Variance Risk Premium: Why Higher Implied Volatility Is Not Enough

Implied volatility can stand above historical volatility while a variance seller still has a negative expected payoff. The missing ingredient is a forecast for the same future window: how much variation might actually occur, with what probability? A second distinction matters just as much. Even a positive expected payoff does not prevent a large realized loss. An observed gap, a modelled risk premium and a cash settlement are three different accounts.

What the historical gap does—and does not—tell you

Suppose the past 20 daily log returns consist of ten pairs of +0.01 and −0.01. Their squared sum is 0.0020. Under the annualization convention used below, that is 252 variance points, equivalent to annualized realized volatility of 15.8745%. Now suppose a new, hypothetical variance exposure can be entered at an agreed variance level of 400, equivalent to 20% volatility.

The subtraction 400 − 252 produces 148 variance points. Subtracting the volatility numbers instead produces 4.1255 percentage points. Neither number is a dollar profit, and the two differences are not interchangeable. More importantly, the 252 describes observations that have already happened; the 400 prices a period that has not yet happened.

Using the past window as the forecast for the next one is a possible modelling choice. Under that explicit choice, 148 would be a premium estimate. It is not a conclusion supplied by the chart itself. A coming event, a changed trading regime or a different estimate of extreme moves can alter the forecast without changing a single historical observation.

Define the future claim before estimating its premium

Here the positive-sign convention for the variance risk premium is the pricing expectation of future variance minus its physical, or statistical, expectation: E^Q[future RV] − E^P[future RV]. Q and P are different probability weightings, not bullish and bearish scenarios. The physical expectation is unobservable and has to be estimated.

For a suitably specified, ideal zero-initial-value variance claim, its fair strike can represent the pricing expectation. An actual executable quote may require adjustments for contract terms, funding, credit and transaction costs. A single at-the-money option’s implied volatility, or a volatility index squared, is not automatically the executable strike of that claim.

Our original example starts before all 20 return observations. There are 21 daily prices, with decimal log returns r_i = ln(P_i/P_i−1). The agreed calculation is:

RV in variance points = 10,000 × (252/20) × Σr_i²

There is no subtraction of the sample mean and no denominator of 19. Those belong to different estimators. Set the strike K to 400 and the aggregate exposure to USD 100 per variance point. With zero interest, no credit adjustment and costs initially excluded, the seller’s terminal gross P&L is 100 × (400 − RV); the buyer has the opposite result. The USD 100 is not invested principal or a claim about any listed contract’s multiplier, so it cannot serve as the denominator for an investment return.

The price finishes unchanged, but the variance seller does not

Consider two hypothetical future paths, both starting at 100 and ending at 100. Path A repeats ten pairs of +0.01 and −0.01 log returns. Path B repeats nine such pairs, then finishes with −0.08 and +0.08. Both are back at 100 on day 18. The last two moves in B are roughly −7.6884% and +8.3287% in ordinary percentage-return terms—not a conventional 8% fall followed by an 8% rise.

For B, the calculation is 10,000 × (252/20) × [18 × 0.01² + 2 × 0.08²] = 1,839.6 variance points. The complete rebound cancels the cumulative log return, not the squares already added to the variance account. Its seller therefore loses USD 143,960 gross, whereas A generates USD 14,800 gross.

This is why an unchanged underlying price does not establish that a short-variance position was safe. The claim pays on the specified path statistic, not solely on the final price. Equally, this example observes daily fixings; it does not capture every intraday move. A different sampling rule would define a different claim.

A probability assumption can reverse the expected result

Now make a separate, deliberately simple forecast for future variance: 252 points with probability 1 − q, and 1,839.6 with probability q. The two price paths demonstrate those variance outcomes; they are not a complete, calibrated, arbitrage-free model of the traded underlying. In particular, q is an assumed forecasting input, not a market-implied probability or a frequency inferred from the past 20 returns.

The forecast is 252 + 1,587.6q variance points. The strike less that forecast is 148 − 1,587.6q. Multiplying the latter by USD 100 gives the seller’s expected gross cash result. To make the cost question visible, also assume a fixed total USD 1,000 cost in either outcome; this is an illustrative cash adjustment, not an exchange fee schedule.

At q = 5%, the cash calculation can also be checked directly: 0.95 × 14,800 + 0.05 × (−143,960) = USD 6,862. This is the probability-weighted expectation, not a third realized path or promised payment. At q = 10%, expected gross P&L is already negative. The past observations and agreed strike have not changed. Only the future-variance forecast has.

The expectation must be taken in variance units. Taking a square root of forecast variance does not turn it into the expected value of volatility: a nonlinear square root cannot simply be moved through an expectation. Averaging two volatility figures and then squaring them would change this model.

Find the assumption that would invalidate the edge

Within this two-outcome model, expected gross P&L is positive only when q < 148/1,587.6 = 9.3222474%. At equality it is zero. The USD 1,000 cost consumes 10 variance points of the assumed USD 100-per-point exposure, moving the positive-net threshold to q < 138/1,587.6 = 8.6923658%.

Those are sensitivity boundaries, not market constants. Changing the high-variance outcome or executable strike changes both thresholds. Changing exposure relative to a fixed dollar cost changes the net threshold; scaling a positive exposure alone does not change the gross threshold. A forecast that only narrowly clears a boundary deserves a different interpretation from one that remains positive under a range of defensible stress assumptions.

Even the 5% forecast still allows the USD 143,960 gross loss. After the assumed cost, the two realized outcomes are +USD 13,800 and −USD 144,960. Positive expectation is compensation for taking uncertain outcomes, not a coupon promised to the seller. Conversely, one bad settlement does not by itself prove that the original expectation was negative; the forecast process and the realized draw must be assessed separately.

Check that the instrument can deliver the payoff you calculated

A short option, a short straddle and a dynamically delta-hedged option book do not inherit this linear variance formula. Directional exposure, gamma, rebalancing, execution costs and discrete hedging can all affect their P&L. A model premium in variance units is not yet an implementable strategy until the actual instrument and executable price are identified.

The window also has to match. This example begins before every observation; a contract that has already accumulated variance contains a known part and a remaining uncertain part. When comparing expiries, aligning the observation windows with forward variance is a separate necessary step. Missing fixings, final settlement prices and market interruptions must be handled under the actual contract, not by deleting inconvenient observations and changing the denominator.

Finally, terminal P&L is not the same as interim funding. Variation margin, collateral requirements or a costly early exit can require cash before the last observation. None is simulated by this zero-interest terminal example. A positive final expectation cannot demonstrate that the position is affordable through a stress path.

Five checks before calling the spread a selling edge

1. Identify the claim. Record the actual instrument, observation dates, price fixings and settlement rule. Confirm whether all returns are still in the future.
2. Reconcile the units. Use the same log-return definition, annualization and variance-point convention. Separate strike, exposure, dollars and return on capital.
3. Write down the forecast. Keep the historical observation beside, not inside, the future estimate. Preserve the data available at the decision time and explain the assumed tail outcomes or probabilities.
4. Translate into executable cash. Use the available quote side, intended size and relevant costs. Recalculate the expectation under probabilities and stress amplitudes that would overturn the conclusion; check funding separately.
5. Close the loop. At settlement, reproduce the realized variance and cash ledger. Test forecasts on subsequent, out-of-sample windows without replacing the original forecast with hindsight. Distinguish forecasting errors from execution costs and contract mismatches.

The useful question is not simply whether implied volatility looks expensive against the last month. It is whether a clearly defined future-variance forecast, an executable claim and a survivable cash path jointly support the expected compensation being claimed.