Research project scientific computing-1

Scientific Computing -I

First Lastname

Title: Numerical Methods for Solving Black-Scholes Implied Volatility

In fulfillment of CISC

Summer 2017

1. Proposal

This proposal explores numerical methods for solving financial option pricing problems. The goal is to identify and develop efficient techniques that converge for European and American options, under uncertain volatility. Objective of the current project is to implement binomial and Black-Scholes models using numerical methods in Python and evaluate the performance

2.Background

In the early 1970s, Fisher Black and Myron Scholes [1] proposed a differential equation to represent financial derivative prices under no-arbitrage conditions. Using this equation, analytical solutions were derived for option pricing. This resulted in rapid growth of financial derivatives creation, usage and across the world. Many enhancements were proposed to the (Black-Scholes Merton) BSM models to incorporate real market conditions. This created a need for reliable option price computation. Due to the non-linear nature of problem, numerical methods are deployed as explicit formulas are not available

Also, apart from option prices, deriving implied volatility from market prices is also of significance for the traders. For deriving implied volatility, even under BSM, numerical methods are required.

As the computing power improved, numerous numerical methods have been tried.

3. Proposed Methodology

We will price European options using BSM and Binomial method. Also, we will compute implied volatility using BSM. We will use market data and evaluate bracketing as well as Newton Raphson method. The methodology may change as the course progresses and my understanding about the problem improves. The implementations will be evaluated on robustness, convergence and speed.

Domain/ Mathematical formulae:

For computing implied volatility using BSM, under constant volatility conditions, we can use Newton Raphson method.

The method requires first derivative of the function being solved and the same can be derived for BSM under constant volatility conditions.

Symbols:[2]

σ	The actual volatility used to price the call option
V (σ)	The actual price of the call option as a function of the actual volatility
σ0	Volatility estimate used to price the call option
V (σ0)	The calculated price of the call option using the volatility estimate (σ0)
V’ (σ0)	First derivative call price with respect to the volatility estimate (σ0)

Option price using the Black-Scholes equation and our volatility estimate is...

Where

* Cumulative normal distribution function of the random variate z

Call price Equation derivative with respect to the volatility estimate is...

Where

Using Taylor Series Expansion of the first order where δ σ^ is the difference between the actual volatility and the volatility estimate

Rearranging the equation gives:

This can be solved iteratively, using newton Raphson method.

Ref:

[1] Black F, Scholes M, \The pricing of options and corporate liabilities," Journal of Political Economy 81 (1973), 637-659.

[2] http://www.appliedbusinesseconomics.com/files/gvsnr02.pdf