Getting the jump on pricing dividend-protected derivatives

Stock dividends can be a boon for investors, especially in stock market swoons like the current one. But for dealers and hedge funds, they can be a big headache. This is because dividend payments directly affect the value of convertible bonds and corporate derivatives, such as over-the-counter options used to manage exposure to equity stakes. These instruments are typically dividend-protected for this reason.

“You want to eliminate the moral hazard of them being able to impact the price of the derivative because they could control the amount of the dividend itself,” explains Amrit Sharma, global head of exotic strategies at Morgan Stanley.

This creates another problem. The dividend-protection feature is itself rather complex, and modelling it incorrectly can have unintended consequences. Using the Black-Scholes model to price these products is ill-advised.

“If you try to price a dividend-protected product using the simplified approach of overriding the market dividends with contractual dividends in the Black-Scholes formula, you will realise that the gamma and theta of the position and of your vega-hedged portfolio do not align,” says François Henneton, an executive director in Sharma’s team.

“If you look at the breakeven volatility implied by the theta/gamma risks of your product, you will see that it differs materially from the volatility at which you are pricing.”

Sharma, Henneton and Benjamin Carton de Wiart (a former colleague who now consults for the bank) have discovered an alternative to the naive Black-Scholes approach. They found that adding a jump component to the pricing framework allowed them to adjust the valuation to perfectly neutralise the ex-date dividend price move, resulting not only in more accurate pricing, but a more robust estimation of the portfolios risk sensitivities, or Greeks.

This is particularly useful for traders and arbitrageurs, whose biggest problem is not so much finding the right price, because that can be matched to the market price, but estimating the correct Greeks to hedge their positions.

“A too elementary model will lead to the wrong Greeks and inappropriate hedge ratios, thereby crippling any arbitrage strategy,” explains Paul Zimmermann, assistant professor of finance at IÉSEG School of Management and former convertible arbitrage quantitative analyst, who has researched the topic from the arbitrageur’s perspective.

A too elementary model will lead to the wrong Greeks and inappropriate hedge ratios, thereby crippling any arbitrage strategy

Paul Zimmermann, IÉSEG School of Management

Arbitrageurs often find that realised volatility deviates significantly from what is priced into the product, resulting in a daily bleed in profit and loss even when the book appears perfectly hedged. “One may end up gaining or losing money without understanding why,” says Zimmerman.

In their paper, the Morgan Stanley quants compare the results from a Black-Scholes models to the jump-dividend approach for a stock paying continuous dividends. In their stylised example, they show that when the contractual dividend yield is lower that what is priced in the market, it results in 15 basis points of bleed over the life of the trade – in other words, the product was overpriced by 15bp. Predictably, they found the opposite when the contractual dividend yield exceeds what is priced-in.

“On short-term options, it has almost no impact, but for five- or 10-year options, the dynamics of dividends has a huge impact on the pricing and the risk management,” explains Zimmermann. “Basically, a long-term equity option should be thought of as a product whose value is essentially driven by the dividends’ dynamics.”

He adds that the jump model “seems the right way to approach the problem and give a better understanding of it, though I cannot say whether it is the definite solution”.

The jump-dividend model does create some calibration problems. One of the assumptions behind the Black-Scholes formula is that dividends are fully proportional to the spot price. The consensus among banks, however, is that short-term dividends should be modelled as fixed amounts, while long-term dividends should be modelled as proportional to the spot price before the ex-date, after which the stock trades without the value of the next dividend payment.

The jump dividend model uses this proportional approach and projects dividends accordingly. But given the diffusion under a jump dividend model differs from Black-Scholes, its outputs will differ from vanilla option prices. As options must be marked-to-market and should have the same price under all models, the volatility term structure of the jump dividend model needs to be recalibrated to line up with Black-Scholes on those options.

At Morgan Stanley, the model is used for two main purposes. “We use it as a rule of thumb to estimate price impact during live prices,” says Sharma. “Traders who tend to use those equations to look at their headline risk are able to adjust quickly.”

The bank also uses the model to explain to clients why the prices for dividend-protected instruments are different from what they might have expected.

“This is an approximation which is very close to more complicated models, but we would not recommend using it to sign off books and records on contractual dividend options,” says Sharma.