Question

(a) Find a stock that pays a dividend and estimate the continuously compounded dividend payment rate (for example, .02). Using the Black/Scholes option pricing model (including dividends), estimate the price of an at the money call option and put option that have the same exercise price and maturity date. Assume r=.005 and use the appropriate S0, t, K. For volatility, use 30%.

(b) Evaluate how well the Black/Scholes model worked by comparing the results to the midpoints of the bid-ask prices.

(c) Find (through trial and error), the implied volatility (i.e., the volatility that equates the (i) midpoint call price and (ii) midpoint put price with the estimates from the Black/Scholes formula).

(d) Are the results of (c) the same? In theory should they be the same? Explain.

Answer 1

Part (a)

I have chosen the stock Accenture.

Data on the options of this stock has been retrieved from https://in.investing.com/equities/accenture-ltd-options.

Let's look at the options with strike price of $ 165. The relevant info have been cased in green color line. These options are expring on June 30th, 2020. Hence time to expiration T = 2.5 months

I have taken the dividend yield as 2.1% from the website: https://investor.accenture.com/stock-information/dividend-history

Part (b)

Snapshot from my model:

Please see the table below. The rows highlighted in yellow contain your answer. Figures in parenthesis, if any, mean negative values. All financials are in $. Adjacent cells in blue contain the formula in excel I have used to get the final output.

Part (c)

Comparison is shown below:

	Bid	Ask	Mid point	Calculated value
Call option	13.50	14.50	14.00	8.71
Put option	10.20	11.20	10.70	9.26

The calculated value is very close to the mid point in case of put option, however the call option has a higher deviation in the calclated price and mid point.

Part (d)

Please see the screenshot, where I did the goal seek to force the price of the options to be same as the mid point price. The implied volatilities are shown in the orange colored cell:

Hence,

(i) the implied volatility for the call option = 47.73%

(ii) and that for the put option = 34.82%

Part (d)

The results of the two are not same.

In theory, the results should have been the same. The volatility denotes the standard deviation in the value of the underlying, which remains the same in case of the two options. Hence, the implied volatility figures should have been the same for the call and the put option, theoretically.