Implied Volatility

Traders who quote options, developers who build valuation systems, and risk managers who measure the risk of positions all work with implied volatility every day.

A price restated as a volatility

Implied volatility is the volatility backed out from an option's price¹. The formula of The Black-Scholes Model takes volatility and other inputs and gives the option price (the premium). Implied volatility runs this in the opposite direction: starting from the price in the market, you look for the volatility at which the formula gives the same price².

Consider a fictional example of a European call option (call) on the shares of Company F, the underlying asset. The underlying price is 100, the strike price is 100, the time to expiry is 1 year, the risk-free rate (a rate with no credit risk or similar risks) is 1% a year (continuously compounded), and there are no dividends. At a volatility of 20%, the Black-Scholes formula gives a price of 8.43. If this call trades at 10.00 in the market, the volatility at which the formula gives 10.00 is 23.97%, and this is the implied volatility.

Using implied volatility does not mean believing that volatility is actually constant. In the same way, quoting a discount bond's price by its yield to maturity (the constant rate that discounts the cash flow back to the price) does not mean believing that interest rates are constant. Implied volatility is another way of stating an option's price³.

Solving backward from the market price

You find implied volatility by solving for the volatility $\sigma$ at which the Black-Scholes price equals the market price. For an underlying with a dividend yield $q$, the price of a call is given by the following formula, using the underlying price $S$, the strike price $K$, the time to expiry $T$ (in years), and the risk-free rate $r$ (continuously compounded)⁴.

$$C_{\text{BS}}(\sigma) = S e^{-qT} N(d_1) - K e^{-rT} N(d_2)$$ $$d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)\,T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$

$N(\cdot)$ is the cumulative distribution function of the standard normal distribution. With the market price written as $C_{\text{mkt}}$, the equation to solve is as follows⁵.

$$C_{\text{BS}}(\sigma) = C_{\text{mkt}}$$

The equation is nonlinear in $\sigma$, so you solve it by repeating a calculation while changing the value⁶. The steps are as follows.

1. Take the market price of the call and check that it lies within the range a price can take (the lower and upper bounds). If it is outside the range, no volatility fits.
2. Set a trial volatility and compute the Black-Scholes price.
3. Compare the formula price with the market price. If a difference remains, update the volatility and return to step 2.
4. When the difference is no larger than the chosen tolerance, take that volatility as the implied volatility.

The answer does not come back from the market; you find it by this inverse calculation. Newton's method and bisection are two ways to do the update.

There is exactly one answer because the Black-Scholes price is continuous in volatility and rises as volatility rises⁷. Higher volatility widens the range of the underlying's price moves and raises the chance that the option ends in the money at expiry, so the price goes up⁸.

As volatility approaches 0, the price approaches the lower no-arbitrage bound, and as volatility grows without limit, the price approaches the upper bound⁹. For a call on an underlying with no dividends, the lower bound is the underlying price minus the present value of the strike price (or 0 if that is negative)¹⁰. Likewise, as volatility grows without limit, $N(d_1)$ in the formula above approaches 1 and $N(d_2)$ approaches 0 for a call on an underlying with no dividends. So the upper bound is the underlying price. If the market price is outside this range, no volatility matches it.

Iterative methods include Newton's method and bisection. Newton's method takes the slope of the price curve at the current volatility, and uses the point where the tangent line meets the market price as the next volatility¹¹. The derivative of the price with respect to volatility is vega $\nu$, so the update is as follows.

$$\sigma_{n+1} = \sigma_n - \frac{C_{\text{BS}}(\sigma_n) - C_{\text{mkt}}}{\nu(\sigma_n)}$$

Vega is $\nu = S e^{-qT} \sqrt{T}\, \phi(d_1)$, where $\phi(\cdot)$ is the probability density function of the standard normal distribution¹². With Newton's method, however, where vega is close to 0, the size of the update (the price difference ÷ vega) becomes large, and the next value can land far from the answer.

Bisection starts from two volatilities that bracket the answer. At each step it checks whether the price at the midpoint is above or below the market price, and narrows the interval to the half that contains the answer¹³. Bisection uses only the fact that the price rises with volatility, and does not use the derivative⁶.

Deep diveHow Newton's method and bisection differ in speed

With bisection, the width of the interval containing the answer halves at each step. If the starting interval has width $b - a$, the width after step $i$ is $(b - a)/2^i$¹³. With Newton's method, near the answer, the next error is proportional to the square of the previous error (quadratic convergence)¹⁴. So if you start from a value close to the answer, Newton's method gets close to the answer in a few steps.

Worked example: a call trading at 10.00

Suppose the Company F call trades at 10.00 in the market. The price of 8.43 at 20% volatility uses the same inputs as the worked example in The Black-Scholes Model. Start from 20% volatility and solve with Newton's method. Vega is shown as the change in price when volatility rises by 1 percentage point (per 1 percentage point), and volatility is also written in %. So the size of each update, in %, is "the price difference ÷ vega per 1%."

In step 1, 20 + 1.566681 ÷ 0.394479 = 20 + 3.9715 = 23.9715%. In step 2, 23.9715 + 0.001333 ÷ 0.393769 = 23.9715 + 0.0034 = 23.9749%. In step 3, the formula price matches 10.000000 to six decimal places, giving an implied volatility of 23.97%¹⁵.

Solving the same problem with bisection shows the difference in how fast each method approaches the answer. The price at 10% volatility is 4.4852 and the price at 40% is 16.2754. The market price of 10.00 lies between them, so start from the interval 10–40%.

After six steps, the interval is still 0.4688% wide, and narrowing it below 0.0001% takes 19 steps (halving 30% 19 times gives about 0.00006%). Newton's method agreed to four decimal places after two steps. Bisection is slower, but it never leaves the interval that brackets the answer.

If a European put option (put) with the same inputs has a market price of 9.0050, the implied volatility from the put is also 23.97%. The value 9.0050 is 10.00 − 100 + 99.0050, where 99.0050 is $100 \times e^{-0.01}$. The put giving the same implied volatility as the call follows from put-call parity¹⁶.

How the market price and vega change the result

The higher the market price, the higher the implied volatility. For the Company F call, each 1.00 rise in the market price raises the implied volatility by about 2.5 percentage points.

How much implied volatility moves for a given price difference depends on the size of vega. If the price change is small and vega changes little over that range, the change in implied volatility is roughly "the price change ÷ vega per 1%," in %. Vega is large near the strike price and approaches 0 as expiry nears¹⁷. The table below uses prices set at an underlying price of 100, a risk-free rate of 1%, and a volatility of 20%, and shows the implied volatility when 0.05 is added to or subtracted from each price.

For the call with a strike price of 100 and 1 year to expiry, a price difference of 0.05 moves the implied volatility by only about 0.13 percentage points. The call with a strike price of 140 and 0.25 years to expiry is worth only 0.0013. Adding 0.05 raises its implied volatility to 28.31%, and subtracting 0.05 makes the price negative, so no implied volatility can be found. For options like this, a small difference in the price makes a large difference in implied volatility. In this row, 0.05 ÷ 0.0009 suggests a rise of about 56 percentage points, but the actual rise is 8.31 percentage points. When vega is this small, the approximation "price change ÷ vega" is far off.

How it differs from historical volatility

Historical volatility is volatility calculated from past moves in the underlying price. For a fixed past period, you calculate the percentage change in price for each business day, take the standard deviation, and convert it to an annual figure¹⁸. Converting to an annual figure is covered in Probability and Statistics Basics. An option's theoretical price is calculated with implied volatility backed out from market prices, not with historical volatility, which is a past value¹⁹.

Historical volatility is used to estimate future volatility from past volatility¹⁸. Implied volatility is calculated from the prices of options trading in the market now. When a major economic indicator changes sharply or when there is political instability, it rises as the market prices in larger moves in share prices²⁰. The two values are not necessarily equal, even for the same underlying²¹.

Where markets use implied volatility

According to Lee's November 2002 manuscript, in the options market traders customarily quote option prices as volatility values (volatility "points")²². The prices themselves vary widely in level by strike price, expiry, and underlying. Converted to implied volatility, options with different strike prices and expiries, different underlyings, and prices taken at different times can be compared on the same scale. Implied volatility also gives a scale for comparing a market price with the price you consider fair²².

According to Haugh's 2013 materials, market participants know that the Black-Scholes model does not match actual price moves, yet talking in the model's terms (implied volatility and the Greeks) is widespread. Trading desks compute the implied volatility surface (the values by strike price and expiry) of the Black-Scholes model, and they also compute and use the Greeks under this model²³. Even products that clearly do not fit the model's assumptions may have their prices quoted in implied volatility. Examples are caplets (the option for one period of an interest rate cap) and swaptions in the interest rate markets, and options on credit default swaps (CDS) in the credit markets²⁴.

There are also volatility indices calculated from option prices, such as the VIX. The same manuscript states that exchanges update implied volatility indices in real time²².

Assumptions and limits: different values by strike and expiry

If the Black-Scholes model were correct, the implied volatility of options on the same underlying would be the same regardless of strike price and expiry. In actual markets, the values differ by strike price and expiry, and the pattern across strike prices is called the volatility smile²⁹.

Most single-stock options are American options³⁰. According to Haugh's 2013 materials, the implied volatilities from calls and puts usually differ for American options³⁰.

Notes

1. Japan Exchange Group, "インプライド・ボラティリティ（IV）（用語集）" (Glossary: implied volatility (IV), in Japanese). ↩

2. Roger W. Lee, "Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation" (manuscript of November 2002; Recent Advances in Applied Probability, Springer, 2004), sections 1.1 and 1.3, pages 1–3. ↩

3. The same source, section 1.1, page 2. It states that using implied volatility does not entail any belief that volatility is actually constant, draws the analogy with quoting a discount bond by its yield to maturity, and describes implied volatility as another way of expressing a price. ↩

4. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), page 3, formula (13). The source writes $\Phi$, $\log$, and $T - t$, which this page writes as $N$, $\ln$, and $T$. ↩

5. The same source, page 4, formula (14). It says the volatility surface is "defined implicitly" by formula (14). ↩

6. 小林健太, "ヨーロピアンオプションとインプライドボラティリティの数値計算" (Numerical computation of European options and implied volatility), Hitotsubashi University, "インプライドボラティリティの計算" (Computing implied volatility) (in Japanese). ↩ ↩² ↩³

7. Martin Haugh, "Black-Scholes and the Volatility Surface," page 4. Note 4 assumes that there are no arbitrage opportunities in the market. ↩

8. Osaka Exchange, "理論" (Theory), Kitahama Investment School (in Japanese), the "理論価格はボラティリティで決まる" (The theoretical price is determined by volatility) section. ↩

9. Roger W. Lee, "Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation," section 1.3, page 3. ↩

10. Martin Haugh, "Forwards, Swaps, Futures and Options" (Columbia University, 2016), page 12, "Model Free Bounds for Option Prices," formula (9) under "Put-Call Parity" and the formula that follows it. ↩

11. 舟木剛, "数値解析 第二回 代数方程式の求解" (Numerical analysis, lecture 2: Solving algebraic equations), Osaka University (October 12, 2016), "ニュートン法" (Newton's method), page 8 (in Japanese). The source's $x_i$ and $f(x)$ are written on this page as $\sigma_n$ and $C_{\text{BS}}(\sigma) - C_{\text{mkt}}$. ↩

12. Martin Haugh, "Black-Scholes and the Volatility Surface," page 9, the definition and formula of vega. ↩

13. 舟木剛, "数値解析 第二回 代数方程式の求解," Osaka University, "2分法" (Bisection), pages 5–7 (in Japanese). ↩ ↩²

14. 舟木剛, "数値解析 第二回 代数方程式の求解," Osaka University, "ニュートン法で求めた解の誤差" (Error of the solution found by Newton's method), page 11 (in Japanese). ↩

15. The values in the table are calculated with the functions in src/lib/simulators/black-scholes.ts and rounded. Recalculating after rounding vega to six decimal places can shift the fifth decimal place and beyond. ↩

16. Martin Haugh, "Black-Scholes and the Volatility Surface," page 4, note 5. ↩

17. The same source, page 9. Together with Figure 5, it states that vega peaks near the strike price and approaches 0 as expiry nears. ↩

18. Japan Exchange Group, "ヒストリカル・ボラティリティ（HV）（用語集）" (Glossary: historical volatility (HV), in Japanese). ↩ ↩²

19. Osaka Exchange, "理論" (Theory), Kitahama Investment School (in Japanese), the "ボラティリティ" (Volatility) section. ↩

20. Osaka Exchange, "理論" (Theory), Kitahama Investment School (in Japanese), the "インプライド・ボラティリティ" (Implied volatility) section. ↩

21. Martin Haugh, "Black-Scholes and the Volatility Surface," page 13 (the implied volatility of an option, the volatility in the model of the underlying, and the actual volatility are not necessarily equal). ↩

22. Roger W. Lee, "Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation" (manuscript of November 2002), section 1.1, page 2. ↩ ↩² ↩³

23. Martin Haugh, "Black-Scholes and the Volatility Surface," page 5. It states that market participants know the model is inaccurate, yet the language of Black-Scholes is widely used. ↩

24. Martin Haugh, "Black-Scholes and the Volatility Surface," page 17. ↩

25. 舟木剛, "数値解析 第二回 代数方程式の求解," Osaka University, "収束判定" (Convergence test), pages 7–8 (in Japanese). ↩

26. Martin Haugh, "Black-Scholes and the Volatility Surface," page 4 and note 6. ↩

27. The same source, pages 11–12, "Delta-Gamma-Vega Approximations to Option Prices." ↩

28. The same source, page 13, "Some Answers to Delta-Hedging Questions." ↩

29. The same source, pages 4 and 5, "Why is there a Skew?" ↩

30. The same source, page 5. It states that single-stock options are generally American, and that in this case the surfaces from calls and puts typically differ. ↩ ↩²