Mathematical and Empirical Finance, 2023– 2024 Assignment 3: Binomial Trees and Option Pricing

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Mathematical and Empirical Finance, 2023–2024

Assignment 3: Binomial Trees and Option Pricing

Note: You should submit your report (written in English) as a PDF file in combination with a code file, via the Assignment upload tool in Canvas. It should be possible to read your report and understand your answers without having to open the code file; but you may refer to the details of particular calculations in the code file. Some questions may require an answer to an earlier question. If you have been unable to fully answer the earlier question, replace this answer by a (reasonable) hypothetical value, and clearly indicate that you will use this guess in the rest of the assignment. Clearly indicate the names of your team (of one, two or three students) on the first page of report, and in the names of both the Excel file and the PDF file (use the format “MEFAssignment3Name1Name2Name3.pdf”). If (and only if) you have problems submitting your work via Canvas, you may e-mail it to [email protected], with “MEF Assignment 3” in the subject line.

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In this assignment, you are tasked to simulate stock prices and value options on the underlying stock, both using binomial trees and the control-variate approach.

For both approaches, we consider the following parameters:

❼ There are N = 5 steps, each of size ∆t = 1/4 , i.e., one quarter

❼ The options to be considered below expire at T = N∆t = 5/4, i.e., 1 year and 3 months

❼ The current stock price is S(0) = 100

❼ The continuously compounded interest rate is r = 0.02

❼ The strike price is K = 80

1. [30 points] As you have noticed, the above parameters do not contain the stock price volatility σ and the dividend yield δ, since in practice they are commonly unknown. Therefore, you are tasked in this ques-tion to compute them using the put-call parity. For this, assume that you have a European call option with price C0 = 25.898301 and a Eu-ropean put option with price P0 = 3.923094.

First, using the put-call parity compute the dividend yield δ. Next, using this result for δ, obtain the implied volatility, i.e., the value of σ such that the Black-Scholes European call option price equals the mar-ket price C0. Report the solutions for δ and σ that you have obtained.

Hint: You may want to use a built-in function such as fzero to find the implied volatility.

2. [30 points] In this question, you are asked to determine the price of a European call option using the binomial tree method, making use of the parameters given above and the previously found dividend yield δ and volatility σ. To this extend, write a program/function that produces and displays:

❼ the stock price tree

❼ the tree of a European call option

Based on a binomial tree, what is the value C0 of the option at time t = 0? Is it different from the price of the call option in part a)? If yes, what could be the reason for it?

Notes:

❼ It is recommended to display the trees in the format of Figure 12.8: as an upper triangular matrix (only elements on or above the main diagonal are considered)

❼ Display (stock and option) prices rounded to two decimal places, to help readability. (This only applies to final results!)

3. [40 points] Now, you want to price an Asian call option on the same stock, where, unlike for classic European call options, the pay-off is determined as A = max{Savg − K, 0}, where Savg denotes the average price of the underlying stock over the sampling period. Using the given parameters and 10,000 Monte Carlo repetitions, simulate the price of this Asian option using the control-variate technique. As a control-variate use the price of a European call-option. Use Table 1 below to present your results. That is, display the Monte Carlo means of the price of the Asian call-option without and with using the control vari-ate, alongside the Monte Carlo standard errors for 100, 500, 1000, 2000 and 10,000 Monte Carlo draws. Comment on the differences between using and not using the control-variate.

Table 1: Question 3 – Control-Variate Results

w/o Control Variate w/ Control Variate

MC Draws Mean StD. Mean StD.

100

500

1000

2000

10000

Hint: For the control-variate approach, use c = 0 for the first Monte Carlo draw and use the optimal value for c for the subsequent Monte Carlo draws!