Risk-Neutral Probability

Risk

• Surprisingly, the option value is independent of q.^a

• Hence it is independent of the expected value of the stock,

qSu + (1 − q) Sd.

• The option value depends on the sizes of price changes, u and d, which the investors must agree upon.

• Then the set of possible stock prices is the same whatever q is.

aMore precisely, not directly dependent on q. Thanks to a lively class discussion on March 16, 2011.

Pseudo Probability

• After substitution and rearrangement,

hS + B =

R−d u−d



C_u +

u−R u−d

 C_d

R .

• Rewrite it as

hS + B = pC_u + (1 − p) C_d

R ,

where

p =^Δ R − d

u − d . (34)

Pseudo Probability (concluded)

• As 0 < p < 1, it may be interpreted as probability.

• Alternatively,

R − d u − d



C_u +

u − R u − d

 C_d

interpolates the value at SR through points (Su, C_u) and (Sd, C_d).

Risk-Neutral Probability

• The expected rate of return for the stock is equal to the riskless rate ˆr under p as

pSu + (1 − p) Sd = RS. (35)

• The expected rates of return of all securities must be the riskless rate when investors are risk-neutral.

• For this reason, p is called the risk-neutral probability.

• The value of an option is the expectation of its

discounted future payoff in a risk-neutral economy.

• So the rate used for discounting the FV is the riskless rate^a in a risk-neutral economy.

aRecall the question on p. 240.

Option on a Non-Dividend-Paying Stock: Multi-Period

• Consider a call with two periods remaining before expiration.

• Under the binomial model, the stock can take on 3 possible prices at time two: Suu, Sud, and Sdd.

– There are 4 paths.

– But the tree combines or recombines; hence there are only 3 terminal prices.

• At any node, the next two stock prices only depend on the current price, not the prices of earlier times.^a

aIt is Markovian.

S

Su

Sd

Suu

Sud

Sdd

Option on a Non-Dividend-Paying Stock: Multi-Period (continued)

• Let C_uu be the call’s value at time two if the stock price is Suu.

• Thus,

C_uu = max(0, Suu − X).

• C_ud and C_dd can be calculated analogously, C_ud = max(0, Sud − X),

C_dd = max(0, Sdd − X).

C

Cu

Cd

Cuu= max( 0, Suu X )

Cud = max( 0, Sud X )

Cdd = max( 0, Sdd X )

Option on a Non-Dividend-Paying Stock: Multi-Period (continued)

• The call values at time 1 can be obtained by applying the same logic:

C_u = pC_uu + (1 − p) Cud

R , (36)

C_d = pC_ud + (1 − p) C_dd

R .

• Deltas can be derived from Eq. (32) on p. 251.

• For example, the delta at C_u is C_uu − C_ud Suu − Sud.

Option on a Non-Dividend-Paying Stock: Multi-Period (concluded)

• We now reach the current period.

• Compute

pC_u + (1 − p) C_d R

as the option price.

• The values of delta h and B can be derived from Eqs. (32)–(33) on p. 251.

Early Exercise

• Since the call will not be exercised at time 1 even if it is American, C_u ≥ Su − X and C_d ≥ Sd − X.

• Therefore,

hS + B = pC^u + (1 − p) Cd

R ≥ [ pu + (1 − p) d ] S − X R

= S − X

R > S − X.

– The call again will not be exercised at present.^a

• So

C = hS + B = pC_u + (1 − p) C_d

R .

aConsistent with Theorem 5 (p. 234).

Backward Induction

• The above expression calculates C from the two successor nodes C_u and C_d and none beyond.

• The same computation happened at C_u and C_d, too, as demonstrated in Eq. (36) on p. 262.

• This recursive procedure is called backward induction.

• C equals

[ p²Cuu + 2p(1 − p) Cud + (1 − p)²Cdd](1/R²)

= [ p² max

0, Su² − X

+ 2p(1 − p) max (0, Sud − X) +(1 − p)² max

0, Sd² − X

]/R².

aErnst Zermelo (1871–1953).

S0

1

*

j

S0u p

*

j

S0d 1 − p

*

j

S0u² p²

S0ud

2p(1 − p)

S0d² (1 − p)²

Backward Induction (continued)

• In the n-period case, C =

_n

j=0

_n

j

p^j(1 − p)^n−j × max

0, Su^jd^n−j − X

Rⁿ .

– The value of a call on a non-dividend-paying stock is the expected discounted payoff at expiration in a

risk-neutral economy.

• Similarly, P =

_n

j=0

_n

j

p^j(1 − p)^n−j × max

0, X − Su^jd^n−j

Rⁿ .

Backward Induction (concluded)

• Note that

p_j =^Δ

_n

j

 p^j(1 − p)^n−j Rⁿ

is the state price^a for the state Su^jd^n−j, j = 0, 1, . . . , n.

• In general,

option price =

j

(p_j × payoff at state j).

aRecall p. 212. One can obtain the undiscounted state price _n

j

p^j(1− p)^n−j—the risk-neutral probability—for the state Su^jd^n−j with (X_M − X_L)⁻¹ units of the butterfly spread where X_L = Su^j−1d^n−j+1, X_M = Su^jd^n−j, and X_H = Su^j−1+1d^n−j−1. See Bahra (1997).

Risk-Neutral Pricing Methodology

• Every derivative can be priced as if the economy were risk-neutral.

• For a European-style derivative with the terminal payoff function D, its value is

e^−ˆrnE^π[D ]. (37) – E^π means the expectation is taken under the

risk-neutral probability.

• The “equivalence” between arbitrage freedom in a model and the existence of a risk-neutral probability is called the (first) fundamental theorem of asset pricing.^a

aDybvig & Ross (1987).

Self-Financing

• Delta changes over time.

• The maintenance of an equivalent portfolio is dynamic.

• But it does not depend on predicting future stock prices.

• The portfolio’s value at the end of the current period is precisely the amount needed to set up the next portfolio.

• The trading strategy is self-financing because there is neither injection nor withdrawal of funds throughout.^a

– Changes in value are due entirely to capital gains.

aExcept at the beginning, of course, when the option premium is paid before the replication starts.

Binomial Distribution

• Denote the binomial distribution with parameters n and p by

b(j; n, p) =^Δ

n j



p^j(1 − p)^n−j = n!

j! (n − j)! p^j(1 − p)^n−j. – n! = 1 × 2 × · · · × n.

– Convention: 0! = 1.

• Suppose you flip a coin n times with p being the probability of getting heads.

• Then b(j; n, p) is the probability of getting j heads.

The Binomial Option Pricing Formula

• The stock prices at time n are

Suⁿ, Suⁿ⁻¹d, . . . , Sdⁿ.

• Let a be the minimum number of upward price moves for the call to finish in the money.

• So a is the smallest nonnegative integer j such that Su^jd^n−j ≥ X,

or, equivalently,

a =

ln(X/Sdⁿ) ln(u/d)

 .

The Binomial Option Pricing Formula (concluded)

• Hence,

C

=

_n

j=a

_n

j

p^j(1 − p)^n−j 

Su^jd^n−j − X

Rⁿ (38)

= S

n j=a

n j

(pu)^j[ (1 − p) d ]^n−j Rⁿ

− X Rⁿ

n j=a

n j



p^j(1 − p)^n−j

= S

n j=a

b (j; n, pu/R) − Xe^−ˆrn

n j=a

b(j; n, p). (39)

Numerical Examples

• A non-dividend-paying stock is selling for $160.

• u = 1.5 and d = 0.5.

• r = 18.232% per period (R = e^0.18232 = 1.2).

– Hence p = (R − d)/(u − d) = 0.7.

• Consider a European call on this stock with X = 150 and n = 3.

• The call value is $85.069 by backward induction.

• Or, the PV of the expected payoff at expiration:

390 × 0.343 + 30 × 0.441 + 0 × 0.189 + 0 × 0.027

(1.2)³ = 85.069.

160

540 (0.343)

180 (0.441)

(0.189)60

(0.027)20 Binomial process for the stock price

(probabilities in parentheses)

(0.49)360

(0.42)120

40 (0.09) (0.7)240

80 (0.3)

85.069 (0.82031)

390

30

0

0 Binomial process for the call price

(hedge ratios in parentheses)

(1.0)235

(0.25)17.5

0 (0.0) 141.458

(0.90625)

10.208 (0.21875)

Numerical Examples (continued)

• Mispricing leads to arbitrage profits.

• Suppose the option is selling for $90 instead.

• Sell the call for $90.

• Invest $85.069 in the replicating portfolio with 0.82031 shares of stock as required by the delta.

• Borrow 0.82031 × 160 − 85.069 = 46.1806 dollars.

• The fund that remains,

90 − 85.069 = 4.931 dollars, is the arbitrage profit, as we will see.

Numerical Examples (continued)

Time 1:

• Suppose the stock price moves to $240.

• The new delta is 0.90625.

• Buy

0.90625 − 0.82031 = 0.08594

more shares at the cost of 0.08594 × 240 = 20.6256 dollars financed by borrowing.

• Debt now totals 20.6256 + 46.1806 × 1.2 = 76.04232 dollars.

Numerical Examples (continued)

• The trading strategy is self-financing because the portfolio has a value of

0.90625 × 240 − 76.04232 = 141.45768.

• It matches the corresponding call value by backward induction!^a

aSee p. 275.

Numerical Examples (continued)

Time 2:

• Suppose the stock price plunges to $120.

• The new delta is 0.25.

• Sell 0.90625 − 0.25 = 0.65625 shares.

• This generates an income of 0.65625 × 120 = 78.75 dollars.

• Use this income to reduce the debt to

76.04232 × 1.2 − 78.75 = 12.5 dollars.

Numerical Examples (continued)

Time 3 (the case of rising price):

• The stock price moves to $180.

• The call we wrote finishes in the money.

• Close out the call’s short position by buying back the call or buying a share of stock for delivery.

• This results in a loss of 180 − 150 = 30 dollars.

• Financing this loss with borrowing brings the total debt to 12.5 × 1.2 + 30 = 45 dollars.

• It is repaid by selling the 0.25 shares of stock for 0.25 × 180 = 45 dollars.

Numerical Examples (concluded)

Time 3 (the case of declining price):

• The stock price moves to $60.

• The call we wrote is worthless.

• Sell the 0.25 shares of stock for a total of 0.25 × 60 = 15

dollars.

• Use it to repay the debt of 12.5 × 1.2 = 15 dollars.

Applications besides Exploiting Arbitrage Opportunities

• Replicate an option using stocks and bonds.

– Set up a portfolio to replicate the call with $85.069.

• Hedge the options we issued.

– Use $85.069 to set up a portfolio to replicate the call to counterbalance its values exactly.^b

• · · ·

• Without hedge, one may end up forking out $390 in the worst case (see p. 275)!^c

aThanks to a lively class discussion on March 16, 2011.

bHedging and replication are mirror images.

cThanks to a lively class discussion on March 16, 2016.

Binomial Tree Algorithms for European Options

• The BOPM implies the binomial tree algorithm that applies backward induction.

• The total running time is O(n²) because there are

∼ n²/2 nodes.

• The memory requirement is O(n²).

– Can be easily reduced to O(n) by reusing space.^a

• To find the hedge ratio, apply formula (32) on p. 251.

• To price European puts, simply replace the payoff.

aBut watch out for the proper updating of array entries.

C[2][0]

C[2][1]

C[2][2]

C[1][0]

C[1][1]

C[0][0]

p

p

p p

p p

max ,

?

D

max ,

?

D

max ,

?

D

max ,

?

D

1 p

1 p

1 p

1 p

1 p

1 p

Further Time Improvement for Calls

0

0 0

All zeros

X

Optimal Algorithm

• We can reduce the running time to O(n) and the memory requirement to O(1).

• Note that

b(j; n, p) = p(n − j + 1)

(1 − p) j b(j − 1; n, p).

Optimal Algorithm (continued)

• The following program computes b(j; n, p) in b[ j ]:

• It runs in O(n) steps.

1: b[ a ] := _n

a

 p^a(1 − p)^n−a;

2: for j = a + 1, a + 2, . . . , n do

3: b[ j ] := b[ j − 1 ] × p × (n − j + 1)/((1 − p) × j);

4: end for

Optimal Algorithm (concluded)

• With the b(j; n, p) available, the risk-neutral valuation formula (38) on p. 273 is trivial to compute.

• But we only need a single variable to store the b(j; n, p)s as they are being sequentially computed.

• This linear-time algorithm computes the discounted expected value of max(S_n − X, 0).

• This forward-induction approach cannot be applied to American options because of early exercise.

• So binomial tree algorithms for American options usually run in O(n²) time.

The Bushy Tree

S

Su

Sd

Su²

Sud

Sdu

Sd²

2ⁿ

n

Suⁿ Su^{n − 1} Su³

Su²d Su²d

Sud² Su²d

Sud² Sud²

Sd³

Su^{n − 1}d

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.

aContinuous-time trading may create arbitrage opportunities in prac- tice (Budish, Cramton, & Shim, 2015)!

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 Δt = τ /n years long.^Δ – 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

n Var

ln S_τ S



denote the variance of that return.

Toward the Black-Scholes Formula (continued)

• Under the BOPM, it is not hard to show that^a μ = q ln(u/d) + ln d,

σ² = q(1 − q) ln²(u/d).

• Assume the stock’s true continuously compounded rate of return over τ years has mean μτ and variance σ²τ .

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

aIt follows the Bernoulli distribution.

Toward the Black-Scholes Formula (continued)

• The BOPM converges to the distribution only if

nμ = n[ q ln(u/d) + ln d ] → μτ, (40) nσ² = nq(1 − q) ln²(u/d) → σ²τ. (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. 285).

– 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

√Δt, d = e^−σ

√Δt, q = 1

2 + 1 2

μ σ

√Δt . (42)

• With Eqs. (42), it can be checked that nμ = μτ,

nσ² =

1 − μ σ

₂ Δt



σ²τ → σ²τ.

• With the above choice, even if σ is not calibrated correctly, the mean is still matched!^a

aRecall Eq. (35) on p. 257. So u and d are related to volatility exclu- sively in the CRR model. Both are independent of r and μ.

Toward the Black-Scholes Formula (continued)

• The choices (42) result in the CRR binomial model.^a – Black (1992), “This method is probably used more

than the original formula in practical situations.”

– OptionMetrics’s (2015) IvyDB uses the CRR model.^b

• The CRR model is best seen in logarithmic price:

ln S →

⎧⎨

⎩

ln S + σ√

Δt, up move, ln S − σ√

Δt, down move.

aCox, Ross, & Rubinstein (1979).

bSee http://www.ckgsb.com/uploads/report/file/201611/02/1478069847635278.pd

Toward the Black-Scholes Formula (continued)

• The no-arbitrage inequalities d < R < u may not hold under Eqs. (42) on p. 296 or Eq. (34) on p. 255.

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

• The problem disappears when n satisfies e^σ√Δt > e^rΔt, i.e., when

n > r²

σ² τ. (43)

– 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, σ²τ ).^a

• So ln S_τ approaches N (μτ + ln S, σ²τ ).

• Conclusion: S_τ has a lognormal distribution in the limit.

aThe normal distribution with mean μτ and variance σ²τ . As our probabilities depend on n, this argument is heuristic.

Toward the Black-Scholes Formula (continued)

Lemma 10 The continuously compounded rate of return ln(S_τ/S) approaches the normal distribution with mean (r − σ²/2) τ and variance σ²τ 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 − σ²/2.

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 is thus the riskless rate r.

– By rate of return we mean (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).

– The latter would give r − σ²/2 by Lemma 10.

aBy Lemma 10 (p. 301) and Eq. (29) on p. 180.

Toward the Black-Scholes Formula (continued)

Theorem 11 (The Black-Scholes Formula, 1973) C = SN (x) − Xe^−rτN (x − σ√

τ ), P = Xe^−rτN (−x + σ√

τ ) − SN (−x), where

x =^Δ ln(S/X) + 

r + σ²/2 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. 273 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

– This holds for the CRR model as well.

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 judicious choices of u and d.^b

aF. Diener & M. Diener (2004); L. 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?

aImplied volatility is hard to compute when τ is small (why?).

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?^b

aRebonato (2004).

bE.g., 1:16:23 of https://www.youtube.com/watch?v=8TJQhQ2GZ0Y

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)

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.

Tackling 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^jd^n−j) and applies backward induction.

• At each intermediate node, compare the payoff if exercised and the continuation value.

• Keep 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 Volatility

• 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 σ²(t) dt rather than σ²τ .

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

 _τ

0 σ²(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)

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

Time-Dependent Short Rates

• Suppose the short rate (i.e., the one-period spot rate or forward rate) changes over time but 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

• Will the binomial tree fail to combine?

aThat is, one-period forward rate.

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.