# Finance: Black-Scholes Model

Questions:

1) On 10/15/2012, identify two call options for Apple (AAPL); one that is slightly in-the-money (ITM) and a second that is slightly out-of-the-money (OTM). Both options should expire on 12/21/2012.

a) Stock close price; b) slightly ITM call option close price; c) slightly OTM call close price.

On 10/15/2012, in addition to these three prices, we need to collect the 12/20/2012 Treasury bill quotes. We only need to collect the T-bill quote only once on 10/15/2012. Use "asked yield" of the T-bill as the risk free rate.

2) Calculate the implied volatilities of the ITM call using the data from 10/15/2012. Attached the Derivagem file)

3) Calculate Black-Scholes model prices for ITM call option for the entire sample period. Construct a table of date, stock price, risk free rate, time to expiration, volatility, strike price and model price. Use the volatility calculated on step 2. Note that the option expire on 12/21/2012.

4) Is the Black-Scholes model price statistically different from the market price? Conduct a paired t-test to determine whether the model prices are same or different from market prices.

5) Repeat question 2 to 4 for the OTM option.

6) Discuss how violations of the Black-Scholes assumptions may have affected your results.

PS. For the stock, use the close price.

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#### Solution Preview

See the attached file. Thanks.

Questions:

1) On 10/15/2012, identify two call options for Apple (AAPL); one that is slightly in-the-money (ITM) and a second that is slightly out-of-the-money (OTM). Both options should expire on 12/21/2012.

a) Stock close price; b) slightly ITM call option close price; c) slightly OTM call close price.

On 10/15/2012, in addition to these three prices, we need to collect the 12/20/2012 Treasury bill quotes. We only need to collect the T-bill quote only once on 10/15/2012. Use "asked yield" of the T-bill as the risk free rate.

The closing price for Apple stock on 15th October 2012 was $634.76. The closest ITM call option is a call with strike price of $630 and the closest OTM call options is a call with strike price of $635. The data is collected and presented in Excel file on sheet Data.

2) Calculate the implied volatilities of the ITM call using the data from 10/15/2012. Attached the Derivagem file)

Used following input values to calculate the implied volatility:

Underlying type: Equity

Stock price: $634.76

Risk free rate: 0.1170% per annum

Time to exercise: 47 Days or 0.1865 year (calculated in terms of trading days using 252 trading days in a year)

Exercise price: $630

Option type: ...

#### Solution Summary

The Black-Scholes Model is examined in finance. The expert idenfies two call options for Apple.