Financial Risk Manager Part 1

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Explanation:

To solve this problem, we follow the risk-neutral pricing framework used in Monte Carlo simulations. The process involves two steps:

1. Calculate the Weighted Average Payoff: Since each analyst used a different number of paths, we must find the expected payoff across all 20,000 paths.

Total Payoff = $(2,000 \times 43) + (4,000 \times 44) + (10,000 \times 46) + (4,000 \times 45) = 902,000$
Average Payoff ( $\bar{x}$ ) = $902,000 / 20,000 = 45.10$

2. Discount to Present Value: In Monte Carlo simulation, the price of the derivative is the present value of the expected payoff, discounted at the risk-free rate ( $r$ ) over the time to maturity ( $T$ ).

$T = 9/12 = 0.75$ years
$r = 0.04$
Price = $\bar{x} \times e^{-rT} = 45.10 \times e^{-(0.04 \times 0.75)} = 45.10 \times e^{-0.03}$
Price $\approx 45.10 \times 0.9704455 \approx \mathbf{43.767}$

Conclusion: The correct answer is B (USD 43.77). Choice D is incorrect because it forgets to discount the payoff to the present. Choice A and C result from simple averages or calculation errors. Always remember: Monte Carlo gives you the future payoff; you must discount it!

Explanation:

To solve this problem, we follow the risk-neutral pricing framework used in Monte Carlo simulations. The process involves two steps:

1. Calculate the Weighted Average Payoff: Since each analyst used a different number of paths, we must find the expected payoff across all 20,000 paths.

Total Payoff = $(2,000 \times 43) + (4,000 \times 44) + (10,000 \times 46) + (4,000 \times 45) = 902,000$
Average Payoff ( $\bar{x}$ ) = $902,000 / 20,000 = 45.10$

$T = 9/12 = 0.75$ years
$r = 0.04$
Price = $\bar{x} \times e^{-rT} = 45.10 \times e^{-(0.04 \times 0.75)} = 45.10 \times e^{-0.03}$
Price $\approx 45.10 \times 0.9704455 \approx \mathbf{43.767}$

Comments (1)

Yash ParwaniMar 22, 2026 7:49 AM

No details have been provided for the question

LeetQuiz .Mar 22, 2026 8:30 AM

Hi Yash, Thanks for your feedback. We've provided the necessary context and table for answering the question. Now, it has been answerable.

A portfolio manager has asked each of four analysts to use Monte Carlo simulation to price a path-dependent derivative contract on a stock. The derivative expires in nine months and the risk-free rate is 4% per year compounded continuously. The analysts generate a total of 20,000 paths using a geometric Brownian motion model, record the payoff for each path, and present the results in the table shown below.

Here's your data formatted as a clean Markdown table:

Analyst Number of Paths Average Derivative Payoff per Path (USD)
1 2,000 43
2 4,000 44
3 10,000 46
4 4,000 45

What is the estimated price of the derivative?

Analyst	Number of Paths	Average Derivative Payoff per Path (USD)
1	2,000	43
2	4,000	44
3	10,000	46
4	4,000	45

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Last updated: March 22, 2026 at 08:29

USD 43.33

8.7%

USD 43.77

39.1%

USD 44.21

26.1%

USD 45.10

26.1%