It is 4:00 am on saturday morning and it is raining outside. Wonderful setting to write something. So, here I go.

BlackScholesandbeyond

This book was referred by Prof Stefanica, the director of Baruch MFE program . I stacked this book in my inventory under the assumption that I will read it at some point of time soon. Few backs , I had to go to Boston to visit Anoop, my childhood friend. I love long journey for it gives me a chance to be away from comp and just read, reflect and think about various things….I managed to go over this book in the journey. Let me attempt to summarize this book :

My first take on this book : This is probably one of the best books where almost every equation is replaced by an intuitive argument. The analogy of hedging with that of bet maker which spans about 4-5 pages is so beautiful that those pages alone make this book worth buying !! Ok , I will summarize in a random order and not in the order the book is organized.

Volatility Smile:

smile logo2

Constant volatility input in BS model is only for discussion purpose. In market , one sees that Lower strikes have high implied vol than higher strikes. A smile , as it is advertised in the finance literature, is when the volatility Vs Strike shows a behavior where OTM puts are expensive than BS Put value and ITM calls are cheaper than BS Call value.
How to incorporate volatility model in to BS world ? Well, there are three types of vol- Historical, model and combination of the the two. It is widely accepted that historical vol is a mean reverting process , meaning, it reverts to a mean, with a specific turn around time, with a specific volatility (yup, this is volatility of volatility - call it meta volatility ). Now how does one go about defining this model and calibrating the model . One way is to look at models like heston model which are mean reverting vol models , the other options is look at the universe of GARCH models like GARCH/AGARCH/EGARCH/GJR-GARCH/TGARCH, well..there is no end to this GARCH life that you can live. So, once you get a hold on mean reversion process, then you can think of pricing , hedging, calibrating and trading on that volatility.

If the correlation between stock prices and vol is +, then returns distribution is generally fat right tailed
If the correlation between stock prices and vol is -, then returns distribution is generally fat left tailed

Another little takeaway from this chapter is the stoppage time for bisection and newton raphson methods. log(sigma/power(2,k))< tolerance , it is pretty self explanatory why the stoppage condition is set up in that manner.

Trees & Basic Option Pricing with Binomial Trees:

bintree

Binomial tree is best described in Joshi - Chap 4. Having read such nice explanation ,it will be difficult to find a book which gives such a clear explanation. It was not a surprise that Option pricing on trees was an okish explanation. What does one need to know in Trees

The whole idea of trees is to build u,d,p in such a way that these prices at each discretization step matches the local volatility Sigma. Well there are a couple of ways to do it, eitherbyassumingp=q=0.5 or by assuming ud=1. The former is robust as forward price always falls between up and down move. A math-fin student should have all the formulae of local vol, u, d on his finger tips !!

  • Bond + Delta shares can replicate an option - Pricing by replication
  • Pricing by Hedging
  • Risk Neutral valuation
  • One comes across Quantization error and Option Specification error while working on trees
  • Best thing about trees is that american option can be easily priced by asking a simple question - Which is greater, the cost of hedging or cost of paying for immediate exercise?
  • Early exercise barrier for American options
  • Tree pricing with continous dividends
  • Tree pricing with discrete lumpy dividends
  • ARROW - DEBREU Price of a node is Value of a security that pays a dollar if the price reaches that node and 0 otherwise
  • Connection between butterfly and arrow-debreu price

Black Scholes World:

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Chapters 1 through 5 talks about BS world, their implications , assumptions, delta hedging , greeks. For a person familiar with the subject , this book is light a good bedtime read with nice intuitive explanations for delta hedging, greeks behavior etc.

In every book , there are things when one needs to carefully understand what the author is trying to say and get the best out of the book. In this book too, there are sections on Implied volatility trees that make an amateur math fin student stretch his thinking. Hence I would carefully summarize my learnings of implied volatility trees now.

volatility poster

What if I were to completely trust the markets and then want to hedge an option that is written. That’s where implied vol trees come in to focus. Thinking is this :

For a given (option price ,strike) points , lets build a price tree , so that for every price point, I know the option price. What do I get from this ? I can calculate all the hedge parameters and hence I am safe after writing an option.

There are a lot of wrinkles that one need to work on , like false probabilities etc. I don’t think I will have time to code up an implied vol tree considering my schedule. Implied Binomial trees are slightly different from implied vol trees where investor preferences are taken as input and all the options for a specific fixed expiration date is taken as input.

Even though I understand the basic concept behind implied vol trees and implied binomial trees, I will not be able to appreciate until I code these pricing techniques.!!! Hmm…Need to somehow find time.

Overall , my takeaway from this book :

A great book which explains intuitively a lot of things from BS world and extends the BS world by looking at its limitations and work arounds.