FIXED-INCOME SECURITIES
Chapter 14
Bonds with Embedded
Options and Options on
Bonds
Outline
• Callable and Putable Bonds
– Institutional Aspects
– Valuation
• Convertible Bonds
– Institutional Aspects
– Valuation
• Options on Bonds
– Institutional Aspects
– Valuation
– Uses
Callable Bonds and Putable Bonds
Bond with Embedded Options
•
Callable bonds
– Issuer may repurchase at a pre-specified call price
– Typically called if interest rates fall
•
A callable bond has two disadvantages for an investor
– If it is effectively called, the investor will have to invest in another bond
yielding a lower rate
– A callable bond has the unpleasant property for an investor to appreciate
less than a normal similar bond when interest rates fall
– Therefore, an investor will be willing to buy such a bond at a lower price
than a comparable option-free bond
•
Examples
– The UK Treasury bond with coupon 5.5% and maturity date 09/10/2012 can
be called in full or part from 09/10/2008 on at a price of pounds 100
– The US Treasury bond with coupon 7.625% and maturity date 02/15/2007
can be called on coupon dates only, at a price of $100, from 02/15/2002 on
– Such a bond is said to be discretely callable
Callable and Putable Bonds
Institutional Aspects
• Putable bond holder may retire at a pre-specified
price
• A putable bond allows its holder to sell the bond at
par value prior to maturity in case interest rates
exceed the coupon rate of the issue
• So, he will have the opportunity to buy a new bond at
a higher coupon rate
• The issuer of this bond will have to issue another
bond at a higher coupon rate if the put option is
exercised
• Hence a putable bond trades at a higher price than a
comparable option-free bond
Callable and Putable Bonds
Yield-to-Worst
• Let us consider a bond with an embedded call option
trading over its par value
• This bond can be redeemed by its issuer prior to
maturity, from its first call date on
– One can compute a yield-to-call on all possible call dates
– The yield-to-worst is the lowest of the yield-to-maturity and all yields-to-call
• Example
– 10-year bond bearing an interest coupon of 5%, discretely callable after 5
years and trading at 102
Yield-to-call
– There are 5 possible call dates before maturity
year 5
4.54%
year 6
4.61%
– Yield-to-worst is 4.54%
year 7
year 8
year 9
4.66%
4.69%
4.72%
year 10
Yield-to-maturity
4.74%
Callable and Putable Bonds
Valuation in a Binomial Model
• Let us assume that a binomial tree has been already
built and calibrated as explained in Chapter 12
• Recursive procedure
– Price cash-flow to be discounted on period n-1 is the minimum value of the
price computed on period n and call price on period n
– And so on until we get the price P of the callable bond
• Example
– We consider a callable bond with maturity two years, annual coupon 5%,
callable in one year at 100
– r0 = 4%, ru = 4.66% and rl = 4.57% (cf. example in Chapter 12)
– We have Pu = 105/1.0466 = 100.32 and Pl = 105/1.0457 = 100.41
– Finally, price of the callable bond
1 ⎛ min (100,100.32 ) + 5 min (100,100.41) + 5 ⎞
P= ⎜
+
⎟ = 100.96
2⎝
1 + 4%
1 + 4%
⎠
Callable and Putable Bonds
Monte Carlo Approach
•
•
•
Step 1: generate a large number of short-term interest rate
paths using some dynamic model (see Chapter 12)
Step 2: along each interest rate path, the price P of the bond
with embedded option is recursively determined
The price of the bond is computed as the average of its prices
along all interest rate paths
Period
1
2
3
4
5
6
7
8
9
10
Path1
4,00%
4,08%
3,83%
4,15%
4,27%
4,69%
4,88%
5,14%
5,24%
5,59%
Path2
4,00%
4,14%
4,02%
3,88%
4,26%
4,49%
5,10%
4,94%
5,47%
5,04%
Path3
4,00%
4,29%
4,35%
4,25%
4,68%
4,33%
5,24%
4,75%
5,15%
5,29%
Path4
4,00%
4,24%
4,27%
3,87%
4,58%
4,29%
5,08%
5,54%
5,26%
5,58%
Path5
4,00%
4,28%
4,24%
4,17%
4,29%
4,47%
5,27%
5,25%
5,43%
5,38%
Path6
4,00%
4,28%
4,23%
4,30%
3,99%
4,32%
4,70%
5,08%
5,64%
5,02%
Callable and Putable Bonds
Monte Carlo Approach - Example
• Price a callable bond with annual coupon 4.57%,
maturity 10 years, redemption value 100 and callable
at 100 after 5 years
• Prices of the bond under each scenario
Price of the callable bond
Path1
100.43
Path2
100.55
Path3
99.90
Path4
99.76
Path5
99.68
Path6
100.55
• Price of the bond is average over all paths
P=1/6(100.43+100.55+99.9+99.76+99.68+100.55)=100.14
• The Monte Carlo pricing methodology can also be
applied to the valuation of all kinds of interest rates
derivatives
Convertible Bonds
Definition
•
•
•
•
Convertible securities are usually either convertible bonds or
convertible preferred shares which are most often
exchangeable into the common stock of the company issuing
the convertible security
Being debt or preferred instruments, they have an advantage to
the common stock in case of distress or bankruptcy
Convertible bonds offer the investor the safety of a fixed income
instrument coupled with participation in the upside of the equity
markets
Essentially, convertible bonds are bonds that, at the holder's
option, are convertible into a specified number of shares
Convertible Bonds
Terminology
• Convertible bonds
– Bondholder has a right to covert bond for pre-specified number of
share of common stock
• Terminology
– Convertible price is the price of the convertible bond
– Bond floor or investment value is the price of the bond if there is no
conversion option
– Conversion ratio is the number of shares that is exchanged for a
bond
– Conversion value = current share price x conversion ratio
– Conversion premium = (convertible price – conversion value) /
conversion value
– Income pickup is the amount by which the yield to maturity of the
convertible bond exceeds the dividend yield of the share
Convertible Bonds
Examples
• Example 1:
–
–
–
–
–
Current bond price = $930
Conversion ratio: 1 bond = 30 shares common
Current stock price = $25/share
Market Conversion Value = (30 shares)x(25) = $750
Conversion Premium = (930 – 750) / 750 = 180 / 750 = 24%
• Example 2: AXA Convertible Bond
– AXA has issued in the € zone a convertible bond paying a 2.5% coupon
rate and maturing on 01/01/2014; the conversion ratio is 4.04
– On 12/13/2001, the current share price was €24.12 and the bid-ask
convertible price was 156.5971/157.5971
– The conversion value was equal to €97.44 = 4.04 x 24.12
– The conversion premium calculated with the ask price 157.5971 was
61.73% = (157.5791 - 97.44)/ 97.44
– The conversion of the bond into 4.04 shares can be executed on any date
before the maturity date
Convertible Bonds
Bloomberg Description
Convertible Bonds
Uses
• For the issuer
– Issuing convertible bonds enables a firm to obtain better financial
conditions
– Coupon rate of such a bond is always lower to that of a bullet bond with the
same characteristics in terms of maturity and coupon frequency
– This comes directly from the conversion advantage which is attached to
this product
– Besides the exchange of bonds for shares diminishes the liabilities of the
firm issuer and increases in the same time its equity so that its debt
capacity is improved
• For the convertible bondholder
– The convertible bond is a defensive security, very sensitive to a rise in the
share price and protective when the share price decreases
– If the share price increases, the convertible price will also increase
– When share price decreases, price of convertible never gets below the
bond floor, i.e., the price of an otherwise identical bullet bond with no
conversion option
Convertible Bonds
Determinants of Convertible Bond Prices
• Convertible bond is similar to a normal coupon bond
plus a call option on the underlying stock
– With an important difference: the effective strike price of the call option will
vary with the price of the bond
• Convertible securities are priced as a function of
–
–
–
–
–
–
–
The price of the underlying stock
Expected future volatility of equity returns
Risk free interest rates
Call provisions
Supply and demand for specific issues
Issue-specific corporate/Treasury yield spread
Expected volatility of interest rates and spreads
• Thus, there is large room for relative mis-valuations
Convertible Bonds
Convertible Bond Price as a Function of Stock Price
Bond
P ric e
C o n v e r ti b l e
Bond
P a rity
S tra ig h t B o n d
S to c k P ric e
Convertible Bonds
Convertible Bond Pricing Model
• A popular method for pricing convertible bonds is the
component model
– The convertible bond is divided into a straight bond component and a call
option on the conversion price, with strike price equal to the value of the
straight bond component
– The fair value of the two components can be calculated with standard
formulas, such as the famous Black-Scholes valuation formula.
• This pricing approach, however, has several
drawbacks
– First, separating the convertible into a bond component and an option
component relies on restrictive assumptions, such as the absence of
embedded options (callability and putability, for instance, are convertible
bond features that cannot be considered in the above separation)
– Second, convertible bonds contain an option component with a stochastic
strike price equal to the bond price
Convertible Bonds
Convertible Bond Pricing Models
• Theoretical research on convertible bond pricing was
initiated by Ingersoll (1977a) and Brennan and
Schwartz (1977), who both applied the contingent
claims approach to the valuation of convertible
bonds.
• In their valuation models, the convertible bond price
depends on the firm value as the underlying variable.
Brennan and Schwartz (1980) extend their model by
including stochastic interest rates.
• These models rely heavily on the theory of stochastic
processes and require a relatively high level of
mathematical sophistication
Convertible Bonds
Binomial Model
• The price of the stock only can go up to a
given value or down to a given value
uS
S
dS
• Besides, there is a bond (bank account) that
will pay interest of r
Convertible Bonds
Binomial Model
•
•
•
•
We assume u (up) > d (down)
For Black and Scholes we will need d = 1/u
For consistency we also need u > (1+r) > d
Example: u = 1.25; d = 0.80; r = 10%
S = 125
S=100
S = 80
Convertible Bonds
Binomial Model
• Basic model that describes a simple world.
• As the number of steps increases, it becomes
more realistic
• We will price and hedge an option: it applies
to any other derivative security
• Key: we have the same number of states and
securities (complete markets)
⇒ Basis for arbitrage pricing
Convertible Bonds
Binomial Model
• Introduce an European call option:
– K = 110
– It matures at the end of the period
S
C (K=110)
uS = 125
Cu = 15
dS = 80
Cd = 0
S=100
Convertible Bonds
Binomial Model
• We can replicate the option with the stock
and the bond
• Construct a portfolio that pays Cu in state u
and Cd in state d
• The price of that portfolio has to be the same
as the price of the option
• Otherwise there will be an arbitrage
opportunity
Convertible Bonds
Binomial Model
• We buy Δ shares and invest B in the bank
• They can be positive (buy or deposit) or negative
(shortsell or borrow)
• We want then,
ΔuS + B (1 + r ) = Cu ⎫
⎬
ΔdS + B (1 + r ) = Cd ⎭
• With solution,
Cu − C d
u × C d − d × Cu
;B =
Δ=
S (u − d )
(u − d )(1 + r )
Convertible Bonds
Binomial Model
• In our example, we get for stock:
Cu − C d
15 − 0
1
Δ=
=
=
S(u − d) 100 × ( 1.25 − 0.8 ) 3
• And, for bonds:
u × C d − d × C u 1.25 × 0 − 0.8 × 15
=
= −24.24
B=
(u − d )(1 + r )
(1.25 − 0.8) × (1.1)
• The cost of the portfolio is,
1
ΔS + B = × 100 − 24.24 = 9.09
3
Convertible Bonds
Binomial Model
• The price of the European call must be 9.09.
• Otherwise, there is an arbitrage opportunity.
• If the price is lower than 9.09 we would buy
the call and shortsell the portfolio
• If higher, the opposite
• We have computed the price and the hedge
simultaneously:
– We can construct a call by buying the stock and borrowing
– Short call: the opposite
Convertible Bonds
Binomial Model
• Remember that
• And
Cu − C d
u × C d − d × Cu
;B =
Δ=
S (u − d )
(u − d )(1 + r )
C = ΔS + B
• Substituting,
Cu − C d u × C d − d × Cu
C=
+
(u − d )
(u − d )(1 + r )
Convertible Bonds
Binomial Model
• After some algebra,
⎡1 + r − d
u − (1 + r ) ⎤
1
C=
Cu +
Cd ⎥
×⎢
1 + r ⎣ (u − d )
(u − d )
⎦
• Observe the coefficients,
1 + r − d u − (1 + r )
,
(u − d ) (u − d )
• Positive
• Smaller than one
• Add up to one
⇒ Like a probability.
Convertible Bonds
Binomial Model
• Rewrite
1
C=
× [ p × Cu + (1 − p) × Cd ]
1+ r
• Where
1+ r − d
u − (1 + r )
,1 − p =
p=
(u − d )
(u − d )
• This would be the pricing of:
– A risk neutral investor
– With subjective probabilities p and (1-p)
Convertible Bonds
Binomial Model
• Suppose the following economy,
u2 S
uS
udS
S
dS
d2S
• We introduce an European call with strike price K that
matures in the second period
Convertible Bonds
Binomial Model
• The price of the option will be:
1
2
2
C=
[
p
max(
0
,
u
S − K)
×
×
2
(1 + r )
+ (1 − p ) 2 × max(0, d 2 S − K )
+ 2 × p × (1 − p ) × max(0, udS − K )]
• There are “two paths” that lead to the intermediate
state (that explains the “2”)
• Suppose we know the volatility σ and the time to
maturity t, we can retrieve u and d (see B&S)
u=e
σ t/n
; d = 1/ u
Convertible Bond
Valuation Methodology
• Given that a convertible bond is nothing but an
option on the underlying stock, we expect to be able
to use the binomial model to price it
• At each node, we test
– a. whether conversion is optimal
– b. whether the position of the issuer can be improved by calling the bonds
• It is a dynamic procedure: max(min(Q1,Q2),Q3)),
where
– Q1 = value given by the rollback (neither converted nor called back)
– Q2 = call price
– Q3 = value of stocks if conversion takes place
Convertible Bond
Example
• Example
– We assume that the underlying stock price trades at $50.00 with a
30% annual volatility
– We consider a convertible bond with a 9 months maturity, a
conversion ratio of 20
– The convertible bond has a $1,000.00 face value, a 4% annual
coupon
– We further assume that the risk-free rate is a (continuously
compounded) 10%, while the yield to maturity on straight bonds
issued by the same company is a (continuously compounded) 15%
– We also assume that the call price is $1,100.00
– Use a 3 periods binomial model (t/n=3 months, or ¼ year)
Convertible Bond
Example
.3 1 / 4
u
e
=
= 1.1618
• We have
d = 1 = .8607
u
1+ r − d
p=
u−d
• Actually (continuously compounded rate)
⎛ 10% ⎞
exp⎜
⎟ − .8607
4 ⎠
⎝
p=
= .547
1.1618 − .8607
Convertible Bond
Example
Bond is Called
G
D
B
A
$58.09
11.03%
$1,191.13
$50.00
12.15%
$1,115.41
C
$43.04
13.51%
$1,006.23
$58.09
10.00%
$1,161.83
looks like a stock: use risk-free rate
conversion: 58.09>1040/20=52
$50.00
12.27% bond should not be converted because 1,073.18>50*20=1,000
$1,073.18
I
F
looks like a stock: use risk-free rate
conversion: 78.42>1040/20=52
$67.49
10.00% calling or converting does not change the bond value because it is already essentially equity
$1,349.86
H
E
$78.42
10.00%
$1,568.31
$43.04
15.00%
$1,040.00
looks like a risky bond: use risky rate
no conversion: 43.04<1040/20=52
$37.04
15.00% bond should not be converted because 1,001.72>50*20=1,000
$1,001.72
J
$31.88
15.00%
$1,040.00
looks like a risky bond: use risky rate
conversion: 31.88<1040/20=52
Convertible Bond
Example
• At node G, the bondholder optimally choose to
convert since what is obtained under conversion
($1,568.31), is higher than the payoff under the
assumption of no conversion ($1,040.00)
• The same applies to node H
• On the other hand, at nodes I and J, the value under
the assumption of conversion is lower than if the
bond is not converted to equity
– Therefore, bondholders optimally choose not to convert, and the payoff is
simply the nominal value of the bond, plus the interest payments, that is
$1,040.00
Convertible Bond
Example
• Working our way backward the tree, we obtain at
node D the value of the convertible bond as the
discounted expected value, using risk-neutral
probabilities of the payoffs at nodes G and H
$1,349.86 = e
-
3
×10%
12
( p ×1,568.31 + (1 − p )×1,161.83)
• At node F, the same principle applies, except that it
can be regarded as a standard bond
• We therefore use the rate of return on a non
convertible bond issued by the same company, 15%
$1,001.72 = e
-
3
×15%
12
( p ×1,040 + (1 − p )×1,040 )
Convertible Bond
Example
•
At node E, the situation is more interesting because the
convertible bond will end up as a stock in case of an up move
(conversion), and as a bond in case of a down move (no
conversion)
•
As an approximate rule of thumb, one may use a weighted
average of the riskfree and risky interest rate in the
computation, where the weighting is performed according to the
(risk-neutral) probability of an up versus a down move
px10% + (1-p)x15% = 12.27%
•
Then the value is computed as
$1,073.18 = e
-
3
×12 . 27 %
12
( p × 1,161 .83 + (1 − p )× 1,040 )
Convertible Bond
Example
•
•
•
•
•
Note that at nodes D, E and F, calling or converting is not
relevant because it does not change the bond value since the
bond is already essentially equity
At node B, it can be shown that the issuer finds it optimal to call
the bond
If the bond is indeed called by the issuer, bondholders are left
with the choice between not converting and getting the call
price ($1,100), or converting and getting $20x58.09=1,161.8$,
which is what they optimally choose
This is less than $1,191.13, the value of the convertible bond if
it were not called, and this is precisely why it is called by the
issuer
Eventually, the value at node A, i.e., the present fair value of
the convertible bond, is computed as $1,115.41
Convertible Bond
Allowing for Stochastic Interest Rates
Common Stock
Price Tree
$9
$8
Interest Rate
Tree
5.0%
4.5%
4.0%
4.0%
3.6%
3.2%
$10
$12
$11
$14
Convertible Bonds
Convertible Arbitrage
• Convertible arbitrage strategies attempt to exploit
anomalies in prices of corporate securities that are
convertible into common stocks
• Roughly speaking, if the issuer does well, the
convertible bond behaves like a stock, if the issuer
does poorly, the convertible bond behaves like
distressed debt
• Convertible bonds tends to be under-priced because
of market segmentation: investors discount securities
that are likely to change types
Convertible Bonds
Convertible Arbitrage
• Convertible arbitrage hedge fund managers typically
buy (or sometimes sell) these securities and then
hedge part or all of the associated risks by shorting
the stock
• Take for example Internet company AOL's zero
coupon converts due Dec. 6, 2019
– These bonds are convertible into 5.8338 shares of AOL stock
– With AOL common stock trading at $34.80 on Dec. 31, 2000, the
conversion value was $203 (=5.8338 x 34.80)
– As the conversion value is significantly below the investment value
(calculated at $450.20), the investment value dominated and the
convertible traded at $474.10
– When, or if, the stock trades above $77.15, the conversion value will
dominate the pricing of the convertible because it will be in excess of the
investment value
Convertible Bonds
Mechanism
•
•
•
•
•
In a typical convertible bond arbitrage position, the hedge fund
is not only long the convertible bond position, but also short an
appropriate amount of the underlying common stock
The number of shares shorted by the hedge fund manager is
designed to match or offset the sensitivity of the convertible
bond to common stock price changes
As the stock price decreases, the amount lost on the long
convertible position is countered by the amount gained on the
short stock position, theoretically creating a stable net position
value
As the stock price increases, the amount gained on the long
convertible position is countered by the amount lost on the
short stock position, theoretically creating a stable net position
value
This is known as delta hedging
Convertible Bonds
Mechanism
Convertible
Bond
Price
Convertible
Bond
Delta =
Change in Price of Conv Bond
Change in Price of Stock
Parity =
Stock Price
Conversion Ratio
Stock Price
Convertible Bonds
Mechanism
•
•
•
•
•
In the AOL example, the delta for the convertible is
approximately 50%
This means that for every $1 change in the conversion value,
the convertible bond price changes by 50 cents
To delta hedge the equity exposure in this bond we need to
short half the number of shares that the bond converts into, for
example 2.9 shares (5.8338\2)
The combined long convertible bond/short stock position should
be relatively insensitive to small changes in the price of AOL's
stock
Over-hedging is sometimes appropriate when there is concern
about default, as the excess short position may partially hedge
against a reduction in credit quality
Convertible Bonds
Risks Involved
•
Because a convertible bond is essentially a bond plus an option
to switch so that these strategies will typically
–
–
–
–
•
The risks involved relate to
–
–
–
–
•
•
•
make money if expected volatility increases (long vega)
make money if the stock price increases rapidly (long gamma)
pay time-decay (short theta)
make money if the credit quality of the issuer improves (short the credit differential)
changes in the price of the underlying stock (equity market risk)
changes in the interest rate level (fixed income market risk)
changes in the expected volatility of the stock (volatility risk)
changes in the credit standing of the issuer (credit risk)
The convertible bond market as a whole is also prone to
liquidity risk as demand can dry up periodically, and bid/ask
spreads on bonds can widen significantly
There is also the risk that the HF manager will be unable to
sustain the short position in the underlying common shares
In addition, convertible arbitrage hedge funds use varying
degrees of leverage, which can magnify both risks and returns
Options on Bonds
Terminology
• An option is a contract in which the seller (writer)
grants the buyer the right to purchase from, or sell to,
the seller an underlying asset (here a bond) at a
specified price within a specified period of time
• The seller grants this right to the buyer in exchange
for a certain sum of money called the option price or
option premium
• The price at which the instrument may be bought or
sold is called the exercise or strike price
• The date after which an option is void is called the
expiration date
– An American option may be exercised any time up to and including the
expiration date
– A European option may be exercised only on the expiration date
Options on Bonds
Factors that Influence Option Prices
• Current price of underlying security
– As the price of the underlying bond increases, the value of a call option
rises and the value of a put option falls
• Strike price
– Call (put) options become more (less) valuable as the exercise price
decreases
• Time to expiration
– For American options, the longer the time to expiration, the higher the
option price because all exercise opportunities open to the holder of the
short-life option are also open to the holder of the long-life option
• Short-term risk-free interest rate
– Price of call option on bond increases and price of put option on bond
decreases as short-term interest rate rises (through impact on bond price)
• Expected volatility of yields (or prices)
– As the expected volatility of yields over the life of the option increases, the
price of the option will also increase
Options on Bonds
Pricing
• Options on long-term bonds
– Interest payments are similar to dividends.
– Otherwise, long-term bonds are like options on stock:
– We can use Black-Scholes as in options on dividend-paying equity
• Options on short-term bonds
– They do not pay dividends
– Problem: they are not like a stock because they quickly converge to par
– We cannot directly apply Black-Scholes
• Other shortcomings of standard option pricing
models
– Assumption of a constant short-term rate is inappropriate for bond options
– Assumption of a constant volatility is also inappropriate: as a bond moves
closer to maturity, its price volatility decline
Options on Bonds
Pricing
• A solution to avoid the problem is to consider an
interest rate model, as described in Chapter 12
– The following figure shows a tree for the 1-year rate of interest (calibrated
to the current TS)
– The figure also shows the values for a discount bond (par = 100) at each
node in the tree
7.5%
7%
Interest rates
6.5%
6.5%
6%
6%
5.5%
5.5%
5%
4.5%
100
Bond prices
93
100
88.2
94
83.97
100
89.8
95
100
Options on Bonds
Pricing
• Consider a 2-year European call on this 3-year bond
struck at 93.5
• Start by computing the value at the end of the tree
– If by the end of the 2nd year the short-term rate has risen to 7% and the
bond is trading at 93, the option will expire worthless
– If the bond is trading at 94 (corresponding to a short-term rate of 6%) the
call option is worth 0.5
– If the bond is trading at 95 (short-term rate = 5%), the call is worth 1.5
• Working our way backward the tree
1
(.5 × 0 + .5 × .5) = .2347
Cu =
1 + 6.5%
1
(.5 × .5 + .5 ×1.5) = .9479
Cl =
1 + 5.5%
1
(.5 × Cu + .5 × Cd ) = .5573
C0 =
1 + 6%
Options on Bonds
Put-Call Parity
•
•
•
Assumption no coupon payments and no premature exercise
Consider a portfolio where we purchase one zero coupon bond,
one put European option, and sell (write) one European call
option (same time to maturity T and the same strike price X)
Payoff at date T
BT < X:
You hold the bond:
The call option is worthless:
The put option is worth:
Thus, your net position is:
BT
0
X - BT
X
You hold the bond:
The call option is worth:
The put option is worthless:
Thus, your net position is:
BT
-(B T - X)
0
X
BT ≥ X:
Options on Bonds
Put-Call Parity – Con’t
• No matter what state of the world obtains at the
expiration date, the portfolio will be worth X
• Thus, the payoff from the portfolio is risk-free, and
we can discount its value at the risk-free rate r
• We obtain the call-put relationship
B0 + C0 − P0 = Xe − rT ⇒ P0 = B0 + C0 − Xe − rT
• For coupon bonds
P0 = B0 + C0 − Xe − rT − PV (Coupons )