### Time-Dependent Instantaneous Volatility

^{a}

*• Suppose the (instantaneous) volatility can change over*
*time but otherwise predictable: σ(t) instead of σ.*

*• In the limit, the variance of ln(S*_{τ}*/S) is*

_{τ}

0 *σ*^{2}*(t) dt*
*rather than σ*^{2}*τ .*

*• The annualized volatility to be used in the Black-Scholes*
formula should now be

_{τ}

0 *σ*^{2}*(t) dt*

*τ* *.*

### Time-Dependent Instantaneous Volatility (concluded)

*• There is no guarantee that the implied volatility is*
constant.

*• For the binomial model,u and d depend on time:*

*u = e*^{σ(t)}

*√**τ /n**,*

*d = e*^{−σ(t)}

*√**τ /n**.*

*• How to make the binomial tree combine?*^{a}

aAmin (1991); Chen (R98922127) (2011).

### Time-Dependent Short Rates

*• Suppose the short rate (i.e., the one-period spot rate)*
changes over time but otherwise predictable.

*• The riskless rate r in the Black-Scholes formula should*
*be the spot rate with a time to maturity equal to τ .*

*• In other words,*

*r =*

_{n−1}

*i=0* *r*_{i}

*τ* *,*

*where r** _{i}* is the continuously compounded, short rate

*measured in periods for period i.*

### 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}

*• How to harmonize these two diﬀerent times into the*
Black-Scholes formula and binomial tree algorithms?^{b}

aFama (1965); French (1980); French and Roll (1986).

bRecall p. 145 about dating issues.

### Trading Days and Calendar Days (continued)

*• Think of σ as measuring the annualized volatility of*
*stock price one year from now.*

*• Suppose a year has m (say 253) trading days.*

*• We can replace σ in the Black-Scholes formula with*^{a}

*σ*

365

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

aFrench (1984).

### Trading Days and Calendar Days (concluded)

*• This works only for European options.*

*• How about binomial tree algorithms?*^{a}

aContributed by Mr. Lu, Zheng-Liang (D00922011) in 2015.

### 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).

### 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. 298 (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}

### The Ad-Hoc Approximation vs. P. 298 (Step 2)

*(S − D/R) + D/R = S*

*

j

*(S − D/R)u*

*

j

*(S − D/R)d*

*

*(S − D/R)u*^{2}

*(S − D/R)ud*

*(S − D/R)d*^{2}

### The Ad-Hoc Approximation vs. P. 298

^{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. 298).

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

*• Note that, as d < R < u,*

*(Su − D) u > (S − D/R)u*^{2}*,*
*(Sd − D) d < (S − D/R)d*^{2}*,*

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

### The Ad-Hoc Approximation vs. P. 298 (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. 686ﬀ).

*• 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.

^{a}

### 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 ),* (29)
*P = Xe*^{−rτ}*N (−x + σ√*

*τ ) − Se*^{−qτ}*N (−x),*

(29* ^{}*)
where

*x ≡* *ln(S/X) +*

*r − q + σ*^{2}*/2*
*τ*
*σ√*

*τ* *.*

*• Formulas (29) and (29** ^{}*) 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* *,* (30)

*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. (30), binomial tree*

*algorithms stay the same as if there were no dividends.*

*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. 280).

*• Recall that*

*N*^{}*(y) =* *e*^{−y}^{2}^{/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.

**– Elementary calculus.**

*• The delta used in the BOPM (p. 226) 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 apparently 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)

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 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 Δ*_{1}

### 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.

### 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 *≡* *∂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 (concluded)

*• Note that if S = X, τ → 0 implies*
Λ *→ 0*

(which answers the question on p. 284 for the Black-Scholes model).

*• The Black-Scholes formula (p. 280) implies*

*C* *→ S,*

*P* *→ Xe*^{−rτ}*,*
*as σ → ∞.*

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 ≡* *∂f*

*∂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.

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,*
*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.*

aPelsser and Vorst (1994).

*S/(ud)*

*S/d*

*S/u*

*Su/d*

*S*

*Sd/u*

*Su*

*Sd*
*Suu/d*

*Sdd/u*

*Suuu/d*

*Suu*

*S*

*Sdd*

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

*• Alternative formulas exist (p. 583).*

### 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*
*f*_{ud}*− f*

*2(τ /n)* *.*

**– In fact, the theta of a European option can be**
derived from delta and gamma (p. 582).

*• For vega and rho, there seems no alternative but to run*
the binomial tree algorithm twice.^{a}

aBut see pp. 943ﬀ.

*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.

### 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 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,*

*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*

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

*• From Theorem 11 (p. 280) 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 = 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 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}

### 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.**

*• 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 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-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. 345 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.

### 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.*

### 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 total strike price of $45,000.

*• n such calls are worth $1,937.5 (p. 345).*

*• So the total market value of the XYZ.com stock is*

### 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 (R86526008, D8852600), Lyuu, and Wang (F95922018) (2010).

### Barrier Options

^{a}

*• Their payoﬀ depends on whether the underlying asset’s*
*price reaches a certain price level H.*

*• 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 HK 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 ),*

(31)

**– x ≡**^{ln(H}^{2}*/(SX))+(r−q+σ*^{2}*/2) τ*
*σ**√*

*τ* .

**– λ ≡ (r − q + σ**^{2}*/2)/σ*^{2}.

*• A European down-and-out call can be priced via the*
in-out parity (see text).

### 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. (29) on p. 307 for*
*the call with H*^{2}*/S.*

*• Equation (31) on p. 356 for the down-and-in call*
*becomes Eq. (29) when r − q = σ*^{2}*/2.*

*• Equation (31) becomes S/H times Eq. (29) when*
*r − q = 0.*

### Interesting Observations (concluded)

*• Replace S in the pricing formula for the down-and-in*
*call, Eq. (31), with H*^{2}*/S.*

*• Equation (31) becomes Eq. (29) when r − q = σ*^{2}*/2.*

*• Equation (31) becomes H/S times Eq. (29) when*
*r − q = 0.*^{a}

*• Why?*

aContributed by Mr. Chou, Ming-Hsin (R02723073) on April 24, 2014.

### Binomial Tree Algorithms

*• Barrier options can be priced by binomial tree*
algorithms.

*• Below is for the down-and-out option.*

0 *H*

*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

### 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 to a node price.**

**– This “eﬀective barrier” changes as n increases.**

*• In fact, the binomial tree is O(1/√*

*n) convergent.*^{b}

*• Solutions will be presented later.*

aBoyle and Lau (1994).

bLin (R95221010) (2008).

### Binomial Tree Algorithms (concluded)

^{a}

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value

aLyuu (1998).