Consider a stock with current price 10 which over the next year can move up to 12 with probability 0.6 and down to 8 with probability 0.4. The annual risk-free rate is 2%.

d) What is the expected payoff of a call option with K = 10 calculated using the real probabilities? Is this larger or smaller than with the risk-neutral probabilities? Explain why it is larger/smaller. (4 marks)

e) What is the appropriate rate to discount the expected payoff from part d) at to give the same price as calculated in part b)? Did you expect this? Explain the relationship between this discount rate and the risk-free rate. (4 marks)

f) Is there a similar relationship for put options on this stock? Repeat the analysis for parts d) and e) for the put option. Comment on your results.