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PierreBoulez
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Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Arbitrage when risk-free portfolio earns less than riskless portfolio

Hi,
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 (as we can easily make a risk-less portfolio that always loses money, or doesn't lose and doesn't earn). Anyone has a clue?
Thank you!
Last edited by PierreBoulez on February 20th, 2019, 9:09 am, edited 1 time in total.

bearish
Posts: 5429
Joined: February 3rd, 2011, 2:19 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Most of us have slightly idiosyncratic ways of explaining this point, so rather than trying to map it into your representation of Paul's, I'll try and give you my own. Consider a world with no transaction costs, no limits to shorting, and a constant interest rate that you can freely borrow or lend at; and a stock with known price dynamics, e.g. binomial. Somebody shows you a derivative contract, e.g. a European call option, written on this stock, along with a price that you can either buy or sell it for. If you can construct a self-financing trading strategy that exactly replicates the cash flow(s) generated by holding the derivative contract; in the case of the European call, max(S(T)-K,0) at the maturity date T, then the initial cost of initiating this replicating strategy is the theoretical (no-arbitrage) value of the derivative. If its quoted price is higher, you sell it and follow the replicating strategy. In this case you will have some cash in hand at time 0 and, regardless of what happens to the stock price, all future cash flows will net out between the derivative and the replicating portfolio. Likewise, if the quoted price is lower than the initial cost of the replicating strategy, you sell (or write, in the lingo) the option and follow the opposite of the replicating strategy. You will, once again, have some cash upfront and no further net gains or losses. In this setting, and this is important, you should not be able to (and certainly not easily) "make a risk-less portfolio that always loses money", since flipping its signs will give you a portfolio that always makes money, which is more or less the definition of an arbitrage opportunity (a free lunch!).

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Thank you bearish!
Construction of a replicating portfolio is actually the one thing I do understand, but my problem is with a risk-free portfolio.
You wrote "you sell (or write, in the lingo) the option and follow the opposite of the replicating strategy. You will, once again, have some cash upfront", and that's my exact problem - how can you know that you will have some cash upfront? The portfolio, by definition, doesn't replicate the cash flow generated by holding the derivative, as the portfolio is risk-free!

bearish
Posts: 5429
Joined: February 3rd, 2011, 2:19 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Ah - I see the problem. The replicating portfolio by itself is not risk free. It is the combination of the replicating portfolio (or hedge, if you want) and the derivative that is risk free, because one offset the other. Does that make sense?

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Thank you for all your replies!
I understand why the replicating portfolio together with the hedge is risk free ("constructs a risk-free portfolio using an option, and a short on the underlying to hedge the risk").
The whole crux of the arbitrage argument in this case lies on the possibility of shorting option + underlying to get money, buy bonds that give the risk free rate (that has a higher return than the portfolio of option + underlying), and then in the next turn you've made "free" money.
So: shorting an option definitely gives you money, but you need to short a short (= buy) the underlying, too. So, you need to expend money for the underlying. Is there a way to prove that no matter what the conditions are, you will need to use less money than what you got for the option? In this case, you will be able to use the "change" and put it in a risk-free investment, and you will have succeeded in creating arbitrage.

Paul
Posts: 10494
Joined: July 20th, 2001, 3:28 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

One thing that's v annoying is how this turns into accountancy. You have some cash, you buy something, you sell something, then you see how there's no risk and so you should get the risk-free rate, but you have to look at the movement in the cash, the option, the stock, it gets irritating.

So what I sometimes do to explain things is to remove one of the moving parts, the cash. I start by assuming that you've inherited from Great Aunt Maud two things, an option and a short stock position. Then look at what happens to that. And ask whether you'd rather have this inheritance or the cash.

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Hi Paul,
Thanks so much for answering, I feel honored!
Even if you inherited an option and a short, because the return of option + short is less than the risk free, we want to show an arbitrage opportunity. You described that arbitrage can be done like so:
1. Sell your European option, get money.
Although you have received an inheritance from Aunt Maud, you need money for step 2.
If step 1 generated strictly more money than needed by 2, invest the change in the risk-free rate and you've succeeded in arbitrage.
But, if step 1 generated strictly less money as needed by 2, you'll need to borrow at the risk free rate.
I have a hard time understanding how, under these circumstances, it is possible to make risk-less profit.
Last edited by PierreBoulez on February 18th, 2019, 10:09 am, edited 1 time in total.

Paul
Posts: 10494
Joined: July 20th, 2001, 3:28 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

No, you didn't inherit any money from Aunt Maud! You inherited an option and a short stock position!

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Thank you so much for keeping an eye on this thread!
I'm not sure I'm following - if you inherited a European option and a short stock position, and will have waited a single turn for the expiration of the option, by the construction of this portfolio you will have made a risk-free return, but under our assumption this return is less than that of the risk-free rate.
I'm trying to understand how under this condition (return < risk-free rate) there is an arbitrage opportunity.

Paul
Posts: 10494
Joined: July 20th, 2001, 3:28 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Slow down!

Try like this: Call with value 1 or 0 tomorrow. Stock worth 100 now, tomorrow worth 101 or 99.

You inherited 1 x option - 0.5 x stock.

Tomorrow you have -99/2 whether stock rises or falls, no?

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Thank you for being patient! I'm trying to slow down but I still don't understand
Yes, tomorrow the value of the portfolio is the same, as we constructed it to be risk-free. I recognize that this is the example from the book, and that under a risk-free rate of 0 assumption it means that the value of the portfolio tomorrow is identical to its value today, in both cases 0.5.
How can I progress from here and add a risk-free rate that is strictly larger than 0?
In my question the risk-free rate is strictly larger than 0, and we work under the assumption that the return from the portfolio is strictly less than the risk-free rate.

Paul
Posts: 10494
Joined: July 20th, 2001, 3:28 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

It doesn't matter whether the interest rate is zero or positive. Either way you can see that tomorrow your inheritance will be worth -99/2. So PV that to today. Your inheritance today will be worth

-99/2 /(1+r dt)

But you know it's worth -0.5 * 100 + V, where V is the market price. So V= 100/2  -99/2  /(1+r dt) is "special."

If you see that V> that special value in the market then you'll be pleased that Great Aunt Maud left you the portfolio and not the cash equivalent. If V< that special value you'll wish she'd left you a cheque not the portfolio.

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Why will my inheritance today be worth that amount? It seems that your answer already assumes that the return on the portfolio equals the risk free rate, but I'm looking for proof that it does!
When assuming it earns more than the risk free rate, there is a contradiction to the no-arbitrage argument:
You can borrow money at the risk-free rate, buy the portfolio, and make a risk-free profit.

Given that:
1) The return on the portfolio is strictly less than r,
2) The amount of stock needed for hedging costs less than the option,
You can also find a contradiction to the no-arbitrage argument.

But, given that:
1) The return on the portfolio is strictly less than r,
2) The amount of stock needed for hedging costs more than the option,
How is it possible to arbitrage?
By shorting the option you get some money, but will need to borrow extra to buy the stock for hedging.

If I understood no-arbitrage correctly (and from this confusion I assume I didn't), the argument in this case (where portfolio return < r) requires you to sell portfolio, thus getting money, and then investing said money and getting the risk-free rate. Then, on the next turn, you've made more money than the portfolio, and can (in that order): sell stock, pay back the short on the option, and still have "left-over" money that you've made at completely 0 risk.
I have scoured the internet for the past few days, looking for papers/books/posts that explain an arbitrage opportunity under these conditions and just didn't. Maybe when assuming point number 2 it is possible to reach some kind of logical contradiction?

Paul
Posts: 10494
Joined: July 20th, 2001, 3:28 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

I only read your first sentence! If I know I'm getting $1 tomorrow then that is worth$1/(1+r dt) today. Forget options and stocks. Are you ok with that concept?

P

PierreBoulez
Topic Author
Posts: 12
Joined: February 17th, 2019, 4:33 pm

### Re: Arbitrage when risk-free portfolio earns less than riskless portfolio

Definitely ok with it! How to proceed now?