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American Options and Callable Bonds: Valuation and Exercise Policies, Exams of Investment Theory

University of Architecture Ho Chi Minh City (UAHCM)Investment Theory

An in-depth analysis of American Options and Callable Bonds, focusing on their valuation, exercise policies, and the impact of intervening payments. concepts such as early exercise, callable bonds, and option-adjusted spread. It also includes class problems for further study.

Typology: Exams

2021/2022

Uploaded on 08/05/2022

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Debt Instruments and Markets Professor Carpenter

American Options and Callable Bonds 1

American)Options)and)

Callable)Bonds)

 American)Options)

 Valuing)an)American)Call)on)

a)Coupon)Bond)

 Valuing)a)Callable)Bond)

 Interest)Rate)Sensitivity)of)a)

Callable)Bond)

 exercise)policy)

 value‐maximization)

 redeem)

 negative)convexity)

 option‐adjusted)

spread)

Concepts)and)Buzzwords)

 Veronesi,)Chapter)12)

 Tuckman,)Chapter)19)

Reading)

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American Options and

Callable Bonds

 American Options

 Valuing an American Call on

a Coupon Bond

 Valuing a Callable Bond

 Interest Rate Sensitivity of a

Callable Bond

 exercise policy

 value‐maximization

 redeem

 negative convexity

 option‐adjusted

spread

Concepts and Buzzwords

 Veronesi, Chapter 12

 Tuckman, Chapter 19

Reading

American Options

 Most corporate bonds, and virtually all mortgages, contain an

embedded option giving the borrower the option to call, or

prepay, the loan at a pre‐specified price on a date of the

borrower’s choosing.

 Valuing these securities amounts to valuing the embedded

American option.

 With an American option, the holder has a choice of when to

exercise.

 Thus, valuing and assessing the risk of an American option

involves determining the holder’s optimal exercise policy.

No Early Exercise of Calls on Assets that Have No Payments Prior to Expiration:

 Prior to expiration, an American call on a “non‐paying"

asset, such as a zero, is worth more than its exercise value:

American call value

≥ European call value

= European put value + V – dTK

> V – dTK

> V ‐ K

= exercise value.

 Therefore, better to sell the call than to exercise it

(assuming the strike price stays constant over time).

Underlying 5.5%‐Coupon Bond

 Each node in the tree below lists

 the current ex‐coupon price of the 5.5%‐coupon bond, and

 the current price of a zero with 6 months to maturity.

 At each node, the bond is priced as a package of zeroes.

Time 0 Time 0.5 Time 1

Time 1.

Class Problems: What are the

time 0 and time 0.5 ex‐coupon

prices of the bond?

Exercise Policy for the American Call To value the call, we need to know its future cash flows at each time and state. At each time and state the option holder must decide whether or not to exercise the option The future cash flows of the option depend on the option holder's exercise policy.

Value‐Maximizing Exercise Policy  We assume that the option holder will choose, at each time and state, to exercise or not depending on which action maximizes the option's market value.  This assumption is reasonable as long as the option holder can freely sell the option.  Then, if ever the option holder wants to get out of his position when the option is worth more than its exercise value, the option holder just sells the option instead of exercising it.  Working backwards in time, decide what the call holder would do if he got to a given point with the call still alive—exercise, or wait?  At time 1.5, the last call date, if the option were still alive, the holder would just exercise it if in the money or else let it expire worthless.  At earlier dates, the holder can either exercise, or else wait one period, pay the coupon, and proceed optimally from there, whichever gives the option greater value. Class Problem: American Call Time 0 Time 0.5 Time 1 Time 1.

 Working backwards, decide what the issuer would do to minimize the cost (pv) of debt if he got to a given point with the bond still outstanding.  The issuer’s choices are to either  call the bond and pay par, or  leave it outstanding for another period, pay another coupon, and then proceed with optimal debt service from there. Class Problem: Callable Bond Time 0 Time 0.5 Time 1 Time 1. Another Way to See Callable = Noncallable – Option: See Refunding Profit as Option Exercise Value

 Suppose the way the firm finances the call is by selling new

noncallable bonds with the same coupon and maturity as the

old debt‐‐a “refunding.”

 The profit from refunding

= proceeds of sale of new debt ‐ call price

= value of the noncallable ‐

= option exercise value

 With this plan in place, a firm that has issued a callable bond

pays the noncallable cash flows until maturity but gets to do a

refunding along the way.

 The firms’ net position = short a callable bond

= short a noncallable bond, long an option.

Class Problems

What are the SR dollar durations and SR durations of

1) the noncallable bond?

2) the call?

3) the callable bond?

Negative Convexity of Callable Bonds

 The value of the embedded call option is a highly convex function

of interest rates.

 So the short call position in the callable bond not only reduces its

duration, it also gives it “negative convexity.”

Option‐Adjusted Spread

 Corporate bonds pay less than promised in the event of default.

 This credit risk makes them worth less than a nondefaultable

bond with same coupon and maturity.

 This price difference results in a higher yield, that is, a credit

spread over the yield of a similar Treasury.

 If the bond is callable, the price is lower because of the

embedded call, which also results in a higher yield to maturity.

 In fact, the call and default risks interact in a complicated way.

 In practice, people try to calculate the component of the spread

of a callable defaultable bond over a noncallable Treasury that

is due to just to credit risk, and not call risk.

 This is the so‐called option‐adjusted spread.

OAS