Estimating European Option Value from an American Option Price
Summary
The response outlines a two-model approach for estimating a European option value from an American option quote. First, use a Cox–Ross–Rubinstein binomial tree to find the volatility that makes the model price of the American option match its observed market price. The tree represents each subperiod with an up or down move, values the option at expiry from its exercise payoff, and works backward through the tree to obtain the initial price. Then use the resulting implied volatility in the Black–Scholes formula for a European option.
The method is an estimation recipe, not a direct conversion formula. It assumes that the inputs needed to price the European contract are known and that the American price can be matched by the chosen tree model. The document does not specify details such as dividends, rates, or contract exercise features for the conversion, and model or market-price differences can affect the estimate. Its midpoint quote convention is described as the market price target for the iterative volatility search.
Key ideas
- A CRR tree can price an American option under a chosen volatility assumption.
- Implied volatility is found by adjusting volatility until the modeled American price matches the quote.
- The calibrated volatility is then used in a European Black–Scholes valuation.
- The approach estimates a European value through models rather than converting prices directly.
- The result depends on model assumptions and the inputs used in both valuations.
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Full text
# Converting an American option to European option
# Converting an American option to European option
I wonder if there are any websites/resources/sample codes/papers on how to convert the American options to European options (when all else are equal). i.e. if given same underlying asset, same expiration date, same exercise price (but these are unknown). Now additionally we are given the price of an American option, are there resources/equations/formula/sample codes directly converting that American option price to European option price?
Hope you can help. Many thanks!
Best
## Answer by phdstudent (score 5)
https://quant.stackexchange.com/a/55195
You can use the standard black-scholes formula to price an european option. The only parameter you do not know to use the formula is the volatility.
If you have the price of an american option then you can use the Cox-Ross-Rubinstein (CRR) model to backout the implied volatility. Then just use black scholes.
The CRR model:
In the framework of the CRR model, the time between now and option expiration is divided into $N$ sub-periods. Over the course of each sub-period, the security price is assumed to move either “up” or “down”. The size of the security price move is determined by the implied volatility and the size of the sub-period. Specifically, the security price at the end of sub-period $i$ is given by one of the following: $$ S_{i+1}^{up} = S_i exp(\sigma \sqrt(h))$$ $$ S_{i+1}^{down} = S_i exp(-\sigma \sqrt(h))$$
where $h \equiv T/N$ is the size of the sub-period, and $S_i$ is the security price at the beginning of the sub-period.
To use the CRR approach to value an option, start at the current security price $S$ and build a “tree” of all the possible security prices at the end of each sub-period, under the assumption that the security price can move only either up or dow.
Next the option is priced at expiration by setting the option expiration value equal to the exercise value: $C = max(S−K,0)$ and $P = max(K−S,0)$. The option price at the beginning of each sub-period is determined by the option prices at the end of the sub-period, using the formula above. Working backwards, the calculated price of the option at time $i=0$ is the theoretical model price.
To compute the implied volatility of an option given its price, the model is run iteratively with new values of $\sigma$ until the model price of the option converges to its market price, defined as the midpoint of the option’s best closing bid and best closing offer prices. At this point, the final value of $\sigma$ is the option’s implied volatility.
Then just use that option implied volatility on the standard european black scholes model.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.