Suppose that a long put with strike $100 is selling for $10, and a long callwith strike $110 (on the same stock with same maturity T) is selling for $11.5. The spot price of the asset is $100, and discount exp^(-rT)=0.9. Does an arbitrage exist? If yes, how to construct it? If no, how to prove it?

Yes an arbitrage exists:From put-call parity the value of 110$ strike put is:Selling 11 put with strike 100 and buying 10 put with strike 110 you get 5$The payoff at maturity T is always >=0 P.

If there aren't 110 strike put quoted on the market you can create them synthetically setting up this strategy:Sell 11 put with strike 100, buy 10 call with strike 110, sell 10 shares:You get immediately 11*10 - 10*11.5 +10* 100=995$Investing 995$ at the risk free rate you obtain at maturity 995/0.9 = 1105.556$The payoff at maturity of this strategy is always >=-1100P.

it seems there is an arbitrage. if I sold the 100 put @ 10 and bought the 110 call @ 11.5 and sold short 1 unit of stock @ 100, I am left with 98.5 which I deposit and get back 109.44. at maturity there are 3 scenarios:1. the stock is below 100, I get assigned on my short put. So I pay 100 and leaving me with 9.44 net. 2. the stock is above 110, I exercise my calls pay 110 and am left short of 0.56. 3. stock is between 100 and 110, I make money but the amount declines with stock price. my interest is 9.44 and lose 1 for every point the stock is above 100 until the stock reaches 109.44 which is my break even. above this my cost is 0.56.