The Fallacy of Fully Dividend-Protected Stock Options and Convertible Bonds
PAUL ZIMMERMANN∗
January 18, 2016
Abstract Full dividend protection via path-dependent re-striking has long been standard market practice for OTC equity options. Since recently, it has become a market standard for convertible bond issuances too. We show that the zero-dividend valuation approach commonly used by market practitioners is structurally biased, and may induce a significant shortfall in protection against dividend payments. We provide the theoretical setting to explain this bias in option pricing. We offer a new low-dimensional valuation approach to quantify the bias, which spares the re-structuring of the original security on a path-dependent basis, and improves the tractability of more exotic securities such as convertible bonds. Keywords: Option Pricing, Convertible Bond, Dividend, Dividend Protection, Path-dependency
∗ Head of Quantitative Analysis at Boussard & Gavaudan Asset Management, 69 Bld Haussmann, 75008, Paris. Email:
[email protected]. Telephone: +33 (0)1 44 90 41 24
1
Because of the stock price decline on the ex-dividend date, discrete dividend payments in cash decrease the value of call options and raise the value of put options. Hence, there is a need to design securities such that market participants are protected against the decline of the stock price on the ex-dividend date. Today, dividend-protected stock options are widely traded in the overthe-counter (OTC) market. Likewise, most convertible bonds and hybrid securities offer protection against dividends since the late 2000s (see Grundy et al. [2014], Grundy and Verwijmeren [2016]). This popular feature of convertible issuances responds to the need of convertible investors to be protected against dividend-induced wealth transfer from bondholders to stockholders1 . On the face of it, dividend-protected convertible bonds provide an enhanced yield advantage to bondholders, and relieve the convertible investor from having to arbitrate between the bond coupon yield and the stock dividend yield. How effective and accurate is the protection in practice? In the case of stock options, the simplest dividend protection mechanism consists in lowering the exercise price by the dividend amount on the ex-dividend date. This basic adjustment provides only imperfect protection against dividend payments because it does not preserve the option’s intrinsic value. A second adjustment, changing the option’s parity level, appears to be needed. In his seminal paper, Merton [1973] laid the ground for the analysis of payout-protected stock options, by deriving a sufficient condition to protect stock options. By adjusting simultaneously the parity upwards and the exercise price downwards by a ratio built from the cum-dividend stock price and the ex-dividend stock price, Merton exhibited the first generic dividend protection mechanism. This method, known as path-dependent re-striking, is now standard market practice when it comes to protecting OTC equity options, warrants or convertible bonds against dividends. Merton’s analysis rests heavily upon the homogeneity in spot and strike of the pricing function. However, homogeneity does vanish in the context of discrete cash dividends. This appears to induce a structural bias in protection: the re-striking method no longer boils down to a mere removal of forthcoming dividends from the stock price process, as it is often assumed in option or convertible markets and known as the “zero-dividend” valuation approach. Grundy and Verwijmeren [2016] go as far as suggesting that the zero-dividend approach was at the root of the introduction of dividend-protected convertible bonds: “Since dividend protection immunizes a convertible bond’s conversion value against changes in dividend policy, dividends have only a muted effect on the value of a dividend-protected convertible, simplifying valuation.” The purpose of this paper is precisely to explore this “muted” effect of dividends. The valuation bias incurred by dividend-protected securities appears to be closely linked to the problem of the modeling of discrete cash dividends. Even though the piecewise geometric Brownian motion (PGBM) has been considered as the modeling framework of choice among practitioners for a long time, it was not until the paper by Vellekoop and Nieuwenhuis [2006] that it was well-established on sound theoretical grounds. Mysona and Zimmermann [2012] further refined its scope by showing how to make it consistent with the well-known escrowed dividend model, i.e. the Black-Scholes-Merton model with the standard correction of subtracting the present value of future dividends from the initial stock price (e.g., Bakstein and Wilmott [2002]). The main theoretical contribution of this paper is therefore to make an extended use of the PGBM setting to elucidate the bias encountered in the re-striking method for dividend protection. 1 Grundy and Verwijemeren [2016] attribute the modification in security design to a change in the US regulatory environment for dividends, with the passage of the Jobs and Growth Tax Relief Reconciliation Act in 2003. However, this rationale doesn’t seem to hold for the European convertible universe, where the dividend-protection feature has been also very popular with issuing corporations.
2
From a practical point of view, we investigate the magnitude of the shortfall in protection which had never been documented, even if incomplete protection against dividends proves to be quite substantial and well known to seasoned convertible bond practitioners. We find that shortfalls in protection are benign for short-lived vanilla options on the typical S&P500 stock, but may amount to several implied volatility points for longer-term equity derivatives. Moreover, we describe the standard numerical technique used to tackle the pricing of dividendprotected contracts: high-dimensional path-dependent finite-difference solvers. Despite performing adequately for stock options, they become rapidly untractable for securities exhibiting additional exotic features such as convertible bonds. The main practical contribution of this paper is therefore to provide an alternative low-dimensional partial differential equation (PDE) approach, encapsulating the path-dependency in a mere adjustment of the local volatility surface. The paper proceeds as follows. First, we review the economics of the full dividend protection of stock options and convertible bonds. Second, we provide the theoretical setting to explain the structural bias induced by the re-striking method, and derive a new low-dimensional method for valuing fully dividend-protected securities. Third, we apply the previous results to measure the incompleteness of the path-dependent re-striking method.
ECONOMICS OF THE FULL DIVIDEND PROTECTION The simplest dividend correction of just subtracting the dividend amount from the option’s exercise price is inadequate for two reasons. First the option’s intrinsic value is not preserved. Second, the strike adjustment is insensitive to the level of the ex-dividend stock price, and the correction will be the same whether the option is in- or out-of-the-money. Corporate issuers as well as market participants have long recognized the necessity of a more effective method of structuring to ensure the protection of financial derivatives against dividend payments in cash.
The path-dependent re-striking method In the case of equity options traded in the OTC market, the standard market practice implies that the “full dividend protection” adjustment is usually carried out in two steps, once the cumdividend fixing price St − is known. The first step consists in lowering the option’s strike on the d ex-dividend date td , while the second step consists in raising the option’s parity simultaneously. These new strike and number of options are given by: Knew =
St − − d d
St −
and
Kold
Nnew =
d
St − d
St − − d
Nold .
(1)
d
ex Note that the same path-dependent ratio Stcum − /S + gets involved in both adjustments, and that td d adjusting the strike and the parity according to Equations (1) preserves the intrinsic value of a call option:
Nold St − − Kold d
+
= Nnew St + − Knew d
+
.
(2)
If n ex-dividend dates t1 < · · · < tn are assumed, the previous adjustments add up in such a way
3
that the final lowered strike K 0 and the final increased parity N 0 are given by: n
K0 = ∏
St − − di i
St −
i=1
St −
n
K
and
N0 = ∏
i
i=1 St − − di
i
(3)
N.
i
It should be clear that the re-striking method of equity options provides perfect protection if the pricing function C(.) is homogeneous of degree one in spot S and strike K. To see that, let us rewrite homogeneity as follows: λ ·C(S, K/λ ) = C(λ · S, K),
(4)
for all λ > 0. This formulation enables to equate the restructured option valued at the ex-dividend stock price (left hand side of (4)) with the original option valued at the cum-dividend stock price (right hand side of (4)). It must be stressed that in the case of a policy of discrete cash dividends (di )1≤i≤n , the homogeneity of the Black-Scholes-Merton [1973] pricing function is no longer guaranteed since the forward price ! F(T ) =
S0 −
∑ die−rti
erT
(5)
ti ≤T
is no longer linear in the spot price S0 . Although it covers most cases when it comes to pricing short-to-medium term stock options, this dividend modeling prevents the re-striking method of stock options from being regarded as providing perfect protection2 . EXHIBIT 1. The path-dependent re-striking method. This figure illustrates the path-dependent re-striking method used for equity options and convertible bonds. Adjusting the exercise price and the parity according to Equations (1) preserves the intrinsic value of a call option (left). Likewise, adjusting the conversion ratio according to Equation (6) decreases the conversion price and preserves the conversion value of a convertible bond (right). Intrinsic value
CB parity
Knew Kold
Stex d
Stcum −
Spot
d
Rnew S
Rold S
Stex Stcum − d
Spot
d
2 By
contrast, the use of a policy of discrete proportional dividends (αi )1≤i≤n guarantees the homogeneity of the Black-Scholes-Merton pricing function since the forward price: F(T ) = S0
∏ (1 − αi )erT
ti ≤T
is linear in the spot price S0 . Hence the re-striking method can be regarded as perfect in this specific case only.
4
Dividend-protected convertible bonds A close inspection of convertible bond indentures issued since the late 2000s shows that dividend protection features may come in various forms. The most standard mechanism intended for protection against dividend pay-outs is the so-called “partial dividend protection”. The bond indenture specifies a schedule of relevant ex-dividend dates (i.e. subject to adjustment) and contractual thresholds of protection. These thresholds are stipulated over specific periods of time (e.g. the financial year), and they apply to the accumulated dividend amount over this period. On each relevant date, the threshold of protection is deflated by the relevant dividend amount, while the current conversion ratio Rt is increased by the dividend amount in excess of the remaining threshold budget as follows: St − i R0 , (6) Rt = ∏ − − (di − ki )+ S ti ≤t t i
where R0 is the initial conversion ratio at bond’s issuance, di is the dividend amount paid on a per-share basis at the relevant date ti , ki is the remaining threshold budget over the period, and St − i is the closing cum-dividend share price on the eve of the relevant date (generally observed through a proxy such as the VWAP on several consecutive dealing days). As a result, the bond conversion price will be adjusted downwards, while the parity of the embedded conversion option will be adjusted upwards3 . The “full dividend protection” of convertible bonds corresponds to the unconditional case in which there are no thresholds for protection against dividends (i.e. ki = 0), and every dividend date is relevant. Since the late 2000s, full dividend protection is a market standard for convertible bond issuers and concerns more than half of European convertible issuances4 . It must be emphasized that fully adjusting the conversion ratio according to Equation (6) amounts to a re-striking of convertible bonds equivalent to the re-striking method of stock options. As shown on Exhibit 1, it decreases the conversion price while fully preserving the conversion value (i.e. the intrinsic value of the conversion rights). However, whether a convertible bond fully protected against dividends is as valuable as an equivalent bond valued in the absence of dividends —in other words perfectly protected —remains an open question at this stage. An important consequence of the dividend protection feature concerns convertible bonds structured with a provisional call feature. In this case, the bond issuer is entitled to early call back its bond, provided that the stock price has traded above a pre-determined stock price level (the call trigger) greater than the call price, thereby forcing the conversion into shares. Incidentally, the original call trigger T0 at bond’s issuance is defined from the initial conversion ratio R0 . In the case of dividend-protected convertible bonds, the running contractual call trigger Tt turns out to be lowered on the same path-dependent basis as the conversion ratio: Tt =
R0 T0 . Rt
3 Recall
(7)
that the conversion price is defined from the bond nominal N as N/Rt , while the parity is defined as Rt St . of November 30, 2015, over a European convertible bond market of roughly USD 100 billion, only a single issue among the 180 outstanding issues had been structured without any dividend protection feature (AXA 33/4 January 2017, issue size: EUR 1.1 billion). 4 As
5
Inhibition of early exercise In the case of fully dividend-protected call options, it turns out that the American style adds no value. Actually, a simple refining of the well-known argument of Geske et al. [1983] shows that the full dividend protection can be thought of as inhibiting American early exercise. Proposition. In the absence of short-selling costs, it is never optimal to early exercise a fully dividend-protected American-style call option. Proof. Without loss of generality, we consider the case of a single ex-dividend date t0 < td < T . No early exercise just prior to td means that for all stock prices S: + CtAm − (S, T, Kold ) > (S − Kold ) .
(8)
d
On the one hand, the dividend-protected call CtAm − (S, T, Kold ) reduces by continuity to: d
S ·CEur + (S − d, T, Knew ), S − d td
(9)
where the American-style call price becomes a European-style call price since there are no dividends left between td+ and T and no short-selling costs. On the other hand, the intrinsic value (S − Kold )+ becomes: S · (S − d − Knew )+ . (10) S−d Consequently, the no-early-exercise condition (8) becomes after simplification for all stock prices superior to d: + (11) CtEur + (S − d, T, Knew ) > (S − d − Knew ) . d
The latter is necessarily true since it is never optimal for a European-style call to be early exercised in the absence of dividends and short-selling costs. The generalization to the case of n ex-dividend dates 0 < t1 < · · · < tn < T requires the same argument to be performed in a recursive way, by starting from tn and then running backwards up to t1 . At each step, the i-th no-early-exercise condition will involve a European call that dominates its intrinsic value, since it cannot be prone to early exercise for any ex-dividend date later than ti . Similarly, the full dividend protection of convertible bonds can be thought of as inhibiting voluntary conversion by the rational bondholder. However, in contrast with equity options, the analysis needs to be performed at the level of the firm value to take into account the credit risk as well as the problem of dilution (see Grundy and Verwijmeren [2016]).
VALUATION OF THE FULL DIVIDEND PROTECTION Preliminary: the piecewise log-normal process To handle discrete cash dividends, we will rely on the PGBM process, where the stock price exhibits jumps downwards at ex-dividend dates and follows a geometric Brownian motion between those dates (for a thorough treatment of the PGBM process, see Vellekoop and Nieuwenhuis 6
[2006]). Its risk-neutral stochastic dynamics is written: ! dSt =
rSt − ∑ di δ (t − ti ) dt + σd (t, St )St dWt ,
(12)
i
where r is the risk-free rate, (di )1≤i≤n is the discrete cash dividend schedule, δ (.) is the Dirac delta function, σd is the instantaneous stock volatility and Wt is a standard Brownian motion. Some specific properties should be kept in mind when dealing with the PGBM model. • Between consecutive ex-dividend dates, the cum-ex ratio St − /St + remains always log-normal. i
i−1
• The PGBM process is consistent at first order with the popular escrowed dividend model, i.e. the Black-Scholes model with the standard correction of subtracting the present value of future dividends from the initial stock price (see Bakstein and Wilmott [2002]), in the sense that the forward price given by Equation (5) remains unique and well defined to avoid arbitrage. • Additional specifications for the volatility parameter are needed to guarantee the consistency of the second moment of the stock price process. Consequently, in order to recover given Black-Scholes European-style option prices, the local volatility parameter σd (t, St ) has to be adjusted to the discrete cash dividend schedule as will be described in the sequel.
Reducing the problem dimension To model the re-striking method of stock options, instead of modifying the option’s structure on a path-dependent basis, we rather suggest preserving its original structure by slightly modifying the random variable at the option’s maturity. Assuming n ex-dividend dates 0 < t1 < · · · < tn < T, the obvious choice for an auxiliary random variable is the fully-protected process defined as follows: ! St − (13) Xt := ∏ S i+ St , X0 = S0. 0
