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## Option Pricing and Implied Volatility

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Preliminary Information

- This case study will focus on the determination of the measure of volatility used when applying the Black-Scholes option pricing formula.
- Historical stock price data and current market prices for selected call option contracts are available.

Preliminary Information

- Two approaches to determining the stock price volatility will be considered:
- estimation from historical stock prices
- implied volatility from market option prices

Preliminary Information

- Skills and background needed:
- standard deviation estimation
- a spreadsheet program with standard normal distribution cdf calculation capability
- Black-Scholes call option pricing formula, described in a report from an assistant

Background to the Problem

- Your employer is a company whose non-dividend paying stock trades actively on a major exchange.
- The company is considering offering a stock option purchase plan to its employees.

Background to the Problem

- The stock options may be regarded as a taxable benefit to the employees and as a deductible expense to the company and must be valued at fair market value.

Background to the Problem

- Currently traded call option contracts have a limited variety of strike prices and expiry dates and do not provide values for some combinations of strike prices and expiry dates being considered.

The Problem

- You are asked to develop valuations for call options on the company’s stock for a range of strike prices and expiry dates.

Information and Data

- The following information is available to you:
- a report from your assistant which summarizes the standard approach for pricing call options using the Black-Scholes option pricing model
- the daily closing price of your company’s stock for the past year up to today’s closing price (in spreadsheet form)

Information and Data

- today’s (Aug. 12, 1998) closing prices for call options currently being traded in the options market
- current Treasury Bill yields for 13 and 26 week T-Bills

Assistant’s Report

- All of the parameters in the formula are readily available except for . There are two approaches that can be taken to determine :
- it can be estimated from historical data
- it can be estimated from prices of options currently traded in the market (the implied volatility)

Today’s Market Information

- Today’s closing stock price - 18.625
- Closing prices on all currently listed call option contracts –

Strike Expiry Market Price

Price Date of Call Option

15 Aug. 21, 1998 3.875

17.5 Aug. 21, 1998 1.5

20 Aug. 21, 1998 0.375

20 Sept. 18, 1998 1

20 Oct. 16, 1998 1.5625

20 Jan. 15, 1999 2.75

22.5 Oct. 16, 1998 0.875

Today’s Market Information

- Today’s Treasury Bill rates

13-week - 5.103% (nominal)

26-week - 5.238% (nominal)

The SolutionEstimating From Historical Data

- From the spreadsheet of daily closing stock prices, we calculate the daily returns for the past year. The natural logs of successive daily returns are used to estimate from historical data. The estimate obtained must be scaled up to an annual measure. Estimates are made using a range of historical periods ending today. The STDEV function in EXCEL can be used.

The SolutionHistorical Estimates of

- Estimated Volatility (Standard Deviation)
- 30-day 0.753168
- 60-day 0.625336
- 90-day 0.603594
- 120-day 0.653869
- 180-day 0.649999
- 1-year 0.728998

The SolutionUsing The Black-Scholes Formula

- The quoted T-Bill rates are nominal rates that must be converted to annual continuously compounded rates.
- The time to maturity T is measured as a fraction of a year the number of trading days to expiry as a fraction of 252
- A spreadsheet function such as NORMDIST in EXCEL can be used for the normal distribution cdf.

The SolutionOption Prices Based on Historical Estimates of

- Exercise Price of 12
- Expiry Date Price
- 1 month 6.700
- 2 month 6.786
- 3 month 7.054
- 4 month 7.132
- 6 month 7.474
- 1 year 8.758

The SolutionOption Prices Based on Historical Estimates of

- Exercise Price of 15
- Expiry Date Price
- 1 month 3.981
- 2 month 4.191
- 3 month 4.738
- 4 month 4.892
- 6 month 5.426
- 1 year 7.163

The SolutionOption Prices Based on Historical Estimates of

- Exercise Price of 18.625
- Expiry Date Price
- 1 month 1.648
- 2 month 1.963
- 3 month 2.714
- 4 month 2.925
- 6 month 3.564
- 1 year 6.657

The SolutionOption Prices Based on Historical Estimates of

- Exercise Price of 22
- Expiry Date Price
- 1 month .578
- 2 month .837
- 3 month 1.531
- 4 month 1.741
- 6 month 2.400
- 1 year 4.555

The SolutionOption Prices Based on Historical Estimates of

- Exercise Price of 25
- Expiry Date Price
- 1 month .197
- 2 month .360
- 3 month .894
- 4 month 1.077
- 6 month 1.670
- 1 year 3.784

The SolutionEstimating As The Implied Volatility

- Using the Black-Scholes formula with the option price known from market data, it is possible to solve for if all other parameters are known. The solution is done by approximation. Trial and error in the spreadsheet, the bisection method or the Newton-Raphson method can be used.

The SolutionImplied Volatility Calculations

Strike Expiry Implied

15 21/8/98 1.193

17.5 21/8/98 .664

20 21/8/98 .695

20 18/9/98 .644

20 16/10/98 .654

20 15/1/99 .650

22.5 16/10/98 .890

The SolutionImplied Volatility Used to Price Options

- The implied volatility seems to be more closely related to the option strike price than the time to maturity. This illustrates the phenomenon of the “volatility smile” seen in market pricing of options. For the four option contracts with strike price of 20 we take the average volatility. Linear interpolation is used for strike prices between those of the market priced options.

The SolutionOption Prices Based on Implied Volatility

- Exercise Price of 12
- Expiry Date Price
- 1 month 7.537
- 2 month 8.530
- 3 month 9.344
- 4 month 10.036
- 6 month 11.169
- 1 year 13.463

The SolutionOption Prices Based on Implied Volatility

- Exercise Price of 15
- Expiry Date Price
- 1 month 4.587
- 2 month 5.427
- 3 month 6.103
- 4 month 6.679
- 6 month 7.644
- 1 year 9.736

The SolutionOption Prices Based on Implied Volatility

- Exercise Price of 18.625
- Expiry Date Price
- 1 month 1.454
- 2 month 2.073
- 3 month 2.552
- 4 month 2.959
- 6 month 3.645
- 1 year 5.192

The SolutionOption Prices Based on Implied Volatility

- Exercise Price of 22
- Expiry Date Price
- 1 month .741
- 2 month 1.453
- 3 month 2.040
- 4 month 2.550
- 6 month 3.419
- 1 year 5.383

The SolutionOption Prices Based on Implied Volatility

- Exercise Price of 25
- Expiry Date Price
- 1 month .697
- 2 month 1.568
- 3 month 2.317
- 4 month 2.976
- 6 month 4.104
- 1 year 6.623

Conclusions

- It appears that estimates of based on historical data may be less appropriate for use in the option pricing formula when the strike price is significantly different from the current stock price. On the other hand, implied volatility values become suspect when extrapolating beyond the range of strike prices currently being traded in the market. Correct volatility values are likely to lie somewhere between the two.

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