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# Toward the Black-Scholes Formula

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### Toward the Black-Scholes Formula

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

• 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.

(2)

### 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

(3)

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

• First, ˆr = rτ/n.

– Each period is τ/n years long.

– The period gross return R = eˆr.

• Let

ln Sτ S

denote the continuously compounded rate of return of the stock.

(4)

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

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

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

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

• Impose

ud = 1.

– It makes nodes at the same horizontal level of the tree have identical price.a

aOther choices are possible (see text).

(5)

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

• Pick

u = eσ

τ /n, d = e−σ

τ /n, q = 1

2 + 1 2

μ σ

τ

n . (12)

• With Eqs. (12), it can be checked that the mean μτ is matched by the BOPM.

• Furthermore, the variance σ2τ is asymptotically matched as well.

(6)

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

• The choices (12) result in the CRR binomial model.a

• The no-arbitrage inequalities d < R < u may not hold under Eqs. (12) on p. 80 or Eq. (8) on p. 56.

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

• The problem disappears if n is large enough.

aCox, Ross, and Rubinstein (1979).

(7)
(8)

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

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

• It approaches N(μτ + ln S, σ2τ ).

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

(9)

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

• In the risk-neutral economy, pick q = R − d

u − d . by Eq. (8) on p. 56.

Lemma 1 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.a

(10)

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

Theorem 2 (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 .

(11)

### 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,

(12)

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).

(13)

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

• The binomial tree algorithms converge reasonably fast.

• The error is O(1/n).a

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

aChang and Palmer (2007).

bSee Exercise 9.3.8 of the textbook.

(14)

### 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.

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

• Implied volatility is

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

• It is often preferred to historical volatility in practice.

aRebonato (2004).

(15)

### 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.

(16)

### 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, it compares the payoﬀ if exercised and the continuation value.

• It keeps the larger one.

(17)

### Extensions of Options Theory

(18)

And the worst thing you can have is models and spreadsheets.

— Warren Buﬀet, May 3, 2008

(19)

### Barrier Options

• Their payoﬀ depends on whether the underlying asset’s price reaches a certain price level H.

• A knock-out option is like an ordinary European option.

• But it ceases to exist if the barrier H is reached by the price of its underlying asset.

(20)

H

Time Price

S Barrier hit

(21)

### Barrier Options (concluded)

• A knock-in option comes into existence if a certain barrier is reached.

• A down-and-in option is a call knock-in option that comes into existence only when the barrier is reached and H < S.

(22)

### A Formula for Down-and-In Calls

a

• Assume X ≥ H.

• The value of a European down-and-in call on a stock paying a dividend yield of q is

Se−qτ

H S



N(x) − Xe−rτ

H S

2λ−2

N(x − σ τ),

(13)

– x ≡ ln(H2/(SX))+(r−q+σ2/2) τ σ

τ .

– λ ≡ (r − q + σ2/2)/σ2.

aMerton (1973).

(23)

### Binomial Tree Algorithms

• Barrier options can be priced by binomial tree algorithms.

• Below is for the down-and-out option.

0 H

(24)

H 8

16

4

32

8

2

64

16

4

1

4.992

12.48

1.6

27.2

4.0

0

58

10

0

0 X

0.0

S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.

Backward-induction: C = (0.5 × Cu + 0.5 × Cd)/1.25.

(25)

### Binomial Tree Algorithms (continued)

• But convergence is erratic because H is not at a price level on the tree.a

– The barrier H is moved to a node price.

– This “eﬀective barrier” changes as n increases.

• In fact, the binomial tree is O(1/√

n) convergent.b

aBoyle and Lau (1994).

bLin (R95221010) (2008).

(26)

### Binomial Tree Algorithms (concluded)

a

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value

aLyuu (1998).

(27)

### Path-Dependent Derivatives

• Let S0, S1, . . . , Sn denote the prices of the underlying asset over the life of the option.

• S0 is the known price at time zero.

• Sn is the price at expiration.

• The standard European call has a terminal value depending only on the last price, max(Sn − X, 0).

• Its value thus depends only on the underlying asset’s

(28)

### Path-Dependent Derivatives (continued)

• Some derivatives are path-dependent in that their terminal payoﬀ depends critically on the path.

• The (arithmetic) average-rate call has this terminal value:

max

 1 n + 1

n i=0

Si − X, 0

 .

• The average-rate put’s terminal value is given by max



X − 1

n + 1

n i=0

Si, 0

 .

(29)

### Path-Dependent Derivatives (concluded)

• Average-rate options are also called Asian options.a

• They are useful hedging tools for ﬁrms that will make a stream of purchases over a time period because the costs are likely to be linked to the average price.

• The averaging clause is also common in convertible bonds and structured notes.

aAs of the late 1990s, the outstanding volume was in the range of 5–10 billion U.S. dollars (Nielsen & Sandmann, 2003).

(30)

### Average-Rate Options

• Average-rate options are notoriously hard to price.

• The binomial tree for the averages does not combine (see next page).

• A naive algorithm enumerates the 2n paths for an n-period binomial tree and then averages the payoﬀs.a

• But the complexity is exponential.

• The Monte Carlo methodb and approximation algorithms are some of the alternatives left.

aDai (B82506025, R86526008, D8852600) and Lyuu (2007) reduce it to 2O(n ). Hsu (R7526001, D89922012) and Lyuu (2004) reduce it to O(n2) given some regularity assumptions.

bSee pp. 142ﬀ.

(31)

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Su

Sd

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Sud

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p

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= 

PD[ 6 6X 6XX ;

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p

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1 p

(32)

### Continuous-Time Financial Mathematics

(33)

A proof is that which convinces a reasonable man;

a rigorous proof is that which convinces an unreasonable man.

— Mark Kac (1914–1984)

(34)

### Brownian Motion

a

• Brownian motion is a stochastic process { X(t), t ≥ 0 } with the following properties.

1. X(0) = 0, unless stated otherwise.

2. for any 0 ≤ t0 < t1 < · · · < tn, the random variables X(tk) − X(tk−1)

for 1 ≤ k ≤ n are independent.b

3. for 0 ≤ s < t, X(t) − X(s) is normally distributed with mean μ(t − s) and variance σ2(t − s), where μ and σ = 0 are real numbers.

aRobert Brown (1773–1858).

bSo X(t) − X(s) is independent of X(r) for r ≤ s < t.

(35)

### Brownian Motion (concluded)

• The existence and uniqueness of such a process is guaranteed by Wiener’s theorem.a

• This process will be called a (μ, σ) Brownian motion with drift μ and variance σ2.

• The (0, 1) Brownian motion is called the Wiener process.

aNorbert Wiener (1894–1964). He received his Ph.D. from Harvard in 1912.

(36)

### Ito Process

a

• A shorthandb is the following stochastic diﬀerential equation for the Ito diﬀerential dXt,

dXt = a(Xt, t) dt + b(Xt, t) dWt. (14) – Or simply

dXt = at dt + bt dWt.

aIto (1944).

bPaul Langevin (1872–1946) in 1904.

(37)

### Ito Process (concluded)

• dW is normally distributed with mean zero and variance dt.

• An equivalent form of Eq. (14) is dXt = at dt + bt

dt ξ, (15)

where ξ ∼ N (0, 1).

(38)

### Modeling Stock Prices

• The most popular stochastic model for stock prices has been the geometric Brownian motion,

dS

S = μ dt + σ dW.

• The continuously compounded rate of return X ≡ ln S follows

dX = (μ − σ2/2) dt + σ dW by Ito’s lemma.a

aConsistent with Lemma 1 (p. 84).

(39)

0.2 0.4 0.6 0.8 1 Time (t) -1

1 2 3 4 5 6 Y(t)

(40)

### Local-Volatility Models

• The more general deterministic volatility model posits dS

S = (rt − qt) dt + σ(S, t) dW,

where instantaneous volatility σ(S, t) is called the local volatility function.a

• One needs to recover σ(S, t) from the implied volatilities.

aDerman and Kani (1994); Dupire (1994).

(41)

0 0.5

1 1.5

2 2.5

3

0 0.2 0.4 0.6 0.8 1 20 30 40 50 60 70 80 90 100 110

Strike (\$)

Implied Vol Surface

Time to Maturity (yr)

Implied Vol (%)

0 0.5

1 1.5

2 2.5

3

0 0.2 0.4 0.6 0.8 1 20 30 40 50 60 70 80 90 100 110

Stock (\$)

Local Vol Surface

Time (yr)

Local Vol (%)

(42)

### Implied Trees

• The trees for the local volatility model are called implied trees.a

• Their construction requires an implied volatility surface.

• An exponential-sized implied tree exists.b

• How to construct a valid implied tree with eﬃciency has been open for a long time.c

aDerman & Kani (1994); Dupire (1994); Rubinstein (1994).

bCharalambousa, Christoﬁdesb, & Martzoukosa (2007).

cRubinstein (1994); Derman & Kani (1994); Derman, Kani, & Chriss (1996); Jackwerth & Rubinstein (1996); Jackwerth (1997); Coleman, Kim, Li, & Verma (2000); Li (2000/2001); Moriggia, Muzzioli, & Torri- celli (2009).

(43)

### Implied Trees (concluded)

• It is solved for separable local volatilities σ.a

– The local-volatility function σ(S, V ) is separableb if σ(S, t) = σ1(S) σ2(t).

• A general solution is close.c

aLok (D99922028) & Lyuu (2015, 2016).

bRebonato (2004); Brace, G¸atarek, & Musiela (1997).

cLok (D99922028) & Lyuu (2016).

(44)

### The Hull-White Model

• Hull and White (1987) postulate the following model, dS

S = r dt +

V dW1, dV = μvV dt + bV dW2.

• Above, V is the instantaneous variance.

• They assume μv depends on V and t (but not S).

(45)

### The SABR Model

• Hagan, Kumar, Lesniewski, and Woodward (2002) postulate the following model,

dS

S = r dt + SθV dW1, dV = bV dW2,

for 0 ≤ θ ≤ 1.

(46)

### The Hilliard-Schwartz Model

• Hilliard and Schwartz (1996) postulate the following general model,

dS

S = r dt + f (S)V a dW1, dV = μ(V ) dt + bV dW2,

for some well-behaved function f (S) and constant a.

(47)

### Heston’s Stochastic-Volatility Model

• Heston (1993) assumes the stock price follows dS

S = (μ − q) dt +

V dW1, (16)

dV = κ(θ − V ) dt + σ√

V dW2. (17) – V is the instantaneous variance, which follows a

square-root process.

– dW1 and dW2 have correlation ρ.

– The riskless rate r is constant.

(48)

### Heston’s Stochastic-Volatility Model (concluded)

• It may be the most popular continuous-time stochastic-volatility model.a

• For American options, we will need a tree for Heston’s model.b

• They are all O(n3)-sized.

aChristoﬀersen, Heston, & Jacobs (2009).

bLeisen (2010); Beliaeva & Nawalka (2010); Chou (R02723073) (2015).

(49)

### Why Are Trees for Stochastic-Volatility Models Diﬃcult?

• The CRR tree is 2-dimensional.a

• The constant volatility makes the span from any node ﬁxed.

• But a tree for a stochastic-volatility model must be 3-dimensional.

– Every node is associated with a pair of stock price and a volatility.

(50)

(51)

(52)

### Why Are Trees for Stochastic-Volatility Models Diﬃcult? (concluded)

• Locally, the tree looks ﬁne for one time step.

• But the volatility regulates the spans of the nodes on the stock-price plane.

• Unfortunately, those spans diﬀer from node to node because the volatility varies.

• So two time steps from now, the branches will not combine!

(53)

### Complexities of Stochastic-Volatility Models

• A few stochastic-volatility models suﬀer from subexponential (cn) tree size.

• Examples include the Hull-White (1987),

Hilliard-Schwartz (1996), and SABR (2002) models.a

aChiu (R98723059) (2012).

(54)

### Trees

(55)

I love a tree more than a man.

— Ludwig van Beethoven (1770–1827)

(56)

### Trinomial Tree

• Set up a trinomial approximation to the geometric Brownian motiona

dS

S = r dt + σ dW.

• The three stock prices at time Δt are S, Su, and Sd, where ud = 1.

• Let the mean and variance of the stock price be SM and S2V , respectively.

aBoyle (1988).

(57)

* - j

pu

pm

pd

Su S Sd S

-



Δt

* - j

* - j

* - j

* - j

(58)

### Trinomial Tree (continued)

• By Eqs. (5) on p. 24,

M ≡ erΔt,

V ≡ M2(eσ2Δt − 1).

• Impose the matching of mean and that of variance:

1 = pu + pm + pd,

SM = (puu + pm + (pd/u)) S,

S2V = pu(Su − SM)2 + pm(S − SM)2 + pd(Sd − SM)2.

(59)

### Trinomial Tree (concluded)

• Use linear algebra to verify that pu = u

V + M2 − M

− (M − 1) (u − 1) (u2 − 1) , pd = u2 

V + M2 − M

− u3(M − 1) (u − 1) (u2 − 1) .

– We must also make sure the probabilities lie between 0 and 1.

(60)

### A Trinomial Tree

• Use u = eλσΔt, where λ ≥ 1 is a tunable parameter.

• Then

pu 1

2 +

r + σ2 √ Δt

2λσ ,

pd 1

2

r − 2σ2 √ Δt

2λσ .

(61)

### Barrier Options Priced by Trinomial Trees

0 50 100 150 200

5.61 5.62 5.63 5.64 5.65 5.66

Down-and-in call value

(62)

### Multivariate Contingent Claims

• They depend on two or more underlying assets.

• The basket call on m assets has the terminal payoﬀ max

 m



i=1

αiSi(τ ) − X, 0

 .

(63)

### Multivariate Contingent Claims (continued)

a

Name Payoﬀ

Exchange option max(S1(τ) − S2(τ), 0) Better-oﬀ option max(S1(τ), . . . , Sk(τ), 0) Worst-oﬀ option min(S1(τ), . . . , Sk(τ), 0)

Binary maximum option I{ max(S1(τ), . . . , Sk(τ)) > X } Maximum option max(max(S1(τ), . . . , Sk(τ)) − X, 0) Minimum option max(min(S1(τ), . . . , Sk(τ)) − X, 0) Spread option max(S1(τ) − S2(τ) − X, 0)

Basket average option max((S1(τ) + · · · + Sk(τ))/k − X, 0) Multi-strike option max(S1(τ) − X1, . . . , Sk(τ) − Xk, 0)

Pyramid rainbow option max(| S1(τ) − X1 | + · · · + | Sk(τ) − Xk | − X, 0)



(64)

### Multivariate Contingent Claims (concluded)

• Trees for multivariate contingent claims typically has size exponential in the number of assets.

• This is called the curse of dimensionality.

(65)

### Numerical Methods

(66)

All science is dominated by the idea of approximation.

— Bertrand Russell

(67)

### Monte Carlo Simulation

a

• Monte Carlo simulation is a sampling scheme.

• In many important applications within ﬁnance and without, Monte Carlo is one of the few feasible tools.

aA top 10 algorithm according to Dongarra and Sullivan (2000).

(68)

### Monte Carlo Option Pricing

• For the pricing of European options, we sample the stock prices.

• Then we average the payoﬀs.

• The variance of the estimator is now 1/N of that of the original random variable.

(69)

• Standard Monte Carlo simulation is inappropriate for American options because of early exercise.

• It is diﬃcult to determine the early-exercise point based on one single path.

• But Monte Carlo simulation can be modiﬁed to price American options with small biases.a

• The LSM can be easily parallelized.b

aLongstaﬀ and Schwartz (2001).

bHuang (B96902079, R00922018) (2013); Chen (B97902046,

(70)

### Delta and Common Random Numbers

• In estimating delta ∂f/∂S, it is natural to start with the ﬁnite-diﬀerence estimate

e−rτ E[ P (S + ) ] − E[ P (S − ) ]

2 .

– P (x) is the terminal payoﬀ of the derivative security when the underlying asset’s initial price equals x.

• Use simulation to estimate E[ P (S + ) ] ﬁrst.

• Use another simulation to estimate E[ P (S − ) ].

• Finally, apply the formula to approximate the delta.

(71)

### Delta and Common Random Numbers (concluded)

• This method is not recommended because of its high variance.

• A much better approach is to use common random numbers to lower the variance:

e−rτ E

 P (S + ) − P (S − ) 2

.

• Here, the same random numbers are used for P (S + ) and P (S − ).

(72)

### Gamma

• The ﬁnite-diﬀerence formula for gamma ∂2f /∂S2 is e−rτ E

 P (S + ) − 2 × P (S) + P (S − ) 2

.

• Choosing an of the right magnitude can be challenging.

– If is too large, inaccurate Greeks result.

– If is too small, unstable Greeks result.

• This phenomenon is sometimes called the curse of diﬀerentiation.a

aA¨ıt-Sahalia and Lo (1998); Bondarenko (2003).

(73)

### Gamma (continued)

• In general, suppose

i

∂θie−rτE[ P (S) ] = e−rτE

 iP (S)

∂θi

holds for all i > 0, where θ is a parameter of interest.

– A common requirement is Lipschitz continuity.a

• Then formulas for the Greeks become integrals.

• As a result, we avoid , ﬁnite diﬀerences, and resimulation.

(74)

### Gamma (concluded)

• This is indeed possible for a broad class of payoﬀ functions.a

• In queueing networks, this is called inﬁnitesimal perturbation analysis (IPA).b

aTeng (R91723054) (2004) and Lyuu and Teng (R91723054) (2011).

bCao (1985); Ho and Cao (1985).

(75)

### Interest Rate Models

(76)

[Meriwether] scoring especially high marks in mathematics — an indispensable subject for a bond trader.

— Roger Lowenstein, When Genius Failed (2000) Bond market terminology was designed less to convey meaning than to bewilder outsiders.

— Michael Lewis, The Big Short (2011)

(77)

### The Vasicek Model

a

• The short rate follows

dr = β(μ − r) dt + σ dW.

• The short rate is pulled to the long-term mean level μ at rate β.

• Superimposed on this “pull” is a normally distributed stochastic term σ dW .

aVasicek (1977).

(78)

### The Cox-Ingersoll-Ross Model

a

• It is the following square-root short rate model:

dr = β(μ − r) dt + σ√

r dW. (18)

• The diﬀusion diﬀers from the Vasicek model by a multiplicative factor

r .

• The parameter β determines the speed of adjustment.

• The short rate can reach zero only if 2βμ < σ2.

aCox, Ingersoll, and Ross (1985).

(79)

### The Ho-Lee Model

a

• The continuous-time limit of the Ho-Lee model is dr = θ(t) dt + σ dW.

• This is Vasicek’s model with the mean-reverting drift replaced by a deterministic, time-dependent drift.

• A nonﬂat term structure of volatilities can be achieved if the short rate volatility is also made time varying,

dr = θ(t) dt + σ(t) dW.

(80)

### The Black-Derman-Toy Model

a

• The continuous-time limit of the BDT model is d ln r =

θ(t) + σ(t)

σ(t) ln r

dt + σ(t) dW.

• This model is extensively used by practitioners.

• The BDT short rate process is the lognormal binomial interest rate process.

• Lognormal models preclude negative short rates.

aBlack, Derman, and Toy (BDT) (1990), but essentially ﬁnished in 1986 according to Mehrling (2005).

(81)

### The Black-Karasinski Model

a

• The BK model stipulates that the short rate follows d ln r = κ(t)(θ(t) − ln r) dt + σ(t) dW.

• This explicitly mean-reverting model depends on time through κ( · ), θ( · ), and σ( · ).

• The BK model hence has one more degree of freedom than the BDT model.

• The speed of mean reversion κ(t) and the short rate volatility σ(t) are independent.

(82)

### The Extended Vasicek Model

a

• The extended Vasicek model adds time dependence to the original Vasicek model,

dr = (θ(t) − a(t) r) dt + σ(t) dW.

• Like the Ho-Lee model, this is a normal model.

• Many European-style securities can be evaluated analytically.

• Eﬃcient numerical procedures can be developed for American-style securities.

aHull and White (1990).

(83)

### The Hull-White Model

• The Hull-White model is the following special case, dr = (θ(t) − ar) dt + σ dW.

(84)

* - j

(0, 0)

* - j

(1, 1)

* - j

(1, 0) 

*

(1, −1) -

* - j

* - j

* - j

* - j

- j R

* - j

* - j

* - j

* - j



* --



Δt

6

?Δr

(85)

### The Extended CIR Model

• In the extended CIR model the short rate follows dr = (θ(t) − a(t) r) dt + σ(t)√

r dW.

• The functions θ(t), a(t), and σ(t) are implied from market observables.

• With constant parameters, there exist analytical solutions to a small set of interest rate-sensitive securities.

(86)

### Finis

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30. – By the

• Appearance: vectorized mathematical code appears more like the mathematical expressions found in textbooks, making the code easier to understand.. • Less error prone: without

• Use the Black-Scholes formula with the stock price reduced by the PV of the dividends.. • This essentially decomposes the stock price into a riskless one paying known dividends and

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

Biases in Pricing Continuously Monitored Options with Monte Carlo (continued).. • If all of the sampled prices are below the barrier, this sample path pays max(S(t n ) −

• But Monte Carlo simulation can be modiﬁed to price American options with small biases (pp..

Biases in Pricing Continuously Monitored Options with Monte Carlo (continued).. • If all of the sampled prices are below the barrier, this sample path pays max(S(t n ) −

• 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