# Option pricing model: Black-Scholes-Merton model

Suppose, that a stock is priced at \$ 400 and a volatility of 0.39. I buy a call option with an exercise price of \$ 400 that expires in 3 months. The risk-free rate is 8%.

Now, The theoretical value of this call is \$ 35.039. Suppose that actual call is selling for \$ 28.81. What would be my strategy but don't worry about hedging the risk. Simply buy or sell 100 calls.

After purchasing the call, I investigate my possible profits. I expect to unwind the position two months later, at which time I expect the call to have converged to its Black-Scholes-Merton value. Of course I don't know what the stock price will be, but I can compute the profits for stock prices over a reasonable range. I expect that the stock will not vary beyond \$ 350 and \$ 450.

Now, how can I determine my profits in increments of \$10 of the stock price?

## 1 Answer

First off, a correction. If you buy 100 calls, at a later date you don't purchase or sell 'the call'.

Next, there's the issue of your historical volatility of .39 versus the reality that your call's implied volatility is .311 . Therefore, this is a hypothetical problem that ASSUMES that implied volatility will revert to historical volatility. If the appropriate action wouldn't be "Simply buy or sell 100 calls." It's a good deal for the buyer, buying undervalued calls (\$6+ discount) but that's hardly the case for the seller.

In order answer your question, you simply need to reduce the time input to one month remaining and calculate the BS value at \$10 intervals from \$350 to \$450 which is your expected range of stock price. FWIW, there are spreadsheet as well as graphical programs that do this.

All of this assumes that the carry cost remains constant and that there is no dividend.

• Would you tell me which spreadsheets and graphical programmes will serve my purpose? Commented Dec 30, 2021 at 16:48
• Your edit of your your questions mucks things up. You inputted 30 days remaining until expiration and calculated what the IV will be at those respective prices (2.65 and 56). What does that prove? To answer your question, if the call has a theoretical value of \$35.04 with an IV 0.39 and it's presently trading at \$28.81 (which is \$6+ less than theoretical), the current IV is lower. If you plug in the numbers, it's .311 Commented Dec 30, 2021 at 16:54
• OK, you've deleted the comment that I replied to and you've replaced it with another one. Excel has templates for determining BS numbers. I have no clue what's out there that's beyond the model's formula but if you can't find it, you can incorporate the formula into a spreadsheet that determines these numbers. Can't help you on the software program side because I use an old non commercial program. OptionVue has been one of the leading option trading programs. Not an inexpensive program. See blurb below Commented Dec 30, 2021 at 17:01
• From the developer: "OptionVue is a program that allows you to analyze option trades of any complexity on any optionable assets including Stocks, Indices and Futures. OptionVue 's Graphic Analysis can show you how your trade will be valued today, tomorrow or at any day up to expiration at any price or volatility. It calculates the break-even points, expected return and even your probability of profit." Commented Dec 30, 2021 at 17:01