Suppose now that we had a continuum of states – say the state could be any positive number.
a. Consider some arbitrary asset with price S(x). Assume that S is strictly increasing and differentiable. In fact, to simplify matters, assume that S(x)=x; that is, we name the state x after the price of our asset in this state, S(x). Let c(k) be the price of a call option on this asset with strike price k. Using the decomposition principle, write down an explicit formula for c(k).
b. What can you say about the second derivative of this price c’’(k)?
c. Suppose you observe the following prices of call options on some stock:
strike 1 2 345 67
price of call 7.01 5.46 4.88 4.46 3.30 1.55 1.02
Can you do arbitrage? How?