On the valuation of derivatives with snapshot reset ...

On the valuation of derivatives with snapshot reset features∗ Eric C K Yu†

William T Shaw‡

August 24, 2007 Provisionally accepted for publication in International Journal of Theoretical and Applied Finance

Abstract We propose a general approach that requires only a simple change of variable that keeps the valuation of call and put options (convertible bonds) with strike (conversion) price resets two-dimensional in the classical Black-Scholes setting. A link between reset derivatives, compound options and “discrete barrier” type options, when there is one reset is then discussed, from which we analyse the risk characteristics of reset derivatives, which can be significantly different from their vanilla counterparts. We also generalise the prototype reset structure and show that the delta and gamma of a convertible bond with reset can both be negative. Finally, we show that the “waviness” property found in the delta and gamma of some reset derivatives is due to the discontinuous nature of the reset structure, which is closely linked to digital options.

1

Introduction

A path-dependent feature in financial derivatives, known as reset or refix, has appeared in the markets since the end of the 1980s. The reset clause allows for a prespecified contractual parameter (reset variable) to be altered (or reset) at some prespecified time (reset dates) according to some prespecified rules (reset structure) with reference to the history realised by a reset date. The common ones are reset calls (hereafter RCOs), reset puts (hereafter RPOs) and reset convertible bonds (hereafter RCBs). We refer to these three types of derivatives with resets collectively as reset derivatives (hereafter RDs) in this article; if we refer only to RCOs and RPOs, we use the term reset options (hereafter ROs).1 With ROs, the reset variable is the strike price; with RCBs, it is the conversion price. If the reset feature can only be to the investor’s interests (downwards only for calls and CBs and upwards only for puts), we would expect the RD to worth more than its vanilla counterpart as there is ∗ The authors would like to thank an anonymous referee and the editors for suggestions and comments that improved the quality of this paper. † Corresponding author. Nomura Centre for Mathematical Finance, OCIAM, Mathematical Institute, Oxford University (email: [email protected]). Generous funding from the United Kingdom Engineering and Physical Sciences Research Council and Nomura International plc in the form of a CASE award is gratefully acknowledged. ‡ Department of Mathematics, King’s College London (email: [email protected]). 1 Some derivatives have a reset clause where the reset is at the liberty of the holder and involve an optimal stopping problem of some sort, shout options for example; see Dai and Kwok [14] and the references therein. These options are different and should not be confused with the RDs that we discuss in this article. Some reset options do not have prespecified reset dates, but instead have prespecified conditions where the reset is triggered when met. These options, which are closely linked to barrier options, are also different and will not be discussed here; refer to, for example, Hsu and Ho [32], Lee and Lin [37, 38] and Brenner et al [7] for details.

1

additional downside protection. For ROs, this feature can be especially appealing to investors if used for portfolio insurance purposes as it saves investors from having to adjust positions in the call or put in the event of adverse market movements. The reset provision in RCBs offers additional downside protection and therefore makes the instrument more attractive to potential investors.2 The issuer also benefits from the restoration of parity as conversion becomes more likely, thereby reducing the odds of having to produce a large sum for redemption or put, which is especially helpful for companies in high-growth industries whose CBs are issued out-of-the-money [44]. RCBs also reduce agency costs as investors are partially protected if the issuer’s management takes actions that reduce the value of the firm [18]. If the reset is against the investor’s interests, the reset clause serves to encourage early exercise/conversion if permitted. The presence of a reset clause can have a significant effect on the value and risk characteristics, as such, fast and accurate methods of valuation are of interest. RDs are clearly path-dependent so we would expect, a priori as with most existing valuation methods, an additional dimension in the valuation process. It turns out that this is often not necessary in the sense that the extent of path-dependency is weak [61, ch 13]. We wish to contribute to the valuation and analysis of reset derivatives with this article. We present a general approach to the valuation of RDs that requires only a simple change of variable leading to a similarity reduction that simplifies valuation in most cases. In addition, we relate RDs to compound options and “discrete barrier” type options from which we analyse the greeks in an attempt to give an explanation of the risk characteristics.3 Numerous examples are discussed to celebrate the fascinating way that resets can significantly alter the risk profile of RDs, which may be of interest to contract design. Although compound options can be dynamically hedged in the classical Black-Scholes setting, market imperfections and uncertainty in volatility render static hedging a more attractive alternative.4 We hope that, by highlighting the link between RDs and compound options, a deeper understanding towards RDs, especially its hedging difficulties, can be gained. The link between RDs and compound options not only allows us to explain the observations of large deltas in RCBs, large gammas in RDs, and negative gammas (even though the vega may be globally positive) in a more elegant manner than simply differentiating the relevant analytic solution, it also permits an attempt at decomposing a RD and the possibility of static replication. As RDs can have risk profiles that are significantly different from their vanilla counterparts, static hedging, with exotics if necessary, may also be of interest to practitioners in evading difficulties specifically associated with RDs. The high absolute value of the gamma means the RD requires more frequent rebalancing to remain delta-neutral and the negativity of the gamma causes the dynamic hedger to lose money. This can be an issue with RCBs with overly generous reset clauses as professional investors initiate downward pressures on the share price by shorting shares in attempts to extract gains from an increase in the conversion ratio [30, 59], the negative gamma aggravates the downward pressures as more shares need to be held short as the share price falls in order to remain delta-hedged, causing the hedger’s carry to be negative and is known as the “short squeeze” [48, 49, 46]. The next section introduces RDs in greater detail. A general approach to the valuation of RDs is 2 There is uncertainty on when RCBs were first issued. According to Ohtake et al [50], it appeared in the Japanese domestic market after the 1996 deregulation when Japanese banks needed to raise new capital; further details can be found in Ammer and Gibson [2]. However, according to Finnerty and Emery [18], the first issue of RCB took place in 1988. This appears to be consistent with the fact that Japanese RCBs first appeared in the Euromarket after the Japanese recession in the early 1990s before they appeared in the domestic market in 1996 [44]. RCBs were popular in Japan in the mid-1990s and have recently gained popularity in the Chinese domestic market [43]; in the United States, RCBs were first publicly issued in 1999 [44]. 3 Lee and Lin [37, 38] decompose ROs where the reset is triggered when the share price hits certain levels into barrier options and compound barrier options with valuation via lattice-based methods. The RDs we consider here have prespecified reset dates and so can be thought of as a special case of Lee and Lin [37, 38] with the monitoring period compressed to a single discrete timepoint; the analysis is, however, very different. 4 As explained by Derman et al [16], static hedging is the hedging of an exotic target option with a portfolio of vanilla options, that does not require rebalancing over time, which matches the value of the target option at maturity and at some critical share price levels at some intermediate times. Although a perfect static replication requires an infinite number of vanilla instruments in the portfolio, a reasonably good match can be attained with a finite portfolio. Schilling [56] and Davis et al [15] discuss the static hedging of compound options.

2

presented in Section 3. Section 4 provides an analysis on the behaviour of the three main RDs and their main greeks. In Section 5, the analysis from the preceding section is supported by means of examples with various sets of reset parameters. A decomposition of RCBs and examples where the vega is not singlesigned everywhere are also presented in this section. In Section 6, we do some explorations in generalising the prototype reset structure from which the preceding sections are based on. We also examine the waviness property found in some RDs in this section. This article ends in Section 7 with an outline of ongoing further research by the authors and closing comments. A supplement to this article, containing additional materials, is available online [67].

2

Reset derivatives

The analysis throughout is under the usual assumptions of Black and Scholes [6] and Merton [47]. The risk-neutral dynamics of the value of the underlying asset, S, is governed by dSt = (r − q)St dt + σSt dWt ,

(1)

where the risk-free short rate r, continuous dividend yield q and volatility of the underlying σ are assumed to be constant and Wt is a Wiener process.5 Unless explicitly stated in context, the payout structure of the underlying is governed solely by the continuous dividend yield q. The value H of a derivative that depends on the value of the underlying and time and is paid for upfront satisfies the well-known Black-Scholes equation ∂H ∂2H 1 ∂H + σ2 S 2 − rH = 0 (2) + (r − q)S ∂t 2 ∂S 2 ∂S for S ∈ [0, ∞) and t < T where T is the expiry date. The equality in (2) becomes a “less than or equal to” inequality for non-European options; refer to, for example, [62, chs 3,6,7] for more details. The payoffs of a call C, put P , and CB V are given by C(S, T )

=

max(S − K, 0),

(3)

P (S, T )

=

max(K − S, 0), and

(4)

V (S, T )

=

max(nS, Z),

(5)

where: K is the strike price for calls and puts; and for CBs, K is the conversion price, n is the conversion ratio and Z is the redemption payment as per the reduced form model of McConnell and Schwartz [45]. The redemption value is not necessarily the same as the face value N unless the CB is redeemable at par; for all CBs, the product of the conversion price and conversion ratio is the face value: N = nK.

(6)

For CBs, we can go further by using the model of Ayache et al [3, 4] where default risk is modelled via a hazard rate h. On default, the share price drops from S to S(1 − l) where 0 ≤ l ≤ 1 and the value of the CB drops from V to R; the pricing equation becomes 1 ∂2V ∂V ∂V + σ2 S 2 + (r − q + hl)S − (r + h)V + U + hR = 0, ∂t 2 ∂S 2 ∂S

(7)

where a coupon term U is included. For non-European, puttable or callable CBs, the equality becomes an inequality; refer to Ayache et al [4] for more details. For some definitions of R, we can use (7) in place of (2) in the forthcoming analysis with negligible additional effort - by adjusting the values of the 5 We can, of course, extend the forthcoming analysis with little additional effort if r, q and σ are time-dependent. In particular, we can make the dividend yield discrete by incorporating delta functions.

3

risk-free short rate and the dividend yield, and regarding hR as a source term where permissible (eg where recovery is given by a fraction of the face value either immediately on default or at maturity). In general, valuation is to be performed numerically as analytic solutions are relatively rare. If the finite difference method is used, upper and lower boundary conditions are to be specified along the Sboundaries. The condition along the lower boundary is simply (2) with S = 0 and the upper boundary condition can be generally specified as ∂2H lim =0 S→∞ ∂S 2 as the payoff is expected to be linear in S when the share price is very high [61, p 624].

2.1

Snapshot resets

A snapshot reset at time t ≤ u ≤ T can be regarded as a mapping g that maps the pair (Ku− −d , Su− −d ) to g(Ku− −d , Su− −d ) = Su where here K is the reset variable, S is the reference variable and d ≥ 0 is the delay. The period [u− − d, u) is the delay period in applying the reset - we know, during this period, what the new value of K is but it is not yet effective. We expect the presence of a delay can complicate valuation - take American options as an example, as such we assume d = 0 throughout for simplicity. Clearly, we need both S and K to be observable at time u− for the function g to be meaningful. We call g a snapshot reset as all the information we need to carry out the reset can be obtained by observing the share price and the strike/conversion price just before the reset is due, the new strike/conversion price takes immediate effect as soon as it is determined as no delays are assumed.6 A reset structure is given by the definition of g, it typically takes the form 8 < αKu− if Su− > aKu− , if bKu− ≤ Su− ≤ aKu− , γSu− (8) g(Ku− , Su− ) = : βKu− if Su− < bKu− , where α, β and γ are strictly positive and finite constants, and 0 ≤ b ≤ a. We also insist that a = b if b = ∞ or a = 0. The values of β when b = 0 and α when a = ∞ are irrelevant. If a = b = 0 or a = b = ∞, the reset would be deterministic in the sense that the reset variable would be the same and known for all values of the share price at the reset time and so is not of much interest to us. An illustration of (8) can be found in the left panel of Figure 1. There is a cap (floor) to the reset in that there is an upper (lower) limit on the reset variable of αK (βK) should the share price exceeds aK (falls below bK) just before the reset time. When a = b = α = β = γ = 1, we effectively have no resets as g is simply the identity function; and when b/β = a/α = 1/γ, the reset, ie the function g, is continuous. The significance of the continuity of the reset will be discussed in a later section. For RCBs with floorless resets, the conversion ratio can potentially be very large thus we have cause to believe dilution to be a significant issue. We bypass this potentially complicated issue by subscribing to the efficient market hypothesis as the post-announcement share price should already have taken dilution into account, adjustments are therefore not required; refer to Handley [28] and the references therein for more details. The special case where b = 0 and a = ∞ is particularly interesting - the reset is both floorless and capless and gives rise to numerous interesting instruments. ROs become forward-start options [53], and RCBs become death spiral or toxic convertibles [30, 59] or variable purchase options [27] if the reset is at maturity. Where there are multiple resets, ROs are similar to ratchet options [8] or cliquets [63]. 6 In practice, the reset variable is reset with reference to some function of the share price attained during a short window period prior to the reset time - typically 15 to 30 days with a delay of up to 15 days [5]. These are window resets and require more effort to model. Note that snapshot resets is a special case of window resets with the length of the window period set to zero. For short window periods, say a few days, it may be valid to simplify the modelling by approximating window resets with their snapshot equivalents.

4

Moneyness on reset

Strike/conversion price on reset

1907

1

1525.6

1525.6

0 0

1907

Share price just before reset

0.8

1

Moneyness just before reset

Figure 1: Illustration of a snapshot reset with Ku− = 1907, γ = α = a = 1 and β = 0.8 = 1 where the primary variable is the share price (left) or moneyness (right). If the strike price is constantly subjected to reset to a fraction of the share price, we have an option to purchase (sell) an asset at a proportional discount (premium), for which valuation is quite straightforward as demonstrated by Gu [26]; if such resets are periodic with a delay, the instrument is similar to fallback options [55]. Note that the reset variable on reset depends only on the share price and reset variable just before the reset date. Hoogland et al [31] and Berger et al [5] mentioned there are resets where the reset variable also depends on a particular numerical value, usually the value of the reset variable at inception, which acts as a bound on the value of the reset variable on reset. Modelling this type of reset is more involved and will not be discussed here, unless there is only one reset. RDs fit nicely into the general framework for path-dependent derivatives of Dewynne and Wilmott [17]. In our context, we have discrete sampling and the new path-dependent quantity is simply the conversion price with the update rule given by the reset structure (8), so the value of a RD H depends on time as well as the prevailing underlying share price and strike/conversion price. By no arbitrage, the value of the RD is continuous across a reset date so that H(S, u− ; Ku− ) = H(S, u; g(Ku− , S)) = H(S, u; Ku ).

(9)

This forms the basis of the bulk of the existing studies on RDs with snapshot resets, which covers the main numerical methods in quantitative finance: • for lattice-based methods, see Gray and Whaley [24, 25], Hsueh and Gou [33], Kwok and Lau [36] and Haug and Haug [29] for ROs; and Ammer and Gibson [2], Connolly [13, ch 9], Nelken [48, 49], Berger et al [5] and Lee [39] for RCBs; • for finite difference methods for RCBs, see Wilmott [61, p 472], Hoogland et al [31] and Yu [65]; • for simulation-based methods, see Shaw and Bennett [58], Kimura and Shinohara [35] and Yagi and Sawaki [64] for RCBs. Analytics is quite straightforward via discounted risk-neutral expectations in the absence of free boundaries; for m resets at times t ≤ t1 < . . . < tm ≤ T , we have Z Z H(S, t; K) = e−r(T −t) . . . Λ(K, St1 , . . . , Stm , ST ) Pr(St1 , . . . , Stm , ST |S)dSt1 . . . dStm dST (10) where Λ is the payoff on expiry and Pr(St1 , . . . , Stm , ST |S) is the joint conditional probability density of St1 , . . . , Stm , ST given S. Existing studies containing analytics on RDs include Gray and Whaley [24, 25], Cheng and Zhang [12], Li and Li [41], Li et al [40] and Liao and Wang [42].

5

The presence (absence) of a cap and floor to the reset is, generally, desirable (undesirable) for numerics and undesirable (desirable) for analytics. Let us take (10) with m = 1, if we regard the RD as a derivative with the reset time u as the expiry date where the payoff is the value on reset, the undiscounted expectation of this payoff at time t ≤ u, ie H(S, t; K)er(u−t) , is given by Z bK Z aK Z ∞ H(Su , u; βK) Pr(Su |S)dSu + H(Su , u; γSu ) Pr(Su |S)dSu + H(Su , u; αK) Pr(Su |S)dSu . 0

bK

aK

The first (last) integral vanishes when there is no floor (cap) to the reset. We would like to note that the above mentioned methods for snapshot resets, except for Hoogland et al [31] and Yu [65] are three-dimensional in essence; the vast majority of the numerical solution procedures surveyed were lattice or simulation -based. The reduction in dimension was first achieved by Hoogland et al who discovered that a similarity reduction is available due to the first degree homogeneity property in the payoff regarded as a function of the share price and the conversion price7 and the identity (6), so a RCB can be weakly path-dependent. Hoogland et al took the conversion ratio as the reset variable, which, when switched to the conversion price, is equivalent to (8) with a = α, b = β and γ = 1. Yu [65] extends the analysis of Hoogland et al by generalising on the parameters and took the conversion price as the reset variable. Refer to the supplement [67] for a detailed review on the above mentioned studies and the literature on RDs with window resets.

3

A general approach

Essentially, all we do is to make a simple change of variable. Following Yu [65], let us define, as usual, the moneyness at time t ≤ T as St Mt = , Kt it can be shown that if S satisfies (1) then so does M except at reset times when K changes. Moreover, the pricing equation (2) transforms to ∂H 1 ∂2H ∂H + σ2 M 2 + (r − q)M − rH = 0. ∂t 2 ∂M 2 ∂M Across a reset date, the moneyness changes: 8 < Mu− /α if Mu− > a, 1/γ if b ≤ Mu− ≤ a, Mu (Mu− ) = : Mu− /β if Mu− < b.

(11)

(12)

After a change of variable from S to M , the reset at time u can be regarded as a discontinuous change in moneyness given by the function f : (0, ∞) → (0, ∞) with f (Mu− ) = Mu defined as per (12). The right panel of Figure 1 shows the corresponding function f . For CBs, the parity is the intrinsic value - the market value of the shares on conversion is nS, but this is (N/K)S = N M and so moneyness is effectively parity and thus the right panel of Figure 1 also shows the intrinsic value of RCBs on reset. How we proceed from here depends on the specific nature of the RD, it hinges on the payoff after the change of variable from S to M - the upper and lower spatial boundary conditions are identical. The payoffs (3)-(5) becomes C(M, T )

=

K max(M − 1, 0),

(13)

P (M, T )

=

K max(1 − M, 0), and

(14)

V (M, T )

=

max(N M, Z).

(15)

7 The first-degree homogeneity property in modern contingent claim analysis was first discovered by Merton [47] for calls and puts.

6

So the similarity reduction is secured for RCBs. For ROs, the similarity reduction requirements are ¯ and P¯ , essentially options on the moneyness which also satisfies (11), where fulfilled by defining C ¯ C(S, t) = Kt C(M, t) and P (S, t) = Kt P¯ (M, t) so that (13)-(14) become ¯ C(M, T) ¯ P (M, T )

=

max(M − 1, 0) and

(16)

=

max(1 − M, 0).

(17)

We can now solve (11) backwards in time, with payoff selected from (15)-(17) as appropriate, and apply a jump condition equivalent to (9) across reset dates. For RCBs, the required jump condition is simply V (M, u− ) = V (f (M ), u);

(18)

and for RCOs (and likewise for RPOs), (9) becomes ¯ ¯ (M ), u) K C(M, u− ) = g(K, M K)C(f which boils down to

8 ¯ u) < αC(M/α, ¯ ¯ γM C(1/γ, u) C(M, u− ) = : ¯ β C(M/β, u)

if M > a, if b ≤ M ≤ a, if M < b.

(19)

The strike/conversion price enters the pricing equation (2) only as the terminal payoff condition (3)(5), and by changing the variable from S to M , it gets implicitly incorporated into the new independent variable. The reset has the effect of changing the moneyness as seen from (12). Once we have changed ¯ for ROs), the payoff is then a function of moneyness and the the variable from S to M (and from H to H no arbitrage condition becomes a jump condition. We recommend using finite difference methods rather than lattice methods for valuation. If we opt for a lattice method, we build the tree in M and solve back as usual, when we reach a reset date, we apply the appropriate jump condition from (18)-(19) and continue. The merit of using the finite difference methods becomes apparent when there are more than one resets or when there is a wide range of possible conversion prices available on reset, ie when a − b or α − β is large. If there is only one reset, we can simply stick with the S − t plane with a modified payoff at reset time. ¯ and P¯ lead to prices Our general approach is quite versatile. For vanilla calls and puts, our resultant C of calls and puts across all strike prices so we need only run at most two calculations to value vertical combinations, and similarly for CBs. Further, we can incorporate floating local volatility [34, ch 18] as we are effectively modelling the dynamics of moneyness so we can have dMt = (r − q)Mt dt + σ(M, t)Mt dWt with little additional effort. We can also accommodate free boundaries. For RCOs (and likewise for RPOs), the early exercise constraint changes from ¯ ≥ max(M − 1, 0). C ≥ max(S − K, 0) to C For RCBs, the conversion constraint transforms from V (S, t) ≥ nS to V (M, t) ≥ N M. If the RCB is soft-callable where the trigger S ∗ is a proportion of the prevailing conversion price as is usually the case so that S ∗ = ζK, the call constraint V (S, t) ≤ Cc for S ≥ S ∗ becomes V (M, t) ≤ Cc for M ≥ ζ

7

where Cc is the call price. The put provision is straightforward to handle likewise. We would expect the numerical solution to be more accurate if the points a/α, b/β and 1/γ coincide with a node; this is easier to achieve with a finite difference grid than with trees - especially with implicit type schemes where unconditional stability permits the avoidance of interpolation through strategic placement of nodes. For multiple resets, the critical points are unchanged on the M − t plane - the transformation from S to M therefore helps in reducing quantisation errors [60, pp 111-113, 161].

4

Analysis

We now analyse the reset structure g given by (8), or equivalently, f as per (12). It is straightforward to show, graphically or otherwise, that: • f is globally monotone non-decreasing if and only if b/β ≤ 1/γ ≤ a/α; • f is surjective if and only if a/α ≤ b/β or 1/γ = a/α = b/β; and • f is continuous if and only if 1/γ = a/α = b/β.

(20)

As a consequence, if the reset is continuous, then we also have a/b = α/β ≥ 1 and thus α ≥ β. Further, the reset is globally monotone non-decreasing and surjective if and only if it is continuous. The continuity of the function f is significant as we shall see shortly. Let us define a reset to be globally moneyness-preserving if f (Mu− ) = Mu ≥ Mu− , ie it can only increase moneyness or leave it unchanged and similarly for moneyness-robbing resets. We would expect a globally moneyness-preserving reset to increase (decrease) the fair value of RCOs and RCBs (RPOs) and vice versa for globally moneyness-robbing resets. It is then a straightforward matter to show that f is: globally moneyness-preserving if and only if β, α ≤ 1 and 1/γ ≥ a ≥ b; and globally moneyness-robbing if and only if 1 ≤ β, α and 1/γ ≤ b ≤ a.8 A direct consequence is that capless (floorless) resets cannot be globally moneyness-preserving (-robbing). Earlier on, we mentioned that the reset is continuous if (20) holds. From inspection of (8), (12) and Figure 1, a continuous reset implies H(bK, u; βK) = H(bK, u; γbK) and H(aK, u; γaK) = H(aK, u; αK)

(21)

for a reset time u. From inspection of the jump conditions (18)-(19), we observe that: for RCBs, we have total equality in (21) in that we also have H(bK, u; γbK) = H(aK, u; γaK) as RCBs with resets that are both floorless and capless have zero delta; this is not true for ROs as forward-start options have constant positive deltas [53] - we have, in fact, the “less than or equal to” inequality.

4.1

Reset derivatives and compound options

Suppose we have a RD with one reset given as per (8), its value at reset time u is given by ˆ b ˆ H(S, u; K) = H(S, u; βK)θ(bK − S) + H(S, u)θ(S − bK)θ(aK − S) + H(S, u; αK)θ(S − aK),

(22)

ˆ b where: H(S, u; K) is the value of the otherwise identical vanilla counterpart, H(S, u) is the floorless and capless reset counterpart and θ is the Heaviside function. The continuity of reset implies (20) so that ˆ ˆ b ˆ ˆ b H(bK, u; βK) = H(bK, u; γbK) = H(bK, u) and H(aK, u; αK) = H(aK, u; γaK) = H(aK, u). 8 We can think of the reset as equivalent to a dividend on the underlying provided it is only the moneyness that matters in the valuation process. It can then be shown that the equivalent dividend amount is: S(1 − 1/β) if S < bK; S − K/γ if bK ≤ S ≤ aK; and S(1 − 1/α) if S > aK. A globally positive (negative) dividend leads to a moneyness-robbing (-preserving) reset.

8

Let us rewrite (22) as

=

H(S, u; K) (23) ˆ b ˆ H(S, u; βK)(1 − θ(S − bK)) + H(S, u)(θ(S − bK) − θ(S − aK)) + H(S, u; αK)θ(S − aK)

=

ˆ ˆ b ˆ b H(S, u; βK) − (H(S, u; βK) − H(S, u))θ(S − bK) + (H(S, u; αK) − H(S, u))θ(S − aK).

b terms as the The second and third terms on the right look like compound options if we regard the H b is S-constant - it is strike prices, however we lack justification as yet since we have no guarantee that H b is not constant over the range [bK, aK], we will have to make the case for RCBs but not so for ROs. If H adjustments. ˆ b ˆ b Proposition 1 The signs of H(S, u; βK) − H(S, u) and H(S, u; αK) − H(S, u) change at only, respectively, S = bK and S = aK. We omit the proof of the above proposition as it is straightforward via a graphical argument - reˆ b fer to the supplement for a rigorous proof. Proposition 1 guarantees that H(S, u; βK) − H(S, u) and ˆ b H(S, u; αK) − H(S, u) have the same sign at both sides of, respectively, S = bK and S = aK. Let us ˜ b = H(bK, ˆ b ˜ a . Note that H ˜ is simply a constant henceforth write H u; βK) = H(bK, u) and likewise for H and would be an ideal candidate for the strike price of compound options as (23) can be rewritten as H(S, u; K)

ˆ β − (H ˆ β − (H ˜b + H b −H ˜ b ))θ(S − bK) + (H ˆ α − (H ˜a + H b −H ˜ a ))θ(S − aK) H ˆ β − (H ˆβ − H ˜ b )θ(S − bK) + (H ˆα − H ˜ a )θ(S − aK) H (24) b −H ˜ b )θ(S − bK) − (H b −H ˜ a )θ(S − aK), +(H

= =

ˆ β = H(S, ˆ ˆ α ) and H b = H(S, b ˜b − H ˆ β and where H u; βK) (and likewise for H u). All we require is for both H ˜b − H b to satisfy Proposition 1 and likewise for H ˜ a . But this is assured as H ˜b = H b =H ˆ β at S = bK by H definition and we can therefore apply arguments similar to the proof of Proposition 1. We can now relate, by no arbitrage, RDs to a package of vanilla instruments and compound options via (24).9 For RCBs, we have e Vˆ (S, t; βK, T ), t; V˜b , u) + C( e Vˆ (S, t; αK, T ), t; V˜a , u) V (S, t; K) = Vˆ (S, t; βK, T ) − C(

(25)

e denotes a compound call option, and the arguments as Vb has zero delta thus V˜b = Vb = V˜a and where C (X1 , X2 ; X3 , X4 ) denotes the underlying X1 , time X2 , strike price X3 and expiry X4 .10 For RCOs, we have C(S, t; K)

=

ˆ e C(S, ˆ ˜b , u) + C( e C(S, ˆ ˜a , u) C(S, t; βK, T ) − C( t; βK, T ), t; C t; αK, T ), t; C b ∂C ˆ ˆ +(C(S, t; bK, u) − C(S, t; aK, u)) t=u ∂S

(26)

b with respect to S is evaluated at time u. Similarly, for RPOs, we have where the derivative of C P (S, t; K)

=

Pˆ (S, t; βK, T ) + Pe(Pˆ (S, t; βK, T ), t; P˜b , u) − Pe(Pˆ (S, t; αK, T ), t; P˜a , u) ∂ Pb ˆ ˆ +(C(S, t; bK, u) − C(S, t; aK, u)) t=u ∂S

(27)

9 For the American vanilla counterparts of RDs: Geske and Johnson [21] value calls and puts via compound options and Gong et al [23] do likewise for CBs. 10 The decomposition for RCBs (25) requires a qualification. As it stands, it is strictly not a compound option as the underlying is a CB - it is more of a compound exchange option which has been studied by Carr [9]. However, as the CB is European and not callable nor puttable, we can regard the CB as a straight bond and n call options struck at ZK/N , so a RCB can indeed be regarded as a package featuring traditional compound options, calls on call in this case.

9

where Pe is a compound put option. Going back to RCBs, as mentioned in Footnote 10, we can rewrite (25) in terms of traditional comˆx for the value of a vanilla European call with pound options. For notational convenience, let us write C strike xZK/N and expiry T , and nx for the conversion ratio when the conversion price is xK. We now have ˆα (aK, u), u) ˆα , t; Fu + nα C ˆβ − C(F e t + nβ C ˆβ , t; Fu + nβ C ˆβ (bK, u), u) + C(F e u + nα C V (S, t; K) = Ft + nβ C where Ft is the S-independent bond part of the CB at time t. The bond part drops out of the compound option terms and we have e C ˆα , t; C ˜a , u). ˆβ − nβ C( e C ˆβ , t; C ˜b , u) + nα C( V (S, t; K) = Ft + nβ C

(28)

Compound options have been studied by Geske [20] who derived the main greeks for a call on call. The result of interest to us is that, for a given strike and expiry of a call on call, the delta, gamma, and vega are positive. Rubinstein [54] extends Geske to cover the cases of call on put, put on call and put on put. Let us define the anchor of a compound option to be the share price that divides the regions of exercise and non-exercise at expiry, the anchors of the compound options in (26)-(28) are either bK or aK by definition of their strikes. The separation of the roles of b and β (and likewise for a and α) becomes clear in that while both b and β determine the strike of the compound option, it is b that determines the anchor and β that determines the strike of the underlying option. An immediate observation is that the delta, gamma and vega of RCBs are guaranteed to be non-negative if the reset is floorless as such resets do not feature short positions in compound options. The same observation cannot readily be made for RCOs and RPOs unless the reset is both floorless and capless. With RCBs, consider its delta eα ˆβ eβ ∂V ∂C ∂C ∂C = nβ − nβ + nα ∂S ∂S ∂S ∂S eβ = C( e C ˆβ , t; C ˜b , u) and likewise for C eα . To convert the delta to market convention terms, ie where C ∂V /∂M , we multiply by K/N so that we have ! ˆβ eβ eα ∂C 1 ∂C 1 ∂C − + , (29) β ∂S ∂S α ∂S which can be arbitrarily large when β (or α) is very small. This is because a low β or α indicates a high ˆβ and C eβ (or C eα ) become in-the-money from a lower share price. In contrast, conversion ratio and that C the delta of RCOs (and likewise for RPOs) is bounded from above and below by, respectively, 2+

b b ∂C ∂C t=u and − 1 − t=u . ∂S ∂S

It is possible to extend the compound option representation to RDs with more than one resets, but it would involve options on some form of forward-starting calls or puts.

4.2

Reset derivatives and discrete barrier options

We may also relate RDs to discrete barrier options [34, pp 206-207], these are barrier options where the barrier is monitored at discrete sets of times in such a way that the option is knocked -in or -out depending on which side the share price is relative to the barrier when monitored. The barrier monitoring dates coincide with the reset dates. The terms on the right side of (22) can be interpreted using the terminology of discrete barrier options. Let us take RCOs as an example, we call the three terms on the right side of (22) respectively:

10

• a “down-and-in” component struck at βK with barrier at bK; • a “double knock out” (down-and-out and up-and-out) forward-start component where the strike is struck at γ times the share price on the monitoring date with barriers set at bK and aK; and • an “up-and-in” component struck at αK with barrier at aK. The separation of the roles of b and β (and likewise for a and α) is again apparent in that b dictates the barrier levels and β determines the strikes. Extension to multiple resets is rather tricky as the strikes and barriers from the second reset date onwards are path-dependent. To see this, consider a simple case with two resets at times u < v. The barrier levels and strikes at time v depends on what happened at the earlier time u. If the strike/conversion price at time u is reset to: • αK, then the upper (and lower) barrier level at time v will be at aαK (and bαK) with corresponding strikes at α2 K (and βαK); • γSu , then the upper (and lower) barrier level at time v will be at aγSu (and bγSu ) with corresponding strikes at αγsu (and βγSu ); • βK, then the upper (and lower) barrier level at time v will be at aβK (and bβK) with corresponding strikes at αβK (and β 2 K). We therefore see that the barrier and strike levels at time v depends on the share price at time u, and hence the path-dependency on the strike and barrier levels.

4.3

Reconciliation

Carr [10] mentioned the similarity between a protected up-and-out call and a portfolio containing an unprotected up-and-in call and short a compound call on the up-and-in call. Although these two portfolios are not equivalent, they inspired us to a way in reconciling the two interpretations of RDs we presented, via compound options, and via “discrete barrier” type options, to one another. Let us write H j , j = 1, 2, 3 to denote, respectively, the “down-and-in”, “up-and-in”, and “double knock out” component of H (with the straight bond value removed for RCBs). We can show the following relationships at time t ≤ u, via an arbitrage argument at time u and (26)-(28), for the three RDs we considered. For RCOs, we have C 1 (S, t; K) 2

C (S, t; K)

= =

ˆβ − C( e C ˆβ , t; C ˜b , u) − C ˜b BC (S, t; bK, u), C ˆα + Pe(C ˆα , t; C ˜a , u) − C ˜a BP (S, t; aK, u), C

(30)

˜a e−r(u−t) + C ˜b BC (S, t; bK, u) + C ˜a BP (S, t; aK, u) C (S, t; K) = −C b ∂C ˆ ˆ +(C(S, t; bK, u) − C(S, t; aK, u)) t=u , ∂S and BP are, respectively, digital call and put options. For RPOs, we have 3

where BC

P 1 (S, t; K) 2

P (S, t; K) 3

P (S, t; K)

= =

Pˆβ + Pe(Pˆβ , t; P˜b , u) − P˜b BC (S, t; bK, u), e Pˆα , t; P˜a , u) − P˜a BP (S, t; aK, u), Pˆα − C(

=

−P˜a e−r(u−t) + P˜b BC (S, t; bK, u) + P˜a BP (S, t; aK, u) ∂ Pb ˆ ˆ +(C(S, t; bK, u) − C(S, t; aK, u)) t=u . ∂S

=

ˆβ − C( e C ˆβ , t; C ˜b , u) − C ˜b BC (S, t; bK, u)), nβ (C ˆ e ˆ ˜ ˜a BP (S, t; aK, u)), nα (Cα + P (Cα , t; Ca , u) − C

(31)

Finally, for RCBs, we have V 1 (S, t; K) 2

V (S, t; K) 3

V (S, t; K)

= =

−Xe

−r(u−t)

+ XBC (S, t; bK, u) + XBP (S, t; aK, u),

11

(32)

where X is the value of a RCB with capless and floorless reset at time u in excess of its straight bond value.

5

Illustrative examples

All examples, unless stated otherwise, are on European RDs and take the following default values: N = Z = 1000000, m = 1, q = 0, r = 0.030945, u = 2.15, T = 6.0548, σ = 0.24345 and K = 1907 as per Shaw and Bennett [58]. Attention is dedicated to the value, delta, gamma and vega; information on rho, theta, lambda and zeta (sensitivity with respect to the reset time) can be found in the online supplement [67]. Where a finite difference method is used, we solve (11) in the x−t plane where x = ln M and apply the relevant jump condition from (18)-(19) across the reset date. We use both the explicit and implicit scheme with lower and upper x-boundaries at -3 and ln 3.25 respectively with 500 spacesteps; the timesteps are set in such a way that the ratio δt/(δx2 ) is kept at 1/6. For the implicit scheme, the SOR tolerance level is set at 10−9 for RCBs and 10−16 for ROs. Where numerical integration is used, the integral e−r(u−t)

Z

∞

−∞

2

√ √ σ2 σ2 e−x /2 ˆ √ H(Se(r−q− 2 )(u−t)+σ u−tx , u; g(K, Se(r−q− 2 )(u−t)+σ u−tx ))dx 2π

(33)

is numerically evaluated over the interval [−5, 5] with 250000 steps using the mid-point rule. Where Monte Carlo simulations are carried out, the discounted expectation (33) is simulated by 400000 paths with antithetic sampling (800000 paths in total) and with the value of the vanilla counterpart as control variate. Where calculation is by decomposition via compound options via (26)-(28) as appropriate, resultant bivariate standard normal integrals „ « Z i Z j 1 x2 − 2pxy + y 2 p exp − dydx 2(1 − p2 ) 2π 1 − p2 −∞ −∞ are evaluated over the range [−5, i] for x and [−5, j] for y with 2500 steps along each direction using the mid-point rule.11 The expected strike/conversion price on reset Z bK Z aK Z ∞ E[Ku |S] = βK Pr(Su |S)dSu + γSu Pr(Su |S)dSu + αK Pr(Su |S)dSu 0

bK

aK

for the four examples in this section are be found in the online supplement. Note that when the reset is continuous, that is when (20) holds, we have e−r(u−t) E[Ku |St ] = e−r(u−t) bK + γ(C(S, t; bK, u) − C(S, t; aK, u)), which exactly generates the strike/conversion price on reset.

5.1

The removal of the floor and cap

Our first example is on the effect of removing the cap and / or floor to the reset. We illustrate and observe characteristics of the RDs across five cases with the below parameters: • Case 0, α = a = γ = β = b = 1 - the vanilla case; • Case 1, β = b = 0.8 and γ = α = a = 1 - floored and capped; 11 This is, of course, not the most efficient method - refer to Aˇ gca and Chance [1] and the references therein for more efficient and advanced methods.

12

• Case 2, b = 0 and γ = α = a = 1 - floorless but capped; • Case 3, β = b = 0.8, γ = 1 and a = ∞ - floored but capless; and • Case 4, b = 0, γ = 1 and a = ∞ - floorless and capless. For want of space, we show only for RCBs, the values and the main greeks in Figure 2; a selection of values from finite difference, numerical integration, Monte Carlo and decomposition via compound options is tabulated in Table 1. Additional figures showing the values and the main greeks as a function of both moneyness and time along with a complete decomposition of values (in excess of its straight bond value for RCBs) and the main greeks via (30)-(32) can be found in the supplement. 160

1

Parity Case 0 Case 1 Case 2 Case 3 Case 4

150 140

0.8

0 1 2 3

0.7

130

0.6

Delta

Value

Case Case Case Case

0.9

120

0.5 0.4

110

0.3

100

0.2 90

0.1

80 0

500

1000

1500

2000

0 0

2500

Share price 0.015

Case Case Case Case

0.01

500

1000

1500

2000

Share price

2500

3000

3500

120

0 1 2 3

Case Case Case Case Case

100

4000

0 1 2 3 4

80

Vega

Gamma

0.005 60

0 40

−0.005

−0.01 0

20

1000

2000

3000

4000

5000

6000

0 0

1000

2000

3000

4000

5000

6000

Share price

Share price

Figure 2: The values and main greeks of reset convertible bonds across the five cases. Going from left to right and from top to bottom, we have the values, deltas, gammas and vegas. Deltas and gammas are not shown for case 4 as they are identically zero. We have satisfactory agreement across the four methods, except near the share price of 94.944 and 6146.169, which are the boundaries of our finite difference grid. In view of this, we have rerun the finite difference calculations with a Dirichlet upper boundary condition with results tabulated in the supplement. As we would expect, we have better agreements near the upper boundary for capless resets using the zero gamma condition. It is reasonable to say that as long as we are interested only at values at share prices away from the boundaries, we can use the more general zero gamma condition with confidence. Extra care should be taken when any of b/β, 1/γ or a/α significantly exceeds the upper boundary of the finite difference grid, in which case we suggest extending the grid accordingly. Inspection of Table 1 and Figure 2 suggests the behaviour of the RCBs (and indeed of RDs in general) to be such that: ˆ β (or H b if the reset is floorless) as the share price • at low share prices, H behaves increasingly like H falls; and ˆ α (or H b if the reset is capless) as the share price • at high share prices, H behaves increasing like H rises.

13

S nS Vˆ 0E 0I 0N 0M 0C 1E 1I 1N 1M 1S 1C 2E 2I 2N 2M 2S 2C 3E 3I 3N 3M 3S 3C 4N 4M 4S

94.944 4.979 82.914 82.914 82.914 82.914 82.914 82.914 82.914 82.914 82.914 82.914 7.2(-7) 82.914 105.527 105.527 105.527 105.527 2.2(-7) 105.527 82.914 82.914 82.914 82.914 3.8(-7) 82.914 105.527 105.527 3.5(-7)

476.395 24.981 83.136 83.136 83.136 83.136 83.136 83.136 83.548 83.549 83.548 83.548 8.9(-4) 83.548 105.527 105.527 105.527 105.529 7.8(-4) 105.527 83.548 83.548 83.548 83.550 9.0(-4) 83.548 105.527 105.529 8.0(-4)

953.276 49.988 87.082 87.082 87.082 87.082 87.082 87.082 90.545 90.545 90.544 90.548 2.0(-3) 90.544 105.824 105.824 105.824 105.839 4.3(-3) 105.824 90.247 90.247 90.247 90.232 4.3(-3) 90.247 105.527 105.531 6.2(-3)

1525.600 80.000 100.670 100.670* 100.670* 100.670 100.670 100.700 105.140* 105.140* 105.140 105.159 5.3(-3) 105.140 110.910* 110.910* 110.910 110.911 1.2(-3) 110.910 99.757* 99.757* 99.757 99.745 1.8(-2) 99.757 105.527 105.533 1.4(-2)

1907.000 100.000 114.002 114.001* 114.001* 114.001 114.002 114.002 117.231* 117.231* 117.232 117.231 2.5(-3) 117.232 119.771* 119.771* 119.772 119.749 4.9(-3) 119.772 102.987* 102.988* 102.987 102.996 2.3(-2) 102.988 105.527 105.539 1.8(-2)

2150.000 112.743 123.713 123.713* 123.713* 123.713 123.713 123.713 126.093* 126.093* 126.094 126.083 3.1(-3) 126.094 127.550* 127.550* 127.550 127.544 5.5(-3) 127.550 104.070* 104.070* 104.070 104.094 2.4(-2) 104.070 105.527 105.635 2.0(-2)

2288.400 120.000 129.566 129.564* 129.564* 129.566 129.566 129.566 131.520* 131.520* 131.522 131.505 3.2(-3) 131.522 132.575* 132.574* 132.577 132.561 5.3(-3) 132.577 104.472* 104.472* 104.472 104.481 2.5(-2) 104.472 105.527 105.560 2.2(-2)

2945.816 154.474 159.569 159.559 159.559 159.569 159.569 159.569 160.234 160.234 160.245 160.239 2.3(-3) 160.245 160.457 160.457 160.468 160.479 3.4(-3) 160.468 105.304 105.304 105.304 105.262 2.8(-2) 105.304 105.527 105.655 2.6(-2)

4664.767 244.613 245.772 245.628 245.628 245.772 245.772 245.772 245.659 245.659 245.803 245.807 5.6(-4) 245.803 245.663 245.663 245.807 245.807 6.4(-4) 245.807 105.522 105.522 105.523 105.553 3.7(-2) 105.523 105.527 105.428 3.7(-2)

6197.750 325.000 325.368 324.799 324.799 325.367 325.368 325.368 324.799 324.799 325.369 325.370 1.2(-4) 325.370 324.799 324.799 325.369 325.370 1.7(-4) 325.370 105.527 105.527 105.527 105.555 4.8(-2) 105.527 105.527 105.612 4.7(-2)

Table 1: The values of reset convertible bonds across the five cases calculated via different methods. The first three rows show the share price, parity and vanilla CB value, which were analytically calculated. Results for the five cases are tabulated in the rows below. For each case, values are given from: finite difference explicit and implicit (E and I), numerical integration (N), Monte Carlo and its standard error (M and S) and decomposition via compound options (C). Values denoted * were calculated by linear interpolation from adjacent grid points. All values are quoted to three decimal places except for standard errors, which are quoted in mantissa and exponent form with the mantissa to one decimal place and exponent in bracket. Our result is consistent with that of Shaw and Bennett [58] who report values, when S = 2150, of 126.093 and 127.546 for cases 1 and 2 respectively using Monte Carlo with 80000 paths and control variates with standard errors of 7.9(-2) and 1.3(-2); and numerical integration values of 126.094 and 127.553. For case 4, an analytic value of 105.527 is available, which is identical to the value via compound options by construction and via finite difference to three decimal places. The Monte Carlo standard error is omitted for case 0 as the vanilla CB is used as control variate. The value of an otherwise identical non-convertible bond is 82.914.

Let us write Hi as the value of a RD with case i reset, readers may like to check for themselves from Table 1 and the supplement that H 2 + H3 = H 1 + H 4 (34) holds at all share prices and at all times. The proof of (34) is straightforward either graphically, via no arbitrage or via (24). The put-call parity relationship between vanilla European calls and puts becomes, for ROs, C − P = Se−q(T −t) − e−r(T −t) E[Ku |S]. (35) The proof of (35) is a straightforward extension of the standard proof of put-call parity; readers may also wish to verify (35) from the supplement.

14

As observed in Section 4.1, negative gammas are possible for RCBs and RCOs (RPOs) if the reset is floored (capped), even though the vega is positive. This is rather unusual as we would expect the gamma Γ and vega ϑ to be of the same sign via ϑ = S 2 σΓ(T − t)

(36)

for standard European options [57, pp 82-84], provided dividends are governed by yields and the payoff is a function of the share price only. For RDs with, say one reset, the relation (36) ceases to hold as the payoff at time u as seen from (33) is not just a function of the share price and thus additional terms are required. We have, in fact, the relation ϑ

=

S 2 σΓ(u − t) Z ∞ −x2 /2 √ √ σ2 σ2 e √ +e−r(u−t) ϑ(Se(r−q− 2 )(u−t)+σ u−tx , u; g(K, Se(r−q− 2 )(u−t)+σ u−tx ))dx 2π −∞

that explains the reason behind a negative gamma with a positive vega as the second term on the right is positive. We have seen from Figure 2 that the vega is positive at all share values. We should, however, be careful as there is nothing from (26)-(28) that guarantees a positive vega unless the reset is both floorless and capless or with RCBs with floorless resets. Figure 3 shows examples where the vega is not single-signed using RCBs and ROs with cases 1 and 3 resets. This should serve as a caution whenever the implied volatility of RDs is discussed. 94.5

Case 3, S=1500 Case 1, S=1450

190

94

Case 3, S=1500 Case 1, S=1420

180

93.5

Value

Value

170

160

93

150 92.5

140 92 0

50 40

0.01

0.02

0.03

0.04

0.05

Volatility

0.06

0.07

0.08

0.09

0

0.1

1000

Case 1, S=1450 Case 3, S=1500

800

30

0.01

0.02

0.03

0.04

0.05

Volatility

0.06

0.07

0.08

0.09

0.1

0.07

0.08

0.09

0.1

Case 1, S=1420 Case 3, S=1500

600

20

400

Vega

Vega

10 0

200 0

−10 −200

−20

−400

−30

−600

−40 −50 0

0.01

0.02

0.03

0.04

0.05

Volatility

0.06

0.07

0.08

0.09

0.1

−800 0

0.01

0.02

0.03

0.04

0.05

Volatility

0.06

Figure 3: The values and vegas of convertible bonds and call options with cases 1 and 3 resets at certain share price levels and varying volatilities. Going from left to right, we have RCBs and RCOs; from top to bottom, we have values and vegas.

15

5.2

From window to snapshot resets

Earlier on in Footnote 6 we mentioned that snapshot resets can be used as an approximation to window resets, especially when the window period is short. This is of practical importance in that provided there are no jump conditions that affect the moneyness within the sampling period, we can simplify valuation by using the snapshot reset counterpart as an approximation. Here we give a first provisional test to this assumption by means of Monte Carlo simulations on cases 1 to 3 RCBs from Section 5.1 with results tabulated in Table 2. 1L 1M 1H 2L 2M 2H 3L 3M 3H

30 94.387 117.248 150.457 106.551 119.724 150.801 93.519 103.343 105.554

20 94.353 117.264 150.460 106.451 119.747 150.810 93.495 103.185 105.530

10 94.349 117.263 150.470 106.503 119.792 150.831 93.482 103.288 105.608

5 94.322 117.272 150.481 106.510 119.821 150.862 93.434 103.226 105.569

2 94.325 117.272 150.481 106.515 119.825 150.855 93.365 103.159 105.476

2h 94.318 117.278 150.480 106.507 119.811 150.854 93.444 103.129 105.557

0 94.322 117.250 150.479 106.438 119.787 150.848 93.411 102.990 105.158

Table 2: Convergence of window to snapshot resets. Tabulated above are the values from Monte Carlo simulations of RCBs with cases 1-3 resets as per Section 5.1 with various number of daily sampling dates - eg the column 30 means 30 daily sampling dates just before the reset is due so the window period is 29 days. Also shown are the RCB values where there are two half-daily samplings (2h) and the snapshot case (0, calculated via numerical integration). We show values for three levels of the current share price: 1117.325 (L), 1907.523 (M) and 2732.372 (H). Given that we are using simulations, the convergence of window to snapshot resets we have in Table 2 is generally satisfactory as the sampling period decreases to zero.

5.3

Connecting convertible bonds and call options with resets

We mentioned earlier the well-known fact that vanilla European non-puttable and non-callable CBs may be decomposed into a package of straight bonds and call options. Inspection of Table 1 and the supplement indicates what holds for vanilla CBs and COs does not hold for RCBs and RCOs. A direct link between RCBs and RCOs is thus not so obvious at first glance, but is readily available from the analysis of Section 4.1. From (26) and (28), we have, assuming without loss of generality that Z = N , that V (S, t; K)

=

e C(S, ˆ ˜a , u) Ft + nβ C(S, t; K) + (nα − nβ )C( t; αK, T ), t; C b ∂C ˆ ˆ −nβ (C(S, t; bK, u) − C(S, t; aK, u)) t=u ∂S

(37)

so a RCB can be decomposed into a portfolio that consists of straight bonds, RCOs, compound options and vanilla calls. We can manipulate (37) to arrive at V1 (S, t; K)

=

Ft + nβ C1 (S, t; K) + (nα − nβ )C2 (S, t; K)

(38)

b ∂C b ˆ ˆ +(nβ − nα )C(S, t) − (nβ C(S, t; bK, u) − nα C(S, t; aK, u)) t=u ∂S for RCBs with case 1 resets. An illustration with the example from Section 5.1 can be found in Table 3. We see, from Table 3, that the decomposition (38) is perfect for the value, delta and gamma but not so

16

for the vega. This is because the numbers of units of the vanilla call options to be held are themselves sensitive to volatility. As such, an adjustment of b ∂2C t=u ∂σ∂S −(655.480 × 761.222 − 524.384 × 498.183) × 0.699 ≈ −166067.571 ˆ ˆ −(nβ C(S, t; bK, u) − nα C(S, t; aK, u))

=

is necessary, which takes the vega to 1049106.564 − 166067.571 = 883038.993 as required. This should serve as a warning that the connection we give here between RCBs and RCOs are unstable in the sense that, unlike put-call parity or vanilla CBs, the decomposition is sensitive to parameters, volatility in particular. Constituent Bond part F Case 1 RCO C1 Case 2 RCO C2 b Forward-start CO C ˆ Vanilla call C(S, t; bK, u) ˆ Vanilla call C(S, t; aK, u)

Unit

Value

1.000 655.480 -131.096 131.096 -158.420 126.736

829140.393 806.319 819.205 519.624 761.222 498.183

Total NI

Delta 829140.393 528525.802 -107394.530 68120.634 -120592.772 63137.694

0.000 0.753 0.724 0.242 0.908 0.758

Gamma 0.000 493.501 -94.870 31.684 -143.778 96.104

0.000 2.541(-4) 3.167(-4) 0.000 2.158(-4) 4.066(-4)

Vega 0.000 0.167 -0.042 0.000 -0.034 0.052

0.000 1580.622 1722.556 1501.915 522.139 983.800

0.000 1036066.133 -225820.161 196895.017 -82717.331 124682.906

1260937.221 126.094

382.641 0.730

0.142 5.177(-3)

1049106.564 104.911

126.094

0.730

5.177(-3)

88.304

Table 3: Decomposing a RCB. We illustrate using the RCB with case 1 reset from Section 5.1 and show the breakdown of its value and main greeks. For each constituent, we tabulate its unit quantity value and total contribution. The double row “Total” displays the quantities in its natural units at the top and in market convention terms at the bottom; the row “NI” recalls the corresponding quantity in market convention terms from the calculations in Section 5.1. The current share price is 2150; nβ = 655.480; nα = 524.384; and the delta of a forward-start RCO at reset time is 0.242. For case 3 resets, we have b ∂C (39) t=u . ∂S The decomposition for RCBs with case 2 resets can be determined via (34) and (38)-(39) given the value of a forward-start CB V4 . Note that although the decomposition presented here is valid up to the time of reset, adjustments may be necessary on reset. Take, for example, V1 from (38). If, just before the reset, we have S < bK so that the conversion price is set to βK, the decomposition is exact; if, on the other hand, that S ≥ bK, the decomposition is correct only in terms of total portfolio value immediately on reset and adjustments are required to maintain an exact decomposition from then on. ˆ V3 (S, t; K) = Ft + nβ C3 (S, t; K) − nβ C(S, t; bK, u)

5.4

The floor, cap and the limits

We would like to see the effect of varying the floor b, cap a and the limits β, α on the values and the main greeks of RDs. For want of space, we show only selected plots for RCBs here, a full range of plots can be found in the supplement. Let us first fix a = α = γ = 1 and vary β = b in decrements of 0.2 from 1 to 0, we show the values and deltas for RCBs in Figure 4. The figures show the effect of lowering the floor with the lower limit, and we see that altering the floor along with the limit can have a significant effect on the RD, especially at low share prices. Two observations from Section 4.1 are apparent - the gamma can be negative for RCBs and RCOs (RPOs) when there is a floor (cap) to the reset, and that the deltas of RCBs with floored resets can be very large whereas the deltas of ROs appear to be bounded.12 As further support to the 12 Note

that α is fixed at 1, if this is much lower, the deltas of RCB with capped resets can also be large.

17

two observations, we have decomposed the deltas of the RDs by their constituents in accordance with (26)-(28) with results in the supplement. 150

140

1

120

Delta

Value

130

1.5

Parity 1.0 0.8 0.6 0.4 0.2 0.0

110 0.5 100

1.0 0.8 0.6 0.4 0.2 0.0

90

80 0

500

1000

1500

2000

Share price

0 0

2500

500

1000

1500

2000

Share price

2500

3000

Figure 4: The values (left) and deltas (right) of reset convertible bonds with case 1 resets as b = β varies from 1 to 0 in decrements of 0.2. To finish off this section, we would like to look at two more examples. Let us remove the cap (floor) of the reset and vary b = β (a = α) from 1 to 0.2 in decrements of 0.2. We show the deltas in Figure 5 for RCBs - our two observations regarding large deltas for RCBs and negative gammas are further reinforced. 1.5

5

1.0 0.8 0.6 0.4 0.2

1.0 0.8 0.6 0.4 0.2

4.5 4 3.5

1

Delta

Delta

3 2.5 2 0.5

1.5 1 0.5

0 0

500

1000

1500

Share price

2000

2500

3000

0 0

500

1000

1500

2000

Share price

2500

3000

3500

4000

Figure 5: The deltas of reset convertible bonds with capless (floorless) resets as b = β (a = α) varies from 1 to 0.2 in decrements of 0.2. We show the capless (floorless) case on the left (right).

6

Further reset structures

We now generalise the prototype reset structure (8) considered in the previous sections and examine discontinuous resets. For want of space, we show only selected plots for RCBs here, a full range of plots can be found in the supplement.

6.1

Generalising the prototype reset

Consider the reset structure: 8 > < αKu− g¯(Ku− , Su− ) = βKu− + > : βK − u

(S−bKu− )(α−β) (a−b)

18

if Su− > aKu− , if bKu− ≤ Su− ≤ aKu− , if Su− < bKu− .

(40)

It differs from (8) in that, unless b = a and β 6= α, g¯ is automatically continuous by construction; g¯ is identical to (8) if b/β = a/α. Unlike g, the values of β and α are required even when b = 0 or α = ∞ as we do not have the parameter γ in g¯. As we shall see shortly, these seemingly innocuous differences can make a significant difference to the value and risk characteristics of RDs. In moneyness terms, the reset (40) corresponds to 8 if Mu− > a, > < Mu− /α (a−b)Mu− if b ≤ Mu− ≤ a, Mu (Mu− ) = (41) β(a−b)+(Mu− −b)(α−β) > : Mu− /β if Mu− < b, with f¯(Mu− ) = Mu defined as per (41). The method of Section 3 can be applied to value RDs with resets given by g¯, the only difference is that for ROs, the second line of (19) changes to ¯ C(M, u− ) =

¯ f¯(M ), u) (a − b)M C( if b ≤ M ≤ a. β(a − b) + (M − b)(α − β)

(42)

We would like to first look at the effect of varying just b or β. Let us set the default values at b = β = 0.6 and a = α = 1 and look at the following cases: • Case 5, β varies from 1 to 0.1 in decrements of 0.2; • Case 6, β varies from 1 to 1.9 in increments of 0.2; and • Case 7, b varies from 1 to 0 in decrements of 0.2. The function f¯ is shown in Figure 6; the expected strike/conversion prices and the functions g¯ and f¯ across these three cases can be found in the supplement. We see a big difference between f and f¯ in that when we have bK ≤ S ≤ aK just before the reset, f resets the moneyness to an uniform constant 1/γ; this, however, is not the case for f¯ where the moneyness on reset over the interval [b, a] is neither constant nor linear. The value and risk characteristic of the RDs exhibit profiles significantly different to their prototype reset counterparts. For want of space, we show the values and of RCBs with cases 5 reset in Figure 7. We have two new observations in that the delta of RCBs can be negative and the delta of RPOs can fall below -1 from. For RCBs, we see from (29) and the figures that although the variation of b has an effect on the delta, its effect is confined in that the delta cannot exceed a maximum delta that is determined by β. This is confirmed by a further calculation with case 7 resets with β set at 0.15 where we show the RCB values in Figure 7. Three-dimensional plots for a RCB can be found in the supplement - note that the vega is clearly not single-signed everywhere. Why should it be the case that once the prototype reset g is extended to g¯, the behaviour of the RDs can significantly differ? We believe this is due to the loss of linearity and monotonicity in f¯ when b ≤ M ≤ a just before reset - compare Figure 6 with the left panel of Figure 1, and (42) with the middle row of (19). The negative delta persists, to a lesser extent, even if the RCB is not simple nor European, refer to the supplement for an example. It is possible to define a generalised capless and floorless reset by setting b = 0, and it can be shown that provided β 6= 0, the loss of linearity in f¯ induces a delta and gamma on RCBs and gamma on ROs. Refer to the supplement for further details.

6.2

Discontinuous resets and the waviness property

Liao and Wang [42] reported the phenomenon of waviness in some RDs in that the delta and gamma oscillate around zero as we approach a reset time with the share price near some specific levels. While it was known from the early literature on RDs that the delta and gamma jump across reset dates [24, 33, 25],

19

3

1.0 0.8 0.6 0.4 0.2

Moneyness on reset

2.5

2

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

Moneyness just before reset

1.2

Figure 6: The function f¯ with b = 0.6 and a = α = 1 as β varies from 1 to 0.2 in decrements of 0.2. 200

400

180

350

300

Parity

Value

160

140

250

200

120

Parity 1.0 0.8 0.6 0.4 0.2

100

80

Parity 1.0 0.8 0.6 0.4 0.2 0.0

500

1000

1500

2000

Share price

2500

3000

150

100

3500

0

500

1000

1500

2000

2500

3000

Share price

3500

4000

4500

5000

Figure 7: The values of reset convertible bonds with case 5 reset (left) and case 7 reset with β = 0.15 (right). the oscillations in the delta and gamma seem intriguing. Liao and Wang [42] considered a reset structure that is critically different to the ones we have seen thus far in that the reset is discontinuous and we now show that it is the discontinuity of the reset that is causing the oscillations.13 Let us restrict attention to just one reset for expositional convenience and consider a reset gˆ with L 13 We have, in fact, seen a discontinuous reset in case 7 with b = 1 in the previous subsection. The waviness property of the delta and gamma should not be confused with the spurious oscillations we may get in solving the pricing equation with discontinuous initial data using certain schemes. Specifically, spurious oscillations arise only in the case of the Crank-Nicolson scheme where the timestep exceeds a certain size and not in fully implicit or stable explicit schemes [52]. As we were using the latter two schemes, the numerical solution methods cannot be the source of the waviness.

20

levels at time u given by 8 < K0 Ki gˆ(Ku− , Su− ) = : KL

if Su− < D1 , if Di ≤ Su− < Di+1 , 1 ≤ i ≤ L − 1, if Su− ≥ DL ,

where D1 < . . . < DL . On reset, the strike/conversion price takes value from one of K0 , K1 , . . . , KL . Note that the strike/conversion price just before the reset Ku− takes no role in gˆ, but since we are considering just one reset, we can define Ki = ki Ku− and Di = di Ku− so that we can apply the method of Section 3 in a straightforward manner in valuation. A sample illustration of gˆ and fˆ can be found in the supplement. We show, for L = 5, the values and main greeks of RCBs at various times prior to the reset in Figure 8. Compared to the continuous resets in previous sections, there are clearly oscillations in the delta, gamma, theta and zeta; and to some extent, the value, vega, rho and lambda. 140

130

2

On reset −0.01 −0.05 −0.15 −1.15 −2.15

1

0

Delta

Value

120

110

−1

100

−2

90

−3

80 0

500

1000

1500

Share price

2000

−4 0

2500

3

120

2

100

1

On reset −0.01 −0.05 −0.15 −1.15 −2.15 500

1000

1500

2000

Share price

2500

On reset −0.01 −0.05 −0.15 −1.15 −2.15

Vega

Gamma

80 0 60

−1 40 −2

On reset −0.01 −0.05 −0.15 −1.15 −2.15

−3

−4 400

600

800

1000

1200

Share price

1400

1600

1800

20

2000

0 0

500

1000

1500

Share price

2000

2500

3000

Figure 8: The values and the main greeks of reset convertible bonds with a discontinuous reset with L = 5 at various times prior to the reset. Going from left to right and from top to bottom, we have values, deltas, gammas and vegas. The oscillation in the delta and gamma is due to the discontinuities in the value of the RD across the reset levels Di s on reset. The value of the RD on reset is given by ˆ k0 θ(D1 − S) + H(S, u) = H

L−1 X

ˆ k θ(S − DL ), ˆ k θ(S − Di )θ(Di+1 − S) + H H i L

(43)

i=1

which, following an argument similar to that of Section 4.1, will lead to a decomposition in terms of the vanilla instrument, compound options, and digital calls.

21

For RCOs, (43) leads to C(S, t)

=

ˆk0 + C

L X e C ˆk , t; C(D ˆ i , t; Ki , T ), u) − C( e C ˆk , t; C(D ˆ i , t; Ki−1 , T ), u)) (C( i i−1 i=1

L X ˆ i , t; Ki , T ) − C(D ˆ i , t; Ki−1 , T ))B(S, t; Di , u) + (C(D

(44)

i=1

where B is a digital call option. Similarly, for RPOs, we have P (S, t)

=

Pˆk0 +

L X ˆ i , t; Ki−1 , T ), u) − Pe(Pˆk , t; Pˆ (Di , t; Ki , T ), u)) (Pe(Pˆki−1 , t; C(D i i=1

+

L X

(Pˆ (Di , t; Ki , T ) − Pˆ (Di , t; Ki−1 , T ))B(S, t; Di , u);

(45)

i=1

and for RCBs, we obtain V (S, t) =

ˆk0 + Ft + nk0 C

L X e C ˆk , t; C(D ˆ i , t; Ki , T ), u) − nk C( e C ˆk , t; C(D ˆ i , t; Ki−1 , T ), u)) (nki C( i i−1 i−1 i=1

L X ˆ i , t; Ki , T ) − nk C(D ˆ i , t; Ki−1 , T ))B(S, t; Di , u). + (nki C(D i−1

(46)

i=1

ˆ i , t; Ki , T )s and Pˆ (Di , t; Ki , T )s are constants. Note that the C(D The decompositions (44)-(46) contain digital calls with strikes at D1 , . . . , DL that expire on reset. As we approach a reset date, the delta of the RD near the share price levels of D1 , . . . , DL therefore approaches the delta function, causing oscillations in the delta; the contributions of the compound options and the vanilla instrument to the delta are relatively insignificant. Note, from (44)-(46), that we are short (long) digital calls with RCBs and RCOs (RPOs) and thus the downwards (upwards) direction of the spikes in the delta. The oscillations in the gamma, theta and zeta are in both directions because the gamma and theta of binary calls are double-signed and looks like a single cycle of a sine wave centered near the strike. These oscillations fade out as the time to reset increases because the “amplitudes” of the delta and gamma of binary calls decreases with increasing time to maturity. The hedging of digital options or portfolios with similar payoffs is known to be problematic [61, pp 102-103] and sensitive to the underlying share price dynamics and volatility specification [19]; further it was shown by Gobet and Temam [22] that the discrete hedging error converges at a slower rate for options with irregular payoffs, such as digitals. When faced with the need to dynamically hedge RDs with discontinuous resets, it may perhaps be preferable to hedge the compound option parts statically and leave the digital parts unhedged and accept a possible but bounded error, or perhaps acquire positions in asset-or-nothing calls to complete the call options.

7

Summary

We have, in this article, proposed a general method to value RDs with snapshot resets. Our method remains two-dimensional and compares well with other established methods, especially when there are multiple resets. We have also offered, in the absence of free boundaries, a decomposition of RDs with one reset as a package of vanilla instruments and compound options and analysed the behaviour of the greeks

22

via extensive examples and extensions to other reset structures. We hope to have made a contribution to both the theoretical and practical aspects of RDs with this article. We can use the framework of Section 3 to value a selection of elementary exotics with strike resets, such as asset-or-nothing and digital options and some barrier options [66]. As the payoff condition for a mandatory convertible differs from that of a CB only in the loss of the option to redeem, we can equally value mandatory convertibles with resets [46] with the same framework. Further work on the valuation of reset derivatives by the authors is ongoing. We are extending the analysis in the present article to window resets, where we speculate that the similarity reduction we have here helps. As we avoided an extra dimension, we could afford to introduce an extra dimension and yet be able to pursue valuation via PDEs, a two-factor CB model for example. We do, however, suspect that the compound option representation may however break down when there are more than one risk factors. In Footnote 8, we mentioned a link between resets and dividends, this may have implications for dividend protections, meriting further attention. There are instances where the similarity reduction is inadequate, for example: a discrete cash dividend on the underlying, multiple interdependent resets reset structures where the strike/conversion price on reset is bounded, or a CB model with credit risk where the hazard rate is given as a function of the share price. The obvious method is, of course, to accept an extra dimension but it may be worthwhile exploring methods that avoid the extra dimension, possibly at the cost of a minor loss of accuracy. We hope to present our findings on these extensions in a follow-up article. Our proposed method is valid only in the classical Black-Scholes world where the parameters are at most t- and M - dependent. It is widely discussed in the literature that exotic options is subject to volatility model risk - take, for example, cliquets [51, 63]. In addition, compound options are second order and care should therefore be taken as their payoffs are in terms of the market value of the underlying option, not the Black-Scholes value [61, p 183-185]; we also have to be confident with our prediction of some future values, eg the strikes of the compound options in (26)-(28). As such, further research on the volatility specification for the valuation of RDs, which we leave for others, would be of interest. We have seen, from the numerous examples presented, that the presence of a reset may pull the value of a RD below its intrinsic value. For RCBs and RCOs in the absence of dividend, we need only look at the moment just before a reset is due if it is moneyness-robbing [47]. An analysis of the exercise boundaries of American RDs, which we also leave for others, may also be of interest.

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