One of the complications in equity derivative modelling is how to treat dividends. On dividend payout, or ex div, dates, the underlying spot drops by the dividend amount, ignoring tax. This introduces discrete deterministic jumps into the geometric Brownian process assumed within the Black-Scholes framework. The underlying stochastic process in the presence of discrete dividend can be expressed as:
Where r is the risk-free rate and D is the discrete dividend. There are various dividend assumptions in equity derivative models which simplify the underlying process. One commonly used assumption is to convert discrete dividends into an equivalent continuous yield q. The simplified underlying stochastic process becomes:
The continuous yield assumption alters the underlying stochastic process. Figure 1 shows two simulated spot paths for a single stock, where the dividend is paid out every six months at an amount of roughly 2.5% of the spot level. One path is generated using the discrete dividends process in equations (1) and the other using the continuous yield process in equations (2). Both paths are generated using exactly the same random sequence, and the difference shown is purely due to different dividend assumptions. As shown in the figure the simplified process is close to the discrete process at the short end. The difference between the two processes becomes large at the long end. The implication is that long-dated derivative products could be significantly mis-priced if the continuous yield process is used improperly.
Pricing Models Incorporating Discrete Dividends
To price discrete dividends accurately, the stochastic process (1) should be incorporated into the model. The process can be included within a smile/skew partial differential equation (PDE) model1. The discrete dividend jumps are treated as if the smile/skew PDE jumps on ex-div dates with the following jump conditions:
Where St+ is the spot just after the ex-div date, St- is the one just before the ex-div date, Vt+ and Vt- are the corresponding option prices. The jump conditions can be implemented by interpolating within the PDE mesh on ex-div dates.
In the following, we compare the pricing difference between the discrete dividend process in equations (1) and the continuous yield process in equations (2). The comparisons are made when the two models use the same market data, including risk-free yield curve and implied volatility surface.
Comparison of Long-Dated European Options
Figure 2 shows the at-the-money (ATM) European call price differences using the two processes for a range of maturities. The underlying is FTSE-100 and the market data is a typical set taken early this year. As can be seen in the figure, the price difference increases dramatically with the maturity, and for a five-year European call, the difference is as large as 75 basis points. Figure 3 shows a similar picture for European puts. Both calls and puts are significantly more expensive when dividends are treated under the process in equations (1), which is in line with put-call parity. The price discrepancy is largely due to the fact that in the discrete dividend case, one effectively uses a larger volatility as the discrete jump process increases the overall variance of the process.
Note that in the above example the underlying is the FTSE-100, whose dividends are fairly spread out, the discrete dividend effects are not deemed to be severe, yet the long-dated vanilla price can be so different.
Comparison of Path-Dependent Options
For American and barrier options the dividend processes also affects the optimum exercise boundary and barrier touching probabilities. In the following we use a single stock to illustrate these effects. The single stock used pays the first dividend in six months and every six months thereafter, of the amount of roughly 2.5% of the current spotequivalent to an annual continuous yield of roughly 5.0%.
American Options
Figure 4 displays the price versus maturity for ATM American calls under different dividend assumptions. The price difference starts to show at the maturity of six months when the first dividend is due. The price difference is substantial. This is due to the fact that discrete dividends alter the optimum exercise boundary of the American option. Under the discrete dividend process, on passing the ex-div date the downwards jump of the stock makes the call option less valuable after the dividend than before it. Clearly, if the option is in the money, one might exercise the option before the dividend. Under the continuous yield process, however, since there is no jump on the ex-div date, the early exercise of an American call does not produce extra gain for the option holder. Consequently American calls under the discrete dividend process are more expensive than their continuous yield counterparts.
Barrier Options
Figure 5 compares barrier one-touch probabilities under the two different dividend processes. The barrier is at 80% of the spot. The difference between the two starts at six months when the first dividend is due. In this example, under the discrete dividend process, the spot drift (µ = r) at non ex-div dates is higher than that in the continuous yield case (µ = rq). The spot on average drifts higher and away from the barrier in the discrete case and hence a lower one-touch probability.
Conclusion
The numerical implementation of discrete dividends within the smile/skew framework is non-trivial. However, the alternative, converting discrete dividends into continuous yield, could produce significant pricing and risk management errors for long-dated and path-dependent option (for example, American and barrier).
In theory, it is possible to adjust partially for the discrete dividend effects by re-calibrating the implied volatility surface at the long end using appropriate models. The implied volatility is then effectively converted from "volatility on spot" into "volatility on forward." In practice, however, given the poorer liquidity of long-dated options, one tends to extrapolate the short end volatility surface into the long end and partially calibrate some points on the implied volatility surface. The short-end volatility is typically well calibrated and it is the "volatility on spot". When this is extrapolated into the long end, it should be treated still as the "volatility on spot". Rigorous option models should therefore use the discrete dividend process described above in equations (1) which is the correct spot process.
Finally, in the case of pricing quanto options, it is necessary to consider the quanto effect with regard to discrete dividends. One technique is to construct an additional equity repo curve if there isn't one and quanto the equity repo curve alone.
1. L. B. G. Andersen & R. Brotherton-Ratcliffe, "The equity option volatility smile: an implicit finite-difference approach", The Journal of Computational Finance (Winter 1997/98), 5-37.
This week's Learning Curve was written by Dong Qu, head equity derivatives quant atAbbey National Treasury Servicesin London.
