Sensitivity Measures (“The Greeks”)

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. 244).

• Note that

N⁰(y) = e^−y2/2

√2π > 0,

the density function of standard normal distribution.

Delta

• Defined as ∆ ≡ ∂f /∂S.

– f is the price of the derivative.

– S is the price of the underlying asset.

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

– Elementary calculus.

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

Delta (concluded)

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

∂C

∂S = N (x) > 0.

• The delta of a European put equals

∂P

∂S = N (x) − 1 < 0.

• The delta of a long stock is 1.

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

0.2 0.4 0.6 0.8 1

Delta (call)

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

-0.8 -0.6 -0.4 -0.2 0

Delta (put)

0 20 40 60 80

Stock price 0

0.2 0.4 0.6 0.8 1

Delta (call)

0 20 40 60 80

Stock price -1

-0.8 -0.6 -0.4 -0.2 0

Delta (put)

Solid curves: at-the-money options.

Delta Neutrality

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

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

• Creating one serves for hedging purposes.

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

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

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

Theta (Time Decay)

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

• For a European call on a non-dividend-paying stock, Θ = −SN⁰(x) σ

2√

τ − rXe^−rτN (x − σ√

τ ) < 0.

– The call loses value with the passage of time.

• For a European put, Θ = −SN⁰(x) σ

2√

τ + rXe^−rτN (−x + σ√ τ ).

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

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 its value with respect to the volatility of the underlying asset Λ ≡ ∂Π/∂σ.

• Volatility often changes over time.

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

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

τ N⁰(x) > 0.

– So higher volatility increases option value.

aVega is not Greek.

0 20 40 60 80 Stock price

0 2 4 6 8 10 12 14

Vega (call/put)

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

2.5 5 7.5 10 12.5 15 17.5

Vega (call/put)

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

Solid curves: at-the-money option.

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

Rho

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

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

Xτ e^−rτN (x − σ√

τ ) > 0.

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

−Xτ e^−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 + ∆S) − f (S − ∆S)

2∆S .

• 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.

• 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

Sddd/u

Numerical Gamma

• At the stock price (Suu + Sud)/2, delta is approximately (f_uu − f_ud)/(Suu − Sud).

• At the stock price (Sud + Sdd)/2, delta is approximately (f_ud − f_dd)/(Sud − Sdd).

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

f_uu−f_ud

Suu−Sud − _Sud−Sdd^fud−fdd (Suu − Sdd)/2 .

• Alternative formulas exist.

Finite Difference Fails for Numerical Gamma

• Numerical differentiation gives

f (S + ∆S) − 2f (S) + f (S − ∆S)

(∆S)² .

• It does not work (see text).

• But why did the binomial tree version work?

Other Numerical Greeks

• The theta can be computed as f_ud − f

2(τ /n) .

– In fact, the theta of a European option can be derived from delta and gamma (see p. 499).

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

Extensions of Options Theory

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

— Joan Robinson (1903–1983)

Pricing Corporate Securities

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

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

• Assume:

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

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

a

Risky Zero-Coupon Bonds and Stock

• Consider XYZ.com.

• Capital structure:

– n shares of its own common stock, S.

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

• What is the value of the bonds, B?

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

Risky Zero-Coupon Bonds and Stock (continued)

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

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

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

V ^∗ ≤ X V ^∗ > X

Bonds V ^∗ X

Stock 0 V ^∗ − X

Risky Zero-Coupon Bonds and Stock (continued)

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

– This call provides the limited liability for the stockholders.

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

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

• Let C stand for a call on V .

Risky Zero-Coupon Bonds and Stock (continued)

• Thus nS = C and B = V − C.

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

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

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

Risky Zero-Coupon Bonds and Stock (continued)

• From Theorem 8 (p. 244) and the put-call parity, nS = V N (x) − Xe^−rτN (x − σ√

τ ), B = V N (−x) + Xe^−rτN (x − σ√

τ ).

– 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 .

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

µ

N (−z) + 1

ω N (z − σ√ τ )

¶ . – ω ≡ Xe^−rτ/V .

– z ≡ (ln ω)/(σ√

τ ) + (1/2) σ√

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

A Numerical Example

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

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

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

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

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

—Call— —Put—

Option Strike Exp. Vol. Last Vol. Last Merck 30 Jul 328 151/4 . . . . . .

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 = 15250 dollars.

• The entire bond issue is worth

B = 44500 − 15250 = 29250 dollars.

– Or $975 per bond.

A Numerical Example (continued)

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

– By the put-call parity.

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

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

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

X B nS V

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

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

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

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

A Numerical Example (continued)

• Suppose the promised payment to bondholders is

$45,000.

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

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

dollars.

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

A Numerical Example (continued)

• There are conflicts between stockholders and bondholders.

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

• But a firm can change its capital structure.

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

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

A Numerical Example (continued)

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

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

• The table on p. 298 says the total market value of the bonds should be $42,562.5.

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

• The remaining stock is worth $1,937.5.

A Numerical Example (continued)

• The stockholders therefore gain

14187.5 + 1937.5 − 15250 = 875 dollars.

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

45 × 42562.5 = 875. (26)

A Numerical Example (continued)

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

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

• The strike price remains X = 30000.

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

• n such calls are worth $1,937.5 (p. 298).

• 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 − 1291.67 = 28375 dollars.

• Hence the stockholders gain

14833.3 + 1291.67 − 15250 ≈ 875 dollars.

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

$28,375, a loss of $875.

Other Examples

• Subordinated debts as bull call spreads.

• Warrants as calls.

• Callable bonds as American calls with 2 strike prices.

• Convertible bonds.

Barrier Options

• 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 student told me on March 26, 2004, that she did not un- derstand why I covered barrier options until she started working in a bank. She is 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 kinds of barrier options.

A Formula for Down-and-In Calls

• Assume X ≥ H.

• The value of a European down-and-in call on a stock paying a dividend yield of q is

Se^−qτ

µH S

¶_2λ

N (x) − Xe^−rτ

µH S

¶_2λ−2

N (x − σ√ τ ),

(27)

– 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).

A Formula for Down-and-Out Calls

• Assume X ≤ H.

• The value of a European up-and-in put is

Xe^−rτ

µH

S

¶_2λ−2

N (−x + σ√

τ ) − Se^−qτ

µH

S

¶_2λ

N (−x).

• A European up-and-out put can be priced via the in-out parity.

aMerton (1973).

Interesting Observations

• Assume H < X.

• Replace S in the pricing formula for the down-and-in call, Eq. (27) on p. 309, with H²/S.

• Equation (27) becomes Eq. (24) on p. 261 when r − q = σ²/2.

• Equation (27) becomes S/H times Eq. (24) on p. 261 when r − q = 0.

• Why?

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

S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.

Binomial Tree Algorithms (concluded)

• But convergence is erratic because H is not at a price level on the tree (see plot on next page).

– Reason: The barrier has to be adjusted to be at a price level.

• Solutions will be presented later.

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value

Daily Monitoring

• Almost all barrier options monitor the barrier only for the daily closing prices.

• In that case, 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.

• This saves time and space?^a

aContributed by Ms. Chen, Tzu-Chun (R94922003) and others on April 12, 2006.

A Heptanomial Tree (6 Periods Per Day)

-

¾

Foreign Currencies

• S denotes the spot exchange rate in domestic/foreign terms.

• σ denotes the volatility of the exchange rate.

• r denotes the domestic interest rate.

• ˆr denotes the foreign interest rate.

• 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.

Foreign Exchange Options

• 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)

• The formulas derived for stock index options in Eqs. (24) on p. 261 apply with the dividend yield equal to ˆr:

C = Se^−ˆrτN (x) − Xe^−rτN (x − σ√

τ ), (28) P = Xe^−rτN (−x + σ√

τ ) − Se^−ˆrτN (−x).

(28⁰) – Above,

x ≡ ln(S/X) + (r − ˆr + σ²/2) τ σ√

τ .

Bar the roads!

Bar the paths!

Wert thou to flee from here, wert thou to find all the roads of the world, the way thou seekst the path to that thou’dst find not[.]

— Richard Wagner (1813–1883), Parsifal

Path-Dependent Derivatives

• Let S₀, S₁, . . . , S_n denote the prices of the underlying asset over the life of the option.

• S₀ is the known price at time zero.

• S_n is the price at expiration.

• The standard European call has a terminal value depending only on the last price, max(S_n − X, 0).

• Its value thus depends only on the underlying asset’s terminal price regardless of how it gets there.

Path-Dependent Derivatives (continued)

• Some derivatives are path-dependent in that their terminal payoff depends explicitly on the path.

• The (arithmetic) average-rate call has this terminal value:

max

à 1

n + 1

Xn i=0

S_i − X, 0

! .

• The average-rate put’s terminal value is given by max

Ã

X − 1

n + 1

Xn i=0

S_i, 0

! .

Path-Dependent Derivatives (continued)

• Average-rate options are also called Asian options.

• They are very popular.^a

• They are useful hedging tools for firms that will make a stream of purchases over a time period because the costs are likely to be linked to the average price.

• They are mostly European.

aAs of the late 1990s, the outstanding volume was in the range of 5–10 billion U.S. dollars according to Nielsen and Sandmann (2003).

Path-Dependent Derivatives (concluded)

• A lookback call option on the minimum has a terminal payoff of S_n − min_0≤i≤n S_i.

• A lookback put on the maximum has a terminal payoff of max_0≤i≤n S_i − S_n.

• The fixed-strike lookback option provides a payoff of – max(max_0≤i≤n S_i − X, 0) for the call;

– max(X − min_0≤i≤n S_i, 0) for the put.

• Lookback calls and puts on the average (instead of a constant X) are called average-strike options.

Average-Rate Options

• Average-rate options are notoriously hard to price.

• The binomial tree for the averages does not combine.

• A straightforward algorithm enumerates the 2ⁿ price paths for an n-period binomial tree and then averages the payoffs.

• But the complexity is exponential.

• As a result, the Monte Carlo method and approximation algorithms are some of the alternatives left.

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