# Call option arbitrage opportunity

I am having trouble wrapping my head around some text provided to us by our lecturer (unfortunately he is currently unavailable). If we let $c$ be the price of a European call option, $S_0$ the current price of an asset (say a stock), $X$ the strike price, $T$ the time to maturity, and $r$ the (static) interest rate. We ignore dividends.

Suppose that $$c = 3, S_0 = 20, X = 18,\\ T = 1, r = 10\%$$ Is there an arbitrage opportunity?

• buy the call, short the stock
• proceeds: $-3+20 =17$; grows to $17e^{0.1} = 18.79 > 18$
• yes!

I don't understand how you can deduce the existence of an arbitrage opportunity from $18.79 > 18$.

• If the stock never moved.. Haha yes.... The risk premium over and above the intrinsic implies a distro of stock returns ... So all joking aside... No. No arb there .. There's risk from hedging, cost of borrowing.. Etc – cdcaveman Apr 15 '13 at 7:15
• The simplest way to show arbitrage opportunities here is the lower bound of the call price which is call > value of underlying asset - PV of strike which should force the call option price to be above about 3.71. Anything below that presents an arbitrage opportunity. Obviously the OP has made clear that he made a lot of simplifying assumptions. Thus, we are talking about a theoretical arbitrage opportunity absent of any transaction costs, liquidity issues, dividends,... – Matt Apr 15 '13 at 8:00
• @Freddy I understand that since $c = 3 < 20 - 18e^{-0.1} = 3.71$ this presents an arbitrage opportunity since by definition a sub-optimal price reflects potential achievable value. Perhaps my original post should have asked how exactly the above quoted arbitrage strategy works. Am I right in saying that if you buy the call and short the stock, you end up with $-3 + 20 = 17$, which you can then invest to get $18.79$ in a year's time; at maturity, you buy the stock for $18$, return it, and pocket $0.79$ instead of getting $-1$, or rather $0$, were you not to engage in arbitrage? – user72180 Apr 15 '13 at 9:06
• Not exactly: You buy back the stock at whatever price it is traded at, you exercise the call if the stock price is above your strike at expiry and you get back your investment of 17 plus interest. – Matt Apr 15 '13 at 9:13
• Let $S_1 = 20$, i.e. the stock price stays the same, how do you "get back" your investment if you have to return the stock to whoever lent it to you for you to have gone short with it in the first place? At maturity you're left with 18.79, a call option, and an obligation to return stock. You exercise the call option buying 20 worth of stock for 18, you return this stock. You now have 0.79 left. Alternatively, you buy 20 worth of stock for 18, sell it for 20, pocket 2, you now have 20.79 but you must still return stock, you buy it for 20, return it, and you are again left with 0.79, no? – user72180 Apr 15 '13 at 11:02