Volatility Term Structure, Option Hedging, and Theta P&L
Summary
The discussion examines whether an option’s quoted price captures the cost of hedging when implied volatility varies across the life of the trade. In an illustrative case, volatility is expected to be concentrated in one part of the option’s life. The answer argues that if the terminal distribution is modeled correctly and implied volatility is updated over time, the option price and delta can remain consistent even as annualized volatility changes. Theta calculated as if volatility were constant can then be misleading: changes in implied volatility contribute vega P&L that offsets the apparent time-decay discrepancy.
If a trader instead holds volatility constant in the model despite a different expected volatility path, the modeled terminal distribution, price, and delta may be wrong. The resulting P&L errors across periods are not guaranteed to cancel, since their size depends on the price path, especially the option’s proximity to its strike. This is a conceptual example, not a general proof or empirical study; conclusions rely on the assumed volatility path and how the position is marked and hedged.
Key ideas
- A correct terminal distribution can support consistent option prices and deltas while implied volatility changes over time.
- Theta computed under a constant-volatility assumption can omit P&L from changing implied volatility.
- Vega P&L can explain an apparent time-decay discrepancy when volatility is updated.
- Using the wrong terminal distribution can misstate option prices and hedge ratios.
- P&L errors from a constant-volatility assumption need not cancel because they depend on the price path.
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Full text
# Is price really the cost of hedging? # Is price really the cost of hedging? Assume a vanilla option with 1y expiry. The total vol in 1yr is 20 bps, the vol in first 6 months is 5 bps. The price is created by BS(20 bps). But is this price the correct cost of hedging? Will I not leak PnL in the first 6 months according to the difference between implied and realized vol, and leak in the opposite direction in the other 6 months? Is there a guarantee that the two will cancel out exactly, and how can we show it? If there is still a PnL leak after pricing the call correctly, then price must not capture the cost of hedging? ## Answer by dm63 (score 3) https://quant.stackexchange.com/a/79485 Let’s make the scenario even more extreme to make the point clear: assume that the 1yr implied vol is $\sigma$, and that zero vol is expected for the next 6 months. Also assume interest rates are zero for simplicity. Then I claim (1) the price of the option at any time is given by the terminal distribution (we don’t care how the vol is distributed over time) (2) during the first 6 months, the quantity $\sigma^2 (T-t)$ is stationary, so the annualized implied vol gradually creeps up as time goes on. (3) thus what happens is that, assuming you update the implied vol every day , your option price and delta are correct (depending on the terminal distribution only) BUT your time decay calculated at any constant vol is wrong. During the first 6 months , the daily time decay should be zero (proof : examine the BS formula when $\sigma^2 (T-t)$ is constant). The missing term is the Vega p/l when implied vol accretes slightly every day. Hence yes, if you naively calculate time decay you will experience positive leak over the first six months. Main point is that as long as you use the correct terminal distribution you will calculate correct deltas and will just have a theta miscalculation. Now, if you were to actually be even more naive and just use a constant vol for the whole period, you will be using the wrong terminal distribtion and therefore you will miscalculate the price and the delta. During the first six months your model shows a time decay which does not occur, so you will show a positive drift relative to that. In the second six months the actual time decay is faster than your model because the vol you are using is too low. But these errors of the first six months and the second six months are not guaranteed to offset because they are path dependent (error is largest when option is close to the strike). Bottom line : if you correctly update the vol daily, the errors offset but if you use a constant vol they don’t.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.