Why Black–Scholes Misprices Options Away from the Money
Summary
The document describes an attempt to price calls and puts with Black–Scholes using historical volatility. Prices near the underlying stock price appear close to observed prices, while estimates farther away diverge. The author calculates implied volatility across strikes and asks what model could improve estimates away from the money.
A table of put prices, strikes, and open-interest changes provides sample market data, but the document includes no fitted volatility curve, market conditions, expiry details, or comparison of alternative models. It therefore frames a calibration problem rather than supplying a solution. The observed difference suggests that one constant historical volatility may not represent option prices across strikes; the data alone do not establish which pricing model or volatility input would be appropriate.
Key ideas
- Using historical volatility as a constant Black–Scholes input can produce prices that differ from observed prices away from the underlying price.
- Implied volatility can be calculated across strikes to examine how market pricing varies.
- The included put data illustrates the question but does not identify a preferred model.
- Reliable model comparison would require context such as expiry and market conditions.
Tags
Full text
# Option pricing formula for deep in-the/out-of money options?
# Option pricing formula for deep in-the/out-of money options?
I am learning option pricing and trying to calculate the call and put price using the Black-Scholes Formula. I have calculated the historical volatility to be 0.232. The formula is gives value close to the black scholes near the current stock price but away from the stock price it diverges significantly. Following is the plot of call/put prices and acutal call/put prices for a particular stock I analysed.
I calculated the implied volatility for all the prices. Here the vertical redline is the stock price and the horizontal line is the historical volatility.
My Question. What model should I use to get accurate estimates for prices away form the stock price?
EDIT Attaching the put data.
```
STRIKE_PR CLOSE OPEN_INT CHG_IN_OI
4961 1380.0 2.00 0 0
4962 1400.0 2.65 0 0
4963 1420.0 3.45 0 0
4964 1440.0 4.45 0 0
4965 1460.0 5.65 0 0
4966 1480.0 7.15 0 0
4967 1500.0 8.90 0 0
4968 1520.0 11.00 0 0
4969 1540.0 13.45 0 0
4970 1560.0 16.30 0 0
4971 1580.0 19.60 0 0
4972 1600.0 23.35 0 0
4973 1620.0 27.65 0 0
4974 1640.0 15.90 800 0
4975 1660.0 12.00 1600 0
4976 1680.0 16.65 6800 800
4977 1700.0 21.75 17200 3600
4978 1720.0 29.85 10800 1200
4979 1740.0 37.70 8400 -1200
4980 1760.0 55.00 4000 1200
4981 1780.0 52.95 5200 0
4982 1800.0 77.00 5600 0
4983 1820.0 100.00 400 0
4984 1840.0 113.85 0 0
4985 1860.0 125.45 0 0
4986 1880.0 137.60 0 0
4987 1900.0 150.35 0 0
4988 1920.0 163.65 0 0
4989 1940.0 177.50 0 0
4990 1960.0 191.80 0 0
4991 1980.0 206.70 0 0
4992 2000.0 221.95 0 0
4993 2020.0 237.55 0 0
4994 2040.0 253.60 0 0
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.