Volatility

Study guide: Exam INV-201

Volatility ( $\sigma$ ) is the annualized standard deviation of an asset's continuously compounded return: under the lognormal model of the Black-Scholes Model, $\ln(S_T/S_0)$ has variance $\sigma^2 T$ . It is the one Black-Scholes input that cannot be observed directly, so it is either estimated from past prices (realized or historical volatility) or backed out of option prices (implied volatility).

$\operatorname{Var}\!\left[\ln\frac{S_T}{S_0}\right] = \sigma^2 T$

$\hat\sigma = \frac{s}{\sqrt{h}}$

$C_{BS}(S_0, K, r, \delta, \sigma_{\text{imp}}, T) = C_{\text{market}}$

$\text{Vega} = S_0 e^{-\delta T}\, n(d_1) \sqrt{T}$

$s$ is the sample standard deviation (divisor $n-1$ ) of the log returns $\ln(S_i/S_{i-1})$ observed every $h$ years ( $h = 1/52$ for weekly, $1/252$ for trading days); volatility scales with $\sqrt{\text{time}}$ , so a daily $1\%$ is about $16\%$ a year
Implied volatility is the $\sigma$ that makes the model price equal the market price; it is solved numerically, and it differs by strike (the smile or skew), which is itself evidence that the constant- $\sigma$ lognormal model is wrong
Vega ( $n(d_1)$ is the standard normal density) is the option price's sensitivity to $\sigma$ ; it is the same for a call and a put, so writing either leaves an insurer short volatility, as a variable annuity guarantee does
Realized versus implied: a delta hedge priced at $\sigma_{\text{imp}}$ earns or loses according to the volatility actually realized (see Delta Hedging); target volatility funds cap a guarantee's cost by setting the equity weight to $w_t = \sigma_{\text{target}} / \hat\sigma_t$

Estimating Volatility from Weekly Prices An equity index closed at $50.00$ , $51.20$ , $49.80$ , $50.60$ , $52.10$ , $51.40$ on six consecutive Fridays. Estimate its annual volatility.
Answer The five weekly log returns are $0.023717$ , $-0.027725$ , $0.015937$ , $0.029213$ , $-0.013527$ , with mean $\bar R = 0.005523$ and $\sum (R_i - \bar R)^2 = 0.0024690$ .
 $\begin{align*} s &= \sqrt{\frac{0.0024690}{4}} \\ &= 0.024844 \\ \hat\sigma &= 0.024844\sqrt{52} \\ &= 0.1792 \end{align*}$ 
The index's estimated volatility is about $17.9\%$ a year. Five returns make this a very noisy estimate.

Backing Out Implied Volatility An index at $100$ has dividend yield $\delta = 0.015$ ; $r = 0.03$ . A 1-year at-the-money European call trades at $9.00$ . Starting from $\sigma = 0.20$ , take one Newton step to estimate the implied volatility.
Answer At $\sigma = 0.20$ , $d_1 = (0.03 - 0.015 + 0.02)/0.20 = 0.175$ and the model price is $C = 8.5437$ .
 $\begin{align*} \text{Vega} &= 100e^{-0.015}\, n(0.175)\sqrt{1} \\ &= 98.5112 \times 0.39288 \\ &= 38.703 \\ \sigma_1 &= 0.20 + \frac{9.00 - 8.5437}{38.703} \\ &= 0.2118 \end{align*}$ 
The model price at $\sigma_1$ is $8.99993$ , so the implied volatility is $21.18\%$ , above the $20\%$ starting guess because the market price is higher.

Equity Weight in a Target Volatility Fund A fund backing a variable annuity targets $10\%$ volatility, holding equities and cash. The equity index's volatility is currently estimated at $16\%$ . Find the equity allocation, and the fund's volatility if the index's volatility turns out to be $24\%$ before the next rebalancing.
Answer
 $\begin{align*} w &= \frac{0.10}{0.16} \\ &= 0.625 \\ \sigma_{\text{fund}} &= 0.625 \times 0.24 \\ &= 0.15 \end{align*}$ 
The fund holds $62.5\%$ equities and $37.5\%$ cash, treating cash as riskless. Volatility rising between rebalancings pushes the fund to $15\%$ , so the target is met only as well as $\hat\sigma$ forecasts the next period.

How it appears on the exam

INV-201 ("realized versus implied volatility"; risks "that can be hedged, including ... volatility"): explain why implied exceeds or differs from realized volatility, and what that gap does to the profit of a delta-hedged book, as a written answer.

INV-201: compute vega and the vega exposure of a guarantee or hedge portfolio, then the trades in options that neutralize it alongside delta and gamma (see Hedging).

INV-201 (target volatility funds and their effect on "fund performance, guarantee cost, and risk control"): compute the equity weight and describe how the mechanism lowers the guarantee's cost and what it gives up in returns.

The Black-Scholes questions on FAM and ALTAM give $\sigma$ as an input; the volatility work there is reading it correctly into $d_1$ and $d_2$ .

Exam traps

Annualizing with the wrong factor. Multiply a periodic standard deviation by $\sqrt{\text{periods per year}}$ , not by the number of periods; weekly $\times 52$ instead of $\times\sqrt{52}$ gives a volatility seven times too large.

Using simple returns or the population divisor. The estimator uses log returns and divides by $n-1$ ; with few observations, dividing by $n$ visibly understates $\hat\sigma$ .

Mixing up $\sigma^2$ and $\sigma$ . A question may give the variance of the annual log return ( $0.04$ ); $\sigma$ is its square root ( $0.20$ ).

Assuming a put's vega is negative. Both calls and puts gain value as volatility rises; a short put (a GMMB) loses when volatility rises.