Volatility Modeling

MIT OCW 18.S096 + 15.450 (CC BY-NC-SA 4.0)

Volatility is not constant. Historical volatility measures past variation; implied volatility extracts the market's forward-looking estimate from option prices. GARCH models capture the clustering of calm and turbulent periods that real markets exhibit.

Historical vs implied volatility

Historical volatility is the standard deviation of log returns over a lookback window. Implied volatility is the sigma that makes Black-Scholes match the observed option price. They measure different things: what happened vs what the market expects.

Scheme

; Historical volatility: stdev of log returns
; Annualize by multiplying by sqrt(252)

(define (log-return s1 s2)
  (log (/ s2 s1)))

; Sample daily prices
(define prices (list 100 101.5 99.8 102.3 101.0 103.5 102.8 104.1))

(define (log-returns prices)
  (if (null? (cdr prices)) '()
      (cons (log-return (car prices) (cadr prices))
            (log-returns (cdr prices)))))

(define (mean lst)
  (/ (apply + lst) (length lst)))

(define (variance lst)
  (let ((m (mean lst)))
    (/ (apply + (map (lambda (x) (* (- x m) (- x m))) lst))
       (- (length lst) 1))))

(define (stdev lst) (sqrt (variance lst)))

(define rets (log-returns prices))
(define daily-vol (stdev rets))
(define annual-vol (* daily-vol (sqrt 252)))

(display "Log returns: ")
(for-each (lambda (r) (display (/ (round (* r 10000)) 10000)) (display " ")) rets)
(newline)
(display "Daily vol: ") (display (/ (round (* daily-vol 10000)) 10000)) (newline)
(display "Annualized vol: ") (display (/ (round (* annual-vol 10000)) 10000))

; Historical volatility: stdev of log returns
; Annualize by multiplying by sqrt(252)

(define (log-return s1 s2)
  (log (/ s2 s1)))

; Sample daily prices
(define prices (list 100 101.5 99.8 102.3 101.0 103.5 102.8 104.1))

(define (log-returns prices)
  (if (null? (cdr prices)) '()
      (cons (log-return (car prices) (cadr prices))
            (log-returns (cdr prices)))))

(define (mean lst)
  (/ (apply + lst) (length lst)))

(define (variance lst)
  (let ((m (mean lst)))
    (/ (apply + (map (lambda (x) (* (- x m) (- x m))) lst))
       (- (length lst) 1))))

(define (stdev lst) (sqrt (variance lst)))

(define rets (log-returns prices))
(define daily-vol (stdev rets))
(define annual-vol (* daily-vol (sqrt 252)))

(display "Log returns: ")
(for-each (lambda (r) (display (/ (round (* r 10000)) 10000)) (display " ")) rets)
(newline)
(display "Daily vol: ") (display (/ (round (* daily-vol 10000)) 10000)) (newline)
(display "Annualized vol: ") (display (/ (round (* annual-vol 10000)) 10000))

GARCH(1,1) model

GARCH(1,1) models volatility clustering: sigma^2(t) = omega + alpha * r(t-1)^2 + beta * sigma^2(t-1). A large return yesterday raises tomorrow's variance estimate. Alpha captures shock impact, beta captures persistence. Alpha + beta < 1 ensures stationarity.

Scheme

; GARCH(1,1): sigma^2(t) = omega + alpha * r(t-1)^2 + beta * sigma^2(t-1)
; Typical equity parameters: omega=0.000002, alpha=0.1, beta=0.85

(define omega 0.000002)
(define alpha 0.1)
(define beta 0.85)

(define (garch-variance prev-var prev-return)
  (+ omega (* alpha (* prev-return prev-return)) (* beta prev-var)))

; Simulate variance path given returns
(define returns (list 0.01 -0.03 0.005 0.02 -0.015 0.008 -0.025 0.012))
(define initial-var (* 0.015 0.015))  ; start at 1.5% daily vol

(define (garch-path var returns)
  (if (null? returns) '()
      (let ((new-var (garch-variance var (car returns))))
        (cons new-var (garch-path new-var (cdr returns))))))

(define vars (garch-path initial-var returns))
(define vols (map (lambda (v) (sqrt v)) vars))

(display "Return    Cond. Vol (daily)") (newline)
(display "------    -----------------") (newline)
(for-each
  (lambda (r v)
    (display (/ (round (* r 10000)) 10000)) (display "    ")
    (display (/ (round (* v 10000)) 10000)) (newline))
  returns vols)

(newline)
(display "alpha+beta = ") (display (+ alpha beta))
(display "  (< 1: stationary)")

; GARCH(1,1): sigma^2(t) = omega + alpha * r(t-1)^2 + beta * sigma^2(t-1)
; Typical equity parameters: omega=0.000002, alpha=0.1, beta=0.85

(define omega 0.000002)
(define alpha 0.1)
(define beta 0.85)

(define (garch-variance prev-var prev-return)
  (+ omega (* alpha (* prev-return prev-return)) (* beta prev-var)))

; Simulate variance path given returns
(define returns (list 0.01 -0.03 0.005 0.02 -0.015 0.008 -0.025 0.012))
(define initial-var (* 0.015 0.015))  ; start at 1.5% daily vol

(define (garch-path var returns)
  (if (null? returns) '()
      (let ((new-var (garch-variance var (car returns))))
        (cons new-var (garch-path new-var (cdr returns))))))

(define vars (garch-path initial-var returns))
(define vols (map (lambda (v) (sqrt v)) vars))

(display "Return    Cond. Vol (daily)") (newline)
(display "------    -----------------") (newline)
(for-each
  (lambda (r v)
    (display (/ (round (* r 10000)) 10000)) (display "    ")
    (display (/ (round (* v 10000)) 10000)) (newline))
  returns vols)

(newline)
(display "alpha+beta = ") (display (+ alpha beta))
(display "  (< 1: stationary)")

Volatility smile and skew

If Black-Scholes were exactly right, implied volatility would be the same for all strikes. It is not. Deep out-of-the-money puts carry higher implied vol than ATM options — that is the skew. When both wings are elevated, you get the smile. The pattern reveals fat tails and jump risk that the lognormal model misses.

Scheme

; Volatility smile: implied vol varies by moneyness (K/S)
; Quadratic approximation: sigma(m) = sigma_atm + a*(m-1)^2 + b*(m-1)
; where m = K/S

(define sigma-atm 0.20)
(define a 0.30)   ; curvature (smile)
(define b -0.10)  ; tilt (skew)

(define (implied-vol moneyness)
  (let ((m (- moneyness 1)))
    (+ sigma-atm (* a m m) (* b m))))

(display "K/S      Implied Vol") (newline)
(display "---      -----------") (newline)
(for-each
  (lambda (m)
    (display m) (display "     ")
    (display (/ (round (* (implied-vol m) 10000)) 10000))
    (newline))
  (list 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20))

(newline)
(display "Skew: OTM puts (low K/S) have higher implied vol")

Term structure of volatility

Implied volatility also varies by expiration. Short-dated options are more sensitive to current conditions; long-dated options revert toward the long-run average. The volatility surface maps implied vol across both strike and maturity — a two-dimensional object that traders must model and hedge.

Scheme

; Term structure: vol mean-reverts over time
; Simple model: sigma(T) = sigma_long + (sigma_short - sigma_long) * exp(-kappa*T)

(define sigma-long 0.20)   ; long-run volatility
(define sigma-short 0.35)  ; current short-term vol (elevated)
(define kappa 2.0)          ; mean-reversion speed

(define (vol-term-structure t)
  (+ sigma-long (* (- sigma-short sigma-long) (exp (* (- kappa) t)))))

(display "Maturity (yr)  Implied Vol") (newline)
(display "-------------  -----------") (newline)
(for-each
  (lambda (t)
    (display t) (display "          ")
    (display (/ (round (* (vol-term-structure t) 10000)) 10000))
    (newline))
  (list 0.083 0.25 0.5 1.0 2.0 5.0))

(newline)
(display "Short-term vol is elevated (35%), mean-reverts to 20%")

Neighbors

📉 Finance 2 Ch.4 — Monte Carlo uses volatility as a key input
📉 Finance 2 Ch.6 — time series methods for estimating volatility dynamics
📊 Statistics Ch.5 — variance estimation and hypothesis testing

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