### BOPM and Black-Scholes Model

*• The Black-Scholes formula needs ﬁve parameters: S, X,*
*σ, τ , and r.*

*• Binomial tree algorithms take six inputs: S, X, u, d, ˆr,*
*and n.*

*• The connections are*
*u = e*^{σ}

*√**τ /n*

*, d = e*^{−σ}

*√**τ /n*

*, ˆr = rτ /n.*

*• The binomial tree algorithms converge reasonably fast.*

*• Oscillations can be dealt with by the judicious choices of*
*u and d (see text).*

5 10 15 20 25 30 35 n

11.5 12 12.5 13

Call value

0 10 20 30 40 50 60 n

15.1 15.2 15.3 15.4 15.5

Call value

*• S = 100, X = 100 (left), and X = 95 (right).*

*• The error is O(1/n).*^{a}

aChang and Palmer (2007).

### Implied Volatility

*• Volatility is the sole parameter not directly observable.*

*• The Black-Scholes formula can be used to compute the*
market’s opinion of the volatility.

* – Solve for σ given the option price, S, X, τ , and r*
with numerical methods.

**– How about American options?**

*• This volatility is called the implied volatility.*

*• Implied volatility is often preferred to historical*
volatility in practice.^{a}

aIt is like driving a car with your eyes on the rearview mirror?

### Problems; the Smile

*• Options written on the same underlying asset usually do*
not produce the same implied volatility.

*• A typical pattern is a “smile” in relation to the strike*
price.

**– The implied volatility is lowest for at-the-money**
options.

**– It becomes higher the further the option is in- or**
out-of-the-money.

*• Other patterns have also been observed.*

### Problems; the Smile (concluded)

*• To address this issue, volatilities are often combined to*
produce a composite implied volatility.

*• This practice is not sound theoretically.*

*• The existence of diﬀerent implied volatilities for options*
on the same underlying asset shows the Black-Scholes
model cannot be literally true.

### Trading Days and Calendar Days

*• Interest accrues based on the calendar day.*

*• But σ is usually calculated based on trading days only.*

**– Stock price seems to have lower volatilities when the**
exchange is closed.^{a}

* – σ measures the volatility of stock price one year from*
now (regardless of what happens in between).

*• How to incorporate these two diﬀerent ways of day*

count into the Black-Scholes formula and binomial tree algorithms?

### Trading Days and Calendar Days (concluded)

*• Suppose a year has 260 trading days.*

*• A quick and dirty way is to replace σ with*^{a}

*σ*

√ 365 260

number of trading days to expiration
number of calendar days to expiration *.*

*• How about binomial tree algorithms?*

aFrench (1984).

### Binomial Tree Algorithms for American Puts

*• Early exercise has to be considered.*

*• The binomial tree algorithm starts with the terminal*
payoﬀs

*max(0, X* *− Su*^{j}*d*^{n}* ^{−j}*)
and applies backward induction.

*• At each intermediate node, it checks for early exercise*
by comparing the payoﬀ if exercised with the

continuation value.

### Bermudan Options

*• Some American options can be exercised only at discrete*
time points instead of continuously.

*• They are called Bermudan options.*

*• Their pricing algorithm is identical to that for American*
options.

*• The only exception is early exercise is considered for*
only those nodes when early exercise is permitted.

### Options on a Stock That Pays Dividends

*• Early exercise must be considered.*

*• Proportional dividend payout model is tractable (see*
text).

**– The dividend amount is a constant proportion of the**
prevailing stock price.

*• In general, the corporate dividend policy is a complex*
issue.

### Known Dividends

*• Constant dividends introduce complications.*

*• Use D to denote the amount of the dividend.*

*• Suppose an ex-dividend date falls in the ﬁrst period.*

*• At the end of that period, the possible stock prices are*
*Su* *− D and Sd − D.*

*• Follow the stock price one more period.*

*• The number of possible stock prices is not three but*
*four: (Su* *− D) u, (Su − D) d, (Sd − D) u, (Sd − D) d.*

**– The binomial tree no longer combines.**

*(Su* *− D) u*

*↗*
*Su* *− D*

*↗* *↘*

*(Su* *− D) d*
*S*

*(Sd* *− D) u*

*↘* *↗*

*Sd* *− D*

*↘*

*(Sd* *− D) d*

### An Ad-Hoc Approximation

*• Use the Black-Scholes formula with the stock price*
reduced by the PV of the dividends (Roll, 1977).

*• This essentially decomposes the stock price into a*
riskless one paying known dividends and a risky one.

*• The riskless component at any time is the PV of future*
dividends during the life of the option.

* – σ equal to the volatility of the process followed by*
the risky component.

*• The stock price, between two adjacent ex-dividend*
dates, follows the same lognormal distribution.

### An Ad-Hoc Approximation (concluded)

*• Start with the current stock price minus the PV of*
future dividends before expiration.

*• Develop the binomial tree for the new stock price as if*
there were no dividends.

*• Then add to each stock price on the tree the PV of all*
*future dividends before expiration.*

*• American option prices can be computed as before on*
this tree of stock prices.

### An Ad-Hoc Approximation vs. P. 263 (Step 1)

*S* *− D/R*

*

j

*(S* *− D/R)u*

*

j

*(S* *− D/R)d*

*

j

*(S* *− D/R)u*^{2}

*(S* *− D/R)ud*

*(S* *− D/R)d*^{2}

### An Ad-Hoc Approximation vs. P. 263 (Step 2)

*(S* *− D/R) + D/R = S*

*

j

*(S* *− D/R)u*

*

j

*(S* *− D/R)d*

*

j

*(S* *− D/R)u*^{2}

*(S* *− D/R)ud*

*(S* *− D/R)d*^{2}

### An Ad-Hoc Approximation vs. P. 263

^{a}

*• The trees are diﬀerent.*

*• The stock prices at maturity are also diﬀerent.*

**– (Su***− D) u, (Su − D) d, (Sd − D) u, (Sd − D) d*
(p. 263).

**– (S***− D/R)u*^{2}*, (S* *− D/R)ud, (S − D/R)d*^{2} (ad hoc).

*• Note that (Su − D) u > (S − D/R)u*^{2} and
*(Sd* *− D) d < (S − D/R)d*^{2} *as d < R < u.*

aContributed by Mr. Yang, Jui-Chung (D97723002) on March 18, 2009.

### An Ad-Hoc Approximation vs. P. 263 (concluded)

*• So the ad hoc approximation has a smaller dynamic*
range.

*• This explains why in practice the volatility is usually*
increased when using the ad hoc approximation.

### A General Approach

^{a}

*• A new tree structure.*

*• No approximation assumptions are made.*

*• A mathematical proof that the tree can always be*
constructed.

*• The actual performance is quadratic except in*
pathological cases.

*• Other approaches include adjusting σ and approximating*
the known dividend with a dividend yield.

aDai (R86526008, D8852600) and Lyuu (2004).

### Continuous Dividend Yields

*• Dividends are paid continuously.*

**– Approximates a broad-based stock market portfolio.**

*• The payment of a continuous dividend yield at rate q*
*reduces the growth rate of the stock price by q.*

**– A stock that grows from S to S*** _{τ}* with a continuous

*dividend yield of q would grow from S to S*

_{τ}*e*

*without the dividends.*

^{qτ}*• A European option has the same value as one on a stock*
*with price Se** ^{−qτ}* that pays no dividends.

### Continuous Dividend Yields (continued)

*• The Black-Scholes formulas hold with S replaced by*
*Se** ^{−qτ}*:

^{a}

*C = Se*^{−qτ}*N (x)* *− Xe*^{−rτ}*N (x* *− σ√*

*τ ),* (25)
*P = Xe*^{−rτ}*N (−x + σ√*

*τ )* *− Se*^{−qτ}*N (−x),*

(25* ^{′}*)
where

*x* *≡* *ln(S/X) +* (

*r* *− q + σ*^{2}*/2*)
*τ*
*σ√*

*τ* *.*

*• Formulas (25) and (25’) remain valid as long as the*
dividend yield is predictable.

aMerton (1973).

### Continuous Dividend Yields (continued)

*• To run binomial tree algorithms, replace u with ue*^{−q∆t}*and d with de*^{−q∆t}*, where ∆t* *≡ τ/n.*

**– The reason: The stock price grows at an expected**
*rate of r* *− q in a risk-neutral economy.*

*• Other than the changes, binomial tree algorithms stay*
the same.

**– In particular, p should use the original u and d.**^{a}

aContributed by Ms. Wang, Chuan-Ju (F95922018) on May 2, 2007.

### Continuous Dividend Yields (concluded)

*• Alternatively, pick the risk-neutral probability as*
*e*^{(r}^{−q) ∆t}*− d*

*u* *− d* *,* (26)

*where ∆t* *≡ τ/n.*

**– The reason: The stock price grows at an expected**
*rate of r* *− q in a risk-neutral economy.*

*• The u and d remain unchanged.*

*• Other than the change in Eq. (26), binomial tree*
algorithms stay the same.

*Sensitivity Analysis of Options*

Cleopatra’s nose, had it been shorter, the whole face of the world would have been changed.

— Blaise Pascal (1623–1662)

### Sensitivity Measures (“The Greeks”)

*• How the value of a security changes relative to changes*
in a given parameter is key to hedging.

**– Duration, for instance.**

*• Let x ≡* *ln(S/X)+(r+σ*^{2}*/2) τ*
*σ**√*

*τ* (recall p. 251).

*• Note that*

*N*^{′}*(y) =* *e*^{−y}^{2}^{/2}

*√2π* *> 0,*

the density function of standard normal distribution.

### Delta

*• Deﬁned as ∆ ≡ ∂f/∂S.*

**– f is the price of the derivative.**

**– S is the price of the underlying asset.**

*• The delta of a portfolio of derivatives on the same*
underlying asset is the sum of their individual deltas.

**– Elementary calculus.**

*• The delta used in the BOPM is the discrete analog.*

### Delta (concluded)

*• The delta of a European call on a non-dividend-paying*
stock equals

*∂C*

*∂S* *= N (x) > 0.*

*• The delta of a European put equals*

*∂P*

*∂S* *= N (x)* *− 1 < 0.*

*• The delta of a long stock is 1.*

0 50 100 150 200 250 300 350 Time to expiration (days) 0

0.2 0.4 0.6 0.8 1

Delta (call)

0 50 100 150 200 250 300 350 Time to expiration (days) -1

-0.8 -0.6 -0.4 -0.2 0

Delta (put)

0 20 40 60 80

Stock price 0

0.2 0.4 0.6 0.8 1

Delta (call)

0 20 40 60 80

Stock price -1

-0.8 -0.6 -0.4 -0.2 0

Delta (put)

Solid curves: at-the-money options.

Dashed curves: out-of-the-money calls or in-the-money puts.

### Delta Neutrality

*• A position with a total delta equal to 0 is delta-neutral.*

* – A delta-neutral portfolio is immune to small price*
changes in the underlying asset.

*• Creating one serves for hedging purposes.*

**– A portfolio consisting of a call and** *−∆ shares of*
stock is delta-neutral.

**– Short ∆ shares of stock to hedge a long call.**

*• In general, hedge a position in a security with delta ∆*^{1}
by shorting ∆_{1}*/∆*_{2} units of a security with delta ∆_{2}.

### Theta (Time Decay)

*• Deﬁned as the rate of change of a security’s value with*
respect to time, or Θ *≡ −∂f/∂τ = ∂f/∂t.*

*• For a European call on a non-dividend-paying stock,*
Θ = *−SN*^{′}*(x) σ*

2*√*

*τ* *− rXe*^{−rτ}*N (x* *− σ√*

*τ ) < 0.*

**– The call loses value with the passage of time.**

*• For a European put,*
Θ = *−SN*^{′}*(x) σ*

2*√*

*τ* *+ rXe*^{−rτ}*N (−x + σ√*
*τ ).*

**– Can be negative or positive.**

0 50 100 150 200 250 300 350 Time to expiration (days) -60

-50 -40 -30 -20 -10 0

Theta (call)

0 50 100 150 200 250 300 350 Time to expiration (days) -50

-40 -30 -20 -10 0

Theta (put)

0 20 40 60 80

Stock price -6

-5 -4 -3 -2 -1 0

Theta (call)

0 20 40 60 80

Stock price -2

-1 0 1 2 3

Theta (put)

Dotted curve: in-the-money call or out-of-the-money put.

Solid curves: at-the-money options.

### Gamma

*• Deﬁned as the rate of change of its delta with respect to*
the price of the underlying asset, or Γ *≡ ∂*^{2}*Π/∂S*^{2}.

*• Measures how sensitive delta is to changes in the price of*
the underlying asset.

*• In practice, a portfolio with a high gamma needs be*
rebalanced more often to maintain delta neutrality.

*• Roughly, delta ∼ duration, and gamma ∼ convexity.*

*• The gamma of a European call or put on a*
non-dividend-paying stock is

*N*^{′}*(x)/(Sσ√*

*τ ) > 0.*

0 20 40 60 80 Stock price

0 0.01 0.02 0.03 0.04

Gamma (call/put)

0 50 100 150 200 250 300 350 Time to expiration (days) 0

0.1 0.2 0.3 0.4 0.5

Gamma (call/put)

Dotted lines: in-the-money call or out-of-the-money put.

Solid lines: at-the-money option.

Dashed lines: out-of-the-money call or in-the-money put.

### Vega

^{a}

### (Lambda, Kappa, Sigma)

*• Deﬁned as the rate of change of its value with respect to*
the volatility of the underlying asset Λ *≡ ∂Π/∂σ.*

*• Volatility often changes over time.*

*• A security with a high vega is very sensitive to small*
changes or estimation error in volatility.

*• The vega of a European call or put on a*
*non-dividend-paying stock is S√*

*τ N*^{′}*(x) > 0.*

**– So higher volatility increases option value.**

aVega is not Greek.

0 20 40 60 80 Stock price

0 2 4 6 8 10 12 14

Vega (call/put)

50 100 150 200 250 300 350 Time to expiration (days) 0

2.5 5 7.5 10 12.5 15 17.5

Vega (call/put)

Dotted curve: in-the-money call or out-of-the-money put.

Solid curves: at-the-money option.

Dashed curve: out-of-the-money call or in-the-money put.

### Rho

*• Deﬁned as the rate of change in its value with respect to*
*interest rates ρ* *≡ ∂Π/∂r.*

*• The rho of a European call on a non-dividend-paying*
stock is

*Xτ e*^{−rτ}*N (x* *− σ√*

*τ ) > 0.*

*• The rho of a European put on a non-dividend-paying*
stock is

*−Xτe*^{−rτ}*N (−x + σ√*

*τ ) < 0.*

50 100 150 200 250 300 350 Time to expiration (days) 0

5 10 15 20 25 30 35

Rho (call)

50 100 150 200 250 300 350 Time to expiration (days) -30

-25 -20 -15 -10 -5 0

Rho (put)

0 20 40 60 80

Stock price 0

5 10 15 20 25

Rho (call)

0 20 40 60 80

Stock price -25

-20 -15 -10 -5 0

Rho (put)

Dotted curves: in-the-money call or out-of-the-money put.

Solid curves: at-the-money option.

### Numerical Greeks

*• Needed when closed-form formulas do not exist.*

*• Take delta as an example.*

*• A standard method computes the ﬁnite diﬀerence,*
*f (S + ∆S)* *− f(S − ∆S)*

*2∆S* *.*

*• The computation time roughly doubles that for*
evaluating the derivative security itself.

### An Alternative Numerical Delta

^{a}

*• Use intermediate results of the binomial tree algorithm.*

*• When the algorithm reaches the end of the ﬁrst period,*
*f*_{u}*and f** _{d}* are computed.

*• These values correspond to derivative values at stock*
*prices Su and Sd, respectively.*

*• Delta is approximated by*

*f*_{u}*− f*^{d}*Su* *− Sd.*

*• Almost zero extra computational eﬀort.*

*S/(ud)*

*S/d*

*S/u*

*Su/d*

*S*

*Sd/u*

*Su*

*Sd*
*Suu/d*

*Sdd/u*

*Suuu/d*

*Suu*

*S*

*Sdd*

*Sddd/u*

### Numerical Gamma

*• At the stock price (Suu + Sud)/2, delta is*
*approximately (f*_{uu}*− f*^{ud}*)/(Suu* *− Sud).*

*• At the stock price (Sud + Sdd)/2, delta is*
*approximately (f*_{ud}*− f*^{dd}*)/(Sud* *− Sdd).*

*• Gamma is the rate of change in deltas between*
*(Suu + Sud)/2 and (Sud + Sdd)/2, that is,*

*f*_{uu}*−f**ud*

*Suu**−Sud* *−* _{Sud}^{f}^{ud}^{−f}_{−Sdd}^{dd}*(Suu* *− Sdd)/2* *.*

*• Alternative formulas exist.*

### Finite Diﬀerence Fails for Numerical Gamma

*• Numerical diﬀerentiation gives*

*f (S + ∆S)* *− 2f(S) + f(S − ∆S)*

*(∆S)*^{2} *.*

*• It does not work (see text).*

*• But why did the binomial tree version work?*

### Other Numerical Greeks

*• The theta can be computed as*
*f*_{ud}*− f*

*2(τ /n)* *.*

**– In fact, the theta of a European option can be**
derived from delta and gamma (p. 517).

*• For vega and rho, there is no alternative but to run the*
binomial tree algorithm twice.

*Extensions of Options Theory*

As I never learnt mathematics, so I have had to think.

— Joan Robinson (1903–1983)

### Pricing Corporate Securities

^{a}

*• Interpret the underlying asset as the total value of the*
ﬁrm.

*• The option pricing methodology can be applied to*
pricing corporate securities.

*• Assume:*

**– A ﬁrm can ﬁnance payouts by the sale of assets.**

**– If a promised payment to an obligation other than**
stock is missed, the claim holders take ownership of
the ﬁrm and the stockholders get nothing.

aBlack and Scholes (1973).

### Risky Zero-Coupon Bonds and Stock

*• Consider XYZ.com.*

*• Capital structure:*

**– n shares of its own common stock, S.**

**– Zero-coupon bonds with an aggregate par value of X.**

*• What is the value of the bonds, B?*

*• What is the value of the XYZ.com stock?*

### Risky Zero-Coupon Bonds and Stock (continued)

*• On the bonds’ maturity date, suppose the total value of*
*the ﬁrm V* ^{∗}*is less than the bondholders’ claim X.*

*• Then the ﬁrm declares bankruptcy, and the stock*
becomes worthless.

*• If V* ^{∗}*> X, then the bondholders obtain X and the*
*stockholders V* ^{∗}*− X.*

*V* ^{∗}*≤ X V* ^{∗}*> X*

Bonds *V* ^{∗}*X*

Stock 0 *V* ^{∗}*− X*

### Risky Zero-Coupon Bonds and Stock (continued)

*• The stock is a call on the total value of the ﬁrm with a*
*strike price of X and an expiration date equal to the*
bonds’.

**– This call provides the limited liability for the**
stockholders.

*• The bonds are a covered call on the total value of the*
ﬁrm.

*• Let V stand for the total value of the ﬁrm.*

*• Let C stand for a call on V .*

### Risky Zero-Coupon Bonds and Stock (continued)

*• Thus nS = C and B = V − C.*

*• Knowing C amounts to knowing how the value of the*
ﬁrm is divided between stockholders and bondholders.

*• Whatever the value of C, the total value of the stock*
*and bonds at maturity remains V* * ^{∗}*.

*• The relative size of debt and equity is irrelevant to the*
*ﬁrm’s current value V .*

### Risky Zero-Coupon Bonds and Stock (continued)

*• From Theorem 8 (p. 251) and the put-call parity,*
*nS* = *V N (x)* *− Xe*^{−rτ}*N (x* *− σ√*

*τ ),*
*B* = *V N (−x) + Xe*^{−rτ}*N (x* *− σ√*

*τ ).*

**– Above,**

*x* *≡* *ln(V /X) + (r + σ*^{2}*/2)τ*
*σ√*

*τ* *.*

*• The continuously compounded yield to maturity of the*
ﬁrm’s bond is

*ln(X/B)*

*τ* *.*

### Risky Zero-Coupon Bonds and Stock (concluded)

*• Deﬁne the credit spread or default premium as the yield*
diﬀerence between risky and riskless bonds,

*ln(X/B)*

*τ* *− r*

= *−*1
*τ* ln

(

*N (−z) +* 1

*ω* *N (z* *− σ√*
*τ )*

)
*.*
**– ω***≡ Xe*^{−rτ}*/V .*

**– z***≡ (ln ω)/(σ√*

*τ ) + (1/2) σ√*

*τ =* *−x + σ√*
*τ .*
**– Note that ω is the debt-to-total-value ratio.**

### A Numerical Example

*• XYZ.com’s assets consist of 1,000 shares of Merck as of*
March 20, 1995.

**– Merck’s market value per share is $44.5.**

*• XYZ.com’s securities consist of 1,000 shares of common*
stock and 30 zero-coupon bonds maturing on July 21,
1995.

*• Each bond promises to pay $1,000 at maturity.*

*• n = 1000, V = 44.5 × n = 44500, and*
*X = 30* *× 1000 = 30000.*

—Call— —Put—

Option Strike Exp. Vol. Last Vol. Last
**Merck** 30 Jul 328 15^{1/4} *. . .* *. . .*

44^{1/2} 35 Jul 150 9^{1/2} 10 ^{1/16}
44^{1/2} 40 Apr 887 4^{3/4} 136 ^{1/16}
441/2 40 Jul 220 51/2 297 1/4

44^{1/2} 40 Oct 58 6 10 ^{1/2}

44^{1/2} 45 Apr 3050 ^{7/8} 100 1^{1/8}
441/2 45 May 462 13/8 50 13/8

44^{1/2} 45 Jul 883 1^{15/16} 147 1^{3/4}

44^{1/2} 45 Oct 367 2^{3/4} 188 2^{1/16}

### A Numerical Example (continued)

*• The Merck option relevant for pricing is the July call*
*with a strike price of X/n = 30 dollars.*

*• Such a call is selling for $15.25.*

*• So XYZ.com’s stock is worth 15.25 × n = 15250 dollars.*

*• The entire bond issue is worth*

*B = 44500* *− 15250 = 29250 dollars.*

**– Or $975 per bond.**

### A Numerical Example (continued)

*• The XYZ.com bonds are equivalent to a default-free*
*zero-coupon bond with $X par value plus n written*
European puts on Merck at a strike price of $30.

**– By the put-call parity.**

*• The diﬀerence between B and the price of the*
default-free bond is the value of these puts.

*• The next table shows the total market values of the*
XYZ.com stock and bonds under various debt amounts
*X.*

Promised payment Current market Current market Current total to bondholders value of bonds value of stock value of firm

*X* *B* *nS* *V*

30,000 29,250.0 15,250.0 44,500

35,000 35,000.0 9,500.0 44,500

40,000 39,000.0 5,500.0 44,500

45,000 42,562.5 1,937.5 44,500

### A Numerical Example (continued)

*• Suppose the promised payment to bondholders is*

$45,000.

*• Then the relevant option is the July call with a strike*
*price of 45000/n = 45 dollars.*

*• Since that option is selling for $1*^{15/16}, the market value
*of the XYZ.com stock is (1 + 15/16)* *× n = 1937.5*

dollars.

*• The market value of the stock decreases as the*
debt-equity ratio increases.

### A Numerical Example (continued)

*• There are conﬂicts between stockholders and*
bondholders.

*• An option’s terms cannot be changed after issuance.*

*• But a ﬁrm can change its capital structure.*

*• There lies one key diﬀerence between options and*
corporate securities.

**– Parameters such volatility, dividend, and strike price**
are under partial control of the stockholders.

### A Numerical Example (continued)

*• Suppose XYZ.com issues 15 more bonds with the same*
terms to buy back stock.

*• The total debt is now X = 45,000 dollars.*

*• The table on p. 309 says the total market value of the*
bonds should be $42,562.5.

*• The new bondholders pay 42562.5 × (15/45) = 14187.5*
dollars.

*• The remaining stock is worth $1,937.5.*

### A Numerical Example (continued)

*• The stockholders therefore gain*

*14187.5 + 1937.5* *− 15250 = 875*
dollars.

*• The original bondholders lose an equal amount,*
29250 *−* 30

45 *× 42562.5 = 875.* (27)

### A Numerical Example (continued)

*• Suppose the stockholders sell (1/3) × n Merck shares to*
fund a $14,833.3 cash dividend.

*• They now have $14,833.3 in cash plus a call on*
*(2/3)* *× n Merck shares.*

*• The strike price remains X = 30000.*

*• This is equivalent to owning 2/3 of a call on n Merck*
shares with a total strike price of $45,000.

*• n such calls are worth $1,937.5 (p. 309).*

*• So the total market value of the XYZ.com stock is*
*(2/3)* *× 1937.5 = 1291.67 dollars.*

### A Numerical Example (concluded)

*• The market value of the XYZ.com bonds is hence*
*(2/3)* *× n × 44.5 − 1291.67 = 28375 dollars.*

*• Hence the stockholders gain*

*14833.3 + 1291.67* *− 15250 ≈ 875*
dollars.

*• The bondholders watch their value drop from $29,250 to*

$28,375, a loss of $875.