Negative Calibrated Call Prices Can Signal Numerical Integration Error

Summary

The document addresses negative prices produced for low-priced, out-of-the-money calls while calibrating a stochastic-volatility model. Its central explanation is numerical: if option prices are computed by numerical integration, approximation error can overwhelm a very small true price and produce an invalid negative estimate. The suggested practical response is to exclude or truncate the far wings of the implied-volatility surface at an appropriate moneyness when the numerical estimates there are unreliable.

A second answer grounds the issue in the payoff at expiration: a call pays the greater of zero and the difference between the underlying price and strike, so its payoff cannot be negative. Under standard pricing assumptions, the present value of a nonnegative payoff is also nonnegative. Thus a negative model output is a warning about calculation or implementation, rather than a meaningful negative call value. The exchange provides no diagnostic procedure, integration settings, or universal cutoff rule; the proposed wing truncation is a pragmatic suggestion, and the suitable boundary depends on the model and numerical accuracy required.

Key ideas

• A call’s expiration payoff is nonnegative, so a negative theoretical price is not economically valid under standard assumptions.
• Numerical integration error can dominate when the true option price is very small.
• Negative model prices should prompt checks of numerical methods and implementation.
• Truncating unreliable far wings may help when building an implied-volatility surface.
• The document gives no universal moneyness cutoff or detailed numerical fix.

Tags

# calibration - negative call price


# calibration - negative call price












Im trying to calibrate a stochastic volatility model to market. I end with an MSE of 2-3 with approximately 500 quotes. Some out of the money options with call-price under 1 dollar ends up being negative. I dont know how to plot the implied volatility surface of the model if some of the prices is negative. Any help is appreciated.

## Answer by JohnDoe (score 1, accepted)

https://quant.stackexchange.com/a/50282

It's obviously no calibration problem. It's just a numerical issue. The error resulting from solving the integral numerically is just to big for your really small option price.

I would suggest to cut the wings of your volatility surface at an appropriate moneyness.

## Answer by A. Angeli (score 0)

https://quant.stackexchange.com/a/50278

There must me something going wrong since the price of a call is non-negative. This is easy to see from the payoff function, $C$ on an asset, $S$, and strike price $K$.

$$C(S) = \max(0, S - K) \geq 0$$

From the formula we can see that a call at expiry would payout non-negative values, so the price could never fall bellow zero.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.