Interpretation of vega out of BS formula

Question

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 1

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.