Volatility, Black-Scholes and skew
By SuperdayReps · Michigan Ross Undergrad | Wharton MBA · ex-Citi | ex-Greenhill · Former Head Mentor at a Leading IB Prep Service
14 min read · updated November 3, 2026
An equity derivatives trader does not really trade calls and puts. She trades volatility. The price of every option she quotes is a statement about how much the stock will move, and the number she argues about with clients and with her risk manager is that volatility, not the dollar premium. Skew, the volatility surface and variance swaps are the vocabulary of that argument.
Interviewers use this topic to separate candidates who memorized "Black-Scholes has five inputs" from those who know which input is the opinion, what each one does to the price and why the model's assumptions fail in exactly the places traders make and lose money. This lesson works through one priced example, then the volatility questions that follow from it.
Of the five Black-Scholes inputs, four are observed and one, volatility, is a forecast. So traders flip the model around and quote options by the volatility their price implies. Every option is then a bet on implied against realized volatility, and the shape of implied volatility across strikes and dates (the surface) is where the model's broken assumptions show up as prices.
What Black-Scholes does
The Black-Scholes model prices a European option. Its central idea is the hedge from the Greeks and delta hedging: if you can hedge an option perfectly and continuously with the stock, the hedged position has no risk, so it must earn the risk-free rate. That pins down the option's price without anyone needing a view on where the stock goes. The result is a formula: the call is worth the stock times one probability-like term, minus the present value of the strike times another.
You will not be asked to derive it. You will be asked what goes in and which way each input pushes.
| Input | If it rises, calls | If it rises, puts | Why |
|---|---|---|---|
| Stock price | Up | Down | The call is closer to paying, the put further |
| Strike | Down | Up | You pay more to buy, receive more to sell |
| Time to expiry | Up | Usually up | More room for a big move |
| Volatility | Up | Up | Bigger moves help the holder; the downside is capped at the premium |
| Risk-free rate | Up | Down | Cost of carry (below) |
| Dividends (the usual sixth) | Down | Up | The stock drops by the dividend and the holder does not get it |
Here is one option priced in full: stock $100, strike $100, one year, risk-free rate 5 percent (continuously compounded), volatility 20 percent, no dividends.
| Case | Call | Put |
|---|---|---|
| Base | $10.45 | $5.57 |
| Stock at $105 | $13.86 | $3.98 |
| Strike $110 | $6.04 | $10.68 |
| Six months instead of a year | $6.89 | $4.42 |
| Volatility 30 percent | $14.23 | $9.35 |
| Rate 2 percent | $8.92 | $6.94 |
| 2 percent dividend yield | $9.23 | $6.33 |
Two checks are worth doing out loud. Raising volatility from 20 to 30 percent raised both the call and the put, by $3.78 each, the same amount, as put-call parity requires. And in the base case the call minus the put is $4.88, which is the stock minus the present value of the strike: $100 minus $100 × e to the minus 0.05, or $4.88.
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Frequently asked
- What are the inputs to Black-Scholes?
- The stock price, the strike, the time to expiry, the risk-free interest rate and the volatility of the stock, plus expected dividends if it pays them. Every input can be looked up except volatility, which has to be estimated. That is why traders turn the model around and quote options by the volatility their price implies.
- What is the difference between implied and historical volatility?
- Historical, or realized, volatility measures how much a stock actually moved over a past period. Implied volatility is the volatility an option's market price assumes, found by solving the pricing model backward. One looks back, the other is the market's price for future movement, and implied usually sits a little above realized because investors pay up for protection.
- Why do out-of-the-money puts have higher implied volatility?
- Because investors who own stocks keep buying downside puts for protection, few natural sellers want to provide it, and stocks really do fall faster than they rise. That pattern of higher implied volatility at low strikes is called skew. It reflects the size of the feared move and who needs insurance, not a forecast that the stock will fall.
- What does the VIX measure?
- The VIX is the market's estimate of how volatile the S&P 500 will be over the next 30 days, worked out from the prices of a wide range of S&P 500 index options and shown as an annualized percentage. It rises when stocks fall because investors rush to buy protection. You cannot buy the VIX itself; you trade it through VIX futures, options and funds that hold the futures.
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