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).
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. 304 (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. 304 (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. 304
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. 304).
– (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. 304 (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 (see pp. 722ff).
• 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 options (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 Sτ with a continuous dividend yield of q would grow from S to Sτeqτ 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 ), (39) P = Xe−rτN (−x + σ√
τ ) − Se−qτN (−x),
(39) where
x ≡ ln(S/X) +
r − q + σ2/2 τ σ√
τ .
• Formulas (39) and (39) 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.
– 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 , (40)
where Δt ≡ 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. (40), binomial tree
algorithms stay the same as if there were no dividends.
Exercise Boundaries of American Options
a• 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 Exercise 15.2.7 of the textbook.
Risk Reversals
a• From formulas (39) and (39) on p. 313, one can verify that C = P when
X = Se(r−q)τ.
• A risk reversal consists of a short out-of-the-money put and a long out-of-the-money call with the same
maturity.b
• Furthermore, the portfolio has zero value.
• A short risk reversal position is also called a collar.c
aNeftci (2008).
bThus their strike prices must be distinct.
cBennett (2014).
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+σ2/2) τ σ√
τ (recall p. 285).
• Recall that
N(y) = e−y2/2
√2π > 0,
the density function of standard normal distribution.
Delta
• Defined as
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. 230) is the discrete analog.
• The delta of a long stock is apparently 1.
aElementary calculus.
Delta (continued)
• 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 = −N (−x) < 0.
• So the deltas of a call and an otherwise identical put cancel each other when N (x) = 1/2, i.e., whena
X = Se(r+σ2/2) τ. (41)
aThe straddle (p. 195) 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 ≡ ln(S/X) +
r − q + σ2/2 τ σ√
τ (42)
(recall p. 313).
• 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
xi ≡ ln(S/Xi) +
r − q + σ2/2 τ σ√
τ . (43)
• Then their deltas sum to zero when x1 = −x2.a
• That implies
S
X1 = X2
S e−(2r−2q+σ2) τ. (44)
aThe strangle (p. 197) C + P then has zero delta!
Delta (continued)
• 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. (41) on p. 323.
• When C(X1)’s delta and P (X2)’s delta sum to zero, does the portfolio C(X1) − P (X2) have zero value?
Delta (concluded)
• This portfolio C(X1) − P (X2) has value
Se−qτN (x1) − X1e−rτN (x1 − σ√ τ )
−X2e−rτN (−x2 + σ√
τ ) + Se−qτN (−x2)
= 2Se−qτN (x1) − X1e−rτN (x1 − σ√
τ ) − X2e−rτN (x1 + σ√ τ )
= 2Se−qτN (x1) − X1e−rτN (x1 − σ√ τ )
− S2
X1e(r−2q+σ2) τN (x1 + σ√ τ ).
• This is not identically zero so not a risk reversal (p. 318).
• E.g., with r = q = 0 and τ large, it is about 2S − (S2/X1) eσ2τ = 2S − X2.
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 ≡ −∂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.
Gamma
• Defined as the rate of change of its delta with respect to the price of the underlying asset, or Γ ≡ ∂2Π/∂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
a(Lambda, Kappa, Sigma)
• Defined as the rate of change of a securitys value with respect to the volatility of the underlying asset
Λ ≡ ∂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)
• If the stock pays a continuous dividend yield of q, then Λ = Se−qτ√
τ N(x), where x is defined in Eq. (42) on p. 325.
• 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.
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 xi are defined in Eq. (43) on p. 326.
• This leads to Eq. (44) on p. 326.
– The same condition leads to zero delta for the strangle C + P (p. 326).
Vega (concluded)
• Note that if S = X, τ → 0 implies Λ → 0
(which answers the question on p. 290 for the Black-Scholes model).
• The Black-Scholes formula (p. 285) implies
C → S,
P → Xe−rτ, as σ → ∞.
• These boundary conditions may be handy for certain 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
a• Defined as the rate of change of a securitys value with respect to the variance (square of volatility) of the underlying asset
V ≡ ∂f
∂σ2 .
• It is easy to verify that
V = Λ 2σ.
aDemeterfi, Derman, Kamal, & Zou (1999).
Rho
• Defined as the rate of change in its value with respect to interest rates
ρ ≡ ∂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
a• Use intermediate results of the binomial tree algorithm.
• When the algorithm reaches the end of the first period, fu and fd are computed.
• These values correspond to derivative values at stock prices Su and Sd, respectively.
• Delta is approximated by
fu − fd Su − Sd.
• Almost zero extra computational effort.
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 (fuu − fud)/(Suu − Sud).
• At the stock price (Sud + Sdd)/2, delta is approximately (fud − fdd)/(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−Sddfud−fdd
(Suu − Sdd)/2 . (45)
• Alternative formulas exist (p. 628).
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.
• But why did the binomial tree version work?
Other Numerical Greeks
• The theta can be computed as fud − f
2(τ /n) .
– In fact, the theta of a European option can be derived from delta and gamma (p. 627).
• For vega and rho, there seems no alternative but to run the binomial tree algorithm twice.a
aBut see pp. 974ff.
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.
– 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).
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 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 calla 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 .
aSee p. 186.
Risky Zero-Coupon Bonds and Stock (continued)
• Thus
nS = C,
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 12 (p. 285) and the put-call parity,a nS = V N (x) − Xe−rτN (x − σ√
τ ), (46) B = V N (−x) + Xe−rτN (x − σ√
τ ). (47) – Above,
x ≡ 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 .
aMerton (1974).
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−rτ/V .
– z ≡ (ln ω)/(σ√
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. (46)–(47) on p. 355 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 ≡ 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.
• 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 = 1, 000, V = 44.5 × n = 44, 500, and X = 30 × 1, 000 = 30, 000.
—Call— —Put—
Option Strike Exp. Vol. Last Vol. Last Merck 30 Jul 328 151/4 . . . . . .
441/2 35 Jul 150 91/2 10 1/16 441/2 40 Apr 887 43/4 136 1/16 441/2 40 Jul 220 51/2 297 1/4
441/2 40 Oct 58 6 10 1/2
441/2 45 Apr 3050 7/8 100 11/8 441/2 45 May 462 13/8 50 13/8
441/2 45 Jul 883 115/16 147 13/4
441/2 45 Oct 367 23/4 188 21/16
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.
aSee p. 209.
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 $115/16, 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-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. 362 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 & Miller (1972).
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 = 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. 362).
• 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:
– Subordinated debts as bull call spreads.
– Warrants as calls.
– Callable bonds as American calls with 2 strike prices.
– Convertible bonds.
• Securities with a complex liability structure must be solved by trees.a
aDai (B82506025, R86526008, D8852600), Lyuu, & C. Wang (F95922018) (2010).
Barrier Options
a• Their payoff depends on whether the underlying asset’s price reaches a certain price level H throughout its life.
• A knock-out option is an ordinary European option
which ceases to exist if the barrier H is reached by the price of its underlying asset.
• A call knock-out option is sometimes called a down-and-out option if H < S.
• A put knock-out option is sometimes called an up-and-out option when H > S.
aA former MBA student in finance told me on March 26, 2004, that she did not understand why I covered barrier options until she started working in a bank. She was working for Lehman Brothers in Hong Kong as of April, 2006.
H
Time Price
S Barrier hit
Barrier Options (concluded)
• A knock-in option comes into existence if a certain barrier is reached.
• A down-and-in option is a call knock-in option that comes into existence only when the barrier is reached and H < S.
• An up-and-in is a put knock-in option that comes into existence only when the barrier is reached and H > S.
• Formulas exist for all the possible barrier options mentioned above.a
aHaug (2006).
A Formula for Down-and-In Calls
a• Assume X ≥ H.
• The value of a European down-and-in call on a stock paying a dividend yield of q is
Se−qτ
H S
2λ
N (x) − Xe−rτ
H S
2λ−2
N (x − σ√ τ ),
(48)
– x ≡ ln(H2/(SX))+(r−q+σ2/2) τ σ√
τ .
– λ ≡ (r − q + σ2/2)/σ2.
• A European down-and-out call can be priced via the in-out parity (see text).
aMerton (1973). See Exercise 17.1.6 of the textbook for a proof.
A Formula for Up-and-In Puts
a• Assume X ≤ H.
• The value of a European up-and-in put is
Xe−rτ
H S
2λ−2
N(−x + σ√
τ) − Se−qτ
H S
2λ
N(−x).
• Again, a European up-and-out put can be priced via the in-out parity.
aMerton (1973).
Are American Options Barrier Options?
a• American options are barrier options with the exercise boundary as the barrier and the payoff as the rebate?
• One salient difference is that the exercise boundary must be derived during backward induction.
• But the barrier in a barrier option is given a priori.
aContributed by Mr. Yang, Jui-Chung (D97723002) on March 25, 2009.
Interesting Observations
• Assume H < X.
• Replace S in the pricing formula Eq. (39) on p. 313 for the call with H2/S.
• Equation (48) on p. 373 for the down-and-in call becomes Eq. (39) when r − q = σ2/2.
• Equation (48) becomes S/H times Eq. (39) when r − q = 0.
Interesting Observations (concluded)
• Replace S in the pricing formula for the down-and-in call, Eq. (48), with H2/S.
• Equation (48) becomes Eq. (39) when r − q = σ2/2.
• Equation (48) becomes H/S times Eq. (39) when r − q = 0.a
• Why?b
aContributed by Mr. Chou, Ming-Hsin (R02723073) on April 24, 2014.
bApply the reflection principle (p. 656), Eq. (38) on p. 278, and Lemma 11 (p. 283).
Binomial Tree Algorithms
• Barrier options can be priced by binomial tree algorithms.
• Below is for the down-and-out option.
0 H
• Princing down-and-in options is subtler.
H 8
16
4
32
8
2
64
16
4
1
4.992
12.48
1.6
27.2
4.0
0
58
10
0
0 X
0.0
S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.
Backward-induction: C = (0.5 × Cu + 0.5 × Cd)/1.25.
Binomial Tree Algorithms (continued)
• But convergence is erratic because H is not at a price level on the tree (see plot on next page).a
– The barrier H is moved lowerb to a node price.
– This “effective barrier” changes as n increases.
• In fact, the binomial tree is O(1/√
n) convergent.c
• Solutions will be presented later.
aBoyle & Lau (1994).
bHigher is an alternative
cJ. Lin (R95221010) (2008).
Binomial Tree Algorithms (concluded)
a100 150 200 250 300 350 400
#Periods 3
3.5 4 4.5 5 5.5
Down-and-in call value
aLyuu (1998).
