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# Estimate the value of the European six-month put option using the Black-Scholes model: Explain to students what happens when the number of steps in the binomial tree increases to a very large number.

QUESTION 1 (5 + 2 + 3 + 5 = 15 marks)

You been invited to the university to do a lecture on option pricing methods (including binomial trees and the Black-Scholes model). Vinay has asked you to:

1. a)  Show students how to estimate option prices using the binomial tree model. Using a live demonstration on Bloomberg you identify the following information: 1 USD = 1.30 AUD, the volatility of the USD/AUD currency pair is 40% per annum, the US risk-free rate is 1.50% per annum with continuous compounding and the AUD risk-free rate is 1.75% per annum with continuous compounding. Using a four-step binomial tree calculate the price of a European six month put option to sell 1 USD for 1.40 AUD.
2. b)  Using the same information in part a) estimate the value of the European six-month put option using the Black-Scholes model. In addition, explain to students what happens when the number of steps in the binomial tree increases to a very large number.
3. c)  Using the same information as part a) estimate the price of an American six month put option to sell 1 USD for 1.40 AUD. In addition, explain why the American option price differs from the European option price.
 Call prices (\$) Call Implied Volatility Put prices (\$) Put Implied Volatility K=10 6.55 0.02 K=15 2 0.40 K=20 0.15 3.80 K=25 0.01 8.80
1. d)  For the final part of the lecture show using Microsoft Excel how students can estimate implied volatility using option information. From Bloomberg you identify that the price of a nine- month silver futures contract is 16.50, the risk-free rate is 1.50% per annum with continuous compounding, and the dividend yield is 3.80% per annum with continuous compounding. Calculate implied volatility for the following futures options on silver with six-months to maturity and explain the importance of implied volatility.