# Why do transaction costs increase the range of the no-arbitrage bounds for an option's price?

I am reading this book by Mark S. Joshi. Can you help me make sense of one of the exercise questions? Here is the question (from page 40 of the book):

Exercise 2.5 Suppose no-arbitrage bounds for an option price show that the price lies between $$L_1$$ and $$L_2$$ in a world without transaction costs. What can we say about the bounds if we take transaction costs into account?

Here are the solutions (from page 474):

Exercise 2.5 Increasing transaction costs can only decrease the number of arbitrage portfoliois, so the bounds will be at least as wide.

I don't understand why the bounds will be at least as wide. I understand that transaction costs make it harder to find an arbitrage. Shouldn't decreasing the number of arbitrage portfolios act to decrease the no-arbitrage price range?

Assume a store is fairly pricing a bottle of water at \$1. Now another store is pricing the same bottle of water for \$1.2. Assuming it is possible, you can buy the water at the first store, end sell it to the second store for a \$0.2 risk-less profit, as a consequence forcing him to lower the price to \$1. At this point you no longer arbitrage. In this example $$L_1=L_2=1.0.$$
Now assume instead that the second store is a long journey away, costing you \$0.1 in petrol (transaction cost). Hence, you will only buy from the cheap store and sell to the expensive store as long as you make money from it. Initially you make 1.2-(1+0.1) = \$0.1 a bottle, but the more you arbitrage the more the second store will lower their price. In fact, they will lower it until you stop, which occurs when the price is \$1.10 (you make zero profit). In this example $$L_1=0.90, \;L_2=1.10.$$ So including costs will widen the bounds for where there is no arbitrage, simply because no one can bother making pennies when they are eaten up by costs. • Thank you so much for all the answers. I really liked this one for its simple and intuitive example. – Peanutlex Dec 15 '20 at 21:35 • Nice answer. When I view the question I see line breaks in the formula but not when I go to the edit screen. Is this happening for anyone else? The$\LaTeX\$ doesn't seem to have these breaks. It happens to me in both Firefox and Chrome on Windows. – Bob Jansen Dec 16 '20 at 15:39