Knock-out options on dividend swaps – as the name suggests – are not easy to price.
For starters, a knock-out option ceases to exist, or ‘knocks out’, if the underlying goes above a certain value. Dividend swaps are equally complex products that pay dividend payments on an underlying equity or equity index in exchange for a floating payment.
Since dividend payments are involved in the pricing of the derivatives, one would have to model stochastic dividends on top of the local volatility model that is typically used to calibrate options to market prices in order to reflect more realistic dynamics of dividends. The trouble is that the popular Dupire’s local volatility model, when used to price these options in the presence of stochastic dividends, would end up producing prices that do not match market prices.
“We were not able to price these kinds of options,” says Pierre Henry-Labordère, a member of the global markets quantitative analytics team at Societe Generale Corporate & Investment Banking in Paris. “It is quite easy to use stochastic dividend models but the main difficulty is the calibration on vanilla options because, for example, let’s assume we are using classical local volatility models, and on top of that we add stochastic dividend models. If you reprice the vanilla options you will find a mismatch between market prices and prices coming from models.”
This is because Dupire’s formula makes a strong assumption that dividends are deterministic and therefore the dynamics become incompatible when one tries to incorporate stochastic dividends into the model.
One crude method that market participants have been using to calibrate options with stochastic dividends is that of approximation techniques, which are accurate enough only for short maturities.
“For the local volatility models people are using the classical Dupire formula and then on top of that they are using stochastic dividend models, and then what you try to do is modify Dupire’s formula using an asymptotic expansion in the case where the maturity is short,” says Henry-Labordère. “If the maturity of the product is quite short we assume the effect of volatility of the dividend will be small on the equity smile.”
In this month’s first technical, Equity modelling with local stochastic volatility and stochastic discrete dividends, Henry-Labordère and his co-author, Hamza Guennoun, a senior quantitative researcher within Henry-Labordère’s team, propose a technique to exactly calibrate exotic option prices in the presence of stochastic dividends.
As a first step, the quants define a model with discrete stochastic dividends. A discrete model is used to reflect the fact that dividend payments are made on a given date, and that when the payment is made, stock prices jump to reflect the payment – in other words, the dividend payment is not a continuous process.
“There are some technical difficulties when we are dealing with dividends, because by construction when you give me a dividend it is something like a jump process. Which means when you cross the dividend date the spot will jump, so it’s a little different from a fixed-income rate where the rate is by construction a continuous process. So the numerical implementation of the calibration method in the case of discrete dividends is a little bit more difficult,” says Henry-Labordère.
The quants fix the calibration problem by applying the so-called particle method, a technique introduced in 2012 by Henry-Labordère and Julien Guyon, his colleague at the time, to calibrate local stochastic volatility models to market smiles.
The particle method is a simulation technique borrowed from physics to study the interaction between particles. The calibration method proposed by Henry-Labordère and Guyon in 2012 converted the particle method into a discrete Monte Carlo simulation that could be solved faster than standard techniques in existence at the time. Applying this to the discrete dividend model allows it to calibrate exactly to market prices for any products that are liquid enough in the market.
“For example, a vanilla option which depends on two strikes can be found in the market but this one is not liquid. So you cannot use it for calibration,” adds Henry-Labordère.
Calibration of local stochastic volatility models for complex products has always created problems in quantitative finance, from poor capture of dynamics at extreme strikes to negative interest rates causing the model to churn out nonsensical values. This makes accurate calibration computationally very intensive.
What Henry-Labordère and Guennoun do is bring an old technique to bear on a new set of products to solve a longstanding problem with calibration. This goes to show that sometimes there is benefit in borrowing techniques from other applications and reformulating them in a way that applies to new solutions.
Editing by Lukas Becker