MFE Lesson 11 slides

# MFE Lesson 11 slides - Lesson 11 The Black-Scholes formula...

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Lesson 11: The Black-Scholes formula: applications and volatility. ACTS 4302 Natalia A. Humphreys October 11, 2012 1 / 25

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Acknowledgement This work is based on the material in ASM MFE Study Manual for Exam MFE/Exam 3F. Financial Economics (7th Edition), 2009, by Abraham Weishaus. 2 / 25
What we’ll study In this section we’ll study how profit evolves on an option as time passes. We’ll also determine how to calculate the volatility that is implied in the BS formula. 3 / 25

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Profit diagrams before maturity As a European option ages, time to expiry decreases. A 91-day European option that is 5 days old is equivalent to an 86-day European option, so we can track the price of options using the BS formula. 4 / 25
Call options and bull spreads Let C T be the price of a call option with expiry time T . Suppose after time t the purchaser decided to sell it. Then the profit they made on this call is the change in call premiums at time T - t , that is the di erence between the final value of the call and the initial value of the call grown with interest:: Profit = C T - t - C T e rt If the time to expiry is measured in days, the formula will change to: Profit = C T - t - C T e rt / 365 5 / 25

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Example 11.1 You are given: (i) The price of a stock is 55. (ii) The stock’s continuous dividend rate is 0.01. (iii) The volatility of the stock is 0.1. (iv) A 182-day European call has strike price 56. (v) The continuously compounded risk-free rate is 0.05. Calculate the 10-day holding period profit on the call if the stock’s price is 56 at the end of 10 days. 6 / 25
Example 11.1. Solution. d 1 = ln 55 56 + (0 . 05 - 0 . 01 + 0 . 5 · 0 . 1 2 ) 182 365 0 . 1 q 182 365 = 0 . 0626 d 2 = 0 . 0626 - 0 . 1 r 182 365 = - 0 . 0080 N ( d 1 ) = N (0 . 06) = 0 . 5239 , N ( d 2 ) = N ( - 0 . 01) = 0 . 4960 e - rt = e - 0 . 05 · 182 365 = 0 . 9754 e - δ t = e - 0 . 01 · 182 365 = 0 . 9950 C = 55 · 0 . 9950 · 0 . 5239 - 56 · 0 . 9754 · 0 . 4960 = 1 . 578 7 / 25

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Example 11.1. Solution (cont.) After 10 days, the BS price is: d 1 = ln 56 56 + (0 . 05 - 0 . 01 + 0 . 5 · 0 . 1 2 ) 172 365 0 . 1 q 172 365 = 0 . 3089 d 2 = 0 . 3089 - 0 . 1 r 172 365 = 0 . 2403 N ( d 1 ) = N (0 . 31) = 0 . 6217 , N ( d 2 ) = N (0 . 24) = 0 . 5948 e - rt = e - 0 . 05 · 172 365 = 0 . 9767 e - δ t = e - 0 . 01 · 172 365 = 0 . 9953 C = 56 · 0 . 9953 · 0 . 6217 - 56 · 0 . 9767 · 0 . 5948 = 2 . 119
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