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# Financial Modelling Tools and Techniques Assignment 2

## Assignment 2

Please submit your assignment online via MyUni as a PDF. Please ensure that your assignment is legible, the answers are clearly indicated, along with your name and student ID number. Print your spreadsheet as a PDF document for submission. All calculations can be made and answers given in the spreadsheet.

Please ensure that all questions are labelled and results clearly indicated, e.g., state “The premium is...”. Please also ensure that all text is readable on your final output document, and that pricing trees appear on a single page.

If you use any spreadsheet tools that don’t show directly all of the calculations performed (e.g., Goal Seek), please clearly label when/where/how you have used those tools.

1. Alumina Limited (AWC), is an Australian company that (partially) owns a number of mining and smelting operations globally; in Australia, this includes a number of Bauxite mines, and refineries that produce alumina and aluminium. At the time of writing, the share price for AWC is \$56. Consider this to be time zero.

You wish to write a European put on AWC shares, with strike price \$1.50, that expires in 4 (monthly) time steps. Assume that the return rate on a bank investment over each time step is R = 1.01.

• Use CRR notation to construct a four-step binomial pricing tree for an AWCshare, with u = 1.25 and d = 1/u.
• Find the premium of the put by calculating the risk neutral probabilities andthen constructing a four-step binomial pricing tree.
• Use put-call parity to find the premium of a European call with the same underlying asset, strike price and expiry as the European put. Use pv0(K) = K/R4 .
• Calculate all state prices at the put’s expiry. That is, calculate all λ(4,j) for j = 0,1,...,
• Use the state prices λ(4,j) to calculate the premium of the European put. Compare this premium to the premium calculated in part (b).
1. com Limited (CAR) is an Australian company that (unsurprisingly) owns the online marketplace www.carsales.com.au. At the time of writing, the share price of CAR is \$18.86. Assume this is time zero.

You wish to compare premium prices of a European call calculated with the BlackScholes model, with premium prices calculated with a binomial model. The call has strike price K = \$20, and expires in 90 days so T = 90/365 years. The yearly volatility of CAR shares is estimated to be σ = 2.55. Assume the continuously compounding interest rate is r = 3% pa.

• Calculate the call premium using the Black-Scholes model.
• Consider a three-step binomial CRR model.
• Assuming interest rates are constant over the life of the call, calculate thereturn R over one time step.
• Calculate the up and down factors u and d in this three-step model.
• Calculate the risk neutral probability π in this three-step model.
• Construct a three-step binomial pricing tree for the call and calculate itspremium.
• Consider a ten-step binomial CRR model.
• Assuming interest rates are constant over the life of the call, calculate thereturn R over one time step.
• Calculate the up and down factors u and d in this ten-step model.
• Calculate the risk neutral probability π in this ten-step model.
• Construct a ten-step binomial pricing tree for the call and calculate its premium.
• Compare the premiums calculated with the three-step and ten-step binomial models with the premium calculated with the Black-Scholes model. You should find that the premium calculated from the ten-step model is closer to the Black-Scholes solution than the premium calculated from the three-step model. Why?
1. 3. A binary call option pays \$1 at expiry if the value of the underlying asset is greater than the strike price, and \$0 otherwise.
• Construct a 4-step binomial tree for the stock price of an asset in CRR notation,with S = \$5, u = 2, d = 1/u, and .
• Work backwards through the tree using the general pricing formula to evaluatethe current price of a binary call option, that expires at time 4, and has strike price \$
• Calculate all state prices at expiry for this 4-step binomial model.
• Use the state prices to directly evaluate the premium of the binary call optionabove (i.e., with expiry at time 4, and strike price \$12).
• Recall that the premium of an Arrow-Debreau security was the state price. Fora strike price K, and an N step binomial model with S,u,d and R defined as normal, what is the premium of the binary call option equal to? Answer with a description, not just a formula.
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