Implied Volatility and the Volatility Smile
(a) What is the implied volatility of an option, and how would you compute it from a market price?
(b) What is the volatility smile (and the volatility skew)? Describe the typical shape for equity index options.
(c) What does the existence of a volatility smile tell you about the Black-Scholes model and its assumptions about the underlying price process?
Hints
- Implied volatility is the $\sigma$ that makes the Black-Scholes price equal the market price; since the price is strictly increasing in $\sigma$ (vega is positive), the root is unique and can be found with Newton's method or bisection.
- Plot implied volatility against strike (or moneyness) for a fixed expiry. If Black-Scholes were right, the plot would be flat.
- A downward-sloping skew means the market pays more for low-strike puts than a lognormal model with a single volatility would justify, which means the risk-neutral distribution has a fatter left tail than the lognormal.
Worked Solution
How to Think About It: Black-Scholes takes five observable inputs and one unobservable one, $\sigma$. Given a market price, invert the formula to back out $\sigma$: that is the implied volatility. A single lognormal model would give the same $\sigma$ for every strike, so any strike dependence in implied volatility is direct evidence about how the market's distribution of $S_T$ deviates from lognormal.
Quick Estimate: For a near-the-money short-dated option, $c \approx 0.4\,\sigma S\sqrt{\tau}$, so $\sigma_{\text{imp}} \approx c/(0.4\,S\sqrt{\tau})$. A 3-month at-the-money call on a $100$ stock trading at $5$ implies $\sigma \approx 5/(0.4 \times 100 \times 0.5) = 25\%$. Away from the money you need the full inversion, but vega is positive so a few Newton steps converge.
Formal Solution:
Part (a): Implied volatility
*Step 1 -- Definition.* The implied volatility $\sigma_{\text{imp}}$ is the value of $\sigma$ such that the Black-Scholes(-Merton) price equals the observed market price: $c_{\text{BS}}(S, K, r, y, \tau; \sigma_{\text{imp}}) = c_{\text{mkt}}$.
*Step 2 -- Computation.* There is no closed form. Because vega $\partial c/\partial\sigma = Se^{-y\tau}\sqrt{\tau}N'(d_1) > 0$, the price is strictly increasing in $\sigma$, so the solution is unique (provided the market price lies within the no-arbitrage bounds). Solve numerically, e.g. by Newton's method $$\sigma_{n+1} = \sigma_n - \frac{c_{\text{BS}}(\sigma_n) - c_{\text{mkt}}}{\text{vega}(\sigma_n)},$$ or by bisection, which is slower but robust when vega is tiny (deep in or out of the money).
Part (b): The volatility smile and skew
*Step 3 -- Definition.* For a fixed maturity, plot $\sigma_{\text{imp}}$ against strike $K$ (or moneyness $K/S$, or delta). The resulting curve is the volatility smile. When it is U-shaped (both wings above the middle, as in currency options) it is called a smile; when it slopes downward, with low strikes carrying higher implied volatility than high strikes, it is called a skew or smirk.
*Step 4 -- Equities.* Since the 1987 crash, equity index options show a pronounced skew: out-of-the-money puts (low $K$) have much higher implied volatility than at-the-money options, which in turn are slightly above out-of-the-money calls. Plotting against maturity as well gives the implied volatility surface, and the skew is usually steepest for short maturities.
Part (c): What the smile says about Black-Scholes
*Step 5 -- Inconsistency.* Black-Scholes assumes the stock follows geometric Brownian motion with a constant (or at least deterministic) volatility, so $\ln S_T$ is normal and one $\sigma$ prices every strike. A non-flat smile is therefore a direct rejection of that assumption: the market's risk-neutral distribution is not lognormal with a single volatility.
*Step 6 -- What the shape reveals.* A downward skew means low-strike puts are expensive relative to lognormal, i.e. the risk-neutral density has a fatter left tail (and negative skewness) than the lognormal. Economically this reflects (i) volatility that is itself random and negatively correlated with returns, the leverage effect, captured by stochastic volatility models such as Heston; (ii) the possibility of jumps, especially downward, captured by jump-diffusion models such as Merton's; (iii) demand for crash protection. A symmetric smile (as in FX) indicates fat tails on both sides, i.e. excess kurtosis, consistent with stochastic volatility or symmetric jumps.
*Step 7 -- Practical consequence.* Traders still use Black-Scholes as a quoting convention: implied volatility is a price in convenient units, and the smile is the correction table. Hedging and exotic pricing, however, must use a model that reproduces the smile (local volatility, stochastic volatility, or jump models), otherwise deltas and vegas are biased.
Answer:
(a) Implied volatility is the $\sigma$ that equates the Black-Scholes price to the market price; it is found numerically (Newton or bisection) and is unique because vega is positive.
(b) The volatility smile is the graph of implied volatility against strike for a fixed expiry; equity index options show a downward-sloping skew with low-strike puts at the highest implied volatilities.
(c) A non-flat smile contradicts the Black-Scholes assumption of a lognormal price with constant volatility: the market-implied distribution has fatter tails (a fat left tail for equities), which points to volatility that is neither constant nor deterministic and to price jumps.
Intuition
Implied volatility is the market's price of an option quoted in a convenient unit: the constant volatility that would justify that price under Black-Scholes. If the model were literally true, every option on the same underlying and expiry would imply the same number. Instead equity index options show a persistent skew, with low-strike puts trading at much higher implied volatility than high-strike calls, because the market prices a fat left tail (crashes), volatility that rises when prices fall, and jumps. The smile is therefore a map of exactly how the real risk-neutral distribution differs from lognormal, and desks quote, hedge and risk-manage in volatility space precisely because the model's assumptions are known to be wrong.