The Complex Structure of Yields with both Default and Call Risk
By
Shengguang Qian University of Oklahoma and Riskmetrics
S. Lakshmivarahan George Lynn Cross Research Professor
and
Duane Stock University of Oklahoma
April 17, 2006 1. Introduction
The valuation of corporate bonds has always been an important topic for finance researchers. Of course, bond issuers want to know what factors affect prices and yields as yields represent their cost of capital. Prospective bond buyers also wish to know how sensitive yields and yield spreads are to various relevant factors (e.g. leverage) as they develop investment strategies. Many of the modern models for valuing corporate bonds and estimating the credit spread over equal maturity risk free bonds can be traced to
Merton (1974) who modeled zero coupon corporate bonds using option pricing theory wherein the stockholders have an option to default at maturity and thus turn the firm over to the bondholders. Some of the more cited and tested theoretical models after Merton are by Geske (1977), Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-
Dufresne and Goldstein (2001).
Of course, there have been numerous empirical studies to test alternative theories of bond valuation and spreads. For example, Elton and Gruber (2001) find that expected default explains a relatively small part of spreads over default risk free rates but stress that state taxation and systematic factors explaining risk premia for common stocks to be very important. Collin-Dufresne, Goldstein, and Martin (2001) also find a strong and common systematic component. Eom, Helwege, and Huang (2004) perform a rigorous test of all five theoretical models listed above. Some (e.g. Merton) of the structural models they test underestimate observed spreads while others overestimate.
The interest in models of corporate bond values and spreads has recently intensified due to the tremendous growth of the credit derivatives market. The growth has been a cause of concern for regulators who fear that participants may not fully understand the
2 derivative instruments. Furthermore, regulators are very concerned the growth is so dramatic, those keeping track of credit derivatives may not have adequate systems in place to document all the trades and positions taken.1 Academic researchers have thus naturally taken great interest in credit derivatives. See, for example, Longstaff, Mithal, and Neis
(2005) and Blanco, Brennan, and Marsh (2005) who have analyzed the credit default swap market. More specifically, they have analyzed the relation between corporate bond spreads and credit default swaps. Of course numerous features of individual firms bonds can affect credit spreads and credit default swaps values. An outstanding example of a credit default swap with dramatic implications is the GM credit swaps. It is easy to imagine specific details of GM’s debt affecting the credit spread to dramatically affect swap value. Of course, small changes in the GM spread have important implications where one example is given below.
Traders of GM credit-default swaps last week demanded upfront payments in addition to annual premiums to protect debt payments by the Detroit-based company. By doing so, the market relegated GM to the same status that Delphi and Delta Air Lines Inc. had just before those companies defaulted. The annual cost of insuring $10 million of GM debt for five years using default swaps rose to a record of $2.35 million upfront plus $500,000 a year, compared with an annual premium of about $1 million early last week, according to Deutsche Bank prices. The debt-insurance contracts changed hands at about $260,000 at the start of this year, according to Bloomberg data.2
Research on corporate bond valuation and credit spreads is clearly not yet complete. The impact of some factors affecting corporate bond valuation and spreads has not been fully considered. For example, Longstaff, Mithal, and Neis (2005) and and
Blanco, Brennan, and Marsh (2005) examine spreads for small samples of individual
1 For example, see “Derivative Firms Tackle Backlog” by Ramez Mikdashi and Mark Whitehouse, March 14 ,2006, Wall Street Journal. 2 Information from Bloomberg.com, March 17, 2006.
3 bonds in great detail but do not consider the impact of a call feature. It is very easy to believe that the existence of a call feature can significantly affect yields. For example, see
Stanhouse and Stock (1999) for analysis of how bond features (e.g. maturity, call protection call price, etc.), economic conditions (e.g. slope of the term structure) can affect yield by many basis points. Duffee (1998) criticizes the empirical test Longstaff and
Schwartz (1995) perform to test their own model. Specifically, the Longstaff and Schwartz
(1995) model suggests that spreads fall when the level of yields rises. However, Duffee
(1998) maintains their tests results are clouded because the call feature (not included in the
Longstaff and Schwartz theoretical model) is clearly affected by the level of interest rates.
Besides the call feature, the impact of the firm’s growth rate, volatility of value, and shape of the term structure on bond valuation and spread have not been fully analyzed.3
The first purpose of this research is to analyze how default risk and call features affect yields at a specific maturity under varying conditions for the firm and the financial environment. The closely related second purpose is to compute the term structure of par coupon corporate bond yields where we include detailed analysis of factors previously ignored or underdeveloped (such as the call feature, and other factors mentioned above).
Firms may wish to examine these term structures to, for example, find a maturity that offers an attractively low yield. Also, we develop parallel analysis spread that is easily derived from the above term structures.
As an example of how one factor can have complex impacts on term structures and spreads, consider the call feature. Call option value (OV) could increase or decrease with credit quality where the impact on credit spread is also unclear. Let us consider ways how
3 Archarya and Carpenter (2002) analyze, among other things, the correlation between interest rates and firm value and the call feature.
4 lower credit quality could affect OV. On one hand, relatively lower credit quality could reduce the value of the call option because the call would disappear if the firm defaults and disappears. On the other hand, credit quality could increase because option values increase with greater volatility of the underlying instrument (the bond) . 4 This is similar to the reasoning that a call has greater value the lower the credit quality because the firm’s credit rating could increase multiple steps whereas, in contrast, a AAA rated bond cannot realize an improvement in credit quality. Along with above, consider that volatility in firm growth can increase OV even though volatility is considered a risk factor that decreases the value of the firm.
2. The Model
Our model has five primary modules as given in Figure 1. The first represents the risk free interest rate process. The second represents the value of the firm and the third represents the probability of default where the probability of default is largely determined by the value of the firm. The fourth represents the recovery rate in case default occurs and the fifth is bond pricing process.
For the risk free process we choose the Heath, Jarrow, Morton (HJM) model
(1990, 1992). The HJM model has been a very popular to model interest rate processes and many variations of it have been developed. In the context of HJM, drift functions of the interest rate have to be restricted and the restriction results in the drift being a function of interest rate volatility. We choose the discrete variation developed by Grant and Vora
(1999). They develop drift adjusted terms (DAT’s) for forward rates that prevent arbitrage.
Their technique assumes normally distributed forward rate functions.
4 King (2002) suggests this relation.
5 The interest rate process for forward rates is
f (t,T) (t,T) [t,T,f (t,T)] z(t)
where f (t,T) is the forward interest rate from T to T + Δt as seen at time t. The discrete change in the forward rates over interval Δt is defined as
Δf (t ,T) = f(t + Δt, T) - f(t,T)
δ (t,T) is the drift of the forward rate and σ [t,T, f(t,T) ] is the volatility of the forward rate which can depend on both time and level of interest rates. Δt is the change in time which, without loss of generality, we assume to be 1 and Δz(t) is normally distributed (0,Δt2). An alternative expression for the forward rate process is
f (t +1 ,T) - f (t ,T) = δ(t,T) + σ(t,T) Δz(t) .
Our process for the value of firm is derived from the process
dV = αV dt + σV dz .
Here αV is the expected growth rate net of firm payouts, σV is the volatility of growth around the trend and z is a standard Gauss-Weiner process. See Merton (1974),Parrino,
Poteshman, and Weisbach (2005) and Acharya and Carpenter (2002) among others for similar models of firm value. It is useful to note that firms with high expected growth frequently have high growth volatilities and we will explore the implications of this on term structure of par coupon bonds and credit spreads. The joint impact of αV and σV upon call option value is interesting. A high αV may be positively related to option value as the option will less likely disappear due to default and a high σV may also increase call option value. Thus, call option values may be especially high for risky, high growth firms.
This could help explain the tendency for firms/bonds of lower credit quality to almost
6 always have a call option as they are perhaps more valuable to the firm (compared to lower risk, higher credit quality firms.)
Our process for recovery rate (RR) in case of default depends on the probability of default (PD) and is a simply the regression from Altman, Resti and Sironi (2005)
RR 0.1457(PD)0.2801
This expression is consistent with the commonly held view that recovery rates are negatively related to the probability of default. The relation is due to the logic that recoverable assets are typically worth less during periods of weak growth or recession.
The fifth module is the bond pricing framework given in Figure1 where this figure shows he bond price depends on the probability of default and the short term risk free rate
(ri-1). We find bond coupon rates that result in the bond selling at par (100). Our term structures are thus par coupon term structures. For more details on the modules, please see
Figure 1 and the symbol definitions given.
3. Analysis of Yields for Different Classes of Bonds
Given the above model we compute bond yields (term structures) for maturities of one to 30 years for four classes of bonds. We calculate the coupon necessary for the bonds to sell at par (100). Class A, the simplest, is bonds with no default risk and no embedded call. Class B has no default risk but has an embedded call whereas as class C has default risk but no embedded call. Finally, class D is the most complex as it has both default risk and an embedded call. The value of bonds with a call feature (PCB) is thus the value of an identical noncallable bond (PNCB) less the value of the embedded call option (OV).
7 PCB = PNCB - OV
As the initial stage, we assume term structure is flat for class A. This makes for easier initial comparisons across models; the shape of the spread from class A is the same as the shape of the par coupon term structure. Also a flat term structure approximates the term structure shape in early 2006. For the base case we also assume σr is 0.01 and f0 is
0.07. The base assumed exercise price for the call feature is 100.
We first perform a cross sectional analysis for all four classes as we examine the impact of model parameters such as volatility of interest rates (σr) and growth in firm value
(αv) upon yield. We typically examine comparative yields at 20 years maturity. Second, we compare the qualitative aspects of the shape of the term structures and spreads. For example, some term structures may be positive, some negative, some humped, and some
U-shaped. It is interesting that practically any shape can occur when the great majority of professional analysts would state that the term structure is flat. Third, we analyze the shapes of the spreads for classes B, C and D. That is, we compare the yields of classes B, C and D to that of class A.
Figure A1 is the flat term structure of forward rates (7%) we assume for a bond with no default risk and no call feature. Also, the par coupon curve is flat even as we vary the volatility of interest rates (σr). Figure A2 illustrates that as the forward rate changes from 7% to either 6% or 8%, the par coupon rates naturally shifts in parallel.
Figures labeled B illustrate how par coupons for bonds with no default risk but including a call feature depend upon σr . Generally, any option value should increase with volatility of the underlying instrument induced by greater σr and Figure B1-1 confirms this as the greatest par coupon yield occurs for the greatest σr (1.5%). More specially, in B1-1,
8 if there is no call protection, the par coupon for a 20 year maturity is about 8 basis points
5 greater than class A if σr is 1.5%. One can argue that even if the bond indenture contains a
5 year call protection, there is effectively little or no call protection because, as Kerins
(2001) suggests, firms can circumvent the call protection period. The value of the embedded call is about $1.4 on 100 par. If the bond has call protection of 5 years, in B1-2, the result is very similar.
Keeping maturity constant at 20, if we increase the call exercise price to 102 or
105, the results are very similar in Figure B1-3 and B1-4 where the par coupon yield is about 8 basis points higher for the greatest volatility. Apparently, the par coupon yield is not very sensitive to the exercise price for these parameters. Figures B3-1 and B3-2 suggest the same where the strike price is varied in increments of 1.
Figures labeled C illustrate par coupons for bond with varying degrees of default risk but no call feature. Figure C1 illustrates that yields are insensitive to σr . With the introduction of default risk, additional factors such as value of the firm at issuance (V 0) affect par coupons. We use a base V0 of 150 so that the base debt to assets ratio is 0.67.
Given we assume initial debt is 100, varying V0 effectively changes leverage at issuance.
Also, we illustrate the sensitivity of par coupon yields to differing growth rates in firm
6 value (αv), base 10% , and volatility in growth (σv), base 10%.
Figure C3-1 illustrates how much differing growth rates affect par coupons. Of course, greater growth reduces default risk and gives lower par coupons. For example, at a maturity of 20, a 12% growth has the same par coupon as the risk free case (7%) but a 6% growth means par coupons are about 24 basis points more (7.24%). A 4% growth has a 5 More specifically, firms may have nonrefundable bonds that purportedly cannot be called with funds from lower cost debt. But, firms may call nonrefundable and not identify the source of funds used to retire the debt. Call protection has thus been less popular since the 1990’s. 6 These numbers are representative of those used by Parrino, Poteshman and Weisbach (2005).
9 much greater par coupon such that it does not fit on the graph. If we change V0 to 190 in
C3-2, for 20 year maturity, the par coupon for 6% growth declines to 7.08% while the 12% growth case remains at 7%. The 4% growth par coupon now fits on the graph and is
7.48%.
The impact of increasing σv is to make the firm and bond more risky as the likelihood of default increases and Figure C4-1 illustrates the sensitivity as σv varies from
4% to 12%. If σv is less than 10%, then par coupons behave like a risk free case. At 20 years, the par coupon premium is about 4 basis points for σv of 12%. At 10 years the premium is much larger at 20 basis points.
We now consider the most complex class, D. Instead of 20 year maturity, we choose to focus on a 15 year maturity as the differences due to parameters is stronger. For each of D1- 1 through D1-6 we present yields for different σr . Unless otherwise stated, the yields and yields spreads discussed below are for σr of 1%. In D1-1 and D1-2 ( call protection versus no call protection) the spread over class A at 15 years 5 and 6 basis points, respectively. In D1-3 and D1-4 (higher strike prices of 102 and 105), the spreads are 3 and 1 basis points, respectively. In D1-5 and D1-6 (increasing V0 from 160 to 190), the spread is 5 basis points for both.
In D3-1 and D3-2 we vary αv. At a maturity of 15 years and αV of 6%, the spreads are 56 and 40 basis points, respectively, for V0 values of 140 and 150. If V0 is as high as
190, in D3-4, the spread shrinks to just 15 basis points.
D4 figures show the spread dependent upon σv where we use the V0 values of D3.
As V0 increases from 140 to 190, the spread declines from 15 to 5 basis points where σv is
12%
10 4. Shapes of Term Structures and Spreads
Even though the shape of the risk free, option free term structure is flat (class A), it is obvious from figures already presented that term structures for classes B, C and D are rarely flat. In fact, the alternative shapes run the gamut: flat (for bonds with trivial default risk and trivial option values), positive throughout, negative throughout, humped, and U- shaped. In some cases, the computed value of the embedded call as maturity varies can clearly help explain the term structure shape. Of course, greater option values help explain high yields. Similarly, option values that grow strongly (decline) with maturity can explain positive (negative) term structures.
First consider class B where we vary σr, strike price, and call protection. For call protection we typically assume 5 years call protection or no call protection. In B1-1, we vary σr where the call price is 100 and there is no call protection. The shape of the term structure is roughly U-shaped for high σr but negative for low σr. Apparently, a strong interaction between high σr and increasing maturity results in high option values whereas lower σr do not generate option values that necessarily increase with maturity. The results are very similar for the call protected case in B1-2.
In B1-3 and B1-4, we have call protection and change the strike price to 102 and
105, respectively. Here the term structure is always positive which is explained by the fact that option value grows more consistently and rapidly with maturity (compared to B1-1 and B1-2).
In Figures B3-1 (no call protection) and B3-2 (call protection) we vary the strike price from 100 to 105 in increments of 1. Here the term structure is always negative for
11 low strike prices as option value grows slowly (or even declines) with maturity. In contrast, the term structure is always positive for high strike prices as option value grows more strongly with increasing maturity.
For bonds with no call feature but default risk, class C, we used a base V0 of 150 and vary αv , V0, and σv . In C1 there is no default premium for maturities over 15 years but the shape is negative for maturities less than 15. In C3-1, αv is varied and the term structure is humped for low αv but negative for high αv. C3-2 increases V0 to 190 but the term structure shape is very similar to C3-1 even though the spreads are very small. In C4-
1 we vary σv and the term structure always has a negative shape. In contrast to C4-1, V0 is increased to 190 in C4-2. The latter figure differs from C4-1 in that there is a hump for high σv.
Of course, class D is the most complex bond. In D1 figures we vary σr and then vary αv, σv, and V0 in later figures. For all D1 figures (D1-1 through D1-6 where call protection, strike price, and V0 are varied along with σr ) the term structure is U-shaped for high σr and negative for low σr. D3 figures vary αv using different values of V0 . If V0 is low at 140, then the term structure is negative but for larger values of V0 ( 150, 160, 190), the term structure is humped. D4 figures vary σv and the term structure is always negative.
In class D, the option values do not provide a simple explanation of term structure because default risk complicates the valuation process.
5. Conclusion
12 The need for precision modeling of yield spreads has recently grown tremendously.
This is largely because of the explosive growth of credit default swaps. An important current case in point is the spread for GM bonds. We build a comprehensive model with five modules representing the process for short term free rates, value of the firm, probability of default, recovery rate, and, finally, bond pricing. To simplify matters, we use a flat risk free, call free term structure. Then we compute par coupon term yields for callable risk free bonds, default risky but noncallable bonds, and default risk callable bonds. We compute the sensitivity of yields to various critical parameters such as volatility of the risk free rate. As expected, option values increase with this volatility although the sensitivity depends on the value of other parameters. Also, we compute numerous term structures for the above classes of bonds. We find that the shape of the term structure can literally be anything even though the risk term structure is flat.
13 References
Acharya, V. V. and J. N. Carpenter. "Corporate Bond Valuation And Hedging With Stochastic Interest Rates and Endogenous Bankruptcy," Review of Financial Studies, 2002, 15,1355-1383.
Blanco, R. S. Brennan. ; and I. Marsh, “An Empirical Analysis of the Dynamic Relation between Investment-Grade Bonds and Credit Default Swaps” Journal of Finance, 2005, 2255-2281.
Collin-Dufresne, P. and R. S. Goldstein “Do Credit Spreads Reflect Stationary Leverage Ratios?” Journal of Finance ,2001, 56, 2117-2207.
Collin-Dufresne, P. , R. S. Goldstein and S. J. Martin “The Determinants of Credit Spreads,” Journal of Finance, 2001, 56, 1929-1957.
Duffee, G. “The Relation Between Treasury Yields and Corporate Bond Yield Spreads”, Journal of Finance, 1998, 54, 2225-2241.
Elton , E. J. , M. J. Gruber, D. Agrawal and C. Mann “Explaining the Rate of Spread on Corporate Bonds,” Journal of Finance, 2001, 56, 247-277.
Eom, Y. H. ; J. Helwege; and J.Z. Huang, “Structural Models of Corporate Bond Pricing: An Empirical Analysis” Review of Financial Studies, 2004, 17499-544.
Geske, R. “ The Valuation of Corporate Liabilities as Compound Options” JFQA, 1977, 12, 541-552.
Heath, D., R. Jarrow and A. Morton. "Bond Pricing And The Term Structure Of Interest Rates: A Discrete Time Approximation," Journal of Financial and Quantitative Analysis, 1990, 25, 419-440.
Heath, D. R. Jarrow and A. Morton. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," Econometrica, 1992, 60, 77-106.
King, T. H. D. “An Empirical Examination of Call Option Values Implicit in U.S. Corporate Bonds,” (JFQA, December 2002, vol. 37 (4), pp. 693- 721).
Longstaff, F. and E. Schwartz, Valuing Risky Debt: A New Approach”, Journal of Finance, 1995, 50, 789-820.
14 Leland, H. and K.Toft. Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, Journal of Finance, 1996, 51, 987-1019.
Longstaff, F.; A. S Mithal, and E. Neis,. “Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market” Journal of Finance, 2005, 2213-2253.
Merton, R. “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, 1974, 29, 449- 470.
Parrino, R. ; A. Poteshman, and M. Weisbach, “Measuring Investment Disortortion when Risk Averse Managers Decide Whether to Undertake Risky Projects, Financial Management, 2005, 21-60.
Stanhouse, B. and D. Stock "Variations in the Impact of Embedded Call Features Upon Par Coupon Yields" Journal of Applied Corporate Finance, 1999, 12, 92-99.
15 Figure 1: A Structural Model of Par Coupons
V V
2 Firm Value HJM Vu 2 p dVt Vu vdt v dWv,t dft,T t,T dt t,T dW f ,t Vt p V t n t V 1 p v t f t,T f 0,T ,T d i i ,T dWi u e 0 0 p i1 1 p t n d e v T Vd vt 1 p t,T i t,T t,d e d t p 2 i1 u d Vd
f r 0 K V D
1 Interest Rate (Heath-Jarrow-Morton) 3 4 Default Probability f W Recovery rate t,T t,T t,T t PD 2 mv v v / 2 f t t 2 0.2801 t,T t,T t,T D RR j 0.1457 PD j ftt,T 2mv log m T i i 2 v v v KV j f t t logL mvTv D i t ,T t,T t,T PDi j,Tv V j v Tv i v Tv T 1 1 2 t,T t,T 2 t,T t, j 2 jt1
Strike Call r PD RR Price Protection 5 Risky Callable Bond Pricing Framework
Si j
ri j Si j 1 PDi j Si j C PDi j (Si j C) RRi j 0.5 Call Schedule Si1 j 1 r j 1 1 i1 Si1 j 1 pi1Si j 1 pi1 Si j 1 1 ri1 j 1 0.5 Si j 1
ri j 1
Bond Value Par Coupon Credit Spread Option Value
A Structural Framework for Evaluating Risky Callable Bonds
16 Definitions f t,T : the continuously compounded forward rate observed at time t for an instantaneous transaction to begin at time T
t,T and t,T are the drift and volatility term of forward rate process
Vt : firm’s value process
v and v are the drift and volatility term of firm’s value process
PDi j,Tv : Probability of default for a firm, with market value Vi j, where the firm is financed by equity and a zero coupon bond with face value K and maturity date Tv
K: face value of zero coupon bond associated with the firm
K L is the initial leverage ratio, V0
D: Default barriers. Whenever the firm’s value falls below D, it goes default immediately
RRi j: Recovery rate when default probability is PDi j
Si j: Price of a risky coupon bond at time i, level j.
17 Initial Parameters
Initial Parameters:
For interest rate process:
f0,0,1 0.07 f0,1,1 0.07 f0,2,1 0.07 f0,3,1 0.07 f0,4,1 0.07
r0,1 f0,0,1 0.07
t,T 0.01
For firm’s value process:
v 0.12
v 0.1 t 1 d e v t 0.904837 u e v t 1.105171
V0,1 100
For default probability and recovery rate lattice:
K 100 K: face value D 20 D: default barrier
v 0.12
v 0.1
TV 1
For bond’s value process:
par 100 r pt,k pt,k p 0.5
T 30 T: Maturity t 1
18 A1 noncallable, risk-free
= 0 . 0 1 f = 0 . 0 7 M a x S p r e a d = 0 . 0 1 3 2 3 2 r 0 8 e t 7 . 5 a R
d
r 7 a w r
o 6 . 5 F
6 5 1 0 1 5 2 0 2 5 3 0 T f = 0 . 0 7 M a t u r i t y = 2 9 C a l l = 0 r i s k = 0 M a x S p r e a d = 0 . 0 3 6 8 3 0 7 . 1 n o
p 7 u = 0 . 7 % o r C
= 0 . 9 % r 6 . 9 r a
P = 1 % r 6 . 8 = 1 . 2 % 5 1 0 1 5 2 0 2 5 3r 0 = 1 . 5 % T r A2 noncallable, risk-free
= 0 . 0 1 f = 0 . 0 7 M a x S p r e a d = 0 . 0 1 3 2 3 2 r 0 8 e t 7 . 5 a R
d
r 7 a w r
o 6 . 5 F
6 5 1 0 1 5 2 0 2 5 3 0 T = 0 . 0 1 M a t u r i t y = 2 9 C a l l = 0 r i s k = 0 M a x S p r e a d = 0 . 0 1 8 7 4 r 9 f = 6 % 0 n
o f = 7 %
p 8 0 u
o f = 8 % 0 C
r 7 a P
6 5 1 0 1 5 2 0 2 5 3 0 T
19 B1-1 callable, risk-free f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 0 ) n o r i s k 0 ) ) e 7 . 6 = 0 . 7 % 7 . 6 e y
r r f k
k s i s
i = 0 . 9 %
r r r
( ,
l l 7 . 4 7 . 4 a n
c = 1 %
r o o p n ( u = 1 . 2 % n r o o 7 . 2 7 . 2 p C u
o = 1 . 5 % r r C
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
0 . 3 2
e 1 . 5 u l
d 0 . 2 a a v
e 1 n r o p i t s
0 . 1 p 0 . 5 o
0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T B1-2 non-callable, risk-free f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) n o r i s k 0 ) e 7 . 6 ) 7 . 6
e = 0 . 7 % y r r f
k k s s i i = 0 . 9 % r
r
r ( , l
l 7 . 4 7 . 4 a n
c = 1 %
r o o p n (
u
n = 1 . 2 % o
o 7 . 2 r 7 . 2 p C u
o = 1 . 5 %
r
C r
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1 5
e 1 . 5 u l d a a 0 . 1 v
e 1 n r o p i t s
0 . 0 5 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
20 B1-3 callable, risk-free f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 2 ( 5 ) n o r i s k 0 ) e 7 . 6 ) 7 . 6
e = 0 . 7 % y r r f
k k s s i i = 0 . 9 % r
r
r ( , l
l 7 . 4 7 . 4 a n
c = 1 %
r o o p n (
u
n = 1 . 2 % o
o 7 . 2 r 7 . 2 p C u
o = 1 . 5 %
r
C r
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1 e 1 . 5 u l d a a v
e 1 n r
0 . 0 5 o p i t s
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T B1-4 callable, risk-free f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 5 ( 5 ) n o r i s k 0 ) e 7 . 6 ) 7 . 6
e = 0 . 7 % y r
f r
k k s s i i
= 0 . 9 % r r
r ( , l
l 7 . 4 7 . 4 a n c
= 1 % o
o r p n (
u
n = 1 . 2 %
o
o 7 . 2 r 7 . 2 p C u
o = 1 . 5 % r
C r
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1
e 1 . 5 u l d a a v
e 1 n r 0 . 0 5 o p i t s
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
21 B3-1 callable, risk-free f = 0 . 0 7 = 0 . 0 1 C a l l P r o t e c t i o n = 0 n o r i s k 0 r )
e 7 . 6 7 . 6
e S t r i k e = 1 0 0 ) r f k
k s
S t r i k e = 1 0 1 i s r i
r
, o l
l S t r i k e = 1 0 2
7 . 4 n 7 . 4 a (
c
S t r i k e = 1 0 3 n o o n ( p S t r i k e = 1 0 4 u n o 7 . 2 o 7 . 2 p
S t r i k e = 1 0 5 C u
o r C a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
0 . 2 2
0 . 1 5 e 1 . 5 u l d a a v
e 0 . 1 1 n r o p i t s
0 . 0 5 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T B3-2 callable, risk-free f = 0 . 0 7 = 0 . 0 1 C a l l P r o t e c t i o n = 5 n o r i s k 0 r )
e 7 . 6 S t r i k e = 1 0 0 7 . 6 e ) r f k
k S t r i k e = 1 0 1 s i s r i
r
, o
l S t r i k e = 1 0 2 l
7 . 4 n 7 . 4 a (
c
S t r i k e = 1 0 3 n o o n ( p S t r i k e = 1 0 4 u n
o 7 . 2 o 7 . 2 p
S t r i k e = 1 0 5 C u
o r C a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1
e 1 . 5 u l d a a v
e 1 n r 0 . 0 5 o p i t s
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
22 B3-2 callable, risk-free f = 0 . 0 7 = 0 . 0 1 C a l l P r o t e c t i o n = 5 n o r i s k 0 r S t r i k e = 1 0 0 )
e 7 . 0 5 7 . 6 S t r i k e = 1 0 1 e ) r f k S t r i k e = 1 0 2 s k i s r i
r
S t r i k e = 1 0 3 o , l l
n 7 . 4 a (
c S t r i k e = 1 0 4
n o
7 o n ( p S t r i k e = 1 0 5
u n o o 7 . 2 S p o t R a t e p C u
o r C a
r P a
P 6 . 9 5 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1
e 1 . 5 u l d a a v
e 1 n r 0 . 0 5 o p i t s
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
23 C1 non-callable, risky
f = 0 . 0 7 n o c a l l r i s k = 1 = 0 . 1 = 0 . 1 V = 1 5 0 0 v v 0 ) e 7 . 6 ) 7 . 6
e = 0 . 7 % y r
f r
k k s s i i
= 0 . 9 % r r
r ( , l
l 7 . 4 7 . 4 a n c
= 1 % o
o r p n (
u
n = 1 . 2 %
o
o 7 . 2 r 7 . 2 p C u
o = 1 . 5 % r
C r
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1
e 1 . 5 u
0 . 0 8 l d a a v
0 . 0 6 e 1 n r o p i t
s 0 . 0 4
p 0 . 5 0 . 0 2 o 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
24 C3-1 non-callable, risky f = 0 . 0 7 = 0 . 0 1 n o c a l l r i s k = 1 = 0 . 1 V = 1 5 0 0 r v 0 ) e 7 . 6 = 4 % ) 1 0
e y
r v f
k k s s i i = 6 % r r v
( , l
l 7 . 4 9 a n
c = 8 %
v o o p n ( = 1 0 % u
n
v o o 7 . 2 8 p C u
o = 1 2 % v r C
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
1 2 . 5
2 e u l d a a 1 . 5 v
e 0 . 5 n r o p 1 i t s p
0 . 5 o 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T C3-2 non-callable, risky f = 0 . 0 7 = 0 . 0 1 n o c a l l r i s k = 1 = 0 . 1 V = 1 9 0 0 r v 0 ) e 7 . 6 ) 1 0
e = 4 %
y r v f
k k s s i i = 6 % r
r
v ( , l
l 7 . 4 9 a n
c = 8 %
v o o p n (
u
n = 1 0 % o o 7 . 2 v 8 p C u
o = 1 2 %
r
C v
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
1 e
0 . 4 u l d a a v
e 0 . 5 n r o p i
0 . 2 t s p o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
25 C4-1 non-callable, risky f = 0 . 0 7 = 0 . 0 1 n o c a l l r i s k = 1 = 0 . 1 V = 1 5 0 0 r v 0 ) e 7 . 6 = 4 % ) 7 . 6 e y r v f
k k s s i i = 6 %
r r
v ( , l
l 7 . 4 7 . 4 a n
c = 8 %
v o o p n (
= 1 0 % u n v o o 7 . 2 7 . 2 p C u
o = 1 2 % v r C
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 6
e 1 . 5 u l d a
a 0 . 4 v
e 1 n r o p i t s
0 . 2 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T C4-2 non-callable, risky f = 0 . 0 7 = 0 . 0 1 n o c a l l r i s k = 1 = 0 . 1 V = 1 9 0 0 r v 0 ) e 7 . 6 = 4 % ) 7 . 6 e y r v f
k k s s i i = 6 %
r r
v ( , l
l 7 . 4 7 . 4 a n
c = 8 %
v o o p n (
= 1 0 % u n v o o 7 . 2 7 . 2 p C u
o = 1 2 % v r C
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2
0 . 0 4 e 1 . 5 u l d a a v
e 1 n r o p
0 . 0 2 i t s
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
26 D1-1 Callable, risky f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 0 ) r i s k = 1 = 0 . 1 = 0 . 1 V = 1 5 0 0 v v 0 ) e 7 . 6 = 0 . 7 % ) 7 . 6 e
r y r f
k k s s i i = 0 . 9 % r r r
( , l
l 7 . 4 7 . 4 a n
c = 1 %
r o o p n (
= 1 . 2 % u
n
r o o 7 . 2 7 . 2 p C u
o = 1 . 5 % r r C
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 3
e 1 . 5 u l d a a 0 . 2 v
e 1 n r o p i t s
0 . 1 p 0 . 5 o
0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D1-2 Callable, risky f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 = 0 . 1 V = 1 5 0 0 v v 0 ) e 7 . 6 ) 7 . 6
e = 0 . 7 % y r r f
k k s s i i = 0 . 9 %
r r
r ( , l
l 7 . 4 7 . 4 a n
c = 1 % r o o p n ( u
n = 1 . 2 % o
o 7 . 2 r 7 . 2 p C u
o = 1 . 5 % r
C r
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 2
e 1 . 5 u 0 . 1 5 l d a a v
e 1 n r o
p 0 . 1 i t s
p 0 . 5 0 . 0 5 o 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
27 D1-3 Callable, risky f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 2 ( 5 ) r i s k = 1 = 0 . 1 = 0 . 1 V = 1 5 0 0 v v 0 ) e 7 . 6 ) 7 . 6 e y
r = 0 . 7 %
f
r k k s s i i r r
= 0 . 9 %
( , l r l 7 . 4 7 . 4 a n c
= 1 % o o r p n (
u n
= 1 . 2 % o o 7 . 2 r 7 . 2 p C u
o
= 1 . 5 % r C
r a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
0 . 1 2 2 0 . 1 e 1 . 5 u l d
0 . 0 8 a a v
e 1 n r 0 . 0 6 o p i t s
0 . 0 4 p 0 . 5 o 0 . 0 2 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D1-4 Callable, risky f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 5 ( 5 ) r i s k = 1 = 0 . 1 = 0 . 1 V = 1 5 0 0 v v 0 ) e 7 . 6 ) 7 . 6 e y r = 0 . 7 % f r
k k s s i i r r = 0 . 9 % ( ,
l r
l 7 . 4 7 . 4 a n c
= 1 % o o r p n ( u n
= 1 . 2 % o o 7 . 2 r 7 . 2 p C u
o
= 1 . 5 % r
C
r a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1
e 1 . 5 u
0 . 0 8 l d a a v 0 . 0 6 e 1 n r o p i t s 0 . 0 4 p 0 . 5 0 . 0 2 o 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
28 D1-5 Callable, risky f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 = 0 . 1 V = 1 6 0 0 v v 0 )
e = 0 . 7 % 7 . 6 r ) 7 . 6 e y r f
k k
= 0 . 9 % s
s i i r r r
( , l
l 7 . 4 7 . 4 a = 1 % n c r
o o p n (
= 1 . 2 % u
n r o
o 7 . 2 7 . 2 p C u = 1 . 5 %
o r r C
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2
0 . 1 5 e 1 . 5 u l d a a v
e 1 n r 0 . 1 o p i t s
p 0 . 5 0 . 0 5 o 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D1-6 Callable, risky f = 0 . 0 7 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 = 0 . 1 V = 1 9 0 0 v v 0 ) e 7 . 6 ) 7 . 6 e
= 0 . 7 % y r f r
k k s s i i r r = 0 . 9 %
(
, r l
l 7 . 4 7 . 4 a n c
= 1 % o
o r p n ( u n
= 1 . 2 % o o 7 . 2 r 7 . 2 p C u
o
= 1 . 5 % r C
r a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 1 5 e 1 . 5 u l d a a 0 . 1 v
e 1 n r o p i t s
0 . 0 5 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
29 D3-1 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 4 0 0 r v 0 ) e 7 . 6 ) 7 . 6 e
= 4 % y r
f v
k k s s i i r r = 6 %
(
, v l
l 7 . 4 7 . 4 a n c
= 8 % o
o v p n (
u
n = 1 0 % o o 7 . 2 v 7 . 2 p C u
o
= 1 2 % r C v
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
6 2
e 1 . 5 u l
d 4 a a v
e 1 n r o p i t s 2 p 0 . 5 o
0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D3-2 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 5 0 0 r v 0 ) e 7 . 6 ) 7 . 6 e y r f
= 4 % k v k s s i i r r
( , l = 6 % l 7 . 4 v 7 . 4 a n c
o
o = 8 % p n v ( u n o o 7 . 2 = 1 0 % 7 . 2
p
v C u
o r C
= 1 2 % a
r v a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 2 . 5
2 e 1 . 5 u l d a a v
1 . 5
e 1 n r o p 1 i t s
p 0 . 5 0 . 5 o 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
30 D3-3 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 6 0 0 r v 0 ) e 7 . 6 ) 7 . 6
e = 4 % y r v f
k k s s i i = 6 % r
r
v ( , l
l 7 . 4 7 . 4 a n
c = 8 % v o o p n ( u
n = 1 0 % o
o 7 . 2 v 7 . 2 p C u
o = 1 2 %
r
C v
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 1 . 5
e 1 . 5 u l d 1 a a v
e 1 n r o p i t s 0 . 5 p 0 . 5 o
0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D3-4 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 9 0 0 r v 0 ) e 7 . 6 = 4 % ) 7 . 6
e y
r v f
k k s s i i = 6 % r r v
( , l
l 7 . 4 7 . 4 a n
c = 8 %
v o o p n (
= 1 0 % u
n
v o o 7 . 2 7 . 2 p C u
o = 1 2 % v r C
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2
e 1 . 5 0 . 4 u l d a a v
e 1 n r o p i t
s 0 . 2
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
31 D4-1 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 4 0 0 r v 0 ) e 7 . 6 = 4 % ) 7 . 6 e y
r v f
k k s s i i = 6 % r r
v ( , l
l 7 . 4 7 . 4 a n
c = 8 %
v o o p n ( = 1 0 % u n v o o 7 . 2 7 . 2 p C u
o = 1 2 % v r C
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 1 . 5
e 1 . 5 u l d 1 a a v
e 1 n r o p i t s 0 . 5 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D4-2 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 5 0 0 r v 0 ) e 7 . 6 = 4 % ) 7 . 6 e
v y r f
k k s s = 6 % i i r r v
( , l
l 7 . 4 7 . 4 a = 8 % n c
v o o p n (
= 1 0 % u n
v o
o 7 . 2 7 . 2 p C u
= 1 2 % o
v r C
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2 0 . 6
e 1 . 5 u l d a a 0 . 4 v
e 1 n r o p i t s
0 . 2 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
32 D4-3 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 6 0 0 r v 0 ) e 7 . 6 ) 7 . 6
e = 4 % y r v f
k k s s i i = 6 % r
r
v ( , l
l 7 . 4 7 . 4 a n
c = 8 % v o o p n ( u
n = 1 0 % o
o 7 . 2 v 7 . 2 p C u
o = 1 2 %
r
C v
a
r a P
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
0 . 3 2
e 1 . 5 u 0 . 2 l d a a v
e 1 n r o p i t s 0 . 1 p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T D4-4 Callable, risky f = 0 . 0 7 = 0 . 0 1 C a l l ( P r o t e c t i o n ) = 1 0 0 ( 5 ) r i s k = 1 = 0 . 1 V = 1 9 0 0 r v 0 ) e 7 . 6 = 4 % ) 7 . 6 e y
r v f
k k s s i i = 6 % r r
v ( , l
l 7 . 4 7 . 4 a n
c = 8 %
v o o p n (
= 1 0 % u n v o o 7 . 2 7 . 2 p C u
o = 1 2 % v r C
a
r P a
P 7 7 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
2
e 1 . 5 u
0 . 1 l d a a v
e 1 n r o p i t s 0 . 0 5
p 0 . 5 o
0 0 5 1 0 1 5 2 0 2 5 3 0 5 1 0 1 5 2 0 2 5 3 0 T T
33 34