −2.798 + 0.24625 = −2.55175, or about −2.55. The call price falls by about 2.55, a loss for the buyer. Compared with the delta-only estimate, the rise is larger when the price goes up and the fall is smaller when it goes down. Both are only estimates, however, and the larger the move in the underlying price, the further they drift from the actual change.
Learn more: The Greeks
Question 3
As a fictional example, you have sold one unit of a European put option (put) with an underlying price of 100, a strike price of 100, a time to expiry of 1 year, a risk-free rate of 1% a year (continuously compounded), no dividends, and a volatility of 20%. The Greeks are found with the Black-Scholes formula. Which combination of signs for this position's delta, gamma, and vega is correct?
Answer and explanation
Answer: D
A sold put has positive delta and negative gamma and vega. A bought put on the same terms has a delta of −0.4404, a gamma of 0.0197, and a vega (the change in price when volatility rises by 1 percentage point) of 0.3945. A sold position has the opposite signs to a bought position in the same option, so the sold put has a delta of 0.4404, a gamma of −0.0197, and a vega of −0.3945.
Negative delta with positive gamma and vega is a bought put. If you overlook that selling reverses the signs, you choose this combination. All three positive is a bought call, and all three negative is a sold call. Gamma and vega have the same values for a call and a put, so a buyer of either has positive gamma and vega, and a seller of either has negative gamma and vega.
For this position, a rise of 1 in the underlying price gives a profit of about 0.44, and a fall of 1 gives a loss of about 0.44. A rise of 1 percentage point in volatility gives a loss of about 0.39, and a fall of 1 percentage point gives a profit of about 0.39.
Learn more: The Greeks
Question 4
Which statement about implied volatility is correct?
Answer and explanation
Answer: B
Implied volatility is the volatility found by putting an option's market price into a pricing formula and solving backward for the volatility at which the formula price matches the market price. Even for the same underlying, it has a value for each strike price and expiry.
The measure that is calculated from past percentage changes in the underlying price and has a single value once the period is fixed is historical volatility. It does not split by strike price and expiry either. It is found by taking the standard deviation of the percentage changes in price for each business day and converting it to an annual figure.
Learn more: Implied Volatility
Question 5
For options on a stock index, when you fix one expiry and line up the implied volatility for each strike price to form the volatility smile, which statement correctly describes the typical shape and how it differs by expiry?
Answer and explanation
Answer: A
For options on stocks and stock indices, looking at each expiry, the typical shape is a downward slope to the right (the skew), in which implied volatility is lower at higher strike prices. The slope is steeper for shorter expiries and flattens as the expiry gets longer. In a fictional stock index example, the gap between the values at strike prices 80 and 120 is 16.0 percentage points for 3 months, 13.0 for 1 year, and 9.0 for 2 years, getting smaller as the expiry gets longer.
An upward slope to the right has the direction reversed. A slope that is steeper for longer expiries has the difference by expiry reversed. A slope that is the same for every expiry overlooks that the slope flattens as the expiry gets longer.
Learn more: Volatility Smiles and Surfaces
Question 6
As a fictional example, when the forward rate is 1.00%, the Black model volatility is 30% and the normal volatility is 30bp, so the standard deviation of the change is 30bp under either model. The forward rate then falls to 0.50% while the two volatilities stay the same. What happens to the standard deviation of the change under the Black model and the Bachelier model?
Answer and explanation
Answer: D
Under the Black model, the standard deviation of the change is the volatility × the forward, so it is proportional to the level of the forward. At 0.50%, it is 30% × 0.50% = 0.15%, or 15bp. Under the Bachelier model, the standard deviation of the change is the normal volatility itself, so it stays at 30bp whatever the level of the forward.
Becoming 15bp under both models assumes that the Bachelier model is also proportional to the level. Staying at 30bp under the Black model while the Bachelier model becomes 15bp mixes up the two models. Staying at 30bp under both models assumes that the Black model also moves by the same width regardless of the level.
Learn more: The Black and Bachelier Models