Implied Volatility





Kerry Back

  • The inputs to the Black-Scholes formulas are: strike, time to maturity, interest rate, and volatility (and dividends).
  • All of these are known except volatility (and dividends, which is easy to estimate).
  • We could input historical volatility, but volatility changes over time and the market is forward looking.
  • A standard thing is to infer volatility from market prices of actively traded options. This is called the implied volatility.

from scipy.optimize import fsolve
def callIV(C, S, K, T, r, q=0):
  return fsolve(
    lambda x: C - callBS(S, K, T, x, r, q),
    x0=0.3
  )

C = 15
S = 100
K = 100
T = 1
r = 0.05
iv = callIV(C, S, K, T, r)
iv[0]
0.32025830955048284

AAPL Call Data on 3/6/2023

  • last AAPL price:
153.22
  • April 21, 2023 calls
contractSymbol lastTradeDate lastPrice bid ask change percentChange volume openInterest impliedVolatility inTheMoney contractSize currency
strike
140.0 AAPL230421C00140000 2023-03-08 16:36:24+00:00 14.82 15.20 15.40 0.799999 5.706129 95 14898 0.334479 True REGULAR USD
145.0 AAPL230421C00145000 2023-03-08 16:45:34+00:00 11.20 11.20 11.40 1.000000 9.803922 257 11747 0.310432 True REGULAR USD
150.0 AAPL230421C00150000 2023-03-08 16:50:30+00:00 8.00 8.00 8.05 1.050000 15.107916 509 32189 0.297370 True REGULAR USD
155.0 AAPL230421C00155000 2023-03-08 16:50:17+00:00 5.15 5.05 5.10 0.850000 19.767439 3199 46486 0.275703 False REGULAR USD
160.0 AAPL230421C00160000 2023-03-08 16:50:01+00:00 2.92 2.91 2.93 0.550000 23.206760 1686 44386 0.259529 False REGULAR USD
165.0 AAPL230421C00165000 2023-03-08 16:49:14+00:00 1.46 1.44 1.45 0.300000 25.862078 918 33471 0.243538 False REGULAR USD
170.0 AAPL230421C00170000 2023-03-08 16:50:01+00:00 0.66 0.67 0.67 0.140000 26.923088 457 43256 0.236092 False REGULAR USD

AAPL Call and Put Implied Vols

Call IV Put IV
strike
140.0 0.334479 0.280036
145.0 0.310432 0.264045
150.0 0.297370 0.247444
155.0 0.275703 0.232063
160.0 0.259529 0.222359
165.0 0.243538 0.209969
170.0 0.236092 0.160653

TSLA Call and Put Implied Vols

Call IV Put IV
strike
165.00 0.694339 0.636722
166.67 0.675723 0.632694
170.00 0.678226 0.618534
175.00 0.668949 0.612858
180.00 0.659671 0.601078
183.33 0.652347 0.604923
185.00 0.651127 0.591313
190.00 0.645389 0.582341
191.67 0.642215 0.581059
195.00 0.637760 0.576420

VIX (Volatility Index)

  • average of implied vols of out-of-the-money 1-month-to-maturity S&P 500 put and call options