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I'm currently reading Paul Wilmott's excellent book on option pricing. Near the beginning, he constructs a risk-free portfolio using an option, and a short on the underlying to hedge the risk. I'm specifically interested in European options.

A no-arbitrage argument follows:

  • If this portfolio earns more than the risk free rate: borrow money at the risk-free rate, buy the portfolio, and make money off the arbitrage.

  • Conversely: short the portfolio, invest money in a risk-free instrument, and again make money off the arbitrage.

I've scoured the internet, but couldn't find an explanation for the second argument, which I have a hard time grasping! By shorting the portfolio, we short an option, and "short a short", meaning we go long on the stock.

So, when we short the portfolio, we might even have to spend additional money, if shorting the option didn't give enough money to buy the stock.

This segment focuses on the binomial model, so I've tried separating this to 3 cases:

  1. When in both the up and down state the option is worth more than 0. In this case, the arbitrage relies on buying the amount of stock that can be had by exercising the option. I have a hard time finding arguments to why in this case the option should be worth more than the stock at the period before expiration.

  2. When in both the up and down state the option is worth 0. I understand this case, the option is worth 0 at the turn before expiration, and the hedging is a degenerate case (longing 0 stocks).

  3. When in the up state the option is worth > 0, and in the down state the option is worth = 0. Like in case 1, I can't find a good argument.

As you can see, I'm out of answers. I don't even understand why a risk-less portfolio must earn the risk-free rate. Anyone has a clue?

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    $\begingroup$ It seems to me you're trying too move to quickly through the material. Try to follow the examples of the binomial model and construct some yourself to see what is and what is not possible in the context of that model. $\endgroup$ – Bob Jansen Feb 17 at 17:34
  • $\begingroup$ @AlexC thank you! The problem is, I don't understand how it is possible to get money from shorting a riskless portfolio that includes shorts, like in my example: how can you be sure that the money you get from selling the option is enough to buy the stock? If I understand correctly, to show an arbitrage would mean that not only is it enough, it is even higher, thus the change can be put in bonds and earn the risk-free rate. $\endgroup$ – user1926887 Feb 17 at 19:59
  • $\begingroup$ @BobJansen thank you! I've tried working through quite a few examples, and still can't figure this out. Have I done something that isn't allowed in the context of the model? Thank you. $\endgroup$ – user1926887 Feb 18 at 13:10
  • $\begingroup$ Could you show how you created a risk less portfolio in this model that always loses money? $\endgroup$ – Bob Jansen Feb 18 at 13:41
  • $\begingroup$ @BobJansen Assuming the risk-free rate is r > 1, then a "portfolio" where you simply do nothing with your money always loses money. Although today and tomorrow you have the same amount, tomorrow it is worth less. Or is it something that the model doesn't allow? $\endgroup$ – user1926887 Feb 18 at 14:55
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Collecting some of the comments as it's getting too long.

  • The binomial model only assumes properties of two assets: the bond and the stock. In the model, it's possible to hedge contingent claims (i.e. options) using a dynamic allocation to the bond and the stock.

  • "where the return on a portfolio of an option + shorts on the underlying is less than the risk-free rate", in the model these don't exist. The proof works by contradiction. Suppose there is some arbitrage: either it's possible to earn money by buying or selling the portfolio. Now show that both are impossible. Hence, arbitrage is not possible.

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  • $\begingroup$ Thanks @BobJansen! 1. Which model are you referring to? CRR show in the beginning of the binomial model paper a construction of a portfolio involving writing 3 calls. 2. Is "these don't exist" referring to option and shorts? Because shorts are mentioned farther down in the CRR binomial model paper. $\endgroup$ – user1926887 Feb 18 at 18:28
  • $\begingroup$ Not sure what CRR is but I meant the binomial model. $\endgroup$ – Bob Jansen Feb 18 at 18:30
  • $\begingroup$ CRR is short for Cox, Ross and Rubinstein, specifically referring to the paper where they first introduced the binomial model in 1979. $\endgroup$ – user1926887 Feb 18 at 18:31
  • $\begingroup$ No portfolio exists (so also not that one) that is expected to yield less than the risk free rate given the right measure. $\endgroup$ – Bob Jansen Feb 18 at 18:37
  • $\begingroup$ What do you mean by "right measure"? That's something I didn't see in CRR's 1979 paper. It seems maybe a bit over the top solution for something that is supposed to be pretty basic. Can you perhaps refer me to a proof showing that no portfolio exists that is expected to yield less? $\endgroup$ – user1926887 Feb 18 at 18:48

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