Black Scholes Pricing Model

Earlier in the semester we looked briefly at the Black Schoels pricing model and how it is used for pricing options. We also discussed this model in my finance class, so I thought it would be interesting to look at it in a bit more depth and do some actual calculations to see exactly how different options are priced. For example, it is very interesting to see how much the price of an option can change simply due to a change in maturity, volatility, or price of the stock.

I created a fairly simple Excel spreadsheet that computes the prices of puts and calls based on changes in different variables. It is able to change the variables: standard deviation, risk free rate, exercise price, price until maturity, stock price, and dividend yield. The blog does not have the capability to attach files, so here are a few screenshots of the Excel sheet and the prices that resulted. I used the respective formulas for determining the price of puts and calls for this model and input them based on the set of factors used in the equations. I think this is a very helpful exercise to do because it showed me how easily the prices of these options are affected. For example, by only changing the time to maturity on a stock, from 1 year to 5 years, the call price increased from under $2 to almost $7.

Additionally, as part of this spreadsheet I priced the option using both the Black Schoeles model as well as using the Put-Call Parity, I thought this was very helpful because it directly showed that these are always equal.

The following depict the price changes due to a change in maturity (all other factors held constant):

The following depict the price changes due to a change in volatility (all other factors held constant):



This shows the large change in options pricing that can result from an increase in volatility. For example, as seen in the images above, from an increase in the standard deviation (from 20% to 70%), the price of a call increased from $2.77 to $13.59, and the price of a put increased from $7.03 to $17.85.

Overall, I think the Black Schoels pricing model is a great way to view the pricing of calls and puts and how changing different variables affects these prices. However, in using this model, individuals definitely need to be careful, as it does have a large number of assumptions within its formulas. For example, in order to use this model the risk free rate must be constant, there must be no transaction costs, the stock does not pay a dividend, are securities are perfectly divisible, and there are no restrictions on short selling (among other things). But, I do feel like this model is a great way to see the interaction between all the different variables in pricing options.

Black-Schoels Model – Wikipedia

Info and Equations for the Black Schoels Model