Tutorial Questions
Question 1 () A stock is currently priced at $20. In any given 4-month period, stock price will either go up by 18.91% or down by 15.9%.1 The riskless rate of interest is 4% per annum continuously compounded. A European-style call option is written on this stock with a $12 strike price and 8 months to expiry. a) Use the delta-hedging approach to price this call option. b) Use the risk-neutral valuation method to price this call option. Work recursively back through the Binomial tree, calculating the call option price on each branch. Check that the option price on each branch matches that calculated in part a. c) Again use the risk-neutral method to value this call option, but this time do not work back recursively. Rather, use the path-probability approach which focuses on the expiry-date distribution of stock price (and the number of paths which lead to each expiry-date stock price). d) Assume a European-style put option is written on this stock. It has 8 months to expiry and a $25 strike price. Focusing on the expiry-date distribution of stock price, value this put option. Question 2 () [European v. American Put Option]
A stock is currently priced at $100. In any given 3-month period, stock price will either go up by 13.31% or down by 11.75%. The riskless rate of interest is 5% per annum continuously compounded. A put option is written on this stock with a $110 strike price and 9 months to expiry.
a) Assume the put option is European-style. Calculate the current value of the put option. Use whatever method you like (replication, delta hedging, risk neutral, path probability), but it will be quickest to focus on the expiry-date distribution of stock price and path probabilities.
Hint: before you start doing calculations, have a careful think about how many steps will be in this Binomial tree.
b) Now assume this is an American-style put option. Calculate the current value of the put option.
1 You can confirm that this follows from σ = 0.3.
Question 3
Assume the same data as in Question 2. Specifically, we used a 3-step tree, with each step representing 3 months. We will now consider a European-style call option with 9 months to expiry and strike price of $100.
Required:
Use the path-probability approach that focuses on the expiry-date distribution of stock price to quickly calculate the current value of the call option.
Question 4
Recall Tutorial 9 Question 4, where we priced an exotic derivative whose payoff was the square of the expiry-date share price. In Tutorial 9, the derivative had six months before expiry and we employed a one-step Binomial tree. A delta-hedging approach was employed. Keep in mind that the Binomial approach only gives an approximate value, and with a 1-step tree, it will be pretty unreliable. We can get a more-accurate approximation if we use a bigger tree with more branches.
Let’s value the same derivative, this time using a multi-step Binomial tree and risk-neutral pricing. Assume that a stock is currently valued at $20. The standard deviation of its return is 0.40 per annum. The riskfree rate of interest is 4% p.a. continuously compounded.
Like Tutorial 9, the exotic derivative has a payoff equal to the square of the share price at expiry.
Required:
Use a six-step Binomial tree and risk-neutral valuation to approximate the value of this exotic derivative. To draw the tree, perhaps use the spreadsheet “call.xls” available on Moodle.
Additional Questions
You can now try all of the questions from Tutorial 9, this time using the risk-neutral method. Your answers will match those given in Tutorial 9. And you’ll appreciate how much easier valuation is using the risk-neutral approach!
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