As a fictional example, a stock that pays no dividends is priced at 2,000 yen today, and the 1-year risk-free rate (a rate with no credit risk or similar risks) is 3% a year. The rate is simple interest for one year, and you can borrow and deposit money at that same rate. There are no transaction costs, and you can borrow the stock and sell it (short selling). The price of a forward contract for delivery of the stock in one year (the forward price) is 2,100 yen. Which trade leaves a profit whatever the stock price at maturity, and how much is left?
Quiz: Valuation Foundations
Last reviewed: September 26, 2026
This is a draft, shown for review before publication.
5 questions to check your understanding of the chapter “Valuation Foundations”. 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: D
Selling the forward, borrowing 2,000 yen and buying the stock leaves 40 yen at maturity. Buying the stock with a loan is a replication that produces the same cash flow as buying the forward, and the forward price found from its cost is 2,000 × 1.03 = 2,060 yen. A forward at 2,100 yen is expensive compared with this combination. If you sell the expensive forward, borrow 2,000 yen and buy the stock, today's cash flows net to 0 yen. At maturity, you deliver the stock to the buyer of the forward and receive 2,100 yen, then repay 2,060 yen on the loan. The 2,100 − 2,060 = 40 yen is left whatever the stock price at maturity.
If the stock price at maturity is 1,800 yen, the sold forward gains 2,100 − 1,800 = 300 yen. The stock and the loan lose 1,800 − 2,060 = −260 yen, for a total of 40 yen. If the stock price is 2,300 yen, the sold forward loses 200 yen. The stock and the loan gain 2,300 − 2,060 = 240 yen, for the same total of 40 yen.
100 yen is the result of forgetting to subtract the 60 yen of interest on the loan (2,100 − 2,000). Buying the forward, short selling the stock and depositing the proceeds is the trade for a forward price below 2,060 yen. If you make this trade when the forward price is 2,100 yen, at maturity you pay 2,100 yen under the forward out of a deposit of 2,060 yen, so you lose 40 yen whatever the stock price.
This arbitrage calculation does not include costs such as trading commissions. If the costs add up to more than 40 yen, no profit is left. If costs can be ignored and the market offers no such arbitrage opportunities (no-arbitrage), the forward price is 2,060 yen.
Learn more: No-Arbitrage and Replication
Question 2
As a fictional example, consider an underlying asset priced at 100 today that will be 120 or 80 after one period (a one-period binomial model). A call option (call) to buy this underlying at a strike price of 100 pays 20 at the end of the period if the underlying is 120, and 0 if it is 80. The one-period risk-free rate is 2% (you discount by dividing by 1.02). If you replicate the payoff of this call by buying the underlying and borrowing, what is the price of the call?
Answer and explanation
Answer: B
The price of the call (the premium) is 10.78. The number of units of the underlying used for replication is the difference between the payoffs (20 − 0 = 20) divided by the difference between the underlying prices (120 − 80 = 40), which is 0.5. If the underlying falls to 80, the 0.5 units are worth 40, so a loan that repays 40 at the end of the period leaves you with 0, the same as the call. The amount you borrow today is 40 ÷ 1.02 = 39.22. The cost of building the combination today is the 50 paid for the underlying (0.5 × 100) minus the 39.22 borrowed, which is 10.78. If the underlying rises to 120, the 0.5 units are worth 60, and 20 is left after repaying 40, matching the payoff of the call.
The value 10.00 is the result of taking the amount borrowed as 40 without discounting it (50 − 40). Money borrowed today accrues interest until the end of the period. To repay 40 at the end, you therefore borrow less than 40 today. The value 19.61 is just the payoff of 20 in the up case divided by 1.02, and it ignores the down case. The value 21.57 is the cost of buying 1 unit of the underlying and borrowing 80 ÷ 1.02 = 78.43 (100 − 78.43). This combination pays 120 − 80 = 40 in the up case, which does not match the call's 20, so it is not a replication.
If you buy the call for 10.78, you receive 20 if the underlying reaches 120. If it falls to 80, you receive nothing and lose the 10.78 you paid.
Learn more: Binomial Trees
Question 3
As a fictional example, consider a two-step binomial tree in which the underlying price starts at 100 and is multiplied by 1.1 or 0.9 at each step. The risk-free rate for one step is 2% (you discount by dividing by 1.02), and the probability p used for pricing is 0.6 for a rise and 0.4 for a fall. A put option (put) with a strike price of 105 is an American option, which can also be exercised before expiry. What is the value of the put at the node at step 1 where the underlying price is 90?
Answer and explanation
Answer: C
The value at the 90 node is 15.00. From this node, the underlying price at the next step is 99 or 81. The payoff of the put at expiry is 105 − 99 = 6 at 99, and 105 − 81 = 24 at 81. The value of holding on is (0.6 × 6 + 0.4 × 24) ÷ 1.02 = 12.94. The value of exercising now is the intrinsic value at this node, 105 − 90 = 15. The value at each node of an American option is the larger of the two, so it is 15.00, and exercising is the better choice at this node.
The value 12.94 is the value of holding on. It would be the value at this node for a European option, which can be exercised only at expiry. The value 13.20 is the probability-weighted sum (3.6 + 9.6) without the division by 1.02. The value 16.47 comes from swapping the probabilities of the up and down sides, as (0.4 × 6 + 0.6 × 24) ÷ 1.02. The p of 0.6 is the probability of the side that rises to 99, and the probability of the 81 side, where the payoff is larger, is 0.4.
At the node at step 1 where the underlying price is 110, the value of exercising now is 0. The value of holding on is (0.6 × 0 + 0.4 × 6) ÷ 1.02 = 2.35, which is larger, so you hold on.
Learn more: Binomial Trees
Question 4
As a fictional example, consider a binomial model in which the underlying is priced at 100 today and will be 110 or 90 after one period. A call with a strike price of 100 pays 10 if the underlying is 110 and 0 if it is 90. The risk-free rate is 2% for the period, and the cost of the replicating combination, made by buying 0.5 units of the underlying and borrowing 44.12, is 5.88. There are no costs, and short selling is possible. Why is the price of the call 5.88 even for an investor who forecasts a probability of 0.8 that the underlying reaches 110?
Answer and explanation
Answer: A
The payoff of the replicating combination at the end of the period is 55 − 45 = 10 at 110 and 45 − 45 = 0 at 90, the same as the call in both cases. These payoffs do not change whatever probability of a rise you forecast, so the cost of producing them stays at 5.88. However, when the real-world probability changes, the underlying's current price and the size of its moves (volatility) usually change with it, and the option price changes through them.
Nor is the price set by the probability the seller forecasts. A seller who forecasts a probability of 0.8 of reaching 110 has an expected payoff of 8, and dividing it by 1.02 gives 7.84. If the seller could sell the call for 7.84, the seller would use 5.88 of it to build the replicating combination and keep the difference of 1.96 whichever way the market moves. Conversely, a seller who forecasts a probability of 0.3 of a rise would value the call at 0.3 × 10 ÷ 1.02 = 2.94. If a buyer could buy the call for 2.94, the buyer could short sell 0.5 units of the underlying and invest 44.12, keeping the difference of 2.94. The only price at which such arbitrage does not work is the cost of replication, 5.88.
5.88 matches the expected payoff of 6, found with a probability of 0.6 for a rise, divided by 1.02. The 0.6 is a computational probability set by the two prices of the underlying and the interest rate (the risk-neutral probability). It is not an average of investors' forecasts.
With 0.6, the expected underlying price is 0.6 × 110 + 0.4 × 90 = 102, an expected return of 2%, the same as the risk-free rate. The real expected return, on the other hand, is the risk-free rate plus an extra return for bearing risk (the risk premium). For an investor who forecasts 0.8, the expected underlying price is 0.8 × 110 + 0.2 × 90 = 106, an expected return of 6%. That the price matches the expected value under 0.6 does not mean that investors in the market require no risk premium.
If you buy the call for 5.88, you receive 10 if the underlying reaches 110. If it falls to 90, you receive nothing and lose the 5.88 you paid.
Learn more: Risk-Neutral Valuation
Question 5
Among calculations involving derivatives, the calculation of value at risk (VaR) uses real-world probabilities based on investors' forecasts and similar inputs, not risk-neutral probabilities. Which of the following is the reason?
Answer and explanation
Answer: B
VaR forecasts the distribution of values on a future date and finds the loss from a point in its tail. So the whole distribution must be estimated with real-world probabilities. The risk-neutral probability used in risk-neutral valuation is the probability that turns a discounted expected value into a price, and it cannot be used to forecast future values.
Fitting parameters to the prices of instruments actively traded in the market (calibration) is done by valuation models, which are used to find prices. Valuation models in practice describe the movements of the underlying and other variables under risk-neutral probabilities from the start, and fit their parameters to market prices. So what is set is not the real-world probability but the risk-neutral probability, and it is set to match market prices.
Paths can also be generated with random numbers under risk-neutral probabilities. Pricing with the Monte Carlo method finds the expected value in risk-neutral valuation with random numbers. You set the underlying's expected return to the risk-free rate, generate many paths of the underlying price with random numbers, discount the payoff on each path, and take the average.
The Monte Carlo method does not need much more computation as the number of underlying assets increases, but this is a property of the method of calculation, not of the probabilities used. The Monte Carlo method can also be used for pricing with risk-neutral probabilities.
Learn more: Risk-Neutral Valuation