Calculating Time to Maturity Discussion Paper
With regard to calculations of the maturity dates of the options, bear in mind that the XEO and OEX option contracts at CBOE mature on the Saturday following the third Friday of the maturity/expiration month of each contract. (Please check the full contract specifications are available under the ‘Products’ tab in the CBOE website,
Interest Rate Data
You also need an annualised risk-free rate to calculate ‘theoretical’ fair prices for the options using the Black and Scholes and the Binomial pricing models. Use the ‘interest rates’ market data provided by Bloomberg ( https://www.bloomberg.com/markets/rates-bonds/government-bonds/us ). Select an appropriate yield for the risk free interest rate. This usually is the yield of a Treasury Bill (or zero curve) that has a maturity closest to that of the option(s) you downloaded. You are advised to have the same maturity for all your options, but note that if you download options with different maturities you may need more than one interest rate to match. If the maturity of your options is in-between the maturities of two listed Treasury Bills, or has a maturity less than that of the shortest Treasury Bill, then you may need to ‘interpolate’ or ‘extrapolate’ the yield to end up with one that matches the maturity of the options.
Alternative source of interest rate data:
https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=billrates. Use the ‘Coupon Equivalent’ rates, and read the comments at the bottom of the table.
Dividends and Volatility and the proper calculation of theoretical option prices
Note that in calculating theoretical option prices by the Black and Scholes as well as the Binomial Model, you will also need a value for the annual dividend yield on the S&P100 index and an estimate of the annual volatility of the index. Read the relevant sections below on how to obtain these.
Estimating Annualised Dividend Yield (Downloading index dividend yield data)
To calculate theoretical option prices you also need an estimate of the annualised dividend yield of the S&P100 index at the time of downloading the price data. Search for a reasonable value, and although this can be difficult, try http://etfdb.com/index/sp-100-index/dividends/ which gives the dividend yield on an Exchange Traded Fund (ETF) that tracks the S&P100.
An alternative source of the dividend yield is to try:
Estimating Volatility of the Index (Downloading index volatility data)
An estimate of sigma (volatility) for the index can be obtained by reading the value of the volatility index that has a symbol ^VXO but this has been decommissioned. Instead, Enter ^VXN on the ‘Search’ button in the Quote Dashboard in cboe (please also see the text in bold typeset below). Note, this must be obtained within the same 15-minute interval during which you download the option data.
CBoE has decommissioned VXO for the S&P100, so use ^VXN as an alternative symbol. It is a volatility index on the NASDAQ 100 stock index which is not the same as the S&P100 but close.
Enriching Your Analysis?
Bid and Ask Prices
As a base case, do your analysis using mid quotes = (bid+ask)/2. Or you can enrich your analysis by performing calculations on bid and on ask prices separately. The difference between the results using the bid prices from those using the ask prices should be a reflection of the effect of ‘transaction costs,’ and hence you can discuss these effects.
You can also enrich your analysis further by redoing the calculations for options with different maturities. In this case you may choose three or four sets of options, with each set having a different maturity (in this sentence a ‘set’ refers to all the options in the matrix mentioned above). The idea is that you may see patterns across maturities as well as exercise prices, and by discussing these patterns across maturities and exercise prices your analysis will be enriched.
Step 2 Course Requirements
Using the VXO estimate of volatility as input evaluate the call and put options using the three-step binomial tree previously constructed for the index level (here you need to have your tree calculations automated so you can evaluate all options). Compare the binomial option prices with the actual option prices observed in the market and discuss the reasons why you do, or do not, observe any differences. (More emphasis and marks will be given to discussion.)
Black and Scholes versus Binomial
Using the VXO estimate of volatility as input calculate the Black-Scholes prices of the put and call options. Compare these values with the actual market prices and with those obtained by the Binomial model. Discuss possible reasons for any differences (i.e., compare Black-Scholes versus Binomial, American versus European, puts versus calls, ATM versus OTM). (More emphasis and marks will be given to the discussion of each of these comparisons)
By trial and error, find the value of the volatility parameter at which the Black and Scholes price equals the observed actual market price for each option. The value of volatility at which the observed actual market price equals the Black and Scholes price is known as ‘implied volatility’. Create plots of the implied volatility of the options against the exercise prices of these options. Compare these values of implied volatility with each other. Also compare them with the value obtained from the VXO or VXN index. Discuss the reasons for any differences from each other and from the VXO or VXN value. (More emphasis and marks will be given to discussion.)
Guide to Step 2 Requirements
Binomial Model Setup Features
Using the ‘binomial model’, build a binomial tree for the index level with three time steps, so that the overall time horizon is equal to the maturity of the options selected (i.e., divide the time remaining to maturity into three equal intervals).
With regard to the binomial calculations choose the upward and downward size of price movement as a function of the volatility of the index level (i.e., function of sigma of the index). You can use the equations provided by Chance and Brooks for up (u) and down (d) parameter movements as functions of sigma, also provided in the lecture material. For estimates of sigma see Estimating Volatility of the Index above. You will also need an estimate of the dividend yield on the index. For these see Estimating Annualised Dividend Yield above.
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