The Taboga Options Pricing Model

Transcription

1 The Taboga Options Pricing Model with applications Gary R. Evans. This slide set by Gary R. Evans is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License

2 Remember with our assumptions we imply a log-normal distribution... which we can show as a true log-linear distribution as shown on the left, or as a Gaussian distribution with a log-linear abscissa.

3 If rolling two dice, this is the probability of of any given sum (multiples of 1/36th) in decimals. This is symmetric of course

4 Two dice: If each roll is worth the face value of the roll times one dollar, this outcome has a $7 expected value. So what is a call option with a strike price of worth? Shown here are the bin values: the probability of being at that value times the value

5 According to our little Python program rollem.py, the value of the distribution above 8 is $3.89. If we split the gamble, that is what the top half is worth. But with a call option with a strike price of 8, we have to pay $8 for the right to accept any value above $

6 But what will an $8 call be worth?? Remember, a call option gives you the right to pay $8 for any value above $8. So although the payoff probability vector is the same as before, for each possible roll (upper abscissa) you have to calculate the net after paying $8 (bottom abscissa). In this case the call is worth $

7 Adjusting to a fair game.. Wikipedia tells us that if the log of X is normally distributed, then we can write X as (Z is a standard normal variable: The expected value of X (arithemetic mean), the mode and the median are: To write a call option model, we have to adjust the distribution such that the expected outcome of any single gamble is zero-sum.... from finutil_stu.py

8 Adding a drift component and value discounter to any model... This value discounter can be used to discount the present value of an anticipated dividend payment from the day of the dividend payment, and stays in the option value until the ex-dividend date.

9 standard deviations (or values) Calculating the value of the call option (brute force): (We will actually have to calculate both sides, partially to check our result, but also to complete the Aruba model. strike 1. Determine where the strike price is in terms of the SNPDF. 2. Calculate/sum all bin values to the right (brown)

10 What otranche does...

11 Of course we have this issue (red), which is going to give us a little bias... I know we can estimate the green with a Fourier process, but I doubt that will be our solution... Are any of you familiar with these tricks of integration?? We know this It would be trivial to estimate this Conclusion of March 5, 2017 (after some experimentation): This is simply not going to be an issue. Once the bin count gets up to, say, 23 (num=24 in binborder) the error drops to under a penny. Ideal binorder seems to be: binborder = np.linspace(0-5, 4.25, num=24) Why 4.25? If you want the bin count to be odd, which will center the middle bin (cause it to straddle the center), then num must be even (as above), the range when doubled and doubled again must produce an odd result.

12 Otranche the core method..

13 The breakdown for the call writer (relevant to Aruba) If you write a covered call starting with a $7 position, you have a bet worth the weighted value of all possible outcomes, which are: % probability 2. $8 at a probability of $3.11, the expected value of and roll between 2 and 7 (sum of blue columns), where the probability is embodied in the calculation (at 0.583) This equals $7, so this is a zerosum game at this call price Payoff is $8 no matter where you land in this region