Breaking barriers in options pricing

After Black Monday in 1987, options implied volatilities started to display a ‘smile’ in relation to different strike prices, which models at the time could not capture.  

In 1994, two solutions were proposed. First, Bruno Dupire published his famous local volatility formula in Risk, in an article entitled Pricing with a smile. This was the first to model a volatility smile satisfactorily. The model, which is widely used to price exotic options, treats volatility as a function of spot price and time. Its partial differential equation can be solved numerically via a finite difference scheme.

Later that year, Emanuel Derman and Iraj Kani presented a completely different solution, also in the pages of Risk, with their Riding on a smile article. Their binomial options pricing model produces a tree of prices that expands with time until maturity and assigns probabilities to prices going up or down, in a discrete setting. Prices are determined by moving backwards through the tree.

The models are essential for banks to avoid mispricings that may generate arbitrage opportunities. They are equally important for buy-side firms looking for cheaper alternatives to vanilla options.

Now, Peter Austing, quantitative strategist at Eisler Capital, has combined these building blocks of modern finance in a new approach that captures the versatility and practicality of local volatility, and the accuracy of binomial trees.  

“I found there is a discrete version of Dupire’s formula, so that vanilla options on the pricing grid are exactly correctly valued, just like on implied trees. Also, just like the trees, probabilities for jumping between the grid points naturally emerge from the formula,” says Austing.

The method relies on a discretisation technique, based on a grid of prices and time, introduced by Jesper Andreasen and Brian Huge in 2011.

Andreasen and Huge used large time stepping and the adjointness property to tackle volatility interpolation and Monte Carlo simulation. “I hoped to use the same approaches to reduce numerical errors pricing exotics by finite difference, but was surprised to run into technical difficulties,” says Austing. “This work is the result of overcoming those difficulties, leading to an anomalous ‘J’ term in Dupire’s formula and an unexpected choice of boundary condition.”

When calibrating an option pricing model, it is standard practice to measure how accurately it prices vanilla options. Often, there is a so-called discretisation error, which arises when a problem is transferred from a continuous setting to a grid setting. In his paper, Austing shows the discretisation error can be largely eliminated, at least for the vanilla options whose strike prices fall on the nodes of his grid. By pricing those vanilla options correctly, one can be sure to have a reliable hedge for barrier options.

I found there is a discrete version of Dupire’s formula, so that vanilla options on the pricing grid are exactly correctly valued, just like on implied trees

Peter Austing, Eisler Capital

The accuracy of the model even surprised Austing, whose initial goal was not to solve Dupire’s equation, but local stochastic volatility problems.

“When I first tried it, and found prices coming back correctly to machine precision, I almost couldn’t believe my eyes. Seeing the numbers snap into line when I stumbled on the correct formula is one of the most exciting moments of my career,” he admits.

Brian Huge, senior quantitative researcher at Danske Bank, vouches for the model. “Austing supplies the explicit formula for the discrete local volatility model that includes the drift in the underlying. Compared with the work that Jesper Andreasen and I published a few years ago, he sets off with the better finite difference scheme for pricing barrier options or American-style options” he says.

Huge adds: “The paper is very detailed, and at the end it even gives a recipe of how to do finite difference schemes, so it makes the model relatively simple to implement.”

Austing himself is more cautious about calling it simple. “You need to follow the prescriptions very carefully. If you get one thing wrong, then it’s not going to work,” he says.

Given the benefits of the model, that should be no barrier to adoption.