An Ad-Hoc Approximation

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.^a

• 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.

– Then, σ is the volatility of the process followed by the risky component.

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

aRoll (1977); Heath & Jarrow (1988).

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.

The Ad-Hoc Approximation vs. P. 324 (Step 1)

S − D/R

*

j

(S − D/R)u

*

j

(S − D/R)d

*

j

(S − D/R)u²

(S − D/R)ud

(S − D/R)d²

The Ad-Hoc Approximation vs. P. 324 (Step 2)

(S − D/R) + D/R = S

*

j

(S − D/R)u

*

j

(S − D/R)d

*

j

(S − D/R)u²

(S − D/R)ud

(S − D/R)d²

The Ad-Hoc Approximation vs. P. 324

• 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. 324).

– (S − D/R)u², (S − D/R)ud, (S − D/R)d² (ad hoc).

• Note that, as d < R < u,

(Su − D) u > (S − D/R)u², (Sd − D) d < (S − D/R)d²,

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

The Ad-Hoc Approximation vs. P. 324 (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 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 (see pp. 789ff).

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

aDai (B82506025, R86526008, D8852600) & Lyuu (2004). Also Arealy

& Rodrigues (2013).

bGeske & Shastri (1985). It works well for American options but not European ones (Dai, 2009).

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 have grown from S to S_τe^qτ without the dividends.

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

aIn pricing European options, only the distribution of Sτ matters.

Continuous Dividend Yields (continued)

• So the Black-Scholes formulas hold with S replaced by Se^−qτ:^a

C = Se^−qτN (x) − Xe^−rτN (x − σ√

τ ), (44) P = Xe^−rτN (−x + σ√

τ ) − Se^−qτN (−x),

(44^) where

x =^Δ ln(S/X) + 

r − q + σ²/2 τ σ√

τ .

• Formulas (44) and (44^) 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^−qΔt and d with de^−qΔt, where Δt = τ /n.^Δ

– 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) Δt} − d

u − d , (45)

where Δt = τ /n.^Δ

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

• The u and d remain unchanged.

• Except the change in Eq. (45), binomial tree algorithms stay the same as if there were no dividends.

Distribution of Logarithmic Returns of TAIEX

Exercise Boundaries of American Options (in the Continuous-Time Model)

• The exercise boundary is a nondecreasing function of t for American puts (see the plot next page).

• The exercise boundary is a nonincreasing function of t for American calls.

aSee Section 9.7 of the textbook for the tree analog.

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 =^Δ ln(S/X)+(r+σ²/2) τ σ√

τ (recall p. 303).

• Recall that

N^(y) = e√^−y2/2

2π > 0,

the density function of standard normal distribution.

Delta

• Defined as

Δ =^Δ ∂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.^a

• The delta used in the BOPM (p. 249) is the discrete analog.

• The delta of a long stock is 1.

aElementary calculus.

Delta (continued)

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

∂C

∂S = N (x) > 0. (46)

• The delta of a European put equals

∂P

∂S = N (x) − 1 = −N (−x) < 0. (47)

• So the deltas of a call and an otherwise identical put cancel each other when N (x) = 1/2, i.e., when^a

X = Se^{(r+σ2/2) τ}. (48)

aThe straddle (p. 213) C + P then has zero delta!

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)

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

Solid curves: at-the-money options.

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

Delta (continued)

• Suppose the stock pays a continuous dividend yield of q.

• Let

x =^Δ ln(S/X) + 

r − q + σ²/2 τ σ√

τ (49)

(recall p. 333).

• Then

∂C

∂S = e^−qτN (x) > 0,

∂P

∂S = −e^−qτN (−x) < 0.

Delta (continued)

• Consider an X₁-strike call and an X₂-strike put, X₁ ≥ X₂.

• They are otherwise identical.

• Let

x_i =^Δ ln(S/X_i) + 

r − q + σ²/2 τ σ√

τ . (50)

• Then their deltas sum to zero when x1 = −x2.^a

• That implies

S

X1 = X₂

S e^{−(2r−2q+σ2) τ}. (51)

aThe strangle (p. 215) C + P then has zero delta!

Delta (concluded)

• Suppose we demand X₁ = X₂ = X and have a straddle.

• Then

X = Se^{(r−q+σ2/2) τ} leads to a straddle with zero delta.

– This generalizes Eq. (48) on p. 343.

• When C(X1)’s delta and P (X2)’s delta sum to zero, does the portfolio C(X1) − P (X2) have zero value?

• In general, no.

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 −Δ shares of stock is delta-neutral.

– Short Δ shares of stock to hedge a long call.

– Long Δ shares of stock to hedge a short call.

• In general, hedge a position in a security with delta Δ₁ by shorting Δ₁/Δ₂ units of a security with delta Δ₂.

Theta (Time Decay)

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

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

2√

τ − rXe^−rτN (x − σ√

τ ) < 0.

– The call loses value with the passage of time.

• For a European put, Θ = −SN^(x) σ

2√

τ + rXe^−rτN (−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.

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

Theta (concluded)

• Suppose the stock pays a continuous dividend yield of q.

• Define x as in Eq. (49) on p. 345.

• For a European call, add an extra term to the earlier formula for the theta:

Θ = −SN^(x) σ 2√

τ − rXe^−rτN (x − σ√

τ ) + qSe^−qτ N (x).

• For a European put, add an extra term to the earlier formula for the theta:

Θ = −SN^(x) σ 2√

τ +rXe^−rτN (−x+σ√

τ )−qSe^−qτ N (−x).

Gamma

• Defined as the rate of change of its delta with respect to the price of the underlying asset, or Γ = ∂^{Δ 2}Π/∂S².

• 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

(Lambda, Kappa, Sigma)

• Defined as the rate of change of a security’s value with respect to the volatility of the underlying asset

Λ =^Δ ∂f

∂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 always increases the option value.

aVega is not Greek.

Vega (continued)

• Note that^a

Λ = τ σS²Γ.

• If the stock pays a continuous dividend yield of q, then Λ = Se^−qτ√

τ N^(x), where x is defined in Eq. (49) on p. 345.

• Vega is maximized when x = 0, i.e., when S = Xe^{−(r−q+σ2/2) τ}.

• Vega declines very fast as S moves away from that peak.

aReiss & Wystup (2001).

Vega (continued)

• Now consider a portfolio consisting of an X1-strike call C and a short X₂-strike put P , X₁ ≥ X₂.

• The options’ vegas cancel out when x1 = −x2,

where x_i are defined in Eq. (50) on p. 346.

• This also leads to Eq. (51) on p. 346.

– Recall the same condition led to zero delta for the strangle C + P (p. 346).

Vega (concluded)

• Note that τ → 0 implies

Λ → 0

(which answers the question on p. 308).

• The Black-Scholes formula (p. 303) implies

C → S,

P → Xe^−rτ, as σ → ∞.

• These boundary conditions are handy for some numerical methods.

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.

Variance Vega

• Defined as the rate of change of a security’s value with respect to the variance (square of volatility) of the

underlying asset

variance vega =^Δ ∂f

∂σ². – Note that it is not defined as ∂²f /∂σ²!

• It is easy to verify that

variance vega = Λ 2σ.

aDemeterfi, Derman, Kamal, & Zou (1999).

Volga (Vomma, Volatility Gamma, Vega Convexity)

• Defined as the rate of change of a security’s vega with respect to the volatility of the underlying asset

volga =^Δ ∂Λ

∂σ = ∂²f

∂σ² .

• It can be shown that

volga = Λ x(x − σ√ τ ) σ

= Λ

σ

ln²(S/X)

σ²τ − σ²τ 4

 , where x is defined in Eq. (49) on p. 345.^a

aDerman & M. B. Miller (2016).

Volga (concluded)

• Volga is zero when S = Xe^{±σ2τ /2}.

• For typical values of σ and τ, volga is positive except where S ≈ X.

• Volga can be used to measure the 4th moment of the underlying asset and the volatility of volatility.^a

aBennett (2014).

Rho

• Defined as the rate of change in its value with respect to interest rates

ρ =^Δ ∂f

∂r .

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

Xτ e^−rτN (x − σ√

τ ) > 0.

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

−Xτe^−rτN (−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.

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

Numerical Greeks

• Needed when closed-form formulas do not exist.

• Take delta as an example.

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

2ΔS .

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

An Alternative Numerical Delta

• 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. (52)

• Essentially zero extra cost.

aPelsser & Vorst (1994).

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,

fuu−fud

Suu−Sud − _Sud−Sdd^fud−fdd

(Suu − Sdd)/2 . (53)

Alternative Numerical Delta and Gamma

• Let  ≡ ln u.

• Think in terms of ln S.

• Then 

f_u − f_d 2

 1 S approximates the numerical delta.

• And 

f_uu − 2f_ud + f_dd

² − f_uu − f_dd 2

 1 S² approximates the numerical gamma.

aSee p. 688.

Finite Difference Fails for Numerical Gamma

• Numerical differentiation gives

f (S + ΔS) − 2f (S) + f (S − ΔS)

(ΔS)² .

• It does not work (see text for the reason).

• In general, calculating gamma is a hard problem numerically.^a

• But why did the binomial tree version work?

aRecall p. 115.

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.^a

• The vega of a European option can be derived from gamma.^b

• For rho, there seems no alternative but to run the binomial tree algorithm twice.^c

aSee p. 687.

bRecall p. 355.

cBut see p. 874 and pp. 1069ff.

Extensions of Options Theory

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

— Joan Robinson (1903–1983)

Pricing Corporate Securities

• Interpret the underlying asset as the firm’s total value.^b

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

• The result is called the structural model.

• Assumptions:

– 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 & Scholes (1973); Merton (1974).

bMore realistic models posit firm value = asset value + tax benefits− bankruptcy costs (Leland & Toft, 1996).

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, S?

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 has the same payoff as a call!

• It 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^a 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 .

aRecall p. 202.

Risky Zero-Coupon Bonds and Stock (continued)

• Thus

nS = C (market capitalization of XYZ.com), 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 11 (p. 303) and the put-call parity,^a nS = V N (x) − Xe^−rτN (x − σ√

τ ), (54) B = V N (−x) + Xe^−rτN (x − σ√

τ ). (55) – Above,

x =^Δ ln(V /X) + (r + σ²/2)τ σ√

τ .

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

ln(X/B)

τ .

aThis is sometimes called Merton’s (1974) structural model.

Risky Zero-Coupon Bonds and Stock (continued)

• 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^{Δ −rτ}/V .

– z = ln ω/(σ^Δ √

τ ) + (1/2) σ√

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

Risky Zero-Coupon Bonds and Stock (concluded)

• In general, suppose the firm has a dividend yield at rate q and the bankruptcy costs are a constant proportion α of the remaining firm value.

• Then Eqs. (54)–(55) on p. 378 become, respectively, nS = V e^−qτN (x) − Xe^−rτN (x − σ√

τ ),

B = (1 − α)V e^−qτN (−x) + Xe^−rτN (x − σ√ τ ).

– Above,

x =^Δ ln(V /X) + (r − q + σ²/2)τ σ√

τ .

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.

• It issues 1,000 shares of XYZ.com common stock and 30 zero-coupon bonds maturing on July 21, 1995.

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

• n = 1, 000, V = 44.5 × n = 44, 500, and X = 30 × 1, 000 = 30, 000.

• As Merck calls are being traded, we do not need formulas to price them.

—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 44^1/2 40 Jul 220 5^1/2 297 ^1/4

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

44^1/2 45 Apr 3050 ^7/8 100 1^1/8 44^1/2 45 May 462 1^3/8 50 1^3/8

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 = 15, 250 dollars.

• The entire bond issue is worth

B = 44, 500 − 15, 250 = 29, 250 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.^a

• 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.

aRecall p. 228.

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 45, 000/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 = 1, 937.5

dollars.

• The market value of the stock decreases as the debt-to-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,^a dividend, and strike price are under partial control of the stockholders or

boards.

aThis is called the asset substitution problem (Myers, 1977).

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. 385 says the total market value of the bonds should be $42,562.5.

• The new bondholders pay

42, 562.5 × (15/45) = 14, 187.5 dollars.

• The remaining stock is worth $1,937.5.

A Numerical Example (continued)

• The stockholders therefore gain

14, 187.5 + 1, 937.5 − 15, 250 = 875 dollars.

• The original bondholders lose an equal amount, 29, 250 − 30

45 × 42, 562.5 = 875.

– This is called claim dilution.^a

aFama & M. H. Miller (1972).

A Numerical Example (continued)

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

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

• The strike price remains X = 30, 000.

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

• n such calls are worth $1,937.5 (p. 385).

• So the total market value of the XYZ.com stock is (2/3) × 1, 937.5 = 1, 291.67 dollars.

A Numerical Example (concluded)

• The market value of the XYZ.com bonds is hence (2/3) × n × 44.5 − 1, 291.67 = 28, 375 dollars.

• Hence the stockholders gain

14, 833.3 + 1, 291.67 − 15, 250 ≈ 875 dollars.

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

$28,375, a loss of $875.

Further Topics

• Other examples:^a

– Stock as compound call when company issues coupon bonds.

– Subordinated debts as bull call spreads.

– Warrants as calls.

– Callable bonds as American calls with 2 strike prices.

– Convertible bonds.

– Bonds with safety covenants as barrier options.

aCox & Rubinstein (1985); Geske (1977).

Further Topics (concluded)

• Securities issued by firms with a complex capital structure must be solved by trees.^a

aDai (B82506025, R86526008, D8852600), Lyuu, & C. Wang (F95922018) (2010).

Distance to Default (DTD)

• Let μ be the total value V ’s rate of expected return.

• From Eq. (54), on p. 378, the probability of default τ years from now equals

N (−DTD), where

DTD =^Δ ln(V /X) + (μ − σ²/2)τ σ√

τ .

• V/X is called the leverage ratio.

aMerton (1974).