Lesson 5.4: Vega (V): The "Jiggle Risk"

Vega is the "odd one out." It's not a real Greek letter ($\mathcal{V}$ is just a script 'V' that looks fancy). It's the only "Greek" that measures risk from a model input rather than a market variable like $S_t$ or $t$.

The Intuitive Meaning

"If the market's expected 'jiggle rate' ($\sigma$) goes up by 1% (e.g., from 20% to 21%), how many dollars does my option price move?"

An option is, at its core, a "bet on volatility."

• When you buy a call or a put, you *win* if the stock moves a lot. You are "long volatility." A spike in $\sigma$ is *good* for you. Your Vega is positive.
• When you sell a call or a put, you *win* if the stock stays calm. You are "short volatility." A spike in $\sigma$ is *bad* for you. Your Vega is negative.

For a professional trader, "Vega risk" is often the most important (and most profitable) risk they manage.

This is the concept that confuses most beginners.

The Question: "Panic is bad, right? If $\sigma$ spikes, people are scared, so the stock should go down. Why would a call option's price go *up*?"

The Answer: Asymmetric Payoffs. This is the *secret* of all option pricing.

Let's review the payoff for a call option you *bought* (Strike = $100):

• If the stock finishes at $90, you lose your premium.
• If the stock finishes at $50, you *still* just lose your premium.
• If the stock finishes at $10, you *still* just lose your premium.

Your loss is "capped." You do not care *how far* the stock goes down.

• If the stock finishes at $110, you make a profit.
• If the stock finishes at $150, you make a *massive* profit.

Your profit is "uncapped" (unlimited).

Now, a "panic" happens. Volatility ($\sigma$) spikes. This means the market believes there is a *much higher chance of a huge price move in either direction*. The "bell curve" of possible outcomes gets wider and flatter.

As an option *holder*, this is fantastic news!

• Higher chance of a massive crash? I don't care. My loss is capped.
• Higher chance of a massive rally? I love this! This is my "jackpot" scenario.

Because a spike in volatility increases your "jackpot" probability without increasing your "loss," it makes your option more valuable. This is why Vega is positive for both calls and puts that you buy.

Just like our other Greeks, we can find a formula for Vega by taking the derivative of the Black-Scholes formula ($C$) with respect to $\sigma$. The result is:

Let's deconstruct this. This formula is beautiful and tells us *exactly* when Vega is high.

• $S_t$: The stock price. A higher stock price means a higher Vega (for a call).
• $N'(d_1)$: The height of the bell curve at $d_1$. This is the *exact same term* that was in our Gamma formula! This means Vega is highest when the option is "at-the-money" ($S_t \approx K$).
• $\sqrt{T-t}$: The square root of time to expiration.

This $\sqrt{T-t}$ term is the key.

• If time $T-t$ is large (a long-term option), $\sqrt{T-t}$ is large, and Vega is HIGH.
• If time $T-t$ is small (a short-term option), $\sqrt{T-t}$ is small, and Vega is LOW.

Example 1: The "Jiggle Budget" (Long-Term vs. Short-Term)

• Situation: You buy two "at-the-money" call options (Stock = $100, Strike = $100).
• Option A (Low Vega): Expires in 1 Week. Its $\sqrt{T-t}$ is very small.
• Option B (High Vega): Expires in 1 Year. Its $\sqrt{T-t}$ is very large.

The Event: A "panic" event happens. The market's expected "jiggle rate" ($\sigma$) doubles from 20% to 40%.

• What happens to Option A (1 Week)? Its "jiggle rate" ($\sigma$) has doubled. But it only has *one week* to use this new, faster rate. The total "jiggle budget" ($\sigma \sqrt{T-t}$) doesn't increase by that much. Its price might go up, but not by a lot.
• What happens to Option B (1 Year)? Its "jiggle rate" ($\sigma$) has doubled, and it has a *full year* to use it. Its *total* "jiggle budget" has exploded. The market now sees a much, much higher chance of it finishing at $150 or $180.

Result: The 1-Year option's price (Option B) will increase dramatically more than the 1-Week option's price.

Example 2: The Volatility Trader

• The Situation: You are a trader. You don't have an opinion on the stock *price* ($S_t$), but you believe you are smarter than the market about *volatility* ($\sigma$).
• Your Belief: You think the market is too calm. You believe the "true" $\sigma$ should be 30%, but the market is only pricing options using a $\sigma$ of 20% ("implied volatility").
• The Trade: You are *not* going to just buy the stock. You are going to "buy the jiggle."
• The Action: You buy call options (or put options, it doesn't matter). You are now "long Vega."
  • You *also* delta-hedge by selling the stock (Lesson 5.2). You are now "delta-neutral."
• Your Position: You have a portfolio that is "delta-neutral" (immune to small $S_t$ moves) but "long Vega" (exposed to $\sigma$).
• How you win: You wait. If you are right and the market "wakes up" and re-prices volatility from 20% to 30%, your options will *instantly* become more valuable, and you will make a profit. This is a "pure" volatility bet.