Friday, 21 December 2007

Put-Call Parity

If you have follow through my blog on Black-Scholes formula to derive the theoretical option price using Excel, you would have noticed I did it only for call option. I wanted to find how I can do it to find the pricing for put option too.

I was out last Friday to have dinner with my friends at City Hall. Since they were not there yet I decided to go to the MPH book shop. I was looking around and two books caught my attention. These two books namely, "Advanced modeling in finance using Excel and Vba" and "Option pricing models and volatility using Excel Vba", teach you how to perform a lot of text book written theories using Microsoft Excel. As I flipped through the books when I was waiting for my friends, I found the answer. It is so simple. It’s simply using the put-call parity. How could I forget about that? I knew I learnt that. As such, I decided to test out the theory and to my surprise, it works.

You may be wondering what is put-call parity? To put it in a very simple manner, it is a relationship between the price of a call option and a put option - both with the identical strike price and expiry. A derivation of the put-call parity is based on the payoffs of two portfolio combinations, a fiduciary call and a protective put.

A fiduciary call is a combination of a pure-discount, riskless bond that pays the exercise price X at maturity and a call with exercise price X. The payoff for a fiduciary call at expiration is X when the call is out of the money, and X + (S - X) = S when the call is in the money. S is the underlying share price of the stock.

A protective put is a share of stock together with a put option on the stock. The expiration date payoff for a protective put is (X - S) + S = X when the put is in the money, and S when the put is out of the money.

When the put is in the money, the call is out of the money, and both portfolios pay X at expiration.

Similarly, when the put is out of the money and the call is in the money, both portfolios pay S at expiration.

Put-call parity holds that portfolios with identical payoffs must sell for the same price to prevent arbitrage. We can express the put-call parity relationship as:

c + X / (1 + RFR) ^T = S + P

where X / (1 + RFR) ^T is the present value of the riskless bond that pays the strike price X at maturity.

Equivalencies for each of the individual securities in the put-call parity relationship can be expressed as:

S = c - p + X / (1 + RFR) ^T
p = c - S + X / (1 + RFR) ^T
c = S + P - X / (1 + RFR) ^T
X / (1 + RFR) ^T = S+ p - c

The single securities on the left-hand side of the equations all have exactly the same payoffs as the portfolios on the right-hand side. The portfolios on the right-hand side are the "synthetic" equivalents of the securities on the left. Note that the options must be European-style and the puts and calls must have the same exercise prices for these relations to hold.

Now that we understand what put-call parity is we can derive the put option price by using the following equation:

p = c - S + X / (1 + RFR) ^T

that is, the put option price is simply buying the call option with strike price X, selling the stock at share price S and buying the riskless bond that pays the exercise price X at maturity. Or using the Black-Scholes formula, the price of a put option is:

p = X*e^(-rT)* N(-d2) - S*N(-d1)

With that idea in mind, we can used what I have done in my previous post on Black-Scholes formula to derive the theoretical call option price using Excel to do it similar for the put option pricing.

I'm going to use Coca cola company put option chain as an example here again. Here is the based data I'm going to use and I recommend you use these data first before you used your spreadsheet to model other option chains. You can use OptionXpress to get most of the information required to compute the Coca cola put option price and do a comparison with what is shown on the Coca cola put option chain in OptionXpress as well.

Open an Excel workbook and on one of the worksheets, type in the following data;
  1. In cell A1, type in "Current Stock Value" and use this link provided to get the last traded Coca cola stock price. At this point of writing, the last traded price was US$62.28. Type in this value in B1.
  2. In cell A2, type in "Implied Volatility". Used the Coca cola put option chain link to get the implied volatility for the put option with symbol KONM - Feb 2008 put option with a strike price of US$65.00. At this point of writing, the implied volatility was 18.8%. Type this value in B2.
  3. In cell A3, type in "6-month CD rate (annualized)". You can use this link here to compare the different 6-month CD rate. What I did was I used the best rate available on the site as my 6-month annualized rate. You should look under the heading Annual Percentage Yield for this information. At this point of writing, the best Annual Percentage Yield was still provided by Country Wide Bank but with an Annual Percentage Yield of 5.5%. Type this value in B3.
  4. In cell A4, type in "Dividend Yield". Use the same link from step 1 to get the information. You need to do a little of computation here since the dividend yield is not provided. However, you can simply take the dividend payout per share and divide that value with the share price in step 1. At this point of writing, the dividend payout is US$0.34 per share and the last traded price was US$62.28. Hence the dividend yield is US$0.34/US$62.28 which gave us an approximate 0.55% dividend yield. Type this value in B4.
  5. In cell A5, type "Days to expiration". Use the same link from step 2 to get the information. At this point of writing, the number of days to expiration for KONM was 57 days. Type in this value in B5.
  6. In cell A6, type "Strike Price". Again, use the link from step 2 to get the strike price information. The put option we are using here is the KONM, which has a strike price of US$65.00. Type this value in B6.
  7. In cell A7, type in "Discounted Share Price". Type the following formula in cell B7. =B1*EXP(-B4*B6/365). Hit the Enter key and you should get a value of US$62.23.
  8. In cell A8, type in "Put Option Price (Approximate):". This is the most complicated formula in the entire process here. I suggest you copy what I have here and paste it in cell B8. The formula you should type in cell B8 is =-B7*NORMSDIST(-SUM(LN(B7/B6),SUM(B3,POWER(B2,2)/2)*B5/365)/(B2*POWER(B5/365,0.5)))+B6*EXP(-B3*B5/365)*NORMSDIST(-SUM(LN(B7/B6),SUM(B3,POWER(B2,2)/2)*B5/365)/(B2*POWER(B5/365,0.5))+B2*POWER(B5/365,0.5)). Hit the Enter key and you should get a value of US$3.20. At this point of writing, the last traded price for KONM is US$3.30 but the Bid price is US$3.20.

I guess it is coincident again, but the model does give a very close estimate. Hope you enjoy the exercise. By the way, anyone knows how to do it for Singapore warrants?

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