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+σ^{2}/2) τ σ√
τ (recall p. 292).
• Recall that
N^{}(y) = e√^{−y}^{2}^{/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. 239) 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 nondividendpaying 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., when^{a}
X = Se^{(r+σ}^{2}^{/2) τ}. (44)
aThe straddle (p. 207) 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: inthemoney call or outofthemoney put.
Solid curves: atthemoney options.
Dashed curves: outofthemoney calls or inthemoney puts.
Delta (continued)
• Suppose the stock pays a continuous dividend yield of q.
• Let
x =^{Δ} ln(S/X) +
r − q + σ^{2}/2 τ σ√
τ (45)
(recall p. 322).
• Then
∂C
∂S = e^{−qτ}N (x) > 0,
∂P
∂S = −e^{−qτ}N (−x) < 0.
Delta (continued)
• Consider an X_{1}strike call and an X_{2}strike put, X_{1} ≥ X_{2}.
• They are otherwise identical.
• Let
x_{i} =^{Δ} ln(S/X_{i}) +
r − q + σ^{2}/2 τ σ√
τ . (46)
• Then their deltas sum to zero when x1 = −x2.^{a}
• That implies
S
X1 = X_{2}
S e^{−(2r−2q+σ}^{2}^{) τ}. (47)
aThe strangle (p. 209) C + P then has zero delta!
Delta (concluded)
• Suppose we demand X_{1} = X_{2} = X and have a straddle.
• Then
X = Se^{(r−q+σ}^{2}^{/2) τ} leads to a straddle with zero delta.
– This generalizes Eq. (44) on p. 332.
• 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 deltaneutral.
– A deltaneutral 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 deltaneutral.
– 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} by shorting Δ_{1}/Δ_{2} units of a security with delta Δ_{2}.
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 nondividendpaying 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: inthemoney call or outofthemoney put.
Solid curves: atthemoney options.
Dashed curve: outofthemoney call or inthemoney put.
Theta (concluded)
• Suppose the stock pays a continuous dividend yield of q.
• Deﬁne x as in Eq. (45) on p. 334.
• For a European call, add an extra term to the earlier formula for the theta:
Θ = −SN^{}(x) σ 2√
τ − rXe^{−rτ}N (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^{−rτ}N (−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/∂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 nondividendpaying 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: inthemoney call or outofthemoney put.
Solid lines: atthemoney option.
Dashed lines: outofthemoney call or inthemoney put.
Vega
^{a}(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 nondividendpaying 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 σS^{2}Γ.
• If the stock pays a continuous dividend yield of q, then Λ = Se^{−qτ}√
τ N^{}(x), where x is deﬁned in Eq. (45) on p. 334.
• Vega is maximized when x = 0, i.e., when S = Xe^{−(r−q+σ}^{2}^{/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 X1strike call C and a short X_{2}strike put P , X_{1} ≥ X_{2}.
• The options’ vegas cancel out when x1 = −x2,
where x_{i} are deﬁned in Eq. (46) on p. 335.
• This also leads to Eq. (47) on p. 335.
– Recall the same condition led to zero delta for the strangle C + P (p. 335).
Vega (concluded)
• Note that if S = X, τ → 0 implies Λ → 0
(which answers the question on p. 297 for the BlackScholes model).
• The BlackScholes formula (p. 292) 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: inthemoney call or outofthemoney put.
Solid curves: atthemoney option.
Dashed curve: outofthemoney call or inthemoney put.
Variance Vega
^{a}• 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
∂σ^{2}. – Note that it is not deﬁned as ∂^{2}f /∂σ^{2}!
• 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 = ∂^{2}f
∂σ^{2} .
• It can be shown that
volga = Λ x(x − σ√ τ ) σ
= Λ
σ
ln^{2}(S/X)
σ^{2}τ − σ^{2}τ 4
, where x is deﬁned in Eq. (45) on p. 334.^{a}
aDerman & M. B. Miller (2016).
Volga (concluded)
• Volga is zero when S = Xe^{±σ}^{2}^{τ /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 smile of implied volatility at the same maturity.^{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 nondividendpaying stock is
Xτ e^{−rτ}N (x − σ√
τ ) > 0.
• The rho of a European put on a nondividendpaying 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: inthemoney call or outofthemoney put.
Solid curves: atthemoney option.
Dashed curves: outofthemoney call or inthemoney put.
Numerical Greeks
• Needed when closedform 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. (48)
• 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 (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−f_{ud}
Suu−Sud − _{Sud−Sdd}^{f}^{ud}^{−f}^{dd}
(Suu − Sdd)/2 . (49)
• Alternative formulas exist (p. 659).
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 S^{2} approximates the numerical gamma.
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. 658).
• The vega of a European option can be derived from gamma (p. 344).
• For rho, there seems no alternative but to run the binomial tree algorithm twice.^{a}
aBut see p. 840 and pp. 1030ﬀ.
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 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).
Risky ZeroCoupon Bonds and Stock
• Consider XYZ.com.
• Capital structure:
– n shares of its own common stock, S.
– Zerocoupon 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 ZeroCoupon 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 ZeroCoupon 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 .
aSee p. 198.
Risky ZeroCoupon 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 ZeroCoupon Bonds and Stock (continued)
• From Theorem 10 (p. 292) and the putcall parity,^{a} nS = V N (x) − Xe^{−rτ}N (x − σ√
τ ), (50) B = V N (−x) + Xe^{−rτ}N (x − σ√
τ ). (51) – Above,
x =^{Δ} 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 .
aMerton (1974).
Risky ZeroCoupon 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 debttototalvalue ratio.
Risky ZeroCoupon 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. (50)–(51) on p. 367 become, respectively, nS = V e^{−qτ}N (x) − Xe^{−rτ}N (x − σ√
τ ),
B = (1 − α)V e^{−qτ}N (−x) + Xe^{−rτ}N (x − σ√ τ ).
– Above,
x =^{Δ} ln(V /X) + (r − q + σ^{2}/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 zerocoupon 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}
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 defaultfree zerocoupon bond with $X par value plus n written European puts on Merck at a strike price of $30.
– By the putcall parity.^{a}
• The diﬀerence between B and the price of the defaultfree 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. 221.
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 debtequity 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 or their boards.
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. 374 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. 374).
• 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 knockout (KO) 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 knockout option is sometimes called a downandout option if H < S.
• A put knockout option is sometimes called an upandout 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 (continued)
• A knockin (KI) option comes into existence if a certain barrier is reached.
• A downandin option is a call knockin option that comes into existence only when the barrier is reached and H < S.
• An upandin is a put knockin 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).
Barrier Options (concluded)
• Knockin puts are the most popular barrier options.^{a}
• Knockout puts are the second most popular barrier options.^{b}
• Knockout calls are the most popular among barrier call options.^{c}
aBennett (2014).
bBennett (2014).
cBennett (2014).
A Formula for DownandIn Calls
^{a}• Assume X ≥ H.
• The value of a European downandin 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),
(52)
– x =^{Δ} ^{ln(H}^{2}/(SX))+(r−q+σ^{2}/2) τ σ√
τ .
– λ = (r − q + σ^{Δ} ^{2}/2)/σ^{2}.
• A European downandout call can be priced via the inout parity (see text).
aMerton (1973). See Exercise 17.1.6 of the textbook for a proof.
A Formula for UpandIn Puts
^{a}• Assume X ≤ H.
• The value of a European upandin put is
Xe^{−rτ}
H S
_{2λ−2}
N(−x + σ√
τ) − Se^{−qτ}
H S
_{2λ}
N(−x).
• Again, a European upandout put can be priced via the inout 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, JuiChung (D97723002) on March 25, 2009.
Interesting Observations
• Assume H < X.
• Replace S in the Merton pricing formula Eq. (42) on p.
322 for the call with H^{2}/S.
• Equation (52) on p. 386 for the downandin call becomes Eq. (42) when r − q = σ^{2}/2.
• Equation (52) becomes S/H times Eq. (42) when r − q = 0.
Interesting Observations (concluded)
• Replace S in the pricing formula for the downandin call, Eq. (52), with H^{2}/S.
• Equation (52) becomes Eq. (42) when r − q = σ^{2}/2.
• Equation (52) becomes H/S times Eq. (42) when r − q = 0.^{a}
• Why?^{b}
aContributed by Mr. Chou, MingHsin (R02723073) on April 24, 2014.
bApply the reﬂection principle (p. 688), Eq. (41) on p. 285, and Lemma 9 (p. 290).
Binomial Tree Algorithms
• Barrier options can be priced by binomial tree algorithms.
• Below is for the downandout option.
0 H
• Pricing downandin 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.
Backwardinduction: C = (0.5 × C_{u} + 0.5 × C_{d})/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 lower (or higher) to a closeby node price.
– This “eﬀective barrier” thus changes as n increases.
• In fact, the binomial tree is O(1/√
n) convergent.^{b}
• Solutions will be presented later.
aBoyle & Lau (1994).
bJ. Lin (R95221010) (2008).
Binomial Tree Algorithms (concluded)
^{a}100 150 200 250 300 350 400
#Periods 3
3.5 4 4.5 5 5.5
Downandin call value
aLyuu (1998).
Daily Monitoring
• Many barrier options monitor the barrier only for daily closing prices.
• If so, only nodes at the end of a day need to check for the barrier condition.
• We can even remove intraday nodes to create a multinomial tree.
– A node is then followed by d + 1 nodes if each day is partitioned into d periods.
• Does this save time or space?^{a}
aContributed by Ms. Chen, TzuChun (R94922003) and others on April 12, 2006.
A Heptanomial Tree (6 Periods Per Day)

_{1 day}
Discrete Monitoring vs. Continuous Monitoring
• Discrete barriers are more expensive for knockout options than continuous ones.
• But discrete barriers are less expensive for knockin options than continuous ones.
• Discrete barriers are far less popular than continuous ones for individual stocks.^{a}
• They are equally popular for indices.^{b}
aBennett (2014).
bBennett (2014).
Data! data! data!
— Arthur Conan Doyle (1892), The Adventures of Sherlock Holmes
Foreign Currencies
• S denotes the spot exchange rate in domestic/foreign terms.
– By that we mean the number of domestic currencies per unit of foreign currency.^{a}
• σ denotes the volatility of the exchange rate.
• r denotes the domestic interest rate.
• ˆr denotes the foreign interest rate.
aThe market convention is the opposite: A/B = x means one unit of currency A (the reference currency or base currency) is equal to x units of currency B (the countervalue currency).
Foreign Currencies (concluded)
• A foreign currency is analogous to a stock paying a known dividend yield.
– Foreign currencies pay a “continuous dividend yield”
equal to ˆr in the foreign currency.
Time Series of the Daily Euro–USD Exchange Rate
Distribution of the Daily Euro–USD Exchange Rate
Time Series of the Minutely Euro–USD Exchange Rate
Distribution of the Minutely Euro–USD Exchange Rate
Time Series of the Daily GBP–USD Exchange Rate
Distribution of the Daily GBP–USD Exchange Rate
Distribution of the Minutely GBP–USD Exchange Rate
Distribution of the Daily JPY–USD Exchange Rate
Foreign Exchange Options
• In 2000 the total notional volume of foreign exchange options was US$13 trillion.^{a}
– 38.5% were vanilla calls and puts with a maturity less than one month.
– 52.5% were vanilla calls and puts with a maturity between one and 18 months.
– 4% were barrier options.
– 1.5% were vanilla calls and puts with a maturity more than 18 months.
– 1% were digital options (see p. 839).
– 0.7% were Asian options (see p. 420).
aLipton (2002).
Foreign Exchange Options (continued)
• Foreign exchange options are settled via delivery of the underlying currency.
• A primary use of foreign exchange (or forex) options is to hedge currency risk.
• Consider a U.S. company expecting to receive 100 million Japanese yen in March 2000.
• Those 100 million Japanese yen will be exchanged for U.S. dollars.
Foreign Exchange Options (continued)
• The contract size for the Japanese yen option is JPY6,250,000.
• The company purchases
100,000,000
6,250,000 = 16
puts on the Japanese yen with a strike price of $.0088 and an exercise month in March 2000.
• This gives the company the right to sell 100,000,000 Japanese yen for
100,000,000 × .0088 = 880,000 U.S. dollars.
Foreign Exchange Options (concluded)
• Assume the exchange rate S is lognormally distributed.
• The formulas derived for stock index options in Eqs. (42) on p. 322 apply with the dividend yield equal to ˆr:
C = Se^{−ˆrτ}N (x) − Xe^{−rτ}N (x − σ√
τ ), (53) P = Xe^{−rτ}N (−x + σ√
τ ) − Se^{−ˆrτ}N (−x).
(53^{}) – Above,
x =^{Δ} ln(S/X) + (r − ˆr + σ^{2}/2) τ σ√
τ .
Distribution of the Logarithmic Euro–USD Exchange Rate
Distribution of the Logarithmic GBP–USD Exchange Rate
Distribution of the Logarithmic GBP–USD Exchange Rate (after the Collapse of Lehman Brothers and before Brexit)
Distribution of the Logarithmic JPY–USD Exchange Rate