Butterflies and probabilities

Dull Inc.’s stock currently trades at $100 in New York, and the interest rate curve is flat at zero. The fair values of three 1-year calls on Dull Inc. are shown below:

(a) Draw the payoff function of a butterfly strategy on Dull Inc. which is long the $90- and $110-strike calls and short 2 at-the-money calls. What is the cost and maximum payoff of this strategy?

(b) Without making any specific assumption on the distribution of Dull Inc.’s stock price in one year, show that its probability to lie within the $90–$110 range is at least 20%. Hint: Superimpose the payoff of a derivative which pays off $10 whenever the final stock price is in the range, and zero outside.

(c) (*) Let f(x) be the distribution density of Dull Inc.’s stock price in one year, which may or may not be lognormal. Assume that you know all the fair values c0(K) of 1-year calls on Dull Inc. with strikes

90 ≤ K ≤ 110. Using a limit argument, establish that:

.

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