### Toward the Black-Scholes Formula

*• The binomial model seems to suﬀer from two unrealistic*
assumptions.

**– The stock price takes on only two values in a period.**

**– Trading occurs at discrete points in time.**

*• As n increases, the stock price ranges over ever larger*
numbers of possible values, and trading takes place
nearly continuously.^{a}

*• Need to calibrate the BOPM’s parameters u, d, and R*
to make it converge to the continuous-time model.

*• We now skim through the proof.*

### Toward the Black-Scholes Formula (continued)

*• Let τ denote the time to expiration of the option*
measured in years.

*• Let r be the continuously compounded annual rate.*

*• With n periods during the option’s life, each period*
*represents a time interval of τ /n.*

*• Need to adjust the period-based u, d, and interest rate*
*r to match the empirical results as n → ∞.*ˆ

### Toward the Black-Scholes Formula (continued)

*• First, ˆr = rτ/n.*

**– Each period is τ/n years long.**

**– The period gross return R = e*** ^{ˆr}*.

*• Let*

*μ* =^{Δ} 1
*n* *E*

ln *S*_{τ}*S*

denote the expected value of the continuously

compounded rate of return per period of the BOPM.

*• Let*

*σ*^{2 Δ}= 1

Var

ln *S*_{τ}

### Toward the Black-Scholes Formula (continued)

*• Under the BOPM, it is not hard to show that*^{a}

*μ = q ln(u/d) + ln d,*

*σ*^{2} = *q(1 − q) ln*^{2}*(u/d).*

*• Assume the stock’s true continuously compounded rate*
*of return over τ years has mean μτ and variance σ*^{2}*τ .*

*• Call σ the stock’s (annualized) volatility.*

aRecall the Bernoulli distribution.

### Toward the Black-Scholes Formula (continued)

*• The BOPM converges to the distribution only if*

*nμ = n[ q ln(u/d) + ln d ] → μτ,* (40)
*nσ*^{2} = *nq(1 − q) ln*^{2}*(u/d) → σ*^{2}*τ.* (41)

*• We need one more condition to have a solution for u, d, q.*

### Toward the Black-Scholes Formula (continued)

*• Impose*

*ud = 1.*

**– It makes nodes at the same horizontal level of the**
tree have identical price (review p. 278).

**– Other choices are possible (see text).**

*• Exact solutions for u, d, q are feasible if Eqs. (40)–(41)*
are replaced by equations: 3 equations for 3 variables.^{a}

aChance (2008).

### Toward the Black-Scholes Formula (continued)

*• The above requirements can be satisﬁed by*

*u = e*^{σ}

*√**τ /n**, d = e*^{−σ}

*√**τ /n**, q =* 1

2 + 1 2

*μ*
*σ*

*τ*

*n* *.* (42)

*• With Eqs. (42), it can be checked that*
*nμ = μτ,*

*nσ*^{2} =

1 *−* *μ*
*σ*

_{2} *τ*
*n*

*σ*^{2}*τ → σ*^{2}*τ.*

### Toward the Black-Scholes Formula (continued)

*• The choices (42) result in the CRR binomial model.*^{a}

*• With the above choice, even if σ is not calibrated*
correctly, the mean is still matched!^{b}

aCox, Ross, & Rubinstein (1979).

b*Recall Eq. (35) on p. 250. So u and d are related to volatility exclu-*
*sively in the CRR model. They do not depend on r.*

### Toward the Black-Scholes Formula (continued)

*• The no-arbitrage inequalities d < R < u may not hold*
under Eqs. (42) on p. 289 or Eq. (34) on p. 249.

**– If this happens, the probabilities lie outside [ 0, 1 ].**^{a}

*• The problem disappears when n satisﬁes*
*e*^{σ}

*√**τ /n* *> e*^{rτ /n}*,*

*i.e., when n > r*^{2}*τ /σ*^{2} (check it).

**– So it goes away if n is large enough.**

**– Other solutions can be found in the textbook**^{b} or will
be presented later.

aMany papers and programs forget to check this condition!

bSee Exercise 9.3.1 of the textbook.

### Toward the Black-Scholes Formula (continued)

*• The central limit theorem says ln(S*_{τ}*/S) converges to*
*N (μτ, σ*^{2}*τ ).*^{a}

*• So ln S*_{τ}*approaches N (μτ + ln S, σ*^{2}*τ ).*

*• Conclusion: S** _{τ}* has a lognormal distribution in the limit.

a*The normal distribution with mean μτ and variance σ*^{2}*τ .*

### Toward the Black-Scholes Formula (continued)

**Lemma 9 The continuously compounded rate of return***ln(S*_{τ}*/S) approaches the normal distribution with mean*
*(r − σ*^{2}*/2) τ and variance σ*^{2}*τ in a risk-neutral economy.*

*• Let q equal the risk-neutral probability*
*p* *= (e*^{Δ} ^{rτ /n}*− d)/(u − d).*

*• Let n → ∞.*^{a}

*• Then μ = r − σ*^{2}*/2.*

aSee Lemma 9.3.3 of the textbook.

### Toward the Black-Scholes Formula (continued)

*• The expected stock price at expiration in a risk-neutral*
economy is^{a}

*Se*^{rτ}*.*

*• The stock’s expected annual rate of return*^{b} is thus the
*riskless rate r.*

aBy Lemma 9 (p. 294) and Eq. (30) on p. 179.

b*In the sense of (1/τ ) ln E[ S**τ**/S ] (arithmetic average rate of return)*
*not (1/τ )E[ ln(S**τ**/S) ] (geometric average rate of return). In the latter*
*case, it would be r − σ*^{2}*/2 by Lemma 9.*

### Toward the Black-Scholes Formula (continued)

^{a}

**Theorem 10 (The Black-Scholes Formula, 1973)**
*C* = *SN (x) − Xe*^{−rτ}*N (x − σ√*

*τ ),*
*P* = *Xe*^{−rτ}*N (−x + σ√*

*τ ) − SN (−x),*
*where*

*x* =^{Δ} *ln(S/X) +*

*r + σ*^{2}*/2*
*τ*
*σ√*

*τ* *.*

aOn a United ﬂight from San Francisco to Tokyo on March 7, 2010, a real-estate manager mentioned this formula to me!

### Toward the Black-Scholes Formula (concluded)

*• See Eq. (39) on p. 266 for the meaning of x.*

*• See Exercise 13.2.12 of the textbook for an interpretation*
*of the probability associated with N (x) and N (−x).*

### BOPM and Black-Scholes Model

*• The Black-Scholes formula needs 5 parameters: S, X, σ,*
*τ , and r.*

*• Binomial tree algorithms take 6 inputs: S, X, u, d, ˆr,*
*and n.*

*• The connections are*

*u = e*^{σ}

*√**τ /n**,*

*d = e*^{−σ}

*√**τ /n**,*
*r = rτ /n.*ˆ

5 10 15 20 25 30 35 n

11.5 12 12.5 13

Call value

0 10 20 30 40 50 60 n

15.1 15.2 15.3 15.4 15.5

Call value

*• S = 100, X = 100 (left), and X = 95 (right).*

### BOPM and Black-Scholes Model (concluded)

*• The binomial tree algorithms converge reasonably fast.*

*• The error is O(1/n).*^{a}

*• Oscillations are inherent, however.*

*• Oscillations can be dealt with by the judicious choices of*
*u and d.*^{b}

aL. Chang & Palmer (2007).

bSee Exercise 9.3.8 of the textbook.

### Implied Volatility

*• Volatility is the sole parameter not directly observable.*

*• The Black-Scholes formula can be used to compute the*
market’s opinion of the volatility.^{a}

* – Solve for σ given the option price, S, X, τ, and r*
with numerical methods.

**– How about American options?**^{b}

a*Implied volatility is hard to compute when τ is small (why?).*

bOptionMetrics’s (2015) IvyDB uses the CRR binomial tree (see

http://www.ckgsb.com/uploads/report/file/201611/02/1478069847635278.pdf).

### Implied Volatility (concluded)

*• Implied volatility is*

the wrong number to put in the wrong formula to
get the right price of plain-vanilla options.^{a}

*• Just think of it as an alternative to quoting option*
prices.

*• Implied volatility is often preferred to historical*
volatility in practice.

**– Using the historical volatility is like driving a car**
with your eyes on the rearview mirror?

aRebonato (2004).

### Problems; the Smile

*• Options written on the same underlying asset usually do*
not produce the same implied volatility.

*• A typical pattern is a “smile” in relation to the strike*
price.

**– The implied volatility is lowest for at-the-money**
options.

**– It becomes higher the further the option is in- or**
out-of-the-money.

*• Other patterns have also been observed.*

### TXO Calls (September 25, 2015)

^{a}

**300**

**14** **200**
**9000**

**8500**
**16**

**8000** **100**

**7500**
**18**

**7000** **0**
**20**

**22**
**24**

ATM = $8132

aThe underlying Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) closed at 8132. Plot supplied by Mr. Lok, U Hou (D99922028) on December 6, 2017.

### Solutions to the Smile

*• To address this issue, volatilities are often combined to*
produce a composite implied volatility.

*• This practice is not sound theoretically.*

*• The existence of diﬀerent implied volatilities for options*
on the same underlying asset shows the Black-Scholes
model cannot be literally true.

### Binomial Tree Algorithms for American Puts

*• Early exercise has to be considered.*

*• The binomial tree algorithm starts with the terminal*
payoﬀs

*max(0, X − Su*^{j}*d** ^{n−j}*)
and applies backward induction.

*• At each intermediate node, it compares the payoﬀ if*
*exercised and the continuation value.*

*• It keeps the larger one.*

### Bermudan Options

*• Some American options can be exercised only at discrete*
time points instead of continuously.

*• They are called Bermudan options.*

*• Their pricing algorithm is identical to that for American*
options.

*• But early exercise is considered for only those nodes*
when early exercise is permitted.

### 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* *.*

aMerton (1973).

### Time-Dependent Instantaneous Volatility (concluded)

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

*u = e*^{σ(t)}

*√**τ /n**,*

*d = e*^{−σ(t)}

*√**τ /n**.*

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

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

### Volatility (1990–2016)

^{a}

2-Jan-90 2-Jan-91 2-Jan-92 4-Jan-93 3-Jan-94 3-Jan-95 2-Jan-96 2-Jan-97 2-Jan-98 4-Jan-99 3-Jan-00 2-Jan-01 2-Jan-02 2-Jan-03 2-Jan-04 3-Jan-05 3-Jan-06 3-Jan-07 2-Jan-08 2-Jan-09 4-Jan-10 3-Jan-11 3-Jan-12 2-Jan-13 2-Jan-14 2-Jan-15 4-Jan-16 0

10 20 30 40 50 60 70 80 90

VIX

CBOE S&P 500 Volatility Index

aSupplied by Mr. Lok, U Hou (D99922028) on July 17, 2017.

### Time-Dependent Short Rates

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

*• The annual 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.*

^{a}

### 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); K. French (1980); K. French & Roll (1986).

bRecall p. 162 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 *.*

aD. French (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); Heath & Jarrow (1988).

### An Ad-Hoc Approximation (concluded)

*• Start with the current stock price minus the PV of*
future dividends before expiration.

*• Develop the binomial tree for the new stock price as if*
there were no dividends.

*• Then add to each stock price on the tree the PV of all*
*future dividends before expiration.*

*• American option prices can be computed as before on*
this tree of stock prices.

### The Ad-Hoc Approximation vs. P. 317 (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. 317 (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. 317

^{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. 317).

**– (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. 317 (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. 769ﬀ).

*• 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 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)

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

^{a}

*C = Se*^{−qτ}*N (x) − Xe*^{−rτ}*N (x − σ√*

*τ ),* (43)
*P = Xe*^{−rτ}*N (−x + σ√*

*τ ) − Se*^{−qτ}*N (−x),*

(43* ^{}*)
where

*x* =^{Δ} *ln(S/X) +*

*r − q + σ*^{2}*/2*
*τ*
*σ√*

*τ* *.*

*• Formulas (43) and (43** ^{}*) remain valid as long as the
dividend yield is predictable.

aMerton (1973).

### Continuous Dividend Yields (continued)

*• To run binomial tree algorithms, replace u with ue*^{−qΔt}*and d with de*^{−qΔt}*, where Δt* *= τ /n.*^{Δ}

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

*• Other than the changes, binomial tree algorithms stay*
the same.

**– In particular, p should use the original u and d!**^{a}

aContributed by Ms. Wang, Chuan-Ju (F95922018) on May 2, 2007.

### Continuous Dividend Yields (concluded)

*• Alternatively, pick the risk-neutral probability as*
*e*^{(r−q) Δt}*− d*

*u − d* *,* (44)

*where Δt* *= τ /n.*^{Δ}

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

*• The u and d remain unchanged.*

*• Other than the change in Eq. (44), binomial tree*

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

### Distribution of Logarithmic Returns of TAIEX

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

^{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 Section 9.7 of the textbook for the tree analog.

*Sensitivity Analysis of Options*

Cleopatra’s nose, had it been shorter, the whole face of the world would have been changed.

— Blaise Pascal (1623–1662)

### Sensitivity Measures (“The Greeks”)

*• How the value of a security changes relative to changes*
in a given parameter is key to hedging.

**– Duration, for instance.**

*• Let x* =^{Δ} *ln(S/X)+(r+σ*^{2}*/2) τ*
*σ**√*

*τ* (recall p. 296).

*• 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. 243) is the discrete*
analog.

*• The delta of a long stock is apparently 1.*

### 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., when*^{a}

*X = Se*^{(r+σ}^{2}^{/2) τ}*.* (45)

a*The straddle (p. 210) 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.

### 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* (46)

(recall p. 326).

*• 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* *.* (47)

*• Then their deltas sum to zero when x*1 = *−x*2.^{a}

*• That implies*

*S*

*X* = *X*_{2}

*S* *e*^{−(2r−2q+σ}^{2}^{) τ}*.* (48)

### 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. (45) on p. 336.**

*• When C(X*1*)’s delta and P (X*2)’s delta sum to zero,
*does the portfolio C(X*1) *− P (X*2) have zero value?

*• In general, no.*

### Delta Neutrality

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

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

*• Creating one serves for hedging purposes.*

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

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

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

*• In general, hedge a position in a security with delta Δ*_{1}

### Theta (Time Decay)

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

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

2*√*

*τ* *− rXe*^{−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.

### Theta (concluded)

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

*• Deﬁne x as in Eq. (46) on p. 338.*

*• 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*
non-dividend-paying stock is

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 security’s value with*
respect to the volatility of the underlying asset

Λ =^{Δ} *∂f*

*∂σ.*

*• Volatility often changes over time.*

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

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

*τ N*^{}*(x) > 0.*

**– So higher volatility always increases the option value.**

### 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. (46) on p. 338.*

*• 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 X*1-strike call
*C and a short X*_{2}*-strike put P , X*_{1} *≥ X*_{2}.

*• The options’ vegas cancel out when*
*x*1 = *−x*2*,*

*where x** _{i}* are deﬁned in Eq. (47) on p. 339.

*• This also leads to Eq. (48) on p. 339.*

**– Recall the same condition led to zero delta for the**
*strangle C + P (p. 339).*

### Vega (concluded)

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

(which answers the question on p. 301 for the Black-Scholes model).

*• The Black-Scholes formula (p. 296) 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 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. (46) on p. 338.*^{a}

### 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 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.* (49)

*• 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,*

*f**uu**−f*_{ud}

*Suu−Sud* *−* _{Sud−Sdd}^{f}^{ud}^{−f}^{dd}

*(Suu − Sdd)/2* *.* (50)

*• Alternative formulas exist (p. 671).*

### 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}

### 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. 670).

*• The vega of a European option can be derived from*
gamma (p. 348).

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