binomial problem-replication strategy, Basic Statistics

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It is 11:30pm on Thursday evening, New York City time. You are still recovering from your long haul flight from Sydney which arrived at JFK only hours ago. Thank your favourite deity the flight attendant tucked you into your first class cabin bed (say no more) otherwise you wouldn't have had the energy to pop up Fifth Avenue to pick up a casual outfit and make it downtown to Cipriani's this evening.

You drape yourself sleekly on the bar stool, in your brand new Dolce and Gabbana boots. Admiring them, your blood begins to boil at the thought of those primordial peasants who almost stepped on them as they scurried down to their subterranean destinations on the subway.

Your anxiety begins to grow. Luckily, the gorgeous Calvin Klein model sitting next to you hands you a Bellini and a Xanax. Intrigued by your Australian accent, the two of you begin talking about Australia's unsophisticated financial system.

You marvel at New York's sophisticated models (no pun intended). In this land of opportunity, the prices of their risky assets can be not just one of two, but one of three prices tomorrow!

Obviously curious about your excitement, two rather large gentlemen walk over to join your conversation. You immediately recognise one of them as the head of global equity research at Goldman-Sachs. You've done your homework. He snaps his fingers and a plate of antipasto arrives, served by the maitre de, no less.

 "We'll tell you how it's done, kid. I was on the phone to Rupert about 5 minutes ago and he told me that News Limited will either go up to $50, stay at $48 or go down to $45 tomorrow. That's greenbacks, pal. Not your damn monopoly money! Anyway, tomorrow's price all depends on whether a court ruling is made about Rupert having to pay those workers he fired. All this money he has spent on lawyers has had such a negative effect on his share price! I mean, I already told that son of a gun that if he employs the illegals, then it will cost him way less and he can fire 'em whenever he damn well wants! And just between you and me, kid, I already know tomorrow's outcome," he winks.

You wipe the drool from the right side of your mouth and attempt to price a News Limited call option expiring tomorrow with a strike price of $48.

a) Using a trinomial tree to represent the 3 possible states of the world tomorrow, can you find an equivalent martingale measure for the discounted News Limited stock price process? Is it unique? Assume that interest rates have been cut so low that r=0.

b) You observe that the market price for the call option is $3. Desperately, you try to set up a strategy which replicates the payoff of the call option. Can you do it?

Show your working. Do not turn the problem into a Binomial problem

c) Suddenly you realise another risky asset is available. Its price today is $5 and tomorrow it will either be $4, $5 or $6 depending on three completely separate states of the world to Rupert's three court outcomes (payout, no ruling, no payout). Let S1 be the News Limited stock and S2 be the other risky asset. Write out the sample space for S1 + S2.

d) Can you construct a replication strategy using the extra asset? If so, compute the price of the call option. Show your working.

e) Now assume that the other risky asset depends completely on Rupert's three court outcomes. So assume that the price today is $5 and it will go down to $4 if News Limited goes down, it will stay the same if News Limited stays the same or it will go up to $6 if News Limited goes up. Write out the joint sample space of the two stocks. Can you construct a replication strategy? If so, compute the price of the call option. Show your working.


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