Black-Scholes Model for American OptionsThis model uses Black's Approximation to price American Options. Black's Approximation is an extension of the traditional Black-Scholes model that allows the price of American Options to be approximated within the Black-Scholes Framework. This is necessary because the traditional Black-Scholes model only works on options that are exercised at expiry, not before; like American Options can be.
Black's Approximation approximates the value of an American option by:
1st. Calculating the theoretical price of a european call or put based on the strike price (K), spot price (S), annual return (sigma), time until expiry (T), times until the next 2 ex-dividend dates (t1 & t2), and the dividend paid out at times t1 and t2 (D1 and D2).
2nd. The theoretical price of an option expiring on the second ex-dividend date (t2) is calculated. This replicates exercising the option early.
3rd. Finally, the highest price of the two theoretical prices calculated in steps 1 & 2 is chosen as the approximated price.
How to use this:
1st. Input your strike price.
2nd. Input the risk-free-rate of the currency the option is based in.
3rd. Input the dividend yield for the next ex-dividend date. For example AAPL's dividend yield is 0.82 and will be paid out on August 7,2020.
4th. Input the time until the next ex-dividend date. For example AAPL's next ex-dividend date is August 7,2020, which is 61 days away. So you'd input 61 (this includes weekends and holidays).
5th. Input the dividend yield for the ex-dividend date after the next one. For example AAPL's dividend yield after the next one is 0.82 and will be paid out on November 6, 2020.
6th. Input the time until the next furthest ex-dividend date. For example AAPL's next ex-dividend date after Aug 7th, is on November 6, 2020, which is 152 days away. So you'd input 152 (this includes weekends and holidays).
7th. Input your time until expiry. You can do so in terms of days, hours, and minutes.
8th. Input your chart time-frame in term of minutes. For example, if you're using the 1 min time-frame enter 1, 4hr time-frame enter 480, daily time-frame enter 1440.
9th. Lastly, pick what type of option you want data for: Long Call or Long Put.
*Disclaimer, because Black's Approximation is mostly geared towards stocks, this will only work for stocks. Also, the time variables: time until expiry and time until the ex-dividend dates; don't automatically update. So you will have to update them each day.
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Binomial Option Pricing ModelA binomial option pricing model is an option pricing model that calculates an option's price using binomial trees. The BOPM method of calculating option prices is different from the Black-Scholes Model because it provides more flexibility in the type of options you want to price. The BOPM, unlike the BS model typically used for European style options, allows you to price options which have the ability to exercise early, such as American or Bermudan options. Although you can use the BOPM for any option style.
This specific model allows you to price both American and European vanilla options.
The way the BOPM calculates option prices is by:
First, dividing up the time until expiry into equal parts called steps. This specific model presented only uses 2 steps. For example, say you have an option with an expiry of 60 days, and your binomial tree has only two steps. Then each step will contain 30 days.
Second, the model will project the expected price of the underlying at the end of each step, called a node. The expected price is calculated by using the underlying's volatility and projecting what the price of the underlying would be if it were to rise and fall. This step is repeated until the terminal node, aka the end of the tree, is reached.
Third, once the terminal node's expected underlying prices are calculated, their expected option prices must be calculated.
Finally, after calculating the terminal option prices, backwards induction must be used to calculate the option prices at the previous nodes, until you reach Node 0, aka the current option price.
In order to use this model:
1st. Enter your option's strike price.
2nd. Enter the risk-free-rate of the currency the option is based in.
3rd. Enter the dividend yield of the underlying if it's a stock, or the foreign risk-free-rate if it's an FX option.
*For example, if you were trading an AAPL stock option, in the risk-free-rate box mentioned in step 2, you would enter the US risk-free-rate because AAPL options are traded in US dollars. In the dividend yield box mentioned in step 3, you would enter the stock's dividend yield, which for AAPL is 0.82.
*If you were, for example, trading an option on the EUR/JPY currency pair, the risk-free-rate mentioned in step 2, would be the Japanese risk-free-rate. Then in the the dividend yield box from step 3, you'd input the Eurozone risk-free-rate.
*If you were trading an options on futures contract, the risk-free-rate mentioned in step 2, would be the risk-free-rate for whatever currency the futures contract is denominated in. For example EUR futures are denominated in USD, so you would input the US risk-free-rate. Meanwhile, something like FTSE futures are denominated in GBP, so you would input the British risk-free-rate. As for the dividend yield box mentioned in step 3, for all options on futures, enter 0.
4th. Pick what type of underlying the option is based on: stock, FX, or futures.
5th. Pick the style of option: American or European.
6th. Pick the type of option: Long Call or Long Put.
7th. Input your time until expiry. You can express this in terms of days, hours, and minutes.
8th. Lastly, input your chart time-frame in term of minutes. For example, if you're using the 1 min time-frame enter 1, 4hr time-frame enter 480, daily time-frame enter 1440.
*Disclaimer, because this particular model only uses 2 steps, it won't work on stocks with high prices (over $100). If you want to use this on stocks with prices greater than $100, you would need to add more steps to the code, shown below. The model in its current form should work for stocks below $100.
