Arbitrage is the act of making risk-free profits in a market. This is usually done by leveraging inefficiencies within the market and making simultaneous trades to exploit this.
You could arbitrage virtually any market. For example, you notice that the vintage vinyl records at your local thrift store are selling for a 50% markup on eBay. You buy the records in-store and flip them on eBay for a risk-free profit. Anything that is priced or demanded differently across markets or is influenced by a risk-free rate of return presents an arbitrage opportunity. The item itself could range from vintage vinyl to complex stock options.
We made two important observations in our example above.
Profit is risk-free and guaranteed
The value of the commodity (vintage vinyl) is different across exchanges
Arbitrage in the securities market is no different. Treasury notes represent a close proxy for risk-free returns and the value of securities can fluctuate across different exchanges. Former Jane Street alum, Sam Bankman Fried famously arbitraged Bitcoin across international exchanges (exploited Bitcoin price differences between American & Korean exchanges).
In most cases, a highly liquid securities market is virtually arbitrage-free. We can use this principle to price instruments like stock options. Let's consider a simple example.
Suppose at t = 0, a stock S trades at $100 /share. A 3-month American call [1] option C has a strike price of K = 75. At t = 0, the risk-free rate is 20%. What is the lowest this option should be worth?
$25 is a great first guess. If C < $25, anyone could immediately exercise the option, buy the share for $75, and sell it at $100. The profit minus a cost < $25 would give you a risk-free return.
Our guess doesn't account for the risk-free rate of return. Observe the arbitrage opportunity below where C = $25.
t = 0 | t = Expiration | if S > $75 | Total at End | |
short stock | -S + $100 | return stock | S - $75 | 0 shares |
buy call option | C - $25 | exercise call | -C | 0 options |
invest remainder @ RFR | $75 | | $75e^(0.20/4) = $78.85 | $78.85 - $75 = $3.85 |
Sell the stock short (+$100) and use the proceeds to buy the option (-$25).
Invest the remainder of $75 @ the risk-free rate of 20%.
At expiration, you will at worst pay $75 to return the short share by exercising the call option. Even if the stock trades > $75 /share.
At this point, the $75 invested at a 20% risk-free rate has become $78.85.
You are left with a minimum risk-free profit of $3.85 per trade
Hence, the lower bound of a call option must be the initial stock price minus the present value of the option's strike price discounted by our risk-free rate of return.
In our case $100 - $75*e^(-0.20 / 4) = $28.66
We set lower bounds on put options [2] similarly. If we consider European [3] options, we can even derive a mathematical relationship between a call and a put option. This is called put-call parity.
Put-Call Parity states the following...
stock S0 initial value
European put p with strike K with a TTE (time to expiration) of T
European call c with strike K with a TTE (time to expiration) of T
the risk-free rate r
Consider the scenario where at (t=0), stock S trades at $90. European call (c) and put (p) options with strike prices of $75 trade at $25 and $10 respectively. Both options expire in 6 months. The risk-free rate is 30%.
$90 + $10 > $25 + $75*e^(-0.30/2)
$100 > $89.55
S + p is overvalued
We could arbitrage this scenario by shorting the stock + put and using the proceeds to buy the call and a $75 6-month bond that returns the risk-free rate of return. By doing so, we guarantee profit despite any value of the stock price after 6 months.
t = 0 | | t = Expiration | if S = $10 | if S = $1000 |
short stock + put | -(S + p) + $100 | return stock and exercise call if needed | S - $10 | S - $75 (call exercised) |
buy call + bond | (c + K) - $100 | Bond pays RFR | $75*e^(0.30/2) | $75*e^(0.30/2) |
Total Position | $0 | | $77.14 | $12.14 |
At expiration, it does not matter if the stock trades at $10 or $1000. We locked in a risk-free profit. Even if our initial position at time = 0 was net $0.
This process of constructing portfolios to identify arbitrage opportunities and discounting them backward to calculate prices or bounds on prices is a popular proof technique.
Assumptions: The above exercises assumed (for simplicity) zero fees on any trade. In practice, shorting stocks and exercising options come with varying fees. Also, with the prevalence of high-frequency trading algorithms, many arbitrage opportunities usually do not last very long.
Lastly, liquidity often determines the feasibility and scale of arbitrage. Imagine that you bought 100 vintage vinyls from your thrift store to arbitrage on eBay. If demand dries up on eBay before you sell all the vinyl, you may have to sell the rest at a loss or perhaps not at all. Hence, your arbitrage exercise's scale would be limited. Then you could move on to options.
Appendix
[1] A call option gives you the right to buy 1 share of stock at the option's strike price k.
[2] A put option gives you the right to sell 1 share of stock at the option's strike price k.
[3] European options can only be exercised on their expiration date, unlike American options which can be exercised at any time before expiration.
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