- Hunter Sanzeri

# Quantum Finance Optimization - Black-Scholes Model

Every investor wants to minimize risk and maximize payout. Famous algorithms in quantum computing like Shor’s & Grover’s show us that quantum computing far exceeds classical in scope, efficiency, and power. Quantum finance will do the same! The average investor has absolutely no chance of keeping up with the momentous pace of price changes in the option market day to day. In the world of classical computing, the Black-Scholes model is a great example of market actors attempting to optimize their risk & returns.

Let’s take a short look into what makes the Black-Scholes model special.

The Black-Scholes model completely eliminates risk. An investor can perfectly hedge risk by trading the option at an optimal price. Therefore, the Black-Scholes model generates the optimal price & allows the investor the flexibility to predict their maximum payout. In this scenario, there is only one true (right) price for the option. The primary limitation of the Black-Scholes model is that it is not applicable to the United States. In the United States, options can be exercised before the expiration date. In Europe, however, this is not the case.

Here are the assumptions (limitations):

Constant Volatility

Efficient Markets

No dividends

Interest Rate is constant

Lognormally distributed returns

European only

No transaction costs

Liquidity

Notation:

Stock price = S

Price of the option as a function of the stock price = V

Time = t

Risk-free interest rate = r

Volatility of the stock = σ

Here is the Black-Scholes formula:

Adjusting the equation leads us to:

When our equation is presented this way, the left hand side shows our first term, theta, us the change in option value with respect to time (or “time decay” in finance terms).

The second term on the left side is the option value with respect to the underlying value is called gamma.

The right side shows us a zero-risk return forever! The conclusion is that over an infinite amount of time, the loss of theta and the gain from gamma offset one another, giving the investor a risk-free return.

How does quantum computing relate to this?

Now that we’ve discussed risk, let’s talk about efficiency. One of the assumptions in the Black-Scholes model is that all markets are efficient. In the real world, this is obviously not the case. Emmannuel Haven’s synthesis of the Schrödinger equation gives us great insight into what can be achieved with quantum finance. Haven argues that the amount of current market arbitrage is a function of the velocity of pricing, information transfer, and wealth disparity. This is an interesting real world perspective that could also be applied to black market theory.

I think the outstanding challenge for quantum finance is financial modeling. As we’ve seen here, quantum algorithms can greatly benefit an investor or firm of any size. The fundamental way finance is modeled comes from hundreds of years of economic development, the inception of the stock market, and the digital era with respect to cryptocurrencies. In my opinion, 3 dimensional financial modeling visualization is the next step. I believe the future will revolve around: visualizing our current models in a 3D environment and applying famous algorithms such as Shors or Schrödinger, and more. The time is now for quantum computing! Quantum Thought is exploring finance optimization.

[Image Credit]: Medium