In recent years both practitioners and academics have realised that traditional discounted cash flow models erroneously consider the option value embedded in firms. Hence equity and debt valuation methodologies based on option theory have recently become quite popular. Such methodologies take inspiration from the Merton (1974) model which was originally introduced to measure the impact of default risk on corporate bonds yields. Thirty years later the Merton model for its simplicity and rigour remains unrivalled and is the basis of some of the most sophisticated credit risk models. In this paper it will be shown how practitioners often improperly adapt the Merton model for aims beyond its original scope.Merton model, option pricing, default risk, corporate bond.

Andrea Gheno

English

Research Papers in Economics

CORPORATE VALUATIONS AND THE MERTON MODEL
Andrea Gheno
Working Paper n° 55, 2005
UNIVERSITÀ DEGLI STUDI ROMA TRE
DIPARTIMENTO DI ECONOMIAUNIVERSITÀ DEGLI STUDI ROMA TRE
DIPARTIMENTO DI ECONOMIA
Working Paper n° 55, 2005
Comitato Scientifico
Alessandra Carleo
Marisa Cenci
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E-mail: dip_eco@uniroma3.itUNIVERSITÀ DEGLI STUDI ROMA TRE
DIPARTIMENTO DI ECONOMIA
CORPORATE VALUATIONS AND THE MERTON MODEL
Andrea Gheno
*
* Dipartimento di Economia, Università Roma Tre, via Ostiense 139, 00154 Roma, Italy (gheno@uniroma3.it).I. Merton Model and Corporate Valuations 5
II. The Asset Volatility 8
III. The Asset Value 10
IV. Synthesis of the Debt 12
V. Final Remarks 14
References 15  5 
Abstract. In recent years both practitioners and academics have realised that traditional 
discounted cash flow models erroneously consider the option value embedded in firms. Hence 
equity and debt valuation methodologies based on option theory have recently become quite 
popular. Such methodologies take inspiration from the Merton (1974) model which was originally 
introduced to measure the impact of default risk on corporate bonds yields. Thirty years later the 
Merton model for its simplicity and rigour remains unrivalled and is the basis of some of the most 
sophisticated credit risk models. In this paper it will be shown how practitioners often improperly 
adapt the Merton model for aims beyond its original scope. 
 
I. Merton Model and Corporate Valuations 
The Merton model is a credit risk structural model and assumes that if assets are not 
sufficient to meet the obligations the firm defaults. Its original aim is to determine the 
appropriate yield for a corporate bond issued by a firm subject to default risk. 
The model is continuous and assumes that the only source of uncertainty that 
influences the firm’s ability to pay in every time t is its asset value  () t V.  I t s  
stochastic evolution is expressed as: 
 
() () () ( ) dz t V σ dt c t V µ t dV V V V + − =                                                   (1) 
where 
V µ   is the expected return rate of  () t V  per unit of time 
V c   is the payment received or remitted by the firm per unit of time    6 
( V c0 <  in the case of new financing ,  V c0 >  in the case of expenditure
1) 
2
V σ   is the instantaneous variance of the asset return per unit of time 
dz  is a standard Wiener process 
Assets can be financed through debts or equity, therefore: 
 
() () () Vt Dt St =+                                                               (2)   
where  () D t  and  () S t  are the value of the debt and the equity at time t. Supposing 
that  () D 0  is the cost at the time t 0 =  of a zero coupon bond with face value B 
and maturity T and the firm is unable to issue new debts or pay coupons or 
dividends ( V c0 = ), the equity value at time T is represented by the payout of a 
European call option on  () V t  with strike price B: 
 
() () () ST m a x0 , VT B =−                                         (3) 
 This value at time 0 can be found using the Black−Scholes (1973) formula as: 
 
() () ( ) ( )
rT
1 2 S0 V0Nd B e Nd − =−                                                    (4) 
where  
r is the risk free rate 
                                                 
1   For example, dividends or coupon payments.   7 
()
x
2 11
Nx e x p y d y
2 2 −∞
 =−  π  ∫  
() 2
V
1
V
V0 1
log r T
B2
d
T
  ++ σ  
  =
σ
 
2 1 V dd T =− σ  
 
The debt value can be found by difference  () () () D0 V0 S0 =−  and the debt yield 
i* by solving the equation  () ( )
T D0 B 1 i*
− =⋅+ . 
It must be stressed that the use of the Merton model is straightforward when the 
debt of the firm is a single zero coupon bond, but otherwise it does not provide 
closed formulae for corporate valuations. 
In Damodaran (1999) it is shown how to use the Merton model in its simplest 
form to value debt and equity of corporations with any capital structure. When 
doing this there are several practical issues that need to be addressed. In particular, 
in this paper the attention will be focused on the three major ones; two are with 
regard to inputs of the model, namely the asset value volatility  V σ  and the asset 
value  () 0 V , and the remainder is in reference to the adjustment procedure used to 
consider the debt as a single zero coupon bond when it is not. 
   8 
II. The Asset Volatility 
The most critical input for options and hence also for real options valuation is the 
underlying asset volatility. In order to estimate the variance of firm returns 
practitioners often use the following relation: 
()
()
()
()
() ()
()
D S SD 2
2
D
2
2
S
2
2
V σ σ ρ
t V
t D t S
2 σ
t V
t D
σ
t V
t S
σ +  


 


+  


 


=                       (5) 
where 
2
S σ   is the instantaneous variance of the equity return per unit of time 
2
D σ   is the instantaneous variance of the debt return per unit of time 
SD ρ   is the correlation coefficient between equity and debt returns 
 
This equation can be misleading. In fact like any other model, the Merton model 
should not be used without taking into account its explicit and implicit 
hypotheses. Considering equation (1) and (2) and the fact that the unique source 
of uncertainty in the model is the asset value  () t V , the implicit dynamics of debt 
and equity values  () t D  and  () t S a r e :  
 
() () () ( ) dz t D σ dt t D , t µ t dD D + =  (6) 
() () () ( ) dz t S σ dt t S , t µ t dS S + =                                                                     (7)   9 
 
where  () () µt , Dt  and  () () µt , St  are the debt and equity value drift terms. 
Since from equation (2) it can be inferred that 
 
() () () t dS t dD t dV + =                                                                             (8)    
 
it follows that the relation between  V σ  and stock and debt volatilities  S σ  and  D σ  
must be: 
 
()
()
()
() t V
t D
σ
t V
t S
σ σ D S V + = .                                                                     (9) 
 
Therefore Merton model’s hypotheses implicitly assume that the correlation 
coefficient  SD ρ  between the random rates of return of variables S and D is 1. This 
gives a definitive answer to one of the “estimation” questions posed in corporate 
finance when dealing with this framework: the correlation coefficient  SD ρ  is not a 
variable but a constant which does not need to be estimated. 
   10 
III. The Asset Value 
An actual estimation issue is about the asset value  () 0 V . For this purpose 
practitioners usually discount expected cash flows at the weighted average cost of 
capital (WACC) which is an increasing function of the asset volatility. 
Therefore if the WACC is used to find the initial asset value  () 0 V , the resulting 
equity value  () 0 S  is a compound function of the asset volatility  V σ : 
 
() () () V S0 f , V0 =σ    where    () ( ) V V0 f =σ                                   (10) 
 
Hence, at any time t, the sensitivity of the equity value  () t S  to variations of  V σ  
can be expressed as: 
() ()
()
()
V V σ
t V
t V
t S
σ
t S
∂
∂
⋅
∂
∂
+
∂
∂
                                                                         (11) 
Equation (11) differs from the vega
2 of a financial option since it includes the 
additional term  ()
()
()
V σ
t V
t V
t S
∂
∂
⋅
∂
∂
. This term is negative since it is the product 
between the call’s delta  ()
() t V
t S
∂
∂
, which is positive, and the sensitivity of the asset 
                                                 
2  The option’s vega is the rate of change of the option value with respect to the underlying asset 
volatility. In this case the option considered is the equity.   11 
value to variations of the asset volatility  ()
V σ
t V
∂
∂
, which is negative. The latter is 
negative because the higher the asset volatility, the higher the equity risk 
premium
3 is (hence the WACC) and the lower is the resulting asset value. 
Therefore when valuing the equity as a call option it is not appropriate to state that 
the equity value is always an increasing function of the underlying asset volatility 
as it is with financial options. 
This is also in contrast with some real options principles which however do not 
seem to be correct from a common sense point of view
4: 
•  The higher the uncertainty of the economic result of a firm (or a project) the 
higher the value of the call option; 
•  Flexibility in the structure of a firm (or a project) always makes the call 
option more valuable. 
The remarks contained in this section provide a more analytical explanation as to 
why the above statements are inaccurate. 
 
                                                 
3   The equity risk premium is defined as the difference between the cost of equity and the risk-
free rate. 
4   For more details see Mayor (2001).   12 
IV. Synthesis of the Debt  
The debt structure of a company is very seldom as simple as a single issue of zero 
coupon bonds.  However in order to use the Black−Scholes formula to find the 
price of the call option held by the stockholders when the debt structure is more 
complex, a transformation method to convert such debt structure into an 
“equivalent” single issue of zero coupon bonds is required. Unfortunately this 
synthesis process is not acceptable from a probabilistic point of view and leads to 
a great discrepancy from the real world. 
In fact considering the transformation methodology proposed in Damodaran 
(1999), the synthetic zero coupon bond has maturity equal to the face-value-
weighted average of the McCauley durations of the different issues and a face 
value which includes all the principal outstanding on the debt and coupons that 
will become due on existing debt. Therefore the default probability of the liability 
portfolio is entirely concentrated on the maturity date of the synthetic zero coupon 
bond instead of being spread among all the actual servicing dates and the resulting 
corporate valuations reflect such distortion. 
In order to show the magnitude of this phenomenon the discrete structural 
valuation model presented in Cenci−Gheno (2005) is used. This model takes into 
account the default risk involved with any debt structure which is not fully 
captured using the Merton model in its simplest form (i.e. the Black−Scholes   13 
formula) when the debt of the firm is not a single zero coupon bond. Considering 
a firm financed through equity and two issues of zero coupon bonds with face 
values of €500 and maturity of 5 and 10 years, a risk-free rate of 5%, equity and 
debt valuations are shown in Table 1 and 2 respectively using the Black−Scholes 
(1973) formula and the Cenci−Gheno (2005) model with exogenous default. 
 
σ σ σ σV S(0) i* S(0) i* S(0) i* S(0) i* S(0) i* S(0) i* S(0) i*
0,1 82,07 6,82% 149,39 6,05% 231,08 5,64% 321,56 5,43% 416,86 5,33% 514,62 5,28% 613,58 5,26%
0,2 156,13 8,78% 222,80 7,86% 297,32 7,20% 377,91 6,72% 463,12 6,37% 551,84 6,11% 643,22 5,91%
0,3 227,52 11,01% 297,22 10,01% 372,02 9,24% 450,94 8,63% 533,20 8,41% 618,21 7,74% 705,45 7,41%
0,4 295,03 13,56% 368,48 12,50% 445,54 11,64% 525,56 10,94% 609,04 10,36% 692,59 9,87% 778,87 9,45%
0,5 357,67 16,46% 434,93 15,33% 514,86 14,41% 597,00 13,64% 680,98 12,99% 766,53 12,42% 853,41 11,93%
0,6 414,71 19,76% 495,61 18,57% 578,51 17,58% 663,08 16,74% 749,05 16,02% 836,23 15,40% 924,43 14,84%
0,7 465,70 23,53% 549,92 22,26% 635,66 21,20% 722,67 20,29% 810,76 19,51% 899,76 18,82% 989,57 18,21%
1100 1200 1300
V(0)
700 800 900 1000
 
Table 1. Black−Scholes (1973)  − Equity values and debt yields for different input of asset 
volatility and initial asset value 
 
σ σ σ σV S(0) i* S(0) i* S(0) i* S(0) i* S(0) i* S(0) i* S(0) i*
0,1 63,41 6,38% 133,13 5,68% 218,80 5,37% 312,14 5,23% 409,27 5,17% 508,06 5,14% 607,59 5,13%
0,2 136,15 8,22% 203,27 7,35% 277,53 6,71% 360,22 6,30% 445,95 5,97% 536,97 5,77% 629,55 5,61%
0,3 203,32 10,21% 272,66 9,26% 345,37 8,47% 425,63 7,93% 508,65 7,49% 593,29 7,10% 682,67 6,84%
0,4 267,80 12,47% 335,95 11,37% 412,97 10,52% 492,65 9,87% 576,26 9,37% 659,86 8,88% 746,64 8,51%
0,5 326,64 14,95% 399,15 13,73% 476,97 12,83% 556,97 12,06% 641,89 11,51% 727,26 11,00% 812,64 10,51%
0,6 380,32 17,68% 457,01 16,43% 537,22 15,44% 618,39 14,57% 702,76 13,88% 790,28 13,36% 877,81 12,86%
0,7 428,83 20,71% 509,57 19,43% 592,27 18,37% 674,98 17,38% 759,08 16,53% 847,32 15,94% 937,01 15,43%
V(0)
700 800 900 1000 1100 1200 1300
 
Table 2. Cenci−Gheno (2005) − Equity values and debt yields for different input of asset volatility 
and initial asset value 
 
 
As shown in Figure 1 the misspecification of the debt servicing, which 
corresponds to a reduction of the credit risk, leads to overestimates of the equity   14 
value. In general for a given level of asset volatility the higher the leverage the 
higher the overestimate. 
 
700
900
1100
1300
0,1 0,2 0,3 0,4 0,5 0,6 0,7
0,00%
5,00%
10,00%
15,00%
20,00%
25,00%
30,00%
() 0 V
S
(
0
)
 
O
V
E
R
E
S
T
I
M
A
T
E
v σ
700
900
1100
1300
0,1 0,2 0,3 0,4 0,5 0,6 0,7
0,00%
5,00%
10,00%
15,00%
20,00%
25,00%
30,00%
() 0 V
S
(
0
)
 
O
V
E
R
E
S
T
I
M
A
T
E
v σ  
Figure 1. Overestimate of equity value (Black−Scholes (1973) vs. Cenci−Gheno (2005)). 
 
V. Final Remarks 
The Merton model is definitely a powerful and versatile model. As with any other 
financial model in order to implement it properly its assumptions must be taken 
carefully into consideration. With this in mind in section II and III some issues 
about asset value and volatility estimations have been considered. Finally in 
section IV it has been shown how valuations of firms with complex debt 
structures are distorted when unrealistic simplifications are made. 
 
   15 
References 
Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities, 
Journal of Political Economy, 81, 637−54. 
Cenci, M. and Gheno, A. (2005) Equity and debt valuation with default risk: a 
discrete structural model, Applied Financial Economics, 15, 875−881. 
Damodaran, A. (1999) The Promise and Peril of Real Options, Working Paper, 
New York University Stern School of Business. 
Mayor, N. (2001) Jackpot?, Wilmott Magazine, 1, 46−53. 
Merton, R. (1974) On the pricing of corporate debt: the risk structure of interest 
rates, Journal of Finance, 29, 449−470. 
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Corporate valuations and the merton model

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