Here’s a copy of the spreadsheet that I created during the section 1 (11-12:15) meeting of Finance 4335 today (you can bring up the spreadsheet by clicking on the screenshot below):In order to calculate the arbitrage-free price of a call option, we need to solve the following equation:

,

where and . The arbitrage-free price for the put may be obtained by applying the put-call parity equation. By doing so, we obtain the put pricing equation for an otherwise identical (same underlying, same exercise price, same time to expiration) put:

,

where and .

For calls and puts, we need five parameter values: *S* (current underlying asset price), *K* (exercise price, (volatility of underlying asset return), *T* (time to expiration, measured in number of years), and *r* (the annualized riskless rate of interest). These parameters all show up in cells B1:B5. The first step to solving call and put prices requires determining the values for and ; these values are listed in cells B7:B12. Once we obtain this information, it’s simply a matter of coding the equations for the BSM call and put prices; these prices show up in cells E1:E2.

Finally, I also include the components of the replicating portfolios for the call and put options. Of course, the values of these portfolios *must* be the same as the call and put option values; otherwise, there would be riskless arbitrage opportunities. As we showed way back during class on Tuesday, October 23, one replicates a call option by buying delta units of the underlying asset on margin, whereas one replicates a put option by shorting delta units of the underlying asset and lending money. In the Black-Scholes-Merton pricing model, the call delta corresponds to , whereas the put delta corresponds to .