Interpreting Black–Scholes Vega in Volatility Units
Summary
This note explains why Black–Scholes vega can look large when volatility is entered as a decimal. Vega measures the option price’s sensitivity to a one-unit change in volatility, equivalent to a 100 percentage point move. The quoted S&P 500 example has vega 188.48, so a one percentage point increase corresponds to about one hundredth of that change in option value. The answer also suggests checking the interpretation by recalculating the option at the higher volatility and comparing prices.
The key lesson is to match sensitivity units to the size of the volatility move being modeled. The numerical examples concern calls on S&P 500 and Nasdaq indexes with short maturities, but the unit conversion applies generally to options priced with this convention. The note does not address other modeling choices, such as volatility surface changes, position scaling, or whether the Black–Scholes assumptions fit a particular option.
Key ideas
- Vega is the option price change for a one-unit change in volatility expressed as a decimal.
- A one percentage point volatility move is 0.01 volatility units, so its approximate price effect is 0.01 times vega.
- A large quoted vega does not by itself imply a calculation error; its units determine how to interpret it.
- Repricing at the shifted volatility provides a practical check on the sensitivity estimate.
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# Interpretation of vega out of BS formula # Interpretation of vega out of BS formula I am comparing Monte Carlo estimates of VaR (using importance sampling) under both the normal and student distributions. I am also considering risk factors other than log-prices; in particular, implied volatility. To undertake simulation one must know the sensitivities of the option price with respect to changes in the underlying risk factor. My problem is when I consider an S&P500 option and a NASDAQ option (using the current closing price and implied volatilities of the respective indexes) I get extremely large values. From my matlab code, the parameters into BS are given below: [p1 t1 d1 g1 v1 r1]= call_fn(2035.73 , 2050 , 0.002 , 0.1914 , 20/365); %S&P500 VIX = 19.14 [p2 t2 d2 g2 v2 r2]= call_fn(4877.49 , 5100 , 0.002 , 0.2161 , 20/365); %NASDAQ VXN = 21.61% The outputted values of vega are 188.48 and 316.197 (SP and NAS, resp.). These seem extremely large to me and have led me to think the numbers need modifying e.g. /365 or /100 etc. Note: I have double and triple checked that the formulas I use to compute vega are correct. Many thanks ## Answer by Alex C (score 1, accepted) https://quant.stackexchange.com/a/19533 Your Vega of 188.48 is correct, in the sense that matches my calculation. What it means is that if the volatility increase by 1 (i.e. by 100 percentage points, from 19.14% to 119.14%) the call will increase by 188 dollars. Obviously that is an unrealistic move. More realistically if the volatility increases by 0.01 (i.e. 1 percentage point, from 19.14% to 20.14%) then the call will increase by one-hundredths of this i.e. 1.8848 dollars. And you can verify this by plugging 20.14% in your calculation, the call price increase by about 1.8. So what is confusing you is just the units.
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