Pricing Options with Truncated Daily Underlying Returns

Summary

The document considers pricing a European option when the underlying asset’s daily return is constrained to a fixed interval. It proposes replacing the usual return density with a two-sided truncated normal risk-neutral density, then valuing the option by discounting the expected payoff under that distribution. Because the payoff expectation generally does not have a convenient closed form under the proposed density, the answer suggests numerical integration.

The volatility parameter is to be calibrated from market option prices. This is a brief modeling suggestion rather than a full pricing framework: it does not explain how to define the return constraint across the option’s life, derive or validate the risk-neutral distribution, or handle consistency with no-arbitrage conditions. Its usefulness depends on whether the truncated distribution and the stated limit are appropriate for the market and contract being modeled.

Key ideas

• A bounded daily-return assumption can be represented with a two-sided truncated normal density.
• Option value is computed as the discounted expected payoff under the proposed risk-neutral distribution.
• Numerical integration may be needed to evaluate the payoff expectation.
• The volatility parameter is calibrated from market prices, while the distributional assumption remains unvalidated in the discussion.

Tags

# option pricing with limitation on the change of underlying daily changes


# option pricing with limitation on the change of underlying daily changes












how are we supposed to price an European option given the fact that the daily return of the underlying is limited within -X% to X%?

For example, if X = 5, the price of the underlying cannot go up 8% on any given day.

## Answer by emcor (score 1)

https://quant.stackexchange.com/a/14982

As Black Scholes model, you can assume a two-sided Truncated Normal Distribution as riskneutral density $f(x)$ (with $\mu=0,\sigma=T-t$) for the returns and then price the option payoff $H$ as usual by:

$$V_t^H=e^{-r(T-t)}E(H_T|F_t)=e^{-r(T-t)}\int_{-X}^{X}H(S_te^{r(T-t)+\sigma W_{T-t}})f(W_{T-t})dW_{T-t}$$

The integral must likely be calculated numerically. The $\sigma$ parameter must be calibrated from market prices.

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.