Calibration and Implementation of Convertible Bond Models* Leif Andersen# Dan Buffum Banc of America Securities First Version: April 2002 This Verson: October 2002 Abstract While convertible bond models recently have come to rest on solid theoretical foundation, issues in model calibration and numerical implementation still remain. This paper highlights and quantifies a number of such issues, demonstrating, among other things, that naïve calibration approaches can lead to highly significant pricing biases. We suggest a number of techniques to resolve such biases. In particular, we demonstrate how applications of the Fokker-Planck PDE allows for efficient joint calibration to debt and option markets, and also discuss volatility smile effects and the derivation of forward PDEs to embed such information into model calibration. Throughout, we rely on modern finite difference techniques, rather than the binomial or trinomial trees that so far have dominated much of the literature. Keywords: convertible bonds, jump-diffusion, Fokker-Planck equation, forward PDE JEL classifications: G12, G13 1. Introduction. While it has long been realized that a framework for pricing convertible bonds should ideally incorporate elements of both equity and debt modeling, practical efforts in this direction have long been somewhat lacking. In particular, there seems to have been considerable confusion and disagreement about how to appropriately and consistently apply a default-adjusted discount operator to cashflows generated by convertible bonds. Early papers with an ad hoc approach to discounting include McConell and Schwarz (1986), Cheung and Nelken (1994), and Ho and Pfeffer (1996). Many of these models do not explicitly model bankruptcy, and as compensation uniformly apply a somewhat arbitrary risky spread to the risk-free discount rate. More recent papers recognize that equity and debt components of convertible bonds are subject to different default risk and *The authors wish to thank Peter Carr, Alex Lipton, Peter Forsyth, Vladimir Piterbarg, and Jesper Andreasen for insights and comments. All errors are our own. # Communicating author: [email protected] 1 attempt more nuanced schemes. An often-quoted example is Tsivioritis and Fernandes (TF) (1998) (later extended by Yigitbasioglu (2001) to multiple factors), which effectively splits the convertible bond into cash and equity components, with only the former being subject to credit risk. A related approach was promoted by Goldman Sachs (1994) and involves careful weighting of risky and risk-free discounting in a binomial lattice. The TF splitting scheme is analyzed in detail in Ayache et al (2002) who conclude that it is inherently unsatisfactory due to its unrealistic assumption of stock prices being unaffected by bankruptcy. With the advances of credit derivatives theory, in particular the reduced-form approachi of Jarrow and Turnbull (1995), the foundation for convertible bond models has recently improved significantly. A key development has been the inclusion of stock price dynamics that explicitly incorporate default events, as well as the explicit modeling of stock and bond recoveries in default. Most commonly, default is modeled as a Poisson event that drives stock prices into some low value and coupon bond prices (and convertible bonds) into a certain, fixed percentage of their notional values. Representative, and quite similar, papers include Davis and Lischka (1999) and Takahashi et al (2001). See Grimwood and Hodges (2002) and Olsen (2002) for comparisons of the approach in these papers against other models in the literature. In recent work, Ayache et al (2002) lay out a solid basis for the numerical computation of convertible bond prices, discussing in detail how modern finite-difference methods can replace the computationally sub-optimal binomial and trinomial trees that pervade most of the literature. With theory and computing techniques now on a relatively solid basis, it remains to be determined how to best parameterize models for convertible bonds. While a number of specific parameterizations have emerged in the literature, e.g. Muromachi (1999), Bloch and Miralles (2002), and Arvanitis and Gregory (2001), these are typically based on empirical observations and do generally not result in a model that will price any particular instrument close to market. In fact, as we shall see, when applied to such simple instruments as stock options and coupon bonds, naively parameterized convertible bond models can yield surprisingly large price biases. In a trading setting where we might be interested in relative value plays, or perhaps want to hedge all or pieces of the convertible bond with options and straight debt (or credit derivatives), this situation is obviously not ideal. In this paper, we will discuss the parameterization and calibration of convertible bond models to quoted prices of straight debt and equity options. That is, in the time- honored tradition of financial engineering we will attempt to “imply” parameters from market quotes on actively traded securities. The treatment of this topic will proceed as follows: in Section 2, we outline our process assumptions and discuss a number of technical issues. Section 3 discusses numerical implementation and analyzes a number of model effects in vanilla options and straight debt. Section 4 discusses forward and 2 backward equations for transition densities, and outlines an algorithm for joint calibration to debt and stock options markets. A number of numerical examples are provided to illustrate typical calibration results. For reference and future tests, Section 5 lists prices for a few standard convertible bonds, while Section 6 discuss certain interesting extensions and avenues for future research. In particular, we briefly illustrate how, in theory, debt markets can be tied together with equity option volatility skews using forward PDEs. Finally, Section 7 concludes the paper. 2. Model. 2.1 Basics Consider the pricing of a convertible bond issued by a company with publicly traded equity S. For most of this paper, we assume that S is the single underlying state variable of our model. (Extensions to stochastic interest rates are straightforward, albeit labor-intensive, and will be discussed in more detail in Section 6). To incorporate the possibility of defaults on the underlying company, we make the standard assumption that default of the underlying company is governed by the first jump of a Cox processii N(t) with a stochastic intensity entirely captured by a functional dependence on the stock price level. Specifically, we let the time t intensity of N(t) be denoted l(,t S) , where l : 2 fi is some well-behaved deterministic function. Before the default time ¡+ ¡+ t =inf{t:N()1}t = we assume that S is a diffusion process driven by a single Brownian motion W(t), independent of N(t), and let the instantaneous diffusion volatility of S be s(t, S) for some smooth, bounded function s : 2 fi . With the risk-free interest rate ¡+ ¡+ r and the instantaneous dividend yield q both assumed deterministic, the risk-neutral stock process can be stated as ( ( )) ( ) dSt(S)t/rtq(t)t(-=)S(-+t)d-+,tt(S),tdW()(td)-(N) tl - s , (1) where t- is defined as the limit of t- e for e fl 0. A few comments to this SDE is in ( ) order. First, notice the drift term l t,S(t- ) which compensates for the expected ( ) ( ) downward drift of the Cox process term: EdN-=t-t S(t)d,t( - ) l , where E ((cid:215)) is t t the time t risk-neutral expectation operator. The drift compensation is required for the ( ) process to satisfy the arbitrage restriction that St(r)ueqxupd[u-(cid:242)()(t)] - be a 0 martingale in the risk-neutral probability measure. Second, notice that we assume that the stock price drops to zero upon default: when N jumps from 0 to 1, dSt(S)(t=-) - and the stock is driven into 0, where it stays. The assumption that equity holders recover essentially nothing on default is reasonable, and consistent with much existing literature; see for instance Davis and Lischka (2001). However, note that if we instead wanted to assume, as in Ayache et al (2002), that some fraction R of the pre-default value of the S ( ) stock is recovered in default, we simply multiply the terms dN(t) and l t,St(dt- ) in 3 (1) by (1- R )iii. For simplicity, however, we throughout use the approximation R » 0. S S As an aside, we notice that the assumption of a bounded s(t, S) ensures that the stock price cannot diffuse to 0 but only reach this value by a default jump. Consider now the pricing of a contingent claim V with maturity T. Writing VV=t S(, ) , the claim value is governed by the following backward PDEiv, subject to a boundary payout condition at T: ¶¶ VV+(rt(q)(t)-+tS(+,S)=(tS,+)S()r¶ (t,tl)S(sV,)lVt(,SR))-t S l1 2 2 2 ( ) . ¶¶ tS ¶ S 2 2 V (2) Here, Rt(S, ) is the recovery value of V in case of default at time t; the recovery value V can be allowed to depend on both time and the pre-default value of the stock price. For securities, such as convertible bonds, paying intermediate coupon cash-flows, additional boundary conditions are obviously needed at each cash-flow date, see Section 5. Further intermediate boundary conditions are needed to capture early exercise options and put/call features, all of which are present in a typical convertible bond. The formulation of such boundary conditions is standard, see for instance Tavella and Randall (2000) for details (see also Section 5). Derivation of the backward PDE (2) is straightforward and follows from the jump-extended Ito lemma for the stochastic differential dV(t), followed by an ( ) application of the standard arbitrage restriction that EdVtr (t)(V)(=t)dt . Its solution t generally requires the application of numerical methods, although it frequently is possible – by the Feynman-Kac Theorem – to state the solution probabilistically, as an expectation. Consider for instance the important special case of a risky zero-coupon bond B(,t T) which pays out $1 at time T if no default takes place before time T, 0 otherwise. In other words, the PDE boundary condition is B(T,T )=1 and the recovery rate in default is 0. From the Feynman-Kac Theorem, the time 0 solution of this PDE is simply BT(0E,)e1E(e0=P,=T)(cid:230)(cid:246)(cid:230)(cid:246)(cid:230)(cid:231)(cid:247)(cid:231)(cid:247)(cid:231) E--+(cid:242)(cid:242)0eT0Tr()u[du(r)u,u(S)u](cid:246)(cid:247),du(u)S udu = - l( (cid:242) ) 0T l( ) , (3) ŁłŁłŁ t>Tł where we have defined (deterministic) default-free zero-cupon bond prices as ( ) P(,tTr)eux=dp-u(cid:242)( ) T , and where 1 denotes the indicator function for the event A. t A As shown in Appendix A, the prices of coupon bonds and credit default swaps can be stated in terms of risky zero-coupon bonds, and vice-versa, making risky zero-coupon bonds an obvious and convenient target for model calibration to non-convertible debt markets. 2.2. Intensity process and specification. 4 An application of Ito’s lemma results in the following pre-default (t <t) dynamics of the intensity process l (suppressing dependency on S): dtld(tS)(t)r=(t)+q(¶t-)tl(dt) +¶ l ( l ) ¶ t ¶ S (4) ¶ 2l ¶ l ++1 St(t)S(d,2)st(S)st(t,)(S)d,2Wt .t < t 2 ¶ S2 ¶ S In particular, we see that the local (log-normal) intensity volatility becomes ss(l,t)S(t,)lS=S¶ ¶ / S- 1 . In general we would expect the local intensity volatility to l be negative (that is, perfectly negatively correlated to the stock) as any reasonable model would have ¶¶ l£ / S 0 to reflect of the fact that companies with high stock prices are less likely to default than are those with low stock valuations. Specific parameterizations suggested in the literature include: l(t,)S/a =b, S+ p l(t,)Slcnd=, S- l(t,)Seexfgp=S(+ ) - , where ab,,..., g and p are constants. The first specification can be found in Takahashi et al (2001) and Davis and Lischka (1999), among others. The second parameterization is discussed in Bloch and Miralles (2002), and the third in Arvanitis and Gregory (2001). While most of the methods developed in this paper are non-parametric and independent of the particular specification of l(,t S) , for many of our numerical experiments we will use the first of these specifications with a=0. Setting a=0 is natural as it implies reasonable asymptotic behavior: lim l = 0 and lim l = ¥ . Sfi¥ Sfl 0 Moreover, for this specification the dynamics of l are particularly straightforward: dtltl(l)/s(p)(r)t(q)t(t=)(-1)Ø,()(-+(,p+)(t+)S-,tsdt .pt SdW)t 1t<t ( )2ø (5) º ß 2 In other words, the volatility of l is just the equity volatility scaled with a factor of –p: s (,t)S(p,t =)S- s . The interpretation of p representing the ratio of equity and spread l volatility makes for particularly convenient estimation of this parameter. In a study on Japanese companies, Muromachi (1999) estimates for p are in the range 1.2 to 2.0 which appears reasonable and consistent with the fact that short-term credit spreads are typically more volatile than stock prices. We notice that the dynamics (5) imply a certain amount of auto-correlation, with the drift of l involving reversion at speed p around a level of qt(r)t(p)-t S(t+1),(1 ) + s ( )2. Appendix C takes a closer look at the long-term 2 properties of (5) and demonstrates that sometimes a stationary distribution exists. In 5 particular, for constant process parameters r, q, and s , the Appendix shows that ( ) lim()0,Et(MlAX)r=- q+ 1s2 . tfi¥ 2 2.3 Hedging. A brief word on hedging in the model above. With two sources of uncertainty (W and N), hedging of contingent claims will involve taking positions in cash and two traded stock-dependent derivatives. For instance, we could take a position in a corporate bond and the stock itself. To develop the specific hedge for this example, let V be the value of the derivative to be hedged, and let H denote the price of the bond used in the hedge. Further let w and w be the hedge positions in stock and bond, respectively, and let P S H be the portfolio of V and its hedges. With the recovery rate on the bond being a constant R , the evolution of this portfolio is H P=+(+t)w()()H(t)w+ StVtc(cid:222) ash H S dtP=d+t(w+)...(),+()( ) ØŒ wSt¶tHSt(tVd)(Wt) t ¶ øœ s ( ) º H ¶ S S ¶ S ß -+-+غ w St(w)(-H)(t)R()V(t()RtdN t ) ( øß) SHH V where for simplicity we have omitted the somewhat cumbersome t- notation. For the hedge to work, the terms in the square brackets must be 0, leading to the following explicit expression for the hedge at time t: ¶ V(t) RtV()(t-) + ww==- V - w ¶ S ; ¶ Vt(H)( t) ¶ HS HR- S t- ¶ HH(t) ( ) ¶ S ¶ S H ¶ S In practice, volatility (“vega”) and interest rate hedges would likely be added to the hedge portfolio. 3. Numerical Implementation. 3.1. Finite difference scheme. We now turn to the solution of equation (2) by finite difference methods. To this end, we introduce z =lnS, set v(,t)z(V, =t) S , R (tz,R)(t,=S) and rewrite (2) as v V ¶ v +=L-vte Rlt(,z z)( , ) , (6) ¶ t V where L is the operator 6 ( ) ¶ ¶ 2 ( ) Lr=tq-+t-te()te()t(e,r)t(t,)e(,)(l)(s,+s- )zzz +1 2 l1 z 2 . 2 ¶ z 2 ¶ z2 Discretizing z-space into buckets of size D z, we can approximate L by the finite difference operator (dropping t and z dependence for brevity) ( ) Lˆr=q- +-+- +lsd1sd2 1l 2r ( ) 2 zzz 2 where d and d are the usual first- and second-order finite difference operators, z zz d fz(f)z[()=(+)]zD--1Dfz z , d fz(f)z[z(f)=zf+z2D-+1(-D)()z] . We z 2D z zz (D z)2 then introduce a time grid 0..=.tt < < t < with D” tt - t and employ a modified 0 1 n ii +1i theta discretization of the PDE (6) in the time-domain: ( ) ( ) D-=DtL- 1+v-tqztLˆ v(t ,)(z1 +)-(,1 ) q ˆ iii i +1 (7) qlq(tel,,(1zR)),, tz(t.e )R+t -z ( z) ( ) iviiv i +1 +1 In general, (7) results in a series of tri-diagonal matrix equations and is stable for q ‡ 1 . 2 For q = 1 (the Crank-Nicholson method), the precision of the scheme is at its maximum 2 (Ot(D+D2 z 2)), making this the preferred choice for most smooth payoff functions. With n time-steps and m z-steps, the total computational effort is Om( n ) for all values of q. 3.2. Numerical example: pricing of European call options and risky bonds. For later use and as an illustration of the scheme above, consider now the pricing of a call option C with time T payout of (ST( )- K )+ (recall the notation x+ =max(,x0) ). In case of default, the stock price drops to 0 and the call becomes worthless, i.e. its recovery value is zerov and Rt(S, ) =0 for all t and S. When we state our computed call C option prices, we follow the market convention of quoting European option prices in terms of their Black-Scholes implied volatilities s (;t, T)K , defined as the solution to imp the equation Ct(S)()t=(e)F-d(K ) - (cid:242)tTqF()u(du)r dudu - (cid:242) tT , (8) + - ln(()S/t[Kr()u(q)]u)(d+u(cid:242)TT) t - – 1s2 - d = t 2 imp – s T - t imp where the right-hand side of (8) is observed in the market. Notice that implied volatilities are always quoted using default-free discounting. 7 Setting l(t,)S/c(0)=S (cid:215)S( )- p and using a constant diffusion volatility of s(t, S)3=0% , Figure 1 shows the term structure of at-the-money (ATM) implied volatilities (from (8)) for different values of c and p. A note: unless otherwise indicated, we use the term “at-the-money” for options with strikes set at the forward value (“at-the- money forward”) rather than at the spot price (“at-the-money spot”). --------------------- Figure 1 here --------------------- Raising c causes an increase in the variance rate of the Cox process N(t) which is ( ) proportional to l (specifically, EdN[(t)E](d)(N,t2t)S=dt ( )=l ). As implied Black- Scholes volatility is an aggregate of diffusion and jump volatility, an increase in c causes an increase in implied at-the-money volatility, as evidenced in Figure 1, panel A. The volatility increase is typically an increasing function of maturity for short- to medium- dated options, yet can be very substantial even for 3-months options. Figure 1, panel B shows that implied at-the-money volatilities fall when the power p is increased. This effect is a consequence of the fact that when p is increased, defaults at high stock prices become increasingly unlikely. As jumps associated with defaults from high stock price levels correspond to a large effective variance, implied volatility will decrease when p is increased, ceteris paribus. In Figure 2 below, we fix the maturity and now consider the effect of default on implied volatilities at different option strikes (the so-called volatility skew). Adding default jumps to a diffusion process will necessarily fatten the lower tail of the distribution, raising prices of low-struck options relative to the pure diffusion setting; this effect is evident in Figure 2. Not surprisingly, the steepness of the jump-induced partvi of the volatility skew increases in c. Comparison of panels A and B in Figure 2 also demonstrate that the effect of default on the volatility skew decreases with option maturity, a typical characteristic of jump-induced skews. --------------------- Figure 2 here --------------------- Finally, to get a feel for the impact on the skew of the stock-dependency of the jump intensity, Figure 3 graphs the 1-year volatility skew for various values of the power p. While increasing the value of p here steepens the smile somewhat, the effect is comparatively mild. --------------------- 8 Figure 3 here --------------------- Having examined call options, we now turn to the pricing of risky bonds. Rather than report directly bond prices B(0,T), we instead prefer to use the concept of a risky term spread s(T), defined as eE- se(TB)T =T (cid:230)(cid:231) P- (cid:242)0Tl(Tu,S(u)d)u(cid:246)(cid:247) = (0,)/(0, ) . (9) Ł ł We note that s(T) is by definition associated with an assumption of zero recovery. For bonds with a recovery rate of R, the quantity s(T)(1 - R) is roughly equal to the bond credit spread (see Duffie and Singleton, 1999). Figure 4 below graph the risky term spread as a function of T, for various scenarios for p and c. --------------------- Figure 4 here --------------------- Broadly speaking, all figures show that risky spreads initially increase in maturity, but ultimately start falling. The former effect is caused by the fact that the local drift of l(,t S) is here typically positive for small t. Indeed, from the term multiplying dt in (5) we see that for small t the drift of l(t,)S/c(0)S= S( )- p becomes approximately ( ) cpp1pr(1)q+ cs2-- +( ) ; this quantity is positive for all cases in Figure 4 (although 2 barely so for the case c=10% in Panel A). Over longer time horizons, mean reversion (see discussion at the end of Section 2.2) will slow down the growth of the expectation of l(,t S) and eventually pull it towards a long-term stationary level of ( ) MAXr 0,q1s2 - + which here amounts to 2.5%. Convexity effects also contribute to 2 risky spreads ultimately falling, as follows from Jensen’s inequality ee- ss(T)TTT >(cid:222) -E(cid:242)0TuEu<(lSS(uud,du(u))) (),( ) -1 (cid:242) T (l( )) , 0 i.e. the risky spreads will be lower than the average intensity, with the discrepancy between the two increasing in T. Increasing volatility will increase the hump in the spread curve, and lowering it will eventually remove the hump altogether; see Figure 5 for an example. --------------------- Figure 5 here 9 --------------------- 4. Calibration to risky bonds and at-the-money options. In Section 2.4, we considered, among other things, the pricing of risky zero- coupon bonds and at-the-money call options in the model (1). Among the conclusions we can draw from our numerical study is that both implied option volatilities and risky credit spreads depend in a complicated way on maturity and the joint parameterization of l(,t S) and s(t, S) . Indeed, even a straightforward parameterization using constant volatility and the time-homogenous intensity l(t,)S/c(0)S= S( )p can give rise to highly non-flat, non-monotonic term structures of implied volatility and credit spreads. In practice, it is extremely unlikely that these term structures will even remotely resemble those observed in the market. We stress that the often-seen practice of simply importing into the model (1) a volatility function s(t, S) maintained on a “usual”, default-free equity option system is highly inappropriate as the resulting model is likely to severely overstate both at-the-money volatilities and the steepness of the volatility skew. In this section, we will dispense with naïve time-homogenous model parameterizations and discuss schemes to explicitly bring the convertible bond model into calibration with risky zero-coupon bonds and at-the-money call options. We will do so by introducing time-dependent functions into both l(,t S) and s(t, S) . Section 6 will discuss generalizations of the procedure that allow for fitting to an entire strike-maturity surface of option prices. 4.1. Fokker-Planck equation. For numerical efficiency, we wish to base our calibration technique on a forward induction technique. For this, we introduce the concept of a (log) state price density ( ) p(,t z, s, y) as the time t price of delivery of a Dirac amount d ln(Ss) - y at time s>t, given that ln(St) =z . The density solves the usual backward Kolmogorov equation (compare with (6)) ¶¶ pp ( ¶ p ) 2 ( ) +-+-+rt(=q)(tt)e(,t)e(t,e+)r(,t)te(l)(s,sp)zzz 1 2 l1 z 2 (10) ¶¶ tz ¶ z 2 2 2 where s and z are considered fixed. The boundary condition is p(s,z,sy,)y( =zd ) - . Notice that we assume that p recovers nothing in default, and consequently associate with p a defect transition density that excludes the singularity at S =0. The formal adjoint to (10) is the Fokker-Planck (or forward Kolmogorov) equation ¶¶ p (( )¶ ) 2 ( ) ( ) ---+-+= +rs()q(s)s(e,)s(e,)p(s,)e(p)lr(,ssse)yyyp1 2 s 1 y 2 l ¶¶ sy 2 ¶ y2 2 10