Binomial options pricing modelFrom Wikipedia, the free encyclopedia

BOPM redirects here; for other uses see BOPM

(disambiguation).

In finance, the binomial options pricing model (BOPM)

provides a generalizable numerical method for the valuation

of options. The binomial model was first proposed

by Cox, Ross and Rubinstein(1979). Essentially, the model

uses a “discrete-time” (lattice based) model of the varying

price over time of the underlying financial instrument. In

general, binomial options pricing models do not have closed-

form solutions.

Contents

  [hide] 

1 Use of the model

2 Method

o 2.1 STEP 1: Create the binomial price tree

o 2.2 STEP 2: Find Option value at each final node

o 2.3 STEP 3: Find Option value at earlier nodes

2.3.1 Discrete dividends

3 Relationship with Black–Scholes

4 See also

5 Notes

6 References

7 External links

o 7.1 Discussion

o 7.2 Variations

o 7.3 Computer implementations

[edit]Use of the model

The Binomial options pricing model approach is widely used

as it is able to handle a variety of conditions for which other

models cannot easily be applied. This is largely because the

BOPM is based on the description of an underlying

instrument over a period of time rather than a single point. As

a consequence, it is used to value American options that are

exercisable at any time in a given interval as well

as Bermudan options that are exercisable at specific

instances of time. Being relatively simple, the model is readily

implementable in computer software (including

a spreadsheet).

Although computationally slower than the Black–

Scholes formula, it is more accurate, particularly for longer-

dated options on securities with dividend payments. For these

reasons, various versions of the binomial model are widely

used by practitioners in the options markets.

For options with several sources of uncertainty (e.g., real

options) and for options with complicated features (e.g., Asian

options), binomial methods are less practical due to several

difficulties, andMonte Carlo option models are commonly

used instead. When simulating a small number of time

steps Monte Carlo simulation will be more computationally

time-consuming than BOPM (cf. Monte Carlo methods in

finance). However, the worst-case runtime of BOPM will be

O(2n), where n is the number of time steps in the simulation.

Monte Carlo simulations will generally have a polynomial time

complexity, and will be faster for large numbers of simulation

steps. Monte Carlo simulations are also less susceptible to

sampling errors, since binomial techniques use discrete time

units. This becomes more true the smaller the discrete units

become.

[edit]Method

The binomial pricing model traces the evolution of the option's

key underlying variables in discrete-time. This is done by

means of a binomial lattice (tree), for a number of time steps

between the valuation and expiration dates. Each node in the

lattice represents a possible price of the underlying at a given

point in time.

Valuation is performed iteratively, starting at each of the final

nodes (those that may be reached at the time of expiration),

and then working backwards through the tree towards the first

node (valuation date). The value computed at each stage is

the value of the option at that point in time.

Option valuation using this method is, as described, a three-

step process:

1. price tree generation,

2. calculation of option value at each final node,

3. sequential calculation of the option value at each

preceding node.

[edit]STEP 1: Create the binomial price tree

The tree of prices is produced by working forward from

valuation date to expiration.

At each step, it is assumed that the underlying instrument will

move up or down by a specific factor (  or  ) per step of the

tree (where, by definition,   and  ). So, if   is

the current price, then in the next period the price will either

be   or  .

The up and down factors are calculated using the

underlying volatility,  , and the time duration of a step,  ,

measured in years (using the day count convention of the

underlying instrument). From the condition that

the variance of the log of the price is  , we have:

The above is the original Cox, Ross, & Rubinstein

(CRR) method; there are other techniques for

generating the lattice, such as "the equal probabilities"

tree. The Trinomial tree is a similar model, allowing for

an up, down or stable path.

The CRR method ensures that the tree is

recombinant, i.e. if the underlying asset moves up and

then down (u,d), the price will be the same as if it had

moved down and then up (d,u) — here the two paths

merge or recombine. This property reduces the

number of tree nodes, and thus accelerates the

computation of the option price.

This property also allows that the value of the

underlying asset at each node can be calculated

directly via formula, and does not require that the tree

be built first. The node-value will be:

Where   is the number of up ticks and   is the

number of down ticks.

[edit]STEP 2: Find Option value at each final node

At each final node of the tree — i.e. at expiration of

the option — the option value is simply its intrinsic,

or exercise, value.

Max [ ( ), 0 ], for a call option

Max [ (  –  ), 0 ], for a put option:

Where   is the strike price and   is the

spot price of the underlying asset at the   

period.

[edit]STEP 3: Find Option value at earlier nodes

Once the above step is complete, the

option value is then found for each node,

starting at the penultimate time step, and

working back to the first node of the tree

(the valuation date) where the calculated

result is the value of the option.

In overview: the “binomial value” is found at

each node, using the risk

neutrality assumption; see Risk neutral

valuation. If exercise is permitted at the

node, then the model takes the greater of

binomial and exercise value at the node.

The steps are as follows:

(1) Under the risk neutrality assumption,

today's fair price of a derivative is equal to

the expected value of its future payoff

discounted by the risk free rate. Therefore,

expected value is calculated using the

option values from the later two nodes

(Option up and Option down) weighted by

their respective probabilities

—“probability” p of an up move in the

underlying, and “probability” (1-p) of a

down move. The expected value is then

discounted at r, the risk free

rate corresponding to the life of the option.

The following formula to compute the expectation value is

applied at each node:

Binomial Value = [ p × Option up + (1-p) × Option down] ×

exp (- r × Δt), or

where

 is the option's value for the   node at time  ,

 is chosen such that the

related binomial distribution simulates the geometric

Brownian motion of the underlying stock with

parameters r and σ,

 is the dividend yield of the underlying corresponding to

the life of the option. It follows that in a risk-neutral world

futures price should have an expected growth rate of zero

and therefore we can consider   for futures.

Note that for   to be in the interval   the following

condition on   has to be satisfied  .

(Note that the alternative valuation approach, arbitrage-

free pricing, yields identical results; see “delta-hedging”.)

(2) This

result is

the

“Binomial

Value”. It

represents

the fair

price of the

derivative

at a

particular

point in

time (i.e. at

each

node),

given the

evolution

in the price

of the

underlying

to that

point. It is

the value

of the

option if it

were to be

held—as

opposed to

exercised

at that

point.

(3) Depen

ding on the

style of the

option,

evaluate

the

possibility

of early

exercise at

each node:

if (1) the

option can

be

exercised,

and (2) the

exercise

value

exceeds

the

Binomial

Value,

then (3)

the value

at the

node is the

exercise

value.

For

a Euro

pean

option,

there is

no

option

of early

exercis

e, and

the

binomi

al

value

applies

at all

nodes.

For

an Am

erican

option,

since

the

option

may

either

be held

or

exercis

ed prior

to

expiry,

the

value

at each

node

is: Max

(Binomi

al

Value,

Exercis

e

Value).

For

a Berm

udan

option,

the

value

at

nodes

where

early

exercis

e is

allowed

is: Max

(Binomi

al

Value,

Exercis

e

Value);

at

nodes

where

early

exercis

e is not

allowed

, only

the

binomi

al

value

applies

.

In

calculating

the value

at the next

time step

calculated

—i.e. one

step closer

to

valuation

—the

model

must use

the value

selected

here, for

“Option

up”/“Optio

n down” as

appropriat

e, in the

formula at

the node.

The

following al

gorithm de

monstrates

the

approach

computing

the price of

an

American

put option,

although is

easily

generalize

d for calls

and for

European

and

Bermudan

options:

function

americanPu

t(T, S, K,

r, sigma,

q, n) {

'

T...

expiration

time

'

S... stock

price

'

K...

strike

price

'

n...

height of

the

binomial

tree

deltaT

:= T / n;

up :=

exp(sigma

*

sqrt(delta

T));

p0 :=

(up *

exp(-r *

deltaT) -

exp(-q *

deltaT)) *

up / (up^2

- 1);

p1 :=

exp(-r *

deltaT) -

p0;

'

initial

values at

time T

for

i := 0 to

n {

p[i] := K

- S *

up^(2*i -

n);

if

p[i] < 0

then

p[i] := 0;

}

' move

to earlier

times

for

j := n-1

down to 0

{

for i := 0

to j {

p[i] := p0

* p[i] +

p1 *

p[i+1];

' binomial

value

exercise :

= K - S *

up^(2*i -

j); '

exercise

value

if p[i] <

exercise

then

p[i] :=

exercise;

}

}

return

americanPu

t := p[0];

}

[

edit]Discrete dividends

In practice,

the use of

continuous

dividend

yield,  , in

the

formula

above can

lead to

significant

mis-pricing

of the

option

near

an ex-

dividend d

ate.

Instead, it

is common

to model

dividends

as discrete

payments

on the

anticipated

future ex-

dividend

dates.

To model

discrete

dividend

payments

in the

binomial

model,

apply the

following

rule:

At each

time

step,  ,

calculat

e 

, for

all 

 

where 

is the

present

value

of the 

-th

dividen

d.

Subtra

ct this

value

from

the

value

of the

securit

y

price   

at each

node (

, ).

[

edit]Relationship with Black–Scholes

Similar ass

umptions u

nderpin

both the

binomial

model and

the Black–

Scholes

model, and

the

binomial

model thus

provides a

discrete

time

approximat

ion to the

continuous

process

underlying

the Black–

Scholes

model. In

fact,

for Europe

an

options wit

hout

dividends,

the

binomial

model

value

converges

on the

Black–

Scholes

formula

value as

the

number of

time steps

increases.

The

binomial

model

assumes

that

movement

s in the

price

follow

a binomial

distributio

n; for

many

trials, this

binomial

distribution

approache

s

the normal

distribution 

assumed

by Black–

Scholes.

In addition,

when

analyzed

as a

numerical

procedure,

the CRR

binomial

method

can be

viewed as

a special

case of

the explicit

finite

difference

method for

the Black–

Scholes

PDE;

see Finite

difference

methods

for option

pricing.[citation needed]

In 2011,

Georgiadis

shows that

the

binomial

options

pricing

model has

a lower

bound on

complexity

that rules

out

a closed-

form

solution

Black–ScholesFrom Wikipedia, the free encyclopedia

The Black–Scholes model (pronounced /ˌblæk ˈʃoʊlz/ [1] ) or Black–Scholes-Merton is

a mathematical model of a financial market containing certain derivative investment

instruments. From the model, one can deduce the Black–Scholes formula, which

gives the price of European-style options. The formula led to a boom in options trading

and the creation of the Chicago Board Options Exchange[dubious – discuss]. lt is widely used

by options market participants.[2]:751 Many empirical tests have shown the Black–Scholes

price is “fairly close” to the observed prices, although there are well-known

discrepancies such as the “option smile”.[2]:770-771

The model was first articulated by Fischer Black and Myron Scholes in their 1973 paper,

“The Pricing of Options and Corporate Liabilities.” They derived a partial differential

equation, now called theBlack–Scholes equation, which governs the price of the

option over time. The key idea behind the derivation was to hedge perfectly the option

by buying and selling the underlying asset in just the right way and consequently

"eliminate risk". This hedge is called delta hedging and is the basis of more complicated

hedging strategies such as those engaged in by Wall Street investment banks. The

hedge implies there is only one right price for the option and it is given by the Black–

Scholes formula.

Robert C. Merton was the first to publish a paper expanding the mathematical

understanding of the options pricing model and coined the term Black–Scholes options

pricing model. Merton and Scholes received the 1997 Nobel Prize in Economics (The

Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) for their

work. Though ineligible for the prize because of his death in 1995, Black was mentioned

as a contributor by the Swedish academy.[3]

Contents

  [hide] 

1 Assumptions

2 Notation

3 The Black–Scholes equation

o 3.1 Derivation

4 Black–Scholes formula

o 4.1 Interpretation

o 4.2 Derivation

4.2.1 Other derivations

5 The Greeks

6 Extensions of the model

o 6.1 Instruments paying continuous yield dividends

o 6.2 Instruments paying discrete proportional dividends

7 Black–Scholes in practice

o 7.1 The volatility smile

o 7.2 Valuing bond options

o 7.3 Interest rate curve

o 7.4 Short stock rate

8 Criticism

9 Remarks on notation

10 See also

11 Notes

12 References

o 12.1 Primary references

o 12.2 Historical and sociological aspects

o 12.3 Further reading

13 External links

o 13.1 Discussion of the model

o 13.2 Derivation and solution

o 13.3 Revisiting the model

o 13.4 Computer implementations

o 13.5 Historical

[edit]Assumptions

The Black–Scholes model of the market for a particular stock makes the following

explicit assumptions:

There is no arbitrage opportunity (i.e., there is no way to

make a riskless profit).

It is possible to borrow and lend cash at a known

constant risk-free interest rate.

It is possible to buy and sell any amount, even fractional, of

stock (this includes short selling).

The above transactions do not incur any fees or costs

(i.e., frictionless market).

The stock price follows a geometric Brownian motion with

constant drift and volatility.

The underlying security does not pay a dividend.[4]

From these assumptions, Black and Scholes showed that “it is possible to create

a hedged position, consisting of a long position in the stock and a short position in the

option, whose value will not depend on the price of the stock.”[5]

Several of these assumptions of the original model have been removed in subsequent

extensions of the model. Modern versions account for changing interest rates (Merton,

1976)[citation needed],transaction costs and taxes (Ingersoll, 1976)[citation needed], and dividend

payout.[6]

[edit]Notation

Let

, be the price of the stock (please note as below).

, the price of a derivative as a function of time and

stock price.

 the price of a European call option and   

the price of a European put option.

, the strike of the option.

, the annualized risk-free interest rate, continuously

compounded.

, the drift rate of  , annualized.

, the volatility of the stock's returns; this is the square

root of the quadratic variation of the stock's log price

process.

, a time in years; we generally use: now=0, expiry=T.

, the value of a portfolio.

Finally we will use   which

denotes the standard

normal cumulative distribution

function,

.

 which denotes the

standard normal probability

density function,

.

[edit]The Black–Scholes equation

Simulated Geometric

Brownian Motions with

Parameters from Market

Data

As above, the Black–

Scholes equation is

a partial differential

equation, which

describes the price of the

option over time. The key

idea behind the equation

is that one can

perfectly hedge the

option by buying and

selling

the underlying asset in

just the right way and

consequently “eliminate

risk". This hedge, in turn,

implies that there is only

one right price for the

option, as returned by the

Black–Scholes formula

given in the next section.

The Equation:

[edit]Derivation

The following

derivation is given

in Hull's Options,

Futures, and Other

Derivatives.[7]:287–288 Th

at, in turn, is based

on the classic

argument in the

original Black–

Scholes paper.

Per the model

assumptions above,

the price of

the underlying

asset (typically a

stock) follows

a geometric Brownian

motion. That is,

where W is Brown

ian motion. Note

that W, and

consequently its

infinitesimal

increment dW,

represents the

only source of

uncertainty in the

price history of the

stock.

Intuitively, W(t) is

a process that

jiggles up and

down in such a

random way that

its expected

change over any

time interval is 0.

(In addition,

its variance over

time T is equal

to T); a good

discrete analogue

for W is a simple

random walk.

Thus the above

equation states

that the

infinitesimal rate

of return on the

stock has an

expected value

of μ dt and a

variance of  .

The payoff of an

option   at

maturity is known.

To find its value at

an earlier time we

need to know

how   evolves as

a function of   

and  . By Itō's

lemma for two

variables we have

Now consider

a certain

portfolio,

called

the delta-

hedge portfoli

o, consisting

of being short

one option

and long   

shares at

time  . The

value of these

holdings is

Over the

time

period 

, the total

profit or

loss from

changes in

the values

of the

holdings

is:

Now

discreti

ze the

equatio

ns

for dS/

S and 

dV by

replaci

ng

differen

tials

with

deltas:

a

n

d

a

p

p

r

o

p

r

i

a

t

e

l

y

s

u

b

s

t

i

t

u

t

e

t

h

e

m

i

n

t

o

t

h

e

e

x

p

r

e

s

s

i

o

n

f

o

r

 

:

Not

ice

tha

t

the 

ter

m

ha

s

va

nis

he

d.

Th

us

un

cer

tai

nty

ha

s

be

en

eli

mi

nat

ed

an

d

the

por

tfoli

o is

eff

ecti

vel

y

risk

les

s.

Th

e

rat

e

of

ret

urn

on

this

por

tfoli

o

mu

st

be

eq

ual

to

the

rat

e

of

ret

urn

on

an

y

oth

er

risk

les

s

inst

ru

me

nt;

oth

er

wis

e,

the

re

wo

uld

be

op

por

tun

itie

s

for

arb

itra

ge.

No

w

ass

um

ing

the

risk

-

fre

e

rat

e

of

ret

urn

is   

we

mu

st

ha

ve

ov

er

the

tim

e

per

iod 

If we

now

equate

our

two

formul

as

for 

 w

e

obtain:

Simplifyin

g, we

arrive at

the

celebrated

Black–

Scholes

partial

differential

equation:

With the

assumptions

of the Black–

Scholes

model, this

second order

partial

differential

equation

holds for any

type of option

as long as its

price

function   is

twice

differentiable

with respect

to   and once

with respect

to  . Different

pricing

formulae for

various

options will

arise from the

choice of

payoff

function at

expiry and

appropriate

boundary

conditions.

[

edit]Black–Scholes formula

Black–Scholes

European call

optionpricing

surface

The Black–

Scholes

formula

calculates the

price

of European p

ut and call

options. This

price

is consistent 

with the

Black–

Scholes

equation as

above; this

follows since

the formula

can be

obtained by

solving the

equation for

the

corresponding

terminal and

boundary

conditions.

The value of a

call option for

a non-

dividend

paying

underlying

stock in terms

of the Black–

Scholes

parameters is:

The price of a

corresponding

t option based

on put-call

parity is:

For both, as above

 is

the cumulative

distribution

function of

the standard

normal distribution

 is the time

to maturity

 is the spot

price of the

underlying asset

 is the strike

price

 is the risk free

rate (annual rate,

expressed in

terms

of continuous

compounding

 is

the volatility

returns of the

underlying asset

[

edit]Interpretation

The

terms 

are the probabilities

of the option expiring

in-the-money 

the equivalent

exponential martingal

e probability measure

(numéraire=stock)

and the equivalent

martingale probability

measure

(numéraire=risk free

asset), respectively.

The risk neutral

probability density for

the stock

price 

is

where 

defined as above.

Specifically, 

the probability that the

call will be exercised

provided one assumes

that the asset drift is the

risk-free rate. 

however, does not lend

itself to a simple

probability

interpretation.

is correctly interpreted as

the present value, using

the risk-free interest rate,

of the expected asset

price at expiration,

that the asset price at

expiration is above the

exercise price.

related discussion – and

graphical representation

– see

section "Interpretation"

nder Datar–Mathews

method for real option

valuation.

The equivalent

martingale probability

measure is also called

the risk-neutral

probability measure

Note that both of these

are probabilities

a measure

theoretic sense, and

neither of these is the

true probability of

expiring in-the-money

under the real probability

measure. To calculate

the probability under the

real (“physical”)

probability measure,

additional information is

required—the drift term

in the physical measure,

or equivalently,

the market price of risk

[edit]Derivation

We now show how to get

from the general Black–

Scholes PDE to a

specific valuation for an

option. Consider as an

example the Black–

Scholes price of a call

option, for which the

PDE above has

conditions

The last condition gives the value of

the option at the time that the option

matures. The solution of the PDE

gives the value of the option at any

earlier time, 

To solve the PDE we transform the

equation into a

equation which may be solved

using standard methods. To this

end we introduce the change-of-

variable transformation

Then the Black–Scholes PDE becomes

a diffusion equation

The terminal

condition 

becomes an initial condition

Using the standard method for solving a diffusion

equation we have

which, after some manipulations, yields

where

Reverting 

stated solution to the Black–Scholes equation.

[edit]Other derivations

See also: Martingale pricing

Above we used the method of arbitrage-free pricing (“

hedging”) to derive the Black–Scholes PDE, and then solved the

PDE to get the valuation formula. It is also possible to derive the

latter directly using a

the price as the

particular probability measure

which differs from the real world measure. For the underlying logic

see section "risk neutral valuation"

section "Derivatives pricing: the Q world

finance; for detail, once again, see Hull.

[edit]The Greeks

“The Greeks” measure the sensitivity to change of the option price

under a slight change of a single parameter while holding the other

parameters fixed. Formally, they are

price with respect to the independent variables (technically, one

Greek, gamma, is a partial derivative of another Greek, called delta).

The Greeks are not only important for the mathematical theory of

finance, but for those actively involved in trading. Any trader worth

his or her salt will know the Greeks and make a choice of which

Greeks to hedge to limit exposure. Financial institutions will typically

set limits for the Greeks that their trader cannot exceed. Delta is the

most important Greek and traders will zero their delta at the end of

the day. Gamma and vega are also important but not as closely

monitored.

The Greeks for Black–Scholes are given in

can be obtained by straightforward differentiation of the Black–

Scholes formula.

Note that the gamma and vega formulas are the same for calls and

puts. This can be seen directly from

In practice, some sensitivities are usually quoted in scaled-down

terms, to match the scale of likely changes in the parameters. For

example, rho is often reported divided by 10,000 (1bp rate change),

vega by 100 (1 vol point change), and theta by 365 or 252 (1 day

decay based on either calendar days or trading days per year).

[edit]Extensions of the model

The above model can be extended for variable (but deterministic)

rates and volatilities. The model may also be used to value

European options on instruments paying dividends. In this case,

closed-form solutions are available if the dividend is a known

proportion of the stock price.

stocks paying a known cash dividend (in the short term, more

realistic than a proportional dividend) are more difficult to value, and

a choice of solution techniques is available (for

example lattices

[edit]Instruments paying continuous yield dividends

For options on indexes, it is reasonable to make the simplifying

assumption that dividends are paid continuously, and that the

dividend amount is proportional to the level of the index.

The dividend payment paid over the time period

modelled as

for some constant

Under this formulation the arbitrage-free price implied by the Black–

Scholes model can be shown to be

and

where now

is the modified forward price that occurs in the terms

and

Exactly the same formula is used to price options on foreign exchange rates, except that

now q plays the role of the foreign risk-free interest rate and

This is the Garman-Kohlhagen model

[edit]Instruments paying discrete proportional dividends

It is also possible to extend the Black–Scholes framework to options on instruments

paying discrete proportional dividends. This is useful when the option is struck on a single

stock.

A typical model is to assume that a proportion

determined times

where   is the number of dividends that have been paid by time

The price of a call option on such a stock is again

where now

is the forward price for the dividend paying stock.

[edit]Black–Scholes in practice

The normality assumption of the Black–Scholes model does not capture extreme movements such as

crashes.

The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely

employed as a useful approximation, but proper application requires understanding its limitations –

blindly following the model exposes the user to unexpected risk.

Among the most significant limitations are:

the underestimation of extreme moves, yielding

money options;

the assumption of instant, cost-less trading, yielding

the assumption of a stationary process, yielding

volatility hedging;

the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged

with Gamma hedging.

In short, while in the Black–Scholes model one can perfectly hedge options by simply Delta hedging,

in practice there are many other sources of risk.

Results using the Black–Scholes model differ from real world prices because of simplifying

assumptions of the model. One significant limitation is that in reality security prices do not follow a

strict stationary

over time). The variance has been observed to be non-constant leading to models such

as GARCH to model volatility changes. Pricing discrepancies between empirical and the Black–

Scholes model have long been observed in options that are far

extreme price changes; such events would be very rare if returns were lognormally distributed, but

are observed much more often in practice.

Nevertheless, Black–Scholes pricing is widely used in practice,

for it is easy to calculate and explicitly models the relationship of all the variables. It is a useful

approximation, particularly when analyzing the directionality that prices move when crossing critical

points. It is used both as a

volatility is not constant, results from the model are often useful in practice and helpful in setting up

hedges in the correct proportions to minimize risk. Even when the results are not completely

accurate, they serve as a first approximation to which adjustments can be made.

One reason for the popularity of the Black–Scholes model is that it is

to deal with some of its failures. Rather than considering some parameters (such as volatility or

interest rates) as

reflected in the

the partial derivatives with respect to these variables), and hedging these Greeks mitigates the risk

caused by the non-constant nature of these parameters. Other defects cannot be mitigated by

modifying the model, however, notably tail risk and liquidity risk, and these are instead managed

outside the model, chiefly by minimizing these risks and by

Additionally, rather than

model to solve for volatility, which gives the

and exercise prices. Solving for volatility over a given set of durations and strike prices one can

construct an implied volatility surface

transformation

prices in terms of dollars per unit (which are hard to compare across strikes and

prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option

markets.

[edit]The volatility smile

Main article: Volatility smile

One of the attractive features of the Black–Scholes model is that the parameters in the model (other

than the volatility) — the time to maturity, the strike, the risk-free interest rate, and the current

underlying price — are unequivocally observable. All other things being equal, an option's theoretical

value is a monotonic increasing function

By computing the implied volatility for traded options with different strikes and maturities, the Black–

Scholes model can be tested. If the Black–Scholes model held, then the implied volatility for a

particular stock would be the same for all strikes and maturities. In practice, the

surface (the three-dimensional graph

The typical shape of the implied volatility curve for a given maturity depends on the underlying

instrument. Equities tend to have skewed curves: compared to

substantially higher for low strikes, and slightly lower for high strikes. Currencies tend to have more

symmetrical curves, with implied volatility lowest

Commodities often have the reverse behavior to equities, with higher implied volatility for higher

strikes.

Despite the existence of the volatility smile (and the violation of all the other assumptions of the

Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively

in practice. A typical approach is to regard the volatility surface as a fact about the market, and use

an implied volatility from it in a Black–Scholes valuation model. This has been described as using

"the wrong number in the wrong formula to get the right price."

values for the hedge ratios (the Greeks).

Even when more advanced models are used, traders prefer to think in terms of volatility as it allows

them to evaluate and compare options of different maturities, strikes, and so on.

[edit]Valuing bond options

Black–Scholes cannot be applied directly to

reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing

its volatility, and the simple Black–Scholes model does not reflect this process. A large number of

extensions to Black–Scholes, beginning with the

phenomenon.

[edit]Interest rate curve

In practice, interest rates are not constant-they vary by

be interpolated to pick an appropriate rate to use in the Black–Scholes formula. Another

consideration is that interest rates vary over time. This volatility may make a significant contribution

to the price, especially of long-dated options.This is simply like the interest rate and bond price

relationship which is inversely related.

[edit]Short stock rate

It is not free to take a

position for a small fee. In either case, this can be treated as a continuous dividend for the purposes

of a Black–Scholes valuation.

[edit]Criticism

Espen Gaarder Haug

existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk," to

make them more compatible with mainstream

were made in an earlier paper by

Wilmott has defended the model.

Jean-Philippe Bouchaud

effects. The Black–Scholes model,

used extensively. But it assumes that the probability of extreme price changes is negligible, when in

reality, stock prices are much jerkier than this. Twenty years ago, unwarranted use of the model

spiralled into the worldwide October 1987 crash; the Dow Jones index dropped 23% in a single day,

dwarfing recent market hiccups