Option Pricing Models

Option Pricing Models

If the world of sense does not fit mathematics, so much the worse for the world of sense.

— Bertrand Russell (1872–1970)

Black insisted that anything one could do with a mouse could be done better with macro redefinitions of particular keys on the keyboard.

— Emanuel Derman, My Life as a Quant (2004)

The Setting

• The no-arbitrage principle is insufficient to pin down the exact option value.

• Need a model of probabilistic behavior of stock prices.

• One major obstacle is that it seems a risk-adjusted interest rate is needed to discount the option’s payoff.

• Breakthrough came in 1973 when Black (1938–1995) and Scholes with help from Merton published their celebrated option pricing model.^a

– Known as the Black-Scholes option pricing model.

Terms and Approach

• C: call value.

• P : put value.

• X: strike price

• S: stock price

• ˆr > 0: the continuously compounded riskless rate per period.

• R ≡ e^rˆ: gross return.

• Start from the discrete-time binomial model.

Binomial Option Pricing Model (BOPM)

• Time is discrete and measured in periods.

• If the current stock price is S, it can go to Su with probability q and Sd with probability 1 − q, where 0 < q < 1 and d < u.

– In fact, d < R < u must hold to rule out arbitrage.

• Six pieces of information will suffice to determine the option value based on arbitrage considerations: S, u, d, X, ˆr, and the number of periods to expiration.

S

Su q

1 q

Sd

Call on a Non-Dividend-Paying Stock: Single Period

• The expiration date is only one period from now.

• C^u is the call price at time one if the stock price moves to Su.

• C^d is the call price at time one if the stock price moves to Sd.

• Clearly,

C_u = max(0, Su − X), C_d = max(0, Sd − X).

C

Cu= max( 0, Su X ) q

1 q

Cd = max( 0, Sd X )

Call on a Non-Dividend-Paying Stock: Single Period (continued)

• Set up a portfolio of h shares of stock and B dollars in riskless bonds.

– This costs hS + B.

– We call h the hedge ratio or delta.

• The value of this portfolio at time one is either hSu + RB or hSd + RB.

• Choose h and B such that the portfolio replicates the payoff of the call,

hSu + RB = C_u,

Call on a Non-Dividend-Paying Stock: Single Period (concluded)

• Solve the above equations to obtain h = C_u − C^d

Su − Sd ≥ 0, (23)

B = uC_d − dC^u

(u − d) R . (24)

• By the no-arbitrage principle, the European call should cost the same as the equivalent portfolio,^a

C = hS + B.

• As uC^d − dC^u < 0, the equivalent portfolio is a levered long position in stocks.

American Call Pricing in One Period

• Have to consider immediate exercise.

• C = max(hS + B, S − X).

– When hS + B ≥ S − X, the call should not be exercised immediately.

– When hS + B < S − X, the option should be exercised immediately.

• For non-dividend-paying stocks, early exercise is not optimal by Theorem 4 (p. 198).

• So C = hS + B.

Put Pricing in One Period

• Puts can be similarly priced.

• The delta for the put is (P^u − P^d)/(Su − Sd) ≤ 0, where P_u = max(0, X − Su),

P_d = max(0, X − Sd).

• Let B = ^uP_(u^d_{−d) R}^−dPu.

• The European put is worth hS + B.

• The American put is worth max(hS + B, X − S).

– Early exercise is always possible with American puts.

Risk

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

• Hence it is independent of the expected gross return of the stock, qSu + (1 − q) Sd.

• It therefore does not directly depend on investors’ risk preferences.

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

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

aMore precisely, not directly dependent on q. Thanks to a lively class

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 ≡ R − d u − d .

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

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.

• 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 in a risk-neutral economy.

Binomial Distribution

• Denote the binomial distribution with parameters n and p by

b(j; n, p) ≡

(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 toss a coin n times with p being the probability of getting heads.

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

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 three possible prices at time two: Suu, Sud, and Sdd.

– There are 4 paths.

– But the tree combines.

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

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 , (25)

C_d = pC_ud + (1 − p) C^dd

R .

• Deltas can be derived from Eq. (23) on p. 214.

• For example, the delta at C^u is C_uu − C^ud

.

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. (23)–(24) on p. 214.

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 = pCu + (1 − p) C^d

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 .

Backward Induction of Zermelo (1871–1953)

• 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. (25) on p. 225.

• This recursive procedure is called backward induction.

• C equals

[ p²C_uu + 2p(1 − p) C^ud + (1 − p)²C_dd](1/R²)

= [ p² max(

0, Su² − X)

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

0, Sd² − X)

]/R².

S₀ 1

*

j

S₀u p

*

j

S₀d 1 − p

*

j

S₀u² p²

S₀ud

2p(1 − p)

S₀d²

Backward Induction (concluded)

• 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 .

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 ].

– 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

Self-Financing

• Delta changes over time.

• The maintenance of an equivalent portfolio is dynamic.

• The maintaining of an equivalent portfolio does not depend on our correctly 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.

a

Hakansson’s Paradox

• If options can be replicated, why are they needed at all?

aHakansson (1979).

Can You Figure Out u, d without Knowing q?

• Yes, you can, under BOPM.

• Let us observe the time series of past stock prices, e.g.,

u is available

z }| {

S, Su, Su², Su³, Su³d

| {z }

d is available

, . . .

• So with sufficiently long history, you will figure out u and d without knowing q.

aContributed by Mr. Hsu, Jia-Shuo (D97945003) on March 11, 2009.

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)⌉ .

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 (26)

= 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

b (j; n, pu/R) − Xe^−ˆrn

∑n

b(j; n, p).

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

3 = 85.069.

160

540 (0.343)

180 (0.441)

60 (0.189)

20 Binomial process for the stock price

(probabilities in parentheses)

360 (0.49)

120 (0.42)

40 (0.09) 240

(0.7)

80 (0.3)

85.069 (0.82031)

390

30

0

0 Binomial process for the call price

(hedge ratios in parentheses)

235 (1.0)

17.5 (0.25)

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 and invest $85.069 in the replicating portfolio with 0.82031 shares of stock required by 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)

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

Numerical Examples (continued)

Time 3 (the case of rising price):

• The stock price moves to $180.

• The call we wrote finishes in the money.

• For a loss of 180 − 150 = 30 dollars, close out the

position by either buying back the call or buying a share of stock for delivery.

• 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

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.

• Hedge the options we issued (the mirror image of replication).

• · · ·

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

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 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

c

h

max

c

h

max

c

h

max

c

h

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 (26) on p. 236 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).

• The above technique cannot be applied to American options because of early exercise.

• So binomial tree algorithms for American options

The Bushy Tree

S

Su

Sd

Su²

Sud

Sdu

Sd²

2ⁿ 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.

• Any proper calibration of the model parameters makes the BOPM 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 ˆ

r to match the empirical results as n goes to infinity.

• First, ˆr = rτ/n.

– The period gross return R = e^rˆ.

Toward the Black-Scholes Formula (continued)

• Use

bµ ≡ 1 n E

[

ln S_τ S

]

and bσ² ≡ 1

n Var [

ln S_τ S

]

to denote the expected value and variance of the continuously compounded rate of return per period.

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

bσ² = q(1 − q) ln²(u/d).

Toward the Black-Scholes Formula (continued)

• 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.

• The BOPM converges to the distribution only if nbµ = n[ q ln(u/d) + ln d ] → µτ, nbσ² = nq(1 − q) ln²(u/d) → σ²τ.

• Impose ud = 1 to make nodes at the same horizontal level of the tree have identical price (review p. 247).

– Other choices are possible (see text).

– Exact solutions for u, d, q are also available.^a

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 . (27)

• With Eqs. (27), it can be checked that nbµ = µτ,

nbσ² = [

1 − (µ σ

)2 τ n

]

σ²τ → σ²τ.

• The choice (27) results in the CRR binomial model.^a

Toward the Black-Scholes Formula (continued)

• The no-arbitrage inequalities d < R < u may not hold under Eqs. (27) on p. 256.

– If this happens, the risk-neutral probability may lie outside [ 0, 1 ].^a

• The problem disappears when n satisfies e^σ

√τ /n

> e^{rτ /n}, i.e., when n > r²τ /σ² (check it).

– So it goes away if n is large enough.

– Other solutions will be presented later.

Toward the Black-Scholes Formula (continued)

• What is the limiting probabilistic distribution of the continuously compounded rate of return ln(S_τ/S)?

• The central limit theorem says ln(S^τ/S) converges to the normal distribution with mean µτ and variance σ²τ .

• So ln S^τ approaches the normal distribution with mean µτ + ln S and variance σ²τ .

• S^τ has a lognormal distribution in the limit.

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 − σ²/2) τ and variance σ²τ in a risk-neutral economy.

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

• Let n → ∞.

Toward the Black-Scholes Formula (continued)

• The expected stock price at expiration in a risk-neutral economy is Se^rτ.^a

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

aBy Lemma 9 (p. 259) and Eq. (21) on p. 154.

bIn 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).

Toward the Black-Scholes Formula (concluded)

Theorem 10 (The Black-Scholes Formula)

C = SN (x) − Xe^−rτN (x − σ√ τ ), P = Xe^−rτN (−x + σ√

τ ) − SN(−x), where

x ≡ 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!

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

, ˆr = rτ /n.

• The binomial tree algorithms converge reasonably fast.

• Oscillations can be dealt with by the judicious choices of

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).

• The error is O(1/n).^a

aChang and Palmer (2007).

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?

• This volatility is called the implied volatility.

• Implied volatility is often preferred to historical volatility^b in practice.

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

b