Call-Option Arbitrage from Put-Call Parity Bounds
Summary
The example explains an arbitrage when a European call is priced too cheaply relative to the stock and the financing rate. The strategy buys the call and shorts the stock, investing the net cash proceeds. At expiry, if the stock is above the strike, exercising the call supplies the share needed to close the short. If the stock is below the strike, the trader buys a share in the market instead. In either case, the invested proceeds cover the share purchase and leave a surplus under the stated prices.
The worked example uses a call priced at 3, a stock at 20, a strike of 18, and a one-year interest rate of 10 percent, producing invested proceeds of 18.79 at expiry. This exceeds the strike, so the described terminal payoff is positive across both price cases. The reasoning assumes the stated financing, ability to short and deliver the stock, no dividends, and no transaction costs or other trading frictions. It illustrates a pricing inconsistency rather than a forecast about the stock.
Key ideas
- A call can be underpriced relative to the stock and financing costs, creating an arbitrage.
- Buying the call and shorting the stock generates cash that can be invested until expiry.
- Exercise above the strike or buy the share in the market below the strike to close the short.
- The example relies on frictionless trading assumptions, including no dividends or transaction costs.
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Full text
# Call option arbitrage opportunity
# 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 $18.79 > 18$.
## Answer by Rian Rizvi (score 9, accepted)
https://quant.stackexchange.com/a/7760
The option is a contract that gives you the right to buy the stock in one year for 18. Today people are trading the stock for 20, so you can sell the stock short for 20 today, meaning, someone gives you 20 cash today in return for a stock IOU, where you are obligated to deliver the stock to them on a later date.
So you get 20 cash upfront but you need to spend 3 of it to buy the option. After one year this net 17 reinvested becomes 18.79. This is more cash than you need to buy back the stock using your option if the price ends up > 18. So, worst case scenario is the stock ends the year > 18, you exercise the option and buy the stock for 18 from the option seller, and you net 18.79-18.0=.79 cash.
Of course the stock could end the year below 18, and then you wouldn't use the option to buy back the stock, you would just buy the stock at whatever price it trades in the market. So either you make .79 or you make more. Arbitrage.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.