Time-Weighted Vega and Volatility Curve Shifts
Summary
The document examines whether option vegas across different expiries can be added directly to estimate the effect of a volatility change. Its example portfolio has positive and negative vegas at three maturities. The questioner notes that Black–Scholes vega already depends on time to expiry and asks why additional expiry weighting would be needed.
The responses explain that the proposed weighting assumes a particular volatility-surface move: shorter-term volatility changes are larger relative to longer-term changes, with sensitivity scaled by the inverse square root of time. Under that assumption, the adjusted vegas represent exposure to a time-weighted shift rather than a parallel change in implied volatility. Another response frames the scaling through total variance, which depends on volatility squared times expiry, but also questions whether that interpretation matches the industry definition of vega. The key caveat is that the appropriate aggregation depends on the assumed volatility shock; ordinary vega measures price sensitivity to a one-percentage-point volatility change at a given expiry.
Key ideas
- Adding raw vegas estimates exposure to a parallel volatility shift only when the shock is defined consistently across expiries.
- Time-weighted vega assumes volatility changes differ by expiry, with relative sensitivity scaled by inverse square-root time.
- Black–Scholes vega already incorporates an option’s time to expiry in its price sensitivity.
- The total-variance explanation implies a specific surface-shift assumption, not a universal vega convention.
- The portfolio’s volatility exposure depends on the shape of the assumed volatility shock.
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Full text
# Why do we need to calibrate vega?
# Why do we need to calibrate vega?
I was going through some paid video on options. The tutor in the video asked the following question:
Person $A$ has the following portfolio at the start of April
- Portfolio of options with vega $20,000$ expiring end of April. Portfolio of options with vega $-40,000$ expiring end of May. Portfolio of options with vega $15,000$ expiring end of June.
Now if the monthly implied volatility increases from $\sigma$ % to $(\sigma+1)$%, is it good for person $A$, what is his exposure.
The naive approach is to add all vega's to get $-5,000$ and say with increase in volatility he makes a loss. The tutor goes on to explain that this approach is not correct and one needs to calibrate vegas as time of expiry is different. He says one can add $20,000 + (-40,000/(\sqrt{2})) + (15,000/\sqrt{3})$.
My doubt is why is the naive approach wrong. Vega means change in options price with $1$% change in implied volatility. Doesn't vega (if obtained from pricing models like Black Scholes) itself incorporate the time to expiry factor ? Would it be wrong to say portfolio of second month changes by $-40,000*\sqrt{252}$ ( taking annualized volatility).
PS : I know I am missing something. Being a beginner please excuse me if I used any wrong terms.
## Answer by LocalVolatility (score 4, accepted)
https://quant.stackexchange.com/a/41815
It seems like he is assuming that the shorter term volatilities change more than the longer term ones and the relatively sensitivity is proportional to $1 / \sqrt{T}$. Thus, this hedge is not against a parallel shift of the surface. This is not an uncommon assumption and the corresponding vegas are often referred to as "time weighted vegas".
## Answer by Kiann (score 2)
https://quant.stackexchange.com/a/41853
Your tutor is calculating the increase in total variance. The black-scholes model has the variance term of sigma^2 * Time-to-expiry.
Hence, when the monthly volatility increases by 1%, the effective increase for the 3mth option is sqrt(3) * 1%, the 2mth option is sqrt(2) * 1% etc. He explicitly assumes the vega is relative to the total variance - i.e. the vega is due to an increase in the sqrt(variance = sigma^2 * T).
I personally don't think he is doing this correctly, as the industry standard of defining vega, is literally the change in price due to a change in % volatility.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.