# Problem+Set+8+-+Questions

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FNCE90011 Derivative Securities 1 FNCE90011 Derivative Securities Problem Set 8 - Questions Relevant Sections in Hull 9E Hull 9E, chapter 12, sections 12.1, 12.3, 12.6, 12.7, 12.8, Question 1 [Time to expiry, branch length and number of steps] Every semester, students seem to get confused by the interplay between: (i) the time remaining before the derivative expires (T), (ii) the length of each branch in the Binomial tree (h), and (iii) the number of steps in the Binomial tree. In particular, mistakes are often made with respect to the branch length (h). This question will hopefully iron out any problems before it's too late. a) A derivative expires one year from now. We will model movements in the underlying share price using a 3-step Binomial tree. What is the branch length (h)? b) A derivative expires one year from now. We will model movements in the underlying share price using a 4-step Binomial tree. What is the branch length (h)? c) A derivative expires 8 months from now. We will model movements in the underlying share price using a 4-step Binomial tree. What is the branch length (h)? d) A derivative expires 15 months from now. We will model movements in the underlying share price using a 3-step Binomial tree. What is the branch length (h)? e) A derivative expires 15 months from now. We will model movements in the underlying share price using a 5-step Binomial tree. What is the branch length (h)?
FNCE90011 Derivative Securities 2 Question 2 [ calculating u and d from σ ] On a Binomial tree, the proportional up and down movements (u and d) on each branch of the tree are a function of the branch length (h) and - more importantly - the volatility (σ) of the underlying asset. In each of the following scenarios, calculate the proportional up (u) and down (d) movements. If the current share price (S 0 ) is \$100, draw a 1-step Binomial tree. a) The volatility of the stock (σ) is 40% pa. The branch length (h) on the tree is 3 months. b) The volatility of the stock (σ) is 60% pa. The branch length (h) on the tree is 3 months. c) The volatility of the stock (σ) is 40% pa. The branch length (h) on the tree is 6 months. Question 3 [one-step tree, two equivalent methods] A stock is currently priced at \$90. In one months ' time, stock price will either rise to \$100 or fall to \$81. The riskfree interest rate is 8% per annum continuously compounded. a) Draw the 1-step Binomial tree. Given the two possible prices that the stock might move to, calculate the proportional up/down movements ( u and d ). b) A European call option with a \$79 strike price and one month to expiry is written on this stock. Calculate the value of this call option by constructing a portfolio of stock and bond (i.e., bank account) to replicate the call option payoff ('replication approach') . c) Rather than using replication, price the call option by constructing a portfolio of stock and option which is riskless ('delta - hedging' method). Question 4 [two-step tree, call and put options] A stock is currently priced at \$100 with a standard deviation (σ) of 13.48% pa. The continuously-compounded riskfree interest rate is 8% per annum. A European call option with \$90 strike price and one year to expiry is written on this stock. a) Use the delta-hedging approach and a two-step Binomial tree to value the call option. b) Use the delta-hedging and a two-step Binomial tree to value a European put option with \$90 strike and one year to maturity. c) Put-call parity says that there is a relationship between the price of a call option and the price of a put option. Of course, these options are based on the same underlying asset, have the same strike price and the same time to expiry. Given your answers in (a) and (b), does put-call parity hold?
FNCE90011 Derivative Securities 3 Question 5 [replicating strategy for exotic derivative] A stock is currently priced at \$20. The volatility of the stock is 40% pa and the riskfree rate of interest is 4% pa with continuous compounding. We model the possible movement of stock price over the next 6 months using a one-step Binomial tree. This means that u = 1.3269 and d = 0.7536. Let S T denote the stock price 6 months from now. Define a new derivative security which has a payoff equal to ( S T ) 2 . That is, if you buy this derivative security today, in 6 months ' time, you will receive a cash payment equal to the square of stock price at the point. How much will this derivative security sell for in the market today? Hint: don't be distracted by the unusual nature of this derivative. It is not an option. It doesn't even have a strike price! Nevertheless, you can still use either of the methods from Lecture 8 to price the derivative (delta-hedging will be easier; try both methods if you are keen :). Question 6 [Binomial price is only an approximation] The current price of Coles Myer is \$13. The standard deviation ( ) of CML's return is 0.24 per annum. A call option with six months to expiry has a strike price of \$11. The riskless rate of interest is 5% per annum continuously compounded. a) Use the Black-Scholes formula to price the call option. The implicit assumptions are that it is a European-style option and that CML will not pay dividends during the next six months. b) For a European call option, the Black-Scholes formula gives the exact price. In contrast, the Binomial method only gives an approximation to the true Black-Scholes price. Use a one-step Binomial tree to approximate the call option price. c) To get a more accurate approximation, we can use more steps in the Binomial tree. Use a two-step tree to approximate the call option price. Hint: in parts b and c, you will have to calculate the proportional up/down movements using the volatility.