It is a sunny Tuesday afternoon in the perfectly manicured harbourside city of Sydney. Dressed in your favourite pin stripe Gucci (Black Label thank you very much) and those hand crafted Italian leather shoes you picked up in Milan on that overnight trip from Belize to Mauritania through the Galapagos, you await casually in the exclusive dining area of Sydney's trendiest club, The Ivy. Coolly, you dust off that irritating $ 100 bill erupting from your right breast pocket and scoff at the thought of ever having worked in that unenviable position as a lecturer at the Australian National University. The room is bustling with bankers and your eyes begin to dart around surreptitiously. Glancing toward the floor, you smile at how large... your shoes are. A man and woman walk over and find a seat close to you. You recognise them as the heads of the derivatives trading desk at two of Australia's rival banks. Intrigued by their meeting, you listen closely.
"Hmmm", they ponder, as they look blankly at each other. "The Australian market really is very small. It only consists of one risky asset, kryptonite, and one risk-free asset. And our models are so unsophisticated!" they complain.
"The price of Kryptonite can only be one of two possibilities tomorrow. And standard European Put and Call options aren't even available on the exchange! How utterly... primitive!"
"We have a client, you know Lex Luthor? "asks the man.
"Well, we wrote a OTC Euro Put for him on Kryptonite today. It expires tomorrow.
He wants to insure against Kryptonite falling below $150 per micron tomorrow."
"Yes, vaguely" says the woman." I think I recall him. Bald chap? Evil looking?
Oh, yes! I do remember now. He came to us as well, and we sold the equivalent Call option to him!"
Afraid that their methodology is inconsistent and that they will appear daft to their client or, worse, appear on the front page of The Daily Planet (instead of the more usual society pages), they begin to argue.
a) Show that put-call parity holds under the 1 period Binomial model.
b) Given that Kryptonite is worth $150 per micron today and has a 50% chance of being either $160 or $140 tomorrow, the risk-free asset is worth $1 today and will be $ 1.05 tomorrow. Compute the price of the European Call option using the following methods:
i) construct a replication strategy
ii) construct a riskless portfolio
iii) find an equivalent martingale measure for the discounted Kryptonite process
iv) find the state prices using the Radon-Nikodym quotient.
It turns out that one of the bankers didn't use the risk free asset to find the discounted Kryptonite price process, but instead used Kryptonite to define the discounted risk-free price process.
c) Let S^{0} be the risk free asset and S^{1} be Kryptonite. Find a unique EMM, p, that makes S^{0}/S^{1} a martingale.
d) Let v_{1} be the payoff of the European Call option tomorrow. Use p to compute Ep[ v_{1}/ S_{1} 1 ].
e) Assume that vt S_{1} t (t = 0, 1) is a martingale. Under this assumption compute the value of the derivative today, v0.
f) Do the methods in b) and c) result in different Call option prices? Should the bankers be worried about inconsistencies? Conclude whether you should use the riskless asset or the risky asset to discount the derivative.