Validating Monte Carlo Prices for Spread Options

Summary

The document presents a Monte Carlo approach to pricing a European call on the difference between two asset prices, then asks how to validate the implementation and reduce simulation error. The response recommends numerical integration under a correlated lognormal model as a comparison and points to Margrabe’s formula as an exact diagnostic when the strike is zero. Reported simulation and integration outputs are close in the examples, and the zero-strike result aligns with the analytical formula; these checks provide evidence for the implementation under the stated assumptions.

The proposed control variate is not developed or assessed in the answers. The numerical method relies on the specified volatility, correlation, rate, and lognormal dynamics, and the integration uses finite bounds. The example therefore checks a particular model setup; it does not establish accuracy for other processes or market conditions, nor does it supply a general error analysis.

Key ideas

• Monte Carlo estimates for spread calls can be checked against numerical integration under the same model.
• Margrabe’s formula provides an analytical benchmark for the zero-strike case.
• The example models correlated asset prices with lognormal dynamics.
• Agreement between estimates and benchmarks supports the implementation for the illustrated assumptions.
• The answers do not evaluate the proposed control variate.

Tags

# Selling an American call option early


# Selling an American call option early












I understand it is never optimal to exercise an American call option early. [1] [2] However, here are my two contradictory thoughts about selling an American call option early.

Assumptions

- I can only buy or sell a call option, never exercise it.

- I am continuously bullish on the underlying stock.

Contradictory thoughts

- The probability of touching is twice the probability of expiring in the money. [3] This implies that the call option is twice as likely to meet a profit target prior to expiration than at expiration. Thus, it is more profitable to sell early.

- On the other hand, because I am continuously bullish on the underlying stock, it would make sense to wait for the stock to appreciate. Thus, it is more profitable to sell the call at the last minute.

Thus, is it ever optimal to sell an American call option early?

## Answer by emcor (score 1, accepted)

https://quant.stackexchange.com/a/14183

If the option is fairly priced, as under the Black-Scholes Model, one cannot gain by selling it early, because the money you get just reflects the fairly expected value.

If you think you have some inside-information to be bullish on the underlying, you would not sell it early, but this is then just your assumption, so as you may aswell assume to be bearish.

The notion that the probability of touching is twice the probability to expire in the money, does not take into account the actual (expected) payoff received: The call provides unlimited upside potential until maturity.

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.