Last quarter I took a class on convex optimization - ECE 273, which was one of the most challenging, yet enjoyable classes I have taken at UCSD. I was planning on writing a blog post on the duality of optimization (which in my opinion, is absolutely beautiful) but I ended up writing on that for my final project (http://www.arkin.xyz/On-Convex-Optimization.pdf). This brief blog post will show that if European call option price $C$ is not a convex function of strike price $K$ then there exists an arbitrage opportunity.

#### What is a call option

A call option gives the buyer the right, but not an obligation to buy the underlying security at a strike pice at a certain date. The difference between an American and European call option is that the former can be exercised on any date before the expiration of the option, whereas a European option must be exercised exactly on the expiration date. Here is the payoff diagram for a European call option: Interpretation: In the above diagram, the premium (or the price) of the option is $200, and the strike price is ~$40. Now, to make profit from this option contract, the stock price must be greater than the price paid for the contract itself. Since there are 100 shares in one contract, if the share price moves up by $2, we break even. After which, the profit becomes positive. On the other hand, if the stock price at expiration is less than or equal to the strike price, we only lose the premium. This is why options are very cost efficient and could be less riskier than buying stocks. #### What is a convex function A function $f$ is convex iff Jensen’s inequality holds, that is $f(\lambda x_1 + (1-\lambda)x_2) \leq \lambda f(x_1) + (1-\lambda)f(x_2)$, where $\lambda \in [0,1]$. Let the function, $V_c$ be the price of the call option which takes in parameter $K$, the strike price. It is simple to see that in order to show $V_c$ is convex, we have to show that $V_c(K + \delta) + V_c(K - \delta) \ge 2V_c(K)$. #### Proof We proceed by contradiction. Let’s assume the opposite is true, that is $V_c(K + \delta) + V_c(K - \delta) \leq 2V_c(K)$. Then one can create a long position of 1 option with strike $K + \delta$ and 1 option with strike $K - \delta$ and short two call options with strike price $K$. Then, there is a risk-free profit made. Let’s look at a more concrete example and understand how. #### Example Let’s suppose there are three call options on the market, with strike prices 10, 20, and 30. Suppose the call option with strike 10 costs$12, the call option with strike 20 costs $7, and the call option with strike 30 costs$1. Using the strategy mentioned in the proof, we buy the call option of strike 10 and 30, costing $13. We finance this purchase by short-selling two contracts at strike 20 for$14 (leaving us $1). Now, let’s evaluate all possible scenarios at contract expiration. 1) Stock price exceeds$30 by $\delta$ - we exercise both our call options, making $20 + 2\delta$, and since the contracts we wrote get called, we lose $20 + 2\delta$. So after expiration, we have made a profit of $1. 2) Stock price drops below$10 by $\delta$ - none of the options are exercised. So after expiration, we have made a profit of $1. 3) Stock price is$10 + $\delta$ where $\delta \leq 10$. We exercise our call option with strike 10, giving us a total profit of $\delta + 1$.
4) Stock price is \$20 + $\delta$ where $% $. We excercise our call option with strike 10, giving us $10 + \delta$, but since the contracts we wrote will get exercised, we lose $2\delta$, but since $\delta \leq 10$, $10 + \delta \leq 2\delta$, giving us a net profit of $1 + 10 - \delta$.
Hence, no matter where the underlying stock price is at expiration, there was an arbitrage opportunity, since the pricing of the options did not follow $V_c(K + \delta) + V_c(K - \delta) \ge 2V_c(K)$, with $\delta = 10$ in this case.

#### Summary

Although this is a fairly straight forward example, Convex optimization is a great tool, and it’s applications are extremely vast. For readers interested in machine learning, I highly recommend checking out my final paper on support vector machines. Although I missed a whole letter grade by a couple of points on the final for this graduate course, I highly recommend taking ECE 273 to anyone looking to learn convex optimization in a robust manner. Next up on the blog, I plan on writing more on the intersection of finance (buy-side), macroeconomics and statistics, which is also where my career is headed.