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

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. 

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)​

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). 

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

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. 

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?

To perform Monte-Carlo simulations and use it to calculate expected price at a given future date.

Monte-Carlo simulations are basically simulations of probability. As per investopedia:

"Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. "

Basically, given a set of parameters and some random variables, we perform thousands of simulations and then seek expected outcome.

In our case, we will make use of Gaussian Process that provides us with the random variable and then use it along with Asset Volatility, Risk-free rate, and time duration to calculate the expected price at a given future date.

This continuous-time stochastic process represents the Brownian motion (aka the Wiener Process). 
​
Primary use of this Gaussian based Monte-Carlo simulations is to generate asset prices is used to calculate American Option prices, which we will go over in another blog. ​

Below is the Python library to generate value from Gaussian distribution.

In order to successfully apply Monte-Carlo simulations on this Gaussian process, we need to get below data:

• Asset Volatility ( sigma )
• Spot Price ( S )
• Risk Free Rate ( Rf )
• Time to maturity (in years) ( T-t )

Asset volatility can be calculated from historical asset prices; hence, we need to download historical prices from one of the web-sources. Risk-free rate in our case will be a 3-month Treasury Rate that can be fetched from FRED database using Quandl API.  

Panda library has a useful function to download these prices from Morningstar, Quandl, Yahoo, etc. Downloads from Yahoo are having some issue recently; hence, in our case we will use combination of Quandl and Morningstar. 

Now that we have historical asset prices, we will go ahead and calculate the asset volatility. 

We do this by first calculating the Log returns for each day. Once, we have log returns for each day, we calculate the standard deviation using the ever helpful Pandas' std function. Note, this will provide us with daily standard deviation; hence, we need to annualize it by multiplying it with 252 (i.e. expected number of trading days in an year). Once we have annualized standard deviation, we take square root of it to calculate annualized volatility. 

Below code reflects how easy it is in Pandas:

We now have all the required inputs and are ready to apply Gaussian Process to calculate expected asset price using Monte-Carlo simulations.  For each simulation, we calculate expected price using below equation:

                   ST = St * exp(  (Rf− 0.5*σ^2)(T−t) + σϵ√(T−t) )
where,
          ST = Expected asset price at time 
          St  = Current asset price at time t (aka Spot price)
          Rf = Risk free rate or 3-month treasury rate
          σ  = Annualized Volatility
      (T−t) = Time to maturity in years
          ϵ  = Random variable based on Gaussian Distribution

Once, we get expected stock price using above equation, we repeat this calculation for N number of simulations. The only variable changing in each simulation is the Gaussian process; thereby, we representing a continuous time stochastic process, i.e. Wiener process. 

Below Python code performs these Monte-Carlo simulations: