Option Pricing Models Page188~Page230

(1)

Option Pricing Models

c

2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 188

If the world of sense does not fit mathematics, so much the worse for the world of sense. — Bertrand Russell (1872–1970)

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.

– _{Known as the Black-Scholes option pricing model.}

c

Terms and Approach

• C: call value. • P : put value. • X: strike price • S: stock price

• r > 0: the continuously compounded riskless rate perˆ period.

• R ≡ eˆr_{: gross return.}

(2)

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 suffice to determine the option

value based on arbitrage considerations: S, u, d, X, ˆr, and the number of periods to expiration.

c

S

Su

q

1 q

Sd

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

• The expiration date is only one period from now. • Cu is the call price at time one if the stock price moves

to Su.

• Cd is the call price at time one if the stock price moves

to Sd. • Clearly,

Cu = max(0, Su − X),

Cd = max(0, Sd − X).

c

C

Cu= max( 0, Su X ) q

1 q

(3)

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 = Cu,

hSd + RB = Cd.

c

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

(concluded)

• Solve the above equations to obtain h = Cu−Cd

Su − Sd ≥0, (19) B = uCd−dCu

(u − d) R . (20)

• By the no-arbitrage principle, the European call should cost the same as the equivalent portfolio, C = hS + B. • As uCd−dCu < 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 3 (p. 183).

• So C = hS + B.

c

Put Pricing in One Period

• Puts can be similarly priced.

• The delta for the put is (Pu−Pd)/(Su − Sd) ≤ 0, where

Pu = max(0, X − Su),

Pd = max(0, X − Sd).

• Let B = uPd_−dPu

(u−d) R.

• 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.}

(4)

Risk

• Surprisingly, the option value is independent of q. • 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 possible stock prices are the same whether under q or p.

c

Pseudo Probability

• After substitution and rearrangement,

hS + B = R−d u−d Cu+ u−R u−d Cd R . • Rewrite it as hS + B = pCu+ (1 − p) Cd 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. • Risk-neutral investors care only about expected returns. • 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.

c

Binomial Distribution

• Denote the binomial distribution with parameters n and p by b(j; n, p) ≡n j pj_{(1 − p)}n−j₌ n! j! (n − j)!p j_{(1 − p)}n−j_.

– _{n! = n × (n − 1) · · · 2 × 1 with the convention 0! = 1.} • Suppose you toss a coin n times with p being the

probability of getting heads.

(5)

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.

– _{Note that the tree combines.}

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

c

S Su Sd Suu Sud Sdd

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

(continued)

• Let Cuu be the call’s value at time two if the stock price

is Suu. • Thus,

Cuu= max(0, Suu − X).

• Cud and Cdd can be calculated analogously,

Cud = max(0, Sud − X),

Cdd = max(0, Sdd − X).

c

C

Cu

Cd

Cuu= max( 0, Suu X )

Cud = max( 0, Sud X )

(6)

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

(continued)

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

Cu = pCuu+ (1 − p) Cud R , (21) Cd = pCud+ (1 − p) Cdd R .

• Deltas can be derived from Eq. (19) on p. 197. • For example, the delta at Cu is

Cuu−Cud

Suu − Sud.

c

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

(concluded)

• We now reach the current period.

• An equivalent portfolio of h shares of stock and $B riskless bonds can be set up for the call that costs Cu

(Cd, resp.) if the stock price goes to Su (Sd, resp.).

• The values of h and B can be derived from Eqs. (19)–(20) on p. 197.

• Or, we can just compute

pCu+ (1 − p) Cd

R as the price.

Early Exercise

• Since the call will not be exercised at time one even if it is American, Cu ≥Su − X and Cd≥Sd − X. • Therefore, hS + B = pCu+ (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. • So

C = hS + B = pCu+ (1 − p) Cd

R .

c

Backward Induction of Zermelo (1871–1953)

• The above expression calculates C from the two successor nodes Cu and Cd and none beyond.

• The same computation happens at Cu and Cd, too, as

demonstrated in Eq. (21) on p. 208.

• This recursive procedure is called backward induction. • Now, C equals [ p2 Cuu+ 2p(1 − p) Cud+ (1 − p) 2 Cdd](1/R 2 ) = [ p2 max 0, Su2 −X + 2p(1 − p) max (0, Sud − X) +(1 − p)2 max 0, Sd2 −X ]/R2 .

(7)

Backward Induction (continued)

• In the n-period case, C = Pn j=0 n j p j_{(1 − p)}n−j_×_{max 0, Su}j_dn−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.

• The value of a European put is P = Pn j=0 n j p j_{(1 − p)}n−j_×_{max 0, X − Su}j_dn−j Rn . c

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−ˆrnEπ[ D ].

– _Eπ _{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.

S0 1 * j S0u p * j S0d 1 − p * j S0u2 p2 S0ud 2p(1 − p) S0d2 (1 − p)2 c

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. – _{Changes in value are due entirely to capital gains.}

(8)

The Binomial Option Pricing Formula

• 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 such that Sua_dn−a_≥_X, or a = ln(X/Sd n₎ ln(u/d) . c

The Binomial Option Pricing Formula (concluded)

Hence, C = Pn j=a n j p j (1 − p)n−j _Suj dn−j₋_X Rn (22) = S n X j=a n j ! (pu)j [ (1 − p) d ]n−j Rn − X Rn n X j=a n j ! pj (1 − p)n−j = S n X j=a bj; n, pue−rˆ −Xe−rnˆ n X j=a 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 (e0.18232_{= 1.2).}

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

• The call value is $85.069 by backward induction. • Also 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.

c

160 540 (0.343) 180 (0.441) 60 (0.189) 20 (0.027) 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)

(9)

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.

c

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 dollars.

c

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 0.25 × 180 = 45 dollars.

(10)

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.

c

Binomial Tree Algorithms for European Options

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

• The total running time is O(n2_).

• The memory requirement is O(n2_).

– _{Can be further reduced to O(n) by reusing space} • To price European puts, simply replace the payoff.

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

0_Sud2 _X

h

max ,0 2 Su d X

c

h

max ,0 3 Su X

c

h

max ,0 3 Sd X

c

h

1 p 1 p 1 p 1 p 1 p 1 p c

Further Improvement for Calls

0

0 0

All zeros

(11)

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

1: b[ a ] := n_a pa_{(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

• It runs in O(n) steps.

c

Optimal Algorithm (concluded)

• With the b(j; n, p) available, the risk-neutral valuation formula (22) on p. 217 is trivial to compute.

• 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(Sn−X, 0).

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

• So binomial tree algorithms for American options usually run in O(n2_{) time.}

On the Bushy Tree

S Su Sd Su2 Sud Sdu Sd2 2n n Sun Sun − 1 Su3 Su2_d Su2_d Sud2 Su2_d Sud2 Sud2 Sd3 Sun − 1_d c