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
Cu +
u−R u−d
Cd
R .
• Rewrite it as
hS + B = pCu + (1 − p) Cd
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
Cu +
u − R u − d
Cd
interpolates the value at SR through points (Su, Cu) and (Sd, Cd).
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 ratea 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 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
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:
Cu = pCuu + (1 − p) Cud
R , (36)
Cd = pCud + (1 − p) Cdd
R .
• Deltas can be derived from Eq. (32) on p. 251.
• For example, the delta at Cu is Cuu − Cud Suu − Sud.
Option on a Non-Dividend-Paying Stock: Multi-Period (concluded)
• We now reach the current period.
• Compute
pCu + (1 − p) Cd 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, 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.a
• So
C = hS + B = pCu + (1 − p) Cd
R .
aConsistent with Theorem 5 (p. 234).
Backward Induction
a• The above expression calculates C from the two successor nodes Cu and Cd and none beyond.
• The same computation happened at Cu and Cd, 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
pj(1 − p)n−j × max
0, Sujdn−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
pj(1 − p)n−j × max
0, X − Sujdn−j
Rn .
Backward Induction (concluded)
• Note that
pj =Δ
n
j
pj(1 − p)n−j Rn
is the state pricea for the state Sujdn−j, j = 0, 1, . . . , n.
• In general,
option price =
j
(pj × payoff at state j).
aRecall p. 212. One can obtain the undiscounted state price n
j
pj(1− p)n−j—the risk-neutral probability—for the state Sujdn−j with (XM − XL)−1 units of the butterfly spread where XL = Suj−1dn−j+1, XM = Sujdn−j, and XH = Suj−1+1dn−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−ˆrnEπ[D ]. (37) – 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.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
pj(1 − p)n−j = n!
j! (n − j)! pj(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 Sujdn−j ≥ X,
or, equivalently,
a =
ln(X/Sdn) ln(u/d)
.
The Binomial Option Pricing Formula (concluded)
• Hence,
C
=
n
j=a
n
j
pj(1 − p)n−j
Sujdn−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
pj(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 = e0.18232 = 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
a• 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 ,
?
0 Sud2 XD
max ,
?
0 Su d X2D
max ,
?
0 Su3 XD
max ,
?
0 Sd3 XD
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
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
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(Sn − 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 Sun − 1 Su3
Su2d Su2d
Sud2 Su2d
Sud2 Sud2
Sd3
Sun − 1d
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 = eˆr.
• Let
μ =Δ 1 n E
ln Sτ S
denote the expected value of the continuously
compounded rate of return per period of the BOPM.
• Let
σ2 Δ= 1
n Var
ln Sτ S
denote the variance of that return.
Toward the Black-Scholes Formula (continued)
• Under the BOPM, it is not hard to show thata 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
nμ = n[ q ln(u/d) + ln d ] → μτ, (40) nσ2 = 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 = eσ
√Δt, d = e−σ
√Δt, q = 1
2 + 1 2
μ σ
√Δt . (42)
• With Eqs. (42), it can be checked that nμ = μτ,
nσ2 =
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 eσ√Δt > erΔt, i.e., when
n > r2
σ2 τ. (43)
– So it goes away if n is large enough.
– Other solutions can be found in the textbookb 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(Sτ/S) converges to N (μτ, σ2τ ).a
• So ln Sτ approaches N (μτ + ln S, σ2τ ).
• Conclusion: Sτ 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(Sτ/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 = (eΔ 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 isa
Serτ.
• The stock’s expected annual rate of return is thus the riskless rate r.
– By rate of return we mean (1/τ ) ln E[ Sτ/S ] (arithmetic average rate of return) not
(1/τ )E[ ln(Sτ/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)
aTheorem 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 = eσ
√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)
a300
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 − Sujdn−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
a• Suppose the (instantaneous) volatility can change over time but otherwise predictable: σ(t) instead of σ.
• In the limit, the variance of ln(Sτ/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 = eσ(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)
a2-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 ri
τ ,
where ri 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 witha
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.