Implied volatility — details
About this app
The volatility smile: market prices (OTM quote board) visibly gape above the flat Black-Scholes line at the wings — below, the smile σ_impl(K) distilled point by point. Presets “BS world”, “equity skew” (crash fear), “FX smile”; skew/curvature/ATM controls. Debunks “one σ per stock is enough”.
The idea
If Black-Scholes were exact, all options on the same underlying with the same maturity would share one implied volatility. In practice implied σ depends on the strike. Plotted against K, equity indices typically show a downward skew: out-of-the-money puts are in demand as crash protection and trade at higher implied volatility. Currency options often show a more symmetric smile. The smile is the market’s correction to the model’s assumptions, such as normally distributed log returns.
What the app shows
The top chart shows market prices per strike as dots next to the Black-Scholes line with constant at-the-money σ; you can view an OTM mix, calls only or puts only, with the gap highlighted. Below, implied σ is backed out of each price by bisection and drawn as a smile over K, with a flat ATM reference. The values panel shows ATM vol, implied σ at K = 60 and 140, the skew σ(90) − σ(110) and the maximum round-trip error.
Try it
Choose the “BS world” preset: market prices sit on the line and the smile is flat. Switch to “equity skew”: put prices on the left lie visibly above the line and implied σ rises towards low strikes. With “FX smile” both wings lift. Toggle the top view between calls and puts: the smile below does not change, because put-call parity forces the same implied volatility for both.
Frequently asked questions
- What is the volatility smile?
- The volatility smile is implied volatility plotted against strike for a fixed maturity. Instead of the flat line Black-Scholes would predict, it shows higher values for options far out of the money. It reveals that the market sees the price distribution differently from the model.
- Why do equity options have a volatility skew?
- Investors protect themselves against sharp declines by buying out-of-the-money puts. That demand, together with the experience that stock markets can fall abruptly, makes such puts more expensive, which shows up as higher implied volatility at low strikes. Since the 1987 crash this downward skew has been a persistent feature of equity index options.
- Do a call and a put with the same strike have the same implied volatility?
- Yes. For European options with the same strike and maturity, put-call parity forces the same implied volatility, otherwise there would be an arbitrage. Market practice therefore uses the more liquid out-of-the-money option at each strike: puts below spot, calls above.
- Which presets does the smile app offer?
- There are three presets: “BS world” with flat volatility, “equity skew” with a downward slope and “FX smile” with symmetric curvature. You can also set ATM vol, skew, curvature and maturity yourself.
Subject: Financial derivatives
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