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- 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 pv
_{0}(*K*) =*K/R*^{4 }. - 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).

- 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.

- Assuming interest rates are constant over the life of the call, calculate thereturn
- 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.

- Assuming interest rates are constant over the life of the call, calculate thereturn
- 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?

- 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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