Estimate the value of a Google (GOOG) call option using the Black-Scholes option pricing model:

(a) Implement the Black-Scholes model for three Google call options that expire on December 16, 2016. You can use either options on A shares of Google (ticker: GOOGL) or options on C shares of Google (ticker: GOOG). The three options to be considered should have the same expiration day but different strike prices such that one call is in the money, one is out of the money, and one is near the money. Near-the-money options (also referred to as nearest?the-money options sometime) refer to options whose strike prices are close to the spot price of the underlying stock and that are thus always either slightly in the money or slightly out of the money.

You should estimate the value of the three options on the same day (after the equity and options markets are closed) in order to use the same inputs (except for the strike) to the Black-Scholes model. This would allow you to see the impact of the strike on the option price.

The spreadsheet ?example-BlackSchole-for-Project2.xls? includes a simple example on the implementation of the Black-Scholes (BS) model. Among other things, the example illustrates how to obtain the risk-free interest and the historical volatility of the underlying stock, two of the inputs to the BS model. You can use ?example-BlackSchole-for-Project2.xls? as a template for your own analysis if you want.

(b) Compare the Black-Scholes model price of the three call options to their corresponding actual market price (the closing price on the day of your analysis) on Yahoo or other websites, and calculate the pricing error of your model for each of the three options. You can use the mid quote (the average of the bid and ask) as the market price for a given option. Pricing error = model price ? actual price; Percentage pricing error = (model price ? actual price) / actual price.

Things to turn in:

One spreadsheet printout that includes, for each of the three options, the implementation of the BS model, the inputs to the model, the pricing error, and brief comments on the performance of the model (e.g., whether the model prices the near-themoney option reasonably well).

Fin 406 - Fall 2016 Professor Jingzhi (Jay) Huang 350 Business Building Pricing the AAPL May 2011 Options using the Black-Scholes model Inputs: today (t=0) expiration date (T) S (t=0) r (risk-free) d (divident yield) s (volatility) 4/14/2011 5/21/2011 332.42 (the date when the valuation is done) (for May options) (closing price of the AAPL stock on 4/14/2011) 0.0005 0 0.3059 (proxied by the 3-month Treasury yield) (Apple does not pay dividends) (AAPL's stock return volatility) 0.1014 (time to expiration) In-the-money 320 0.4402 0.3428 0.6701 0.6341 near the money 330 0.1242 0.0268 0.5494 0.5107 out-of-the-money 340 -0.1823 -0.2797 0.4277 0.3899 19.845 0.6701 14.119 0.5494 9.623 0.4277 Outputs: T-t (yr) X (strike) d1 d2 N(d1) N(d2) BS model call price BS model call delta Bid and ask are from Yahoo ITM NTM OTM Strike price 320 330 340 Black-Scholes Option price 19.845 14.119 9.623 Observed Option bid and ask prices bid ask mid quote 20.00 20.35 20.18 14.00 14.20 14.10 9.10 9.30 9.20 Pricing Errors (Model--Actual) Model--Actual -0.33 0.02 0.42 Conclusion: The BS model prices the near the money AAPL call reasonably well, but underestimates the ITM call and significantly overestimates the OTM call The 3-month Treasury yield is available on Bloomberg or Google Finance. https://www.google.com/finance https://www.bloomberg.com/markets/rates-bonds/government-bonds/us Instructions on how to get this number: (1) Download AAPL's weekly prices over the past 6 months; (2) Calculate APPL's weekly returns; (3) Estimate APPL's return std. dev. using weekly returns obtained in Step 2; (4) Annualize the std. dev. obtained in Step 3. The resulting number is one estimate of AAPL's stock return volatility. (Please refer to Question 2 of Project 1 on how to estimate a stock's return volatility.) ricing Errors (Model--Actual) Actual -1.63% 0.13% 4.60%