There is evidence that some desks are still risk managing dividend protected options using a simplified Black-Scholes approach. François Henneton, Amrit Sharma and Benjamin Carton de Wiart demonstrate how this leads to a one-sided daily bleed, and they present some simple estimates of the valuation adjustment under a cash dividend model that, under idealised scenarios, exactly compensates for the bleed

Equity derivatives brokers use different models for dividend modelling. This has a significant impact on many derivatives, but the effect on the prices the client receives is mitigated in most cases: vanilla products are calibration instruments and therefore have the same price under all models;11 1 Ignoring effects such as interpolation and extrapolation. variance swaps and other flow derivatives have the basis to absorb model differences, while exotic products have overhedges. Options on contractual dividends are generally traded on corporate and single-name derivatives desks, where the products are simple with limited adjustment factors. As a result, differences in dividend models are reflected in the client price. There is evidence that some brokers are still using the simplified approach of taking the Black-Scholes volatility and overriding the market dividends with contractual dividends. This approach of simultaneously changing the option’s parity level and exercise price was first proposed by Merton (1973). Even though many authors since have pointed out that Merton’s homogeneity does not hold with discrete cash dividends (see, for example, Zimmermann (2016) and the references therein), the practice of pricing dividend-protected options using the Black-Scholes formula persists. The main reason for this is that most practitioners have a good grasp of Black-Scholes pricing and its corresponding dynamics. This creates a market distortion, where products underpriced by the naive model trade with negative profit and loss (P&L) expectation for the seller. This negative P&L is realised over the life of the trade, via a daily bleed that may not be noticed by the trader.

This paper is organised as follows. We first briefly describe the dividend-protected vanilla options issued by corporate desks.22 2 Similar derivatives on a single-name desk are a subset. We show in some detail why the naive Black-Scholes approach leads to mispricing. We then extend this analysis under a richer jump dividend model and show how different models affect pricing and risk management. We compute the mispricing under Black-Scholes in two different ways for representative products, first as a day-zero adjustment to the jump dividend model and then as an expected P&L bleed over the life of the trade. We then approximate those two terms as the product of the square root of the variance and the difference between contractual and absolute market dividends, demonstrating that under specific assumptions they offset each other at first order and that pricing under a jump dividend model compensates for the Black-Scholes hidden carry. After reading this paper, we hope that the practitioner will have gained enough intuition with our simple formulas to understand the P&L of risk managing dividend-protected options.

## Dividend-protected vanilla options

The amount and timing of dividends are key components of derivative pricing and, given they directly affect the forward, a change in either the amount or the ex-dividend date can have a greater effect than, say, a change in volatility. Dividend-protecting a derivative consists in adapting the term-sheet definition of the product in order to limit, as much as possible, its exposure to the change in market dividends.

Dividend protection is now a standard practice for convertible securities and contributes to nullifying the delays in calling the bonds. On the corporate desks, derivatives usually involve long-dated European options. These options do not exhibit exotic features but are traded in such sizeable amounts that the main unhedgeable risk is dividend variance. Counterparties and issuers try to insulate themselves from this risk by making these options dividend protected. The dividend protection will be even more justified if the derivative is traded directly with the owner of the underlying security, who has a direct say in the dividend election. Finally, in March 2020, structured note issuers incurred losses stemming from cuts in previously announced dividends and are now looking into autocall variations that would finally eliminate long-term dividend exposure.

There are different flavours of dividend-protected options; we describe here the two main conventions. We denote by $D$ the dividend going ex at $\tau$, by $\smash{\tilde{D}}$ the corresponding contractual dividend, and by $\smash{S_{\tau_{-}}}$ the spot price at the previous close.33 3 Note that either $D$ or $\smash{\tilde{D}}$ can be zero. The strike and notional adjustments, in the spirit of ‘total return’ assets, consist in multiplying (respectively, dividing) the strike and contractual dividends (respectively, notional) by an $r$-factor defined as $(S_{\tau_{-}}-D)/(S_{\tau_{-}}-\smash{\tilde{D}})$. Alternatively, the cash pass-through protection keeps the product parameters unchanged but involves a payment between the two parties to exactly compensate for the discrepancy between $D$ and $\smash{\tilde{D}}$.

Given the adjustments are driven by the difference between realised and contractual dividends, the payoffs are path dependent. To simplify our analysis, we assume a constant relative-dividend volatility $\sigma$ and ignore rates and the repo curve. Given the impact of dividends on the early exercise boundary of American options has been extensively covered in the literature and that most corporate derivatives are European, we shall consider a European dividend-protected option. We shall also assume that the dividend protection is cash pass-through. We shall also show why our flat volatility assumption ensures that our results also hold for the strike and notional dividend adjustment.

## Mispricing under Black-Scholes

Under the Black-Scholes assumptions, volatility is calibrated in order to target the market prices, assuming that dividends are fully relative. While the proportionality curve of the dividends does not affect pricing, it does affect the spot-related Greeks, as the expected dividends and the forward will be different after a spot perturbation.

A computationally cheap adjustment for dividend proportionality is made by observing that any terminal payoff depends only on the forward. Denote a relative-dividend model calibrated to market prices by:

 $V^{\mathrm{rel\,div}}(t,F_{t}(T),T)=V^{\mathrm{mkt}}(t,F_{t}(T),T).$

Now introduce a proportional-dividend model44 4 This is a model where some dividends are purely fixed and others are proportional. It is usually defined by a proportionality curve, where a curve of 0 denotes purely fixed dividends and a curve of 1 denotes purely relative dividends. such that:

 $V^{\mathrm{prop\,div}}(t,F_{t}(T),T)=V^{\mathrm{rel\,div}}(t,F_{t}(T),T).$

These models have exactly the same valuation, as we must mark to market, but different Greeks. We can map the Greeks via:

 $\displaystyle\varDelta_{\mathrm{prop\,div}}$ $\displaystyle\approx\varDelta_{\mathrm{rel\,div}}\frac{S}{F}\frac{\partial F^{% \mathrm{prop\,div}}}{\partial S}$

This approximation becomes an equality if:

 $\frac{\partial V^{\mathrm{prop\,div}}}{\partial F^{\mathrm{prop\,div}}}=\frac{% \partial V^{\mathrm{rel\,div}}}{\partial F^{\mathrm{rel\,div}}}$

which is not always true because a proportional-dividend model would require the volatility to be recalibrated to ensure $V^{\mathrm{prop\,div}}(t,F_{t}(T),T)=V^{\mathrm{rel\,div}}(t,F_{t}(T),T)$. In our case (European options under flat volatility), $V$ refers to the Black-Scholes pricing formula and the equality holds. A direct consequence of the above is:

 $\displaystyle\varGamma_{\mathrm{prop\,div}}=\frac{\partial\varDelta_{\mathrm{% prop\,div}}}{\partial S}\approx\varGamma_{\mathrm{rel\,div}}\bigg{(}\frac{S}{F% }\frac{\partial F^{\mathrm{prop\,div}}}{\partial S}\bigg{)}^{\!2}$ (1)

We will limit our study to the two corner cases where market dividends are fully absolute or fully relative and introduce $\gamma=1$ (respectively, $\gamma=0$) if dividends are absolute (respectively, relative) to ease the notational burden. In what follows, we focus on call options and ignore rates and repo. Having written the gamma adjustment properly under proportional dividends, the following derivatives with respect to volatility and time at order $1$ and spot up to order $2$ under Black-Scholes are provided to aid the explanation:

 $\displaystyle d_{t}^{+}$ $\displaystyle=\frac{\ln(F_{t}(T)/K)}{\sigma\sqrt{T-t}}+\tfrac{1}{2}\sigma\sqrt% {T-t}$ $\displaystyle\nu_{t}$ $\displaystyle=F_{t}(T)\sqrt{T-t}\phi(d_{t}^{+})$ $\displaystyle\theta_{t}$ $\displaystyle=-\frac{1}{2}\frac{F_{t}(T)\sigma}{\sqrt{T-t}}\phi(d_{t}^{+})$ $\displaystyle\varDelta_{t}$ $\displaystyle=\frac{F_{t}(T)}{S_{t}}\varPhi(d_{t}^{+})\bigg{(}1-\gamma+\gamma% \frac{S_{t}}{F_{t}(T)}\bigg{)}$ $\displaystyle\varGamma_{t}$ $\displaystyle=\frac{F_{t}(T)}{S_{t}^{2}\sigma\sqrt{T-t}}\phi(d_{t}^{+})\bigg{(% }1-\gamma+\gamma\frac{S_{t}^{2}}{F_{t}^{2}(T)}\bigg{)}$

The vega expression $\nu_{t}$ is not a straight equality under absolute dividends ($\gamma=1$) as it requires us to ignore the higher-order terms:

 $\displaystyle\nu_{t}$ $\displaystyle=\frac{\mathrm{d}V}{\mathrm{d}\sigma}=F_{t}(T)\sqrt{T-t}\phi(d_{t% }^{+})+\frac{\partial V}{\partial F_{t}(T)}\frac{\partial F_{t}(T)}{\partial\sigma}$

Since the forward is floored at zero, the second term can be nonzero. We assume in what follows that the second term is small compared with the first and can be safely ignored.

We denote by $\smash{\tilde{F}_{t}(T)}$ the forward between $t$ and $T$ computed using the contractual dividends. The Black-Scholes approach for dividend-protected options simply requires us to calculate the value using the Black-Scholes valuation formula by overriding the forward with the contractual dividends: $P_{\mathrm{BS}}(t,K,T,\sigma,\smash{\tilde{F}_{t}(T)})$. In this setting, we can show that the dividend-protection flavour will be handled by obtaining the volatility at different strikes: for cash pass-through (respectively, strike and notional), we should use the strike vol $\sigma(K,T)$ (respectively, the moneyness adjusted strike vol $\smash{\sigma(KF_{t}(T)/\tilde{F}_{t}(T),T)}$). Given we are in a flat volatility setting, this subtlety is ignored.

We adopt the standpoint of a dealer who sells a European dividend-protected call and continuously vega hedges using vanilla options with the same expiry. By definition, the dividends are contractual amounts and are thus fully absolute.

Denote the Greeks of the dividend-protected option (respectively, of the vanilla option hedge) with the superscript ‘O’ (respectively, ‘H’). Then the number of vanilla options at time $t$ is given by $x_{t}$ such that the portfolio $\varPi$ satisfies: $\nu_{t}^{\varPi}=-\nu_{t}^{\mathrm{O}}+x_{t}\nu_{t}^{\mathrm{H}}=0$, ie:

 $\displaystyle x_{t}=\frac{\tilde{F}_{t}(T)\phi(\tilde{d}_{t}^{+})}{F_{t}(T)% \phi(d_{t}^{+})}$

The portfolio is vega-hedged and, similarly, fully theta-hedged:

 $\displaystyle\theta_{t}^{\varPi}$ $\displaystyle=-\frac{1}{2}\frac{\tilde{F}_{t}(T)\sigma}{\sqrt{T-t}}\phi(\tilde% {d}_{t}^{+})+\frac{\tilde{F}_{t}(T)\phi(\tilde{d}_{t}^{+})}{F_{t}(T)\phi(d_{t}% ^{+})}\frac{1}{2}\frac{F_{t}(T)\sigma}{\sqrt{T-t}}\phi(d_{t}^{+})=0$

We apply the same reasoning to assess the residual gamma of the portfolio $\varGamma^{\varPi}$. To compute $\varGamma_{t}^{\mathrm{O}}$ we shall use $\gamma=0$, given contractual dividends are absolute. We then see that:

 $\displaystyle\varGamma_{t}^{\varPi}$ $\displaystyle=-\varGamma_{t}^{\mathrm{O}}+x_{t}\varGamma_{t}^{\mathrm{H}}$ $\displaystyle=\frac{\nu_{t}^{\mathrm{O}}}{\sigma(T-t)}\bigg{(}\frac{1-\gamma}{% S_{t}^{2}}+\frac{\gamma}{F_{t}^{2}(T)}-\frac{1}{\tilde{F}_{t}^{2}(T)}\bigg{)}$ (2)

Hence, the residual portfolio has no vega, no theta, no delta (as it is delta-hedged) but a gamma, which is always one-sided. This of course is implicit in the structure, as Bergomi (2016) shows that, under a Black-Scholes model, vega and gamma are constrained by $\nu=S^{2}\varGamma\sigma(T-t)$. However, this equivalence between vega and gamma does not hold by hedging the adjusted gamma for mixed dividends from (1).

Let us consider the two situations. Under absolute market dividends ($\gamma=1$), we have:

 $\displaystyle\varGamma_{t}^{\varPi}=\frac{\nu_{t}^{\mathrm{O}}}{\sigma(T-t)}% \bigg{(}\frac{1}{F_{t}^{2}(T)}-\frac{1}{\tilde{F}_{t}^{2}(T)}\bigg{)}$

and the residual gamma is positive (respectively, negative) if $F_{t}(T)<\smash{\tilde{F}_{t}(T)}$ (respectively, $F_{t}(T)>\smash{\tilde{F}_{t}(T)}$), ie, if the option is protected below (respectively, above) the market dividends. The portfolio will constantly gain in the first case and lose in the latter. This means that the price quoted using the Black-Scholes approach for the dividend-protected option is too high (respectively, too low).

Alternatively, for relative market dividends ($\gamma=0$), we have:

 $\displaystyle\varGamma_{t}^{\varPi}=\frac{\nu_{t}^{\mathrm{O}}}{\sigma(T-t)}% \bigg{(}\frac{1}{S_{t}^{2}}-\frac{1}{\tilde{F}_{t}^{2}(T)}\bigg{)}$

and unless the contractual dividends are zero the residual gamma is always negative and the portfolio will constantly incur a daily loss. This cost was not captured in the Black-Scholes pricing at inception, meaning that the dividend-protected option was sold too cheaply.

Hence, depending on the proportionality of the market dividends and on the magnitude of the contractual dividends, the Black-Scholes approach misprices dividend-protected options. Furthermore, risk-managing the options under this model exposes the broker to a hidden carry that will constantly bleed or gain over the life of the trade. The authors have encountered multiple situations in which this model was used by counterparties or other dealers to price or challenge valuations.

We show here that a jump dividend model55 5 This model is described as model (3) in Frishling (2002) or as a piecewise lognormal model in Vellekoop & Nieuwenhuis (2006) and piecewise geometric Brownian motion in Mysona & Zimmermann (2012). is more appropriate to risk manage those options. As a reminder, under this process, the stock $S_{t}$ jumps down by the dividend amount $D_{i}$ at time $\tau_{i}$ and follows a geometric Brownian motion with volatility $\sigma^{\mathrm{JD}}$ between dividends:

 $\displaystyle\mathrm{d}S_{t}=-D_{t}\mathrm{d}t+\sigma^{\mathrm{JD}}S_{t}% \mathrm{d}W_{t}\quad\text{with }D_{t}=\sum_{\forall i}D_{i}\delta_{\tau_{i}}$

This model naturally supports the proportionality curve of dividends. In the two extremes, dividends are either absolute amounts independent of $S$ ($\gamma=1$) or fully relative, expressed as a percentage of the spot before dividends, and this model is equivalent to Black-Scholes ($\gamma=0$). In practice, dividend-protected options and their hedges should be risk managed under a ‘local vol-jump dividend’ model, with local volatility $\sigma^{\mathrm{JD}}(t,S_{t})$ in the above diffusion. Given our assumption of flat volatility, we shall be able to leverage a simplified approach to illustrate why this dividend dynamics is better adapted than Black-Scholes to trade those products. We denote by $P_{\mathrm{JD}}(t,K,T,\sigma_{t}^{\mathrm{JD}}(T),F_{t}(T))$ the price at time $t$ of a vanilla option struck at $K$ under the jump dividend model.

Similarly to the Black-Scholes model, $\sigma_{t}^{\mathrm{JD}}(T)$ should be calibrated to these prices. We start by deriving an approximation of $\sigma_{t}^{\mathrm{JD}}(T)$.

If the dividends are fully relative, we end up in the Black-Scholes world and will naturally have $\sigma_{t}^{\mathrm{JD}}(T)=\sigma$, where $\sigma$ is the Black-Scholes implied volatility calibrated to market prices of European vanillas. If they are fully absolute, we should calibrate $\sigma_{t}^{\mathrm{JD}}(T)$ such that:

 $\displaystyle P_{\mathrm{JD}}(t,K,T,\sigma_{t}^{\mathrm{JD}}(T),F_{t}(T))=P_{% \mathrm{BS}}(t,K,T,\sigma,F_{t}(T))$

The authors acknowledge that, even with a flat Black-Scholes surface, ie, $\sigma(K,T)=\sigma$, the equivalent jump dividend surface will exhibit some skew and kurtosis. We focus here on the at-the-money forward volatility adjustment.

Here, we shall approximate our jump dividend model by reusing the mixed model proposed by Bos & Vandermark (2002). As far as the authors are aware, this approximation is leveraged by practitioners across multiple banks:

 $\displaystyle P_{\mathrm{JD}}(t,K,T,\sigma_{t}^{\mathrm{JD}}(T),F_{t}(T))$ $\displaystyle\qquad\qquad\qquad{}\approx P_{\mathrm{BS}}(t,K+X_{t}^{\mathrm{f}% }(T),T,\sigma_{t}^{\mathrm{JD}}(T),S_{t}-X_{t}^{\mathrm{n}}(T))$

with:

 $\displaystyle X_{t}^{\mathrm{n}}(T)=\sum_{t<\tau_{i}

We calibrate $\sigma_{t}^{\mathrm{JD}}(T)$ to have the same prices for an option struck at-the-money forward (ATMF) and set $K=F_{t}(T)=S_{t}-X_{t}^{\mathrm{n}}(T)-X_{t}^{\mathrm{f}}(T)$:

 $\displaystyle P_{\mathrm{BS}}(t,K+X_{t}^{\mathrm{f}}(T),T,\sigma_{t}^{\mathrm{% JD}}(T),S_{t}-X_{t}^{\mathrm{n}}(T))$ $\displaystyle\qquad\qquad\qquad\qquad{}=P_{\mathrm{BS}}(t,K,T,\sigma,S_{t}-X_{% t}^{\mathrm{n}}(T)-X_{t}^{\mathrm{f}}(T))$

Using the Brenner & Subrahmanyam (1988) approximation for an ATMF struck vanilla option, we end up with the jump dividend volatility approximation for absolute dividends:66 6 The above logic can easily be leveraged to derive reasonably accurate approximations of the at-the-money skew and kurtosis adjustments.

 $\displaystyle\sigma_{t}^{\mathrm{JD}}(T)\approx\sigma\bigg{(}1-\frac{X_{t}^{% \mathrm{f}}(T)}{S_{t}-X_{t}^{\mathrm{n}}(T)}\bigg{)}$

We acknowledge that, under relative market dividends, $X_{t}^{\mathrm{n}}(T)$ and $X_{t}^{\mathrm{f}}(T)$ are not defined, but because in this case $\gamma=0$ and no volatility adjustment is required, we leverage $\gamma$ to write the jump-dividend-to-Black-Scholes price equivalence:

 $\displaystyle P_{\mathrm{BS}}(t,K,T,\sigma,F_{t}(T))$ $\displaystyle\qquad\qquad{}\approx P_{\mathrm{JD}}\bigg{(}t,K,T,\sigma\bigg{(}% 1-\gamma\frac{X_{t}^{\mathrm{f}}(T)}{S_{t}-X_{t}^{\mathrm{n}}(T)}\bigg{)},F_{t% }(T)\bigg{)}$ (3)

Dividend-protected options are valued as the jump dividend price of an option, where the dividends are overridden with the absolute contractual dividends, by using the market implied jump dividend volatility: $P_{\mathrm{JD}}(t,K,T,\sigma_{t}^{\mathrm{JD}}(T),\smash{\tilde{F}_{t}(T)})$. Unlike market dividends, contractual dividends are always absolute and their near and far decompositions are clearly defined as:

 $\displaystyle\tilde{X}_{t}^{\mathrm{n}}(T)=\sum_{t<\tau_{i}

We now derive an approximation for the jump dividend valuation adjustment, denoted by $\Delta P_{t}$:

 $\displaystyle P_{\mathrm{JD}}(t,K,T,\sigma_{t}^{\mathrm{JD}}(T),\tilde{F}_{t}(% T))-P_{\mathrm{BS}}(t,K,T,\sigma,\tilde{F}_{t}(T))$

by converting the jump dividend price into Black-Scholes as follows:

 $\displaystyle P_{\mathrm{JD}}(t,K,T,\sigma_{t}^{\mathrm{JD}}(T),\tilde{F}_{t}(% T))$ $\displaystyle~{}{}\sim P_{\mathrm{JD}}\bigg{(}t,K,T,\sigma\frac{1-\gamma X_{t}% ^{\mathrm{f}}(T)/(S_{t}-X_{t}^{\mathrm{n}}(T))}{1-\tilde{X}_{t}^{\mathrm{f}}(T% )/(S_{t}-\tilde{X}_{t}^{\mathrm{n}}(T))}\bigg{(}1-\frac{\tilde{X}_{t}^{\mathrm% {f}}(T)}{S_{t}-\tilde{X}_{t}^{\mathrm{n}}(T)}\bigg{)},$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad% \qquad\qquad\quad~{}~{}~{}{}\tilde{F}_{t}(T)\bigg{)}$ $\displaystyle~{}{}\approx P_{\mathrm{BS}}\bigg{(}t,K,T,\sigma\frac{1-\gamma X_% {t}^{\mathrm{f}}(T)/(S_{t}-X_{t}^{\mathrm{n}}(T))}{1-\tilde{X}_{t}^{\mathrm{f}% }(T)/(S_{t}-\tilde{X}_{t}^{\mathrm{n}}(T))},\tilde{F}_{t}(T)\bigg{)}$

The last approximation is obtained by observing that contractual dividends are fully absolute and applying the equivalence between Black-Scholes and jump dividend prices (see (3)). The jump dividend valuation adjustment can then be expressed in terms of the Black-Scholes dividend protected option vega:

 $\displaystyle\Delta P_{t}\approx\nu_{t}^{\mathrm{O}}\sigma\frac{\tilde{X}_{t}^% {\mathrm{f}}(T)/(S_{t}-\tilde{X}_{t}^{\mathrm{n}}(T))-\gamma X_{t}^{\mathrm{f}% }(T)/(S_{t}-X_{t}^{\mathrm{n}}(T))}{1-\tilde{X}_{t}^{\mathrm{f}}(T)/(S_{t}-% \tilde{X}_{t}^{\mathrm{n}}(T))}$ (4)

with $\nu_{t}^{\mathrm{O}}=\nu_{\mathrm{BS}}(t,K,T,\sigma,\smash{\tilde{F}_{t}(T)})$. We once again consider the two corner cases.

### Relative market dividends ($\gamma=0$)

The jump dividend valuation adjustment in this case is given by:

 $\displaystyle\Delta P_{t}\approx\nu_{t}^{\mathrm{O}}\sigma\frac{\tilde{X}_{t}^% {\mathrm{f}}(T)}{\tilde{F}_{t}(T)}$

In the case where contractual dividends are split continuously across the life of the trade (for all $i\in[t,T]$, $\smash{\tilde{D}_{i}}=\smash{\tilde{D}_{t}(T)}/(T-t)$), $\smash{\tilde{X}_{t}^{\mathrm{f}}(T)}$ can be approximated as $\tilde{D}_{t}(T)/2$ and:

 $\displaystyle\Delta P_{t}\approx\nu_{t}^{\mathrm{O}}\sigma\frac{\tilde{D}_{t}(% T)}{2\tilde{F}_{t}(T)}$ (5)

We see here that the jump dividend valuation adjustment is positive as soon as the trade is not protected at zero, in line with our gamma observation under the Black-Scholes approach.

### Absolute market dividends ($\gamma=1$)

The jump dividend valuation adjustment in this case is given by:

 $\displaystyle\Delta P_{t}\approx\nu_{t}^{\mathrm{O}}\sigma\frac{\tilde{X}_{t}^% {\mathrm{f}}(T)-X_{t}^{\mathrm{f}}(T)[(S_{t}-\tilde{X}_{t}^{\mathrm{n}}(T))/(S% _{t}-X_{t}^{\mathrm{n}}(T))]}{\tilde{F}_{t}(T)}$

In the case where both the contractual and the market dividends are split continuously across the life of the trade, we have, as before:

 $\tilde{X}_{t}^{\mathrm{f}}(T)=\tilde{X}_{t}^{\mathrm{n}}(T)=\frac{\tilde{D}_{t% }(T)}{2}\quad\text{and}\quad X_{t}^{\mathrm{f}}(T)=X_{t}^{\mathrm{n}}(T)=\frac% {D_{t}(T)}{2}$

Hence:

 $\displaystyle\Delta P_{t}\approx\nu_{t}^{\mathrm{O}}\sigma\frac{\tilde{D}_{t}(% T)-D_{t}(T)}{2\tilde{F}_{t}(T)(1-D_{t}(T)/(2S_{t}))}$ (6)

We see that the jump dividend valuation adjustment is positive whenever the option is protected above the market dividends and negative elsewhere. This is also in line with our gamma observation under the Black-Scholes approach.

The intuition behind these results is as follows. The Black-Scholes approach is equivalent to a jump dividend price whose volatility is adjusted downwards based on the contractual dividends, while the jump dividend approach is equivalent to a jump dividend price whose volatility is adjusted downwards based on the market dividends and proportionality.

## From Black-Scholes bleed to jump dividend valuation adjustment

Here, we show that under the same assumptions as above the jump dividend valuation adjustment at pricing time ($t=0$) exactly compensates the expected gamma bleed under Black-Scholes:

 $\Delta P_{0}\approx\mathbb{E}\bigg{(}\int_{0}^{T}-\tfrac{1}{2}\sigma^{2}S_{t}^% {2}\varGamma_{t}^{\varPi}\mathrm{d}t\bigg{)}$

where $\varGamma_{t}^{\varPi}$ is the residual gamma under the Black-Scholes model. Let $\text{P\&L}^{\mathrm{BS}}_{t}=\frac{1}{2}\sigma^{2}S_{t}^{2}\varGamma_{t}^{\varPi}$ be the gamma bleed P&L at time $t$. Using (2):

 $\displaystyle\text{P\&L}^{\mathrm{BS}}_{t}=\frac{\sigma S_{t}^{2}\nu_{t}^{% \mathrm{O}}}{2(T-t)}\bigg{(}\frac{1-\gamma}{S_{t}^{2}}+\frac{\gamma}{F_{t}^{2}% (T)}-\frac{1}{\tilde{F}_{t}^{2}(T)}\bigg{)}$

Under the assumption that both contractual and market dividends are $\mathcal{O}(S_{t})$ and continuously distributed until expiry, we can approximate the full P&L coming from the gamma bleeding as:

 $\displaystyle\text{P\&L}^{\mathrm{BS}}=\mathbb{E}\bigg{(}\int_{0}^{T}\text{P\&% L}^{\mathrm{BS}}_{t}\mathrm{d}t\bigg{)}\approx-\sigma\frac{\tilde{D}-\gamma D}% {T}\int_{0}^{T}\mathbb{E}\bigg{(}\frac{\nu_{t}^{\mathrm{O}}}{S_{t}}\bigg{)}% \mathrm{d}t$

We now have to address the integral to simplify the above expression. We show that for an option struck at-the-money, under a flat volatility and ignoring the market and contractual dividends, computing the expectation term $\smash{\int_{0}^{T}\mathbb{E}(\nu_{t}^{\mathrm{O}}/S_{t})\mathrm{d}t}$ yields:

 $\displaystyle\int_{0}^{T}\mathbb{E}\bigg{(}\frac{\nu_{t}^{\mathrm{O}}}{S_{t}}% \bigg{)}\mathrm{d}t$ $\displaystyle=\frac{T}{2\sigma}\bigg{(}\varPhi\bigg{(}\frac{\sigma\sqrt{T}}{2}% \bigg{)}-\varPhi\bigg{(}-\frac{\sigma\sqrt{T}}{2}\bigg{)}\bigg{)}\approx\frac{% T\sqrt{T}}{2\sqrt{2\pi}}$

Ignoring both market and contractual dividends in the expectation is a major assumption. However, computing this term analytically with a daily discretisation of the integral shows the impact of the dividends is small as long as both market and contractual dividends remain $\mathcal{O}(S)$. Especially, the sensitivity with respect to $T$ and $\sigma$ is unchanged. The P&L bleed under Black-Scholes becomes:

 $\displaystyle\text{P\&L}^{\mathrm{BS}}\approx-\frac{\sigma\sqrt{T}}{2\sqrt{2% \pi}}(\tilde{D}-\gamma D)$ (7)

We now compare this expression with the jump dividend valuation adjustment at the pricing time. As we have already assumed that dividends were continuously distributed and $\mathcal{O}(S)$, we can combine the previously derived expressions (5) and (6) as follows:

 $\displaystyle\Delta P_{0}$ $\displaystyle\approx\nu_{\mathrm{BS}}(0,K,T,\sigma,\tilde{F}_{0}(T))\sigma% \frac{\tilde{D}-\gamma D}{2\tilde{F}_{0}(T)(1-\gamma(D/2S_{0}))}$

which, for an option struck at-the-money, becomes:77 7 We approximated $\phi(\frac{1}{2}\sigma\sqrt{T})\approx\phi(0)=1/\sqrt{2\pi}$ for $\smash{\frac{1}{2}\sigma\sqrt{T}}$ small.

 $\displaystyle\Delta P_{0}$ $\displaystyle\approx\frac{\sigma\sqrt{T}}{2\sqrt{2\pi}}(\tilde{D}-\gamma D)$ (8) $\displaystyle\Delta P_{0}$ $\displaystyle\approx-\text{P\&L}^{\mathrm{BS}}$

## Results

In this section, the jump dividend valuation and risks are computed using a finite-difference method as detailed in Vellekoop & Nieuwenhuis (2006). Similarly, the jump dividend volatility assumed by the model is numerically calibrated to match the at-the-money forward Black-Scholes option price.

To illustrate the quality of the approximation we derived in the previous section, we first run simulations of a lognormal process with a 20% flat volatility and 3% dividend yield under the assumption that both market and contractual dividends are split weekly. We consider a short at-the-money dividend-protected European option. The Black-Scholes implied volatility used for pricing and risk managing is flat and equal to $\sigma=20\%$. Finally, to estimate $\text{P\&L}^{\mathrm{BS}}$, the position is delta hedged and vega-hedged daily by a long vanilla option, whose strike and expiry match those of the dividend-protected option.

Table A displays $\text{P\&L}^{\mathrm{BS}}$, the approximation derived in (8) and the jump dividend valuation adjustment at inception ($\Delta P_{0}$), computed numerically. The results are rescaled by the Black-Scholes vega at inception and are then expressed in volatility points. For expiries up to one year, the P&L from risk managing under Black-Scholes is within five basis points of our approximation and would be fully offset by the upfront P&L from pricing using a jump dividend model. For longer expiries, the behaviour is still as expected but the difference between the jump dividend adjustment and the Black-Scholes bleed is larger (9bp (respectively, 18bp) when contractual dividends are 6% per year and market dividends are absolute (respectively, relative)).

We now show our results still apply in real-life situations by looking at the cumulative P&L of a short vega-hedged dividend-protected option risk managed under a Black-Scholes and jump dividend model. Additional assumptions are:

• The trade expires after one year and there are only two market dividends of 3% of the initial spot, which fall after the first and second quarters. There is no re-marking of those dividends.

• The contractual dividend dates match the market dividend dates, and those dividends are expressed as a percentage of the absolute value of market dividends (3% of the initial spot).

• The Black-Scholes implied volatility is still equal to 20%. When market dividends are fully absolute, the equivalent jump dividend implied volatility at inception is 19.6%; our approximation derived in (3) would give $\sigma_{0}^{\mathrm{JD}}(T)\approx 19.54\%$.

The results are averaged over 100 paths and are expressed in volatility points based on the Black-Scholes vega at inception.

Under a Black-Scholes approach, the cumulative P&L is positive (respectively, negative) when market dividends are fully absolute and the contractual dividends are below (respectively, above) the market’s. When the market dividends are relative, the cumulative P&L is always negative as soon as the protected dividends are nonzero.

Regarding the jump dividend model, two observations should be made. First, the jump dividend correction is always roughly the opposite of the cumulative P&L under a Black-Scholes approach; this is exactly what we expected. Second, the cumulative P&L over the life of the trade under this model is small and does not exhibit large bias as under Black-Scholes.

Let us focus on a particular example and break down the P&L. Figure 1 analyses the case where market dividends are absolute and the dividends are protected above the market dividends. The total P&L and its standard deviation are displayed alongside the cumulative gamma, vega, theta and residual P&L as a function of time. The results are scaled for a day-one spot notional of 100 million, equivalent to a day-one short vega of approximately 400,000.

• Since the option is vega-hedged and the hedge is risk-managed under the same model as the trade, there is no vega P&L.

• Under Black-Scholes, the vega and theta P&L are constantly zero but the overall portfolio bleeds through the negative gamma P&L.

• The valuation adjustment embedded in the jump dividend price and charged upfront (25bp) almost exactly compensates the realised Black-Scholes gamma bleed ($-$27bp).

• Under the jump dividend model, the cumulative gamma P&L is also one-sided because of the different forwards used by the option and the hedge. However, this is constantly offset by the theta P&L, which is in the opposite direction. Overall, the cumulative P&L is almost zero.

• After the last dividend (at $t=0.5$), the dividend-protected option becomes a vanilla option and matches exactly its hedge.

• The residual P&L is negligible, meaning that a Greek attribution solely based on gamma, theta and vega is enough.

## Conclusion

In this paper we highlighted the pitfalls associated with pricing and risk managing a dividend-protected book using inappropriate but widely used models. The ubiquitous practice of overriding the marked dividends with the contractual dividends and then adjusting the first- and second-order Greeks for dividend dynamics can result in a severe carry bleed. Using the results from Bergomi (2016) we demonstrated that there is no scenario in which delta, gamma, vega and theta can be hedged. The bleed term was written explicitly using the models proposed in Frishling (2002), Bos & Vandermark (2002) and Bos et al (2003). This led to a potential compensatory mechanism on the option’s carry that can realign a book’s valuation and theta with more sophisticated models. Though not exact, this mechanism presents a trade-off between a computationally expensive model and a simple model with an adjustment for its weaknesses when pricing dividend-protected options. Furthermore, the simplicity of the equations and the succinctness of the example will aid practitioners with extrapolating their thinking from the well-understood Black-Scholes model. That is, the question ‘Where does my P&L go?’ is answered through the contractual dividend bleed.

François Henneton is an executive director, while Amrit Sharma and Benjamin Carton de Wiart are managing directors at Morgan Stanley. They are all based in London.

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