Option Pricing (Part 1) - American

Purpose

To apply Monte-Carlo simulations and calculate price of American style options.

Use of Monte-Carlo Simulations

In order to calculate Option prices, we need to calculate the expected price at Expiry date of the underlying asset. One of the ways to do this is by running Monte-Carlo simulations as I had pointed out in my previous blog. 

Calculate Expected Payoffs

Once we have expected prices calculated, next step is to calculate the Option payoffs for those prices. 

• For Call Options, the expected payoff is:
  • If Expected Price is less than Strike Price, then 0
  • Else (Expected Price - Strike Price)
• For Put Options, the expected payoff is:
  • If Expected Price is greater than Strike Price, then 0
  • Else (Strike Price - Expected Price)

 ​In mathematical terms, for each expected price we calculated using Monte-Carlo simulations, the option payoffs are as follows:

• Call Option Payoff = max(0,ExpectedPrice−Strike)
• Put Option Payoff = max(0,Strike-ExpectedPrice)​

Present Value of expected Payoffs

Now, we have the expected Options payoffs; so the next step is to calculate their Present Value using the discount factor dependent on risk-free rate and time to expiry (in years). 

Discount Factor 

We use the above discount factor to calculate the present-value of option payoffs returns through the Monte-Carlo simulations. 

                Ct = PV(E[max(0,PriceAtExpiry−Strike)])
                Pt = PV(E[max(0,Strike−PriceAtExpiry)])

Below code performs this exact calculation and returns the American Call and Put Option prices

Calculate Present Value of Expected Option Payoffs

Running for an extremely volatile asset - TSLA

We have everything setup, so lets start these simulations and calculations on TSLA stock to calculate prices for Call and Put option expiring Sept 7th, 2018 (i.e. 1 month from now) and with Strike price of $370. 

The output we receive is Call Option priced at $11.162 and Put option prices at $25.102. Given the current Spot price is $355.49 (as of EOD Aug 10th), this price somewhat looks fine; but let's confirm it with other sites. 

Output for TSLA Strike 370 expiring September 7th, 2018

As we can see from above Yahoo screenshots, our calculated option prices are pretty close to the ones found in Yahoo. I wonder if the recent news of Tesla going private adds additional volatility to the stock; causing this minor increase in both Call and Put option prices. What do you think?