Answer-first summary for fast verification
Answer: USD 43.77
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!
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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?
A
USD 43.33
B
USD 43.77
C
USD 44.21
D
USD 45.10