Black-Scholes & life choices

We are our choices.

– Jean-Paul Sartre

In this post I’ll talk about how one might apply the Black-Scholes option pricing model to real-life choices. The BSM can provide a structured way to think about decision-making under uncertainty, and time constraints.

The Black-Scholes equation for the price of a European call option is:

$C = N(d_1) S_t - N(d_2) K e^{-rt}$

where:

$d_1 = \frac{\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right) t}{\sigma \sqrt{t}}$

$d_2 = d_1 - \sigma \sqrt{t}$

where:

$C$ = call option price
$N(d)$ = CDF of the normal distribution
$S_t$ = spot price of asset
$K$ = strike price
$r$ = risk-free interest rate
$t$ = time to maturity
$\sigma$ = volatility of asset

Some of the model’s ideas can be transferred to everyday scenarios:

1. Valuing Flexibility and Optionality

In real-life choices, having the option to act (without obligation) can be valuable. For example:

Job Offers or Career Changes: When you receive a job offer, it’s similar to having an option. You have the choice (but not the obligation) to accept the offer at a certain “strike price” (salary/benefits). The value of this “option” increases if you expect the job market (the underlying asset) to become more favorable.
Investment in Education or Skills: Deciding to invest in a new skill or education is like buying an option to improve your future job prospects. The more uncertain the future job market is (analogous to higher volatility in Black-Scholes), the more valuable this option might be.

2. Risk and Uncertainty (Volatility)

The Black-Scholes model emphasizes that volatility (uncertainty) of the underlying asset increases the value of the option. In real-life choices:

Entrepreneurship vs. Employment: Entrepreneurship can be seen as a higher volatility option compared to a stable job. The uncertainty (volatility) of starting a business can increase its potential upside (the payoff if successful), much like how higher volatility makes financial options more valuable. People who tolerate or manage higher uncertainty may have greater potential rewards.
Home Buying vs. Renting: The decision to buy a home versus renting is another example. Renting can be thought of as maintaining flexibility (an option) because you aren’t locked into a long-term asset, which is useful in volatile housing markets or if your life circumstances may change.

3. Time Value of Decisions

In Black-Scholes, time remaining until the option’s expiration affects its value, with longer time increasing the option’s value due to the potential for future favorable changes. In real life:

Delaying Decisions: Sometimes it’s advantageous to delay a decision because more time allows you to gather information, wait for better opportunities, or see how uncertain events unfold. The longer you have before a deadline, the more valuable this flexibility can be, just like in options.
Commitment Decisions: For life choices like marriage, home buying, or long-term contracts, understanding the time value of the option (the longer you wait, the more clarity you gain) can help you make decisions when you’re better informed.

4. Hedging Risk

The Black-Scholes model assumes that investors continuously hedge their position (delta-neutral). In real life:

Risk Mitigation: You can “hedge” your real-life risks by diversifying your activities or investments. For example, in career choices, learning multiple skills or working on side projects can hedge the risk of job loss or market downturns, much like how financial hedging protects against downside risk.

5. Pricing the Decision to Wait (Opportunity Cost)

In finance, the Black-Scholes model helps you understand the cost of waiting versus taking immediate action. In real life:

Choosing to Delay: Delaying a decision has costs, which can be understood in terms of opportunity cost or the potential gain you miss by not acting now. For example, waiting too long to invest in an asset, like stocks or real estate, might mean losing out on potential growth (the asset appreciating).

6. Risk-Free Rate Analogy

In Black-Scholes, the risk-free rate is used to discount future payoffs. In real life, the concept can be applied to decisions where safety or certainty is favored:

Guaranteed Outcomes: Choosing a “risk-free” option in life, such as a stable job with a fixed salary, can be compared to the risk-free rate. While this option may not have as much upside as riskier alternatives, it provides security and predictability.

Practical Examples:

Deciding to Relocate: When considering a move to a new city for work, you might weigh the “option” of staying where you are (your current situation as the underlying asset) against the potential of a new job or market (a volatile but potentially rewarding move). If the potential for growth or better opportunities in the new location is high, the option to move becomes more valuable, especially if you can delay the decision until more information about the market or job prospects becomes clear.
Entrepreneurship: Deciding to start a business can be viewed as purchasing a call option. You have to invest time and resources upfront (the option price) for the possibility of future success (profit). The more uncertain the market is, the greater the potential payoff—but also the greater the risk.

In short, by viewing real-life decisions through the lens of the Black-Scholes model, you can better understand how flexibility, uncertainty, and timing impact decision-making. This helps in assessing when to commit to a choice and when to maintain optionality to maximize future opportunities.

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