December 2019
Beginner
912 pages
28h 36m
English
Content preview from Derivatives
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.
CHAPTER 48Black–Scholes PDE
Aims
- To show how a no-arbitrage ‘replication portfolio’ can be constructed from stocks and options which results in a deterministic partial differential equation (PDE) for the dynamics of an option price.
- Solving this PDE for a European option gives the Black–Scholes closed-form solutions for call and put premia.
- To show how European options can be priced by solving the Black–Sholes PDE numerically, using finite difference methods.
We show how a stochastic differential equation (SDE) for 
and for the option premium 
, can be ‘combined’ to give a purely deterministic partial differential equation (PDE) for the option premium 
– this is the ‘Black–Scholes PDE’. This PDE can be solved (by various methods) to give an analytic solution for European option prices – this is the famous Black–Scholes–Merton closed-form solution for call (or put) premia.
Then we show how finite difference methods can be used to solve the PDE numerically and provide an estimate of call and put premia. After completing this chapter, the reader should have a good basic knowledge of the continuous time approach and feel confident in consulting more advanced texts in this area.
48.1 ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Read now
Unlock full access
More than 5,000 organizations count on O’Reilly
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.Julian F.
I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.Addison B.
I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.