Toward the Black-Scholes Formula

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.

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 τ/n years long.

– The period gross return R = e^ˆr.

• Let

ln S_τ S

denote the continuously compounded rate of return of the stock.

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).

Toward the Black-Scholes Formula (continued)

• Pick

u = e^σ

√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.

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).

Toward the Black-Scholes Formula (continued)

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

• It approaches N(μτ + ln S, σ²τ ).

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

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(S_τ/S) approaches the normal distribution with mean (r − σ²/2) τ and variance σ²τ in a risk-neutral economy.^a

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 .

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,

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 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.

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).

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.

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

• It keeps the larger one.

Extensions of Options Theory

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

— Warren Buffet, May 3, 2008

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.

H

Time Price

S Barrier hit

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.

A Formula for Down-and-In Calls

• 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

_2λ

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).

Binomial Tree Algorithms

• Barrier options can be priced by binomial tree algorithms.

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

0 H

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 × C_u + 0.5 × C_d)/1.25.

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).

Binomial Tree Algorithms (concluded)

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value

aLyuu (1998).

Path-Dependent Derivatives

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

• S0 is the known price at time zero.

• S_n is the price at expiration.

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

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

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

S_i − X, 0

 .

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



X − 1

n + 1

n i=0

S_i, 0

 .

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).

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 method^b and approximation algorithms are some of the alternatives left.

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

bSee pp. 142ff.

S

Su

Sd

Suu

Sud

Sdu

p

1− ^p

⎟⎠

⎜ ⎞

⎝

⎛ + + −

= 

PD[ 6 6X 6XX ;

&_::

⎟⎠

⎜ ⎞

⎝

⎛ + + −

= 

PD[ 6 6X 6XG ;

&_:/

⎟⎠

⎜ ⎞

⎝

⎛ + + −

= 

PD[ 6 6G 6GX ;

&_/:

⎞

⎛6 +6G +6GG

( )

7 :/

: ::

H

&

S S&

& = + −

( )

7 //

/ /:

H

&

S S&

& = + −

( )

7 /

:

H

&

S S&

&= + − p

1− ^p p

1− ^p p

1− ^p

p

1− ^p p

1− ^p

Continuous-Time Financial Mathematics

A proof is that which convinces a reasonable man;

a rigorous proof is that which convinces an unreasonable man.

— Mark Kac (1914–1984)

Brownian Motion

• 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 < · · · < t_n, the random variables X(t_k) − X(t_k−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.

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.

Ito Process

• A shorthand^b is the following stochastic differential equation for the Ito differential dX_t,

dX_t = a(X_t, t) dt + b(X_t, t) dW_t. (14) – Or simply

dX_t = a_t dt + b_t dW_t.

aIto (1944).

bPaul Langevin (1872–1946) in 1904.

Ito Process (concluded)

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

• An equivalent form of Eq. (14) is dX_t = a_t dt + b_t√

dt ξ, (15)

where ξ ∼ N (0, 1).

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).

0.2 0.4 0.6 0.8 1 Time (t) -1

1 2 3 4 5 6 Y(t)

Local-Volatility Models

• The more general deterministic volatility model posits dS

S = (r_t − q_t) 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).

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 (%)

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).

Implied Trees (concluded)

• It is solved for separable local volatilities σ.^a

– The local-volatility function σ(S, V ) is separable^b 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).

The Hull-White Model

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

S = r dt + √

V dW₁, dV = μ_vV dt + bV dW₂.

• Above, V is the instantaneous variance.

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

The SABR Model

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

dS

S = r dt + S^θV dW₁, dV = bV dW₂,

for 0 ≤ θ ≤ 1.

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.

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.

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).

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.

Why Are Trees for Stochastic-Volatility Models

Difficult: Binomial Case?

Why Are Trees for Stochastic-Volatility Models

Difficult: Trinomial Case?

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!

Complexities of Stochastic-Volatility Models

• A few stochastic-volatility models suffer from subexponential (c^√n) tree size.

• Examples include the Hull-White (1987),

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

aChiu (R98723059) (2012).

Trees

I love a tree more than a man.

— Ludwig van Beethoven (1770–1827)

Trinomial Tree

• Set up a trinomial approximation to the geometric Brownian motion^a

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).

* - j

pu

pm

pd

Su S Sd S

-



Δt

* - j

* - j

* - j

* - j

Trinomial Tree (continued)

• By Eqs. (5) on p. 24,

M ≡ e^rΔt,

V ≡ M²(e^σ2Δt − 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.

Trinomial Tree (concluded)

• Use linear algebra to verify that p_u = u

V + M² − M

− (M − 1) (u − 1) (u² − 1) , p_d = u² 

V + M² − M

− u³(M − 1) (u − 1) (u² − 1) .

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

A Trinomial Tree

• Use u = e^λσ√Δt, where λ ≥ 1 is a tunable parameter.

• Then

p_u → 1

2λ² +

r + σ² √ Δt

2λσ ,

p_d → 1

2λ² −

r − 2σ² √ Δt

2λσ .

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

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

α_iS_i(τ ) − X, 0

 .

Multivariate Contingent Claims (continued)

Name Payoff

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

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

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

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



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.

Numerical Methods

All science is dominated by the idea of approximation.

— Bertrand Russell

Monte Carlo Simulation

• 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).

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.

How about American Options?

• 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,

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.

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 − ).

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).

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.

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).

Interest Rate Models

[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)

The Vasicek Model

• 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).

The Cox-Ingersoll-Ross Model

• 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).

The Ho-Lee Model

• 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.

The Black-Derman-Toy Model

• 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).

The Black-Karasinski Model

• 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.

The Extended Vasicek Model

• 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).

The Hull-White Model

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

* - j

(0, 0)

* - j

(1, 1)

* - j

(1, 0) 

*

(1, −1) -

* - j

* - j

* - j

* - j

- j R

* - j

* - j

* - j

* - j



* --



Δt

6

?^Δr

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.

Finis