CQG Integrated Client Options User Guide. November 14, 2012 Version 13.5

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CQG Integrated

Client Options User

Guide

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Table of Contents

About this Document ... 1

About this Document

This document is one of several user guides for CQG Integrated Client (CQG IC). This guide details options-specific tools in CQG.

You can navigate the document in several ways:

• Click a bookmark listed on the left of the page.

• Click an item in the Table of Contents.

• Click a blue, underlined link that takes you to another section of the document. To go back, use Adobe Reader Page Navigation items (View menu).

If you are looking for a particular term, it may be easier for you to search the document for it. There are two ways to do that:

• Right-click the page, and then click Find.

• Press Ctrl+F on your keyboard.

This document is intended to be printed double-sided, so it includes blank pages before new chapters.

Please note that images are examples only and are meant to demonstrate and expose system behavior. They do not represent actual situations.

To ensure that you have the most recent copy of this guide, please go to the user guide page on CQG’s website.

Options in CQG

CQG IC includes five options applications:

• Options Window

• Options Calculator

• Options Graph

• Volatility Workshop

• Strategy Analysis

All CQG IC users have access to the Options Window and the Options Graph. If you would like to learn more about our advanced options offering, which includes Options Calculator, Options Strategy, and Volatility Workshop, please contact CQG.

CQG offers seven basic option models that serve as the framework for valuing options: Black, Black-Scholes, Bourtov, Cox-Ross-Rubinstein, Garman-Kohlhagen, Merton, and Whaley.

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Entering Options Symbols

The format for options on futures is: C.<symbol><month code><year><strike price> for calls and or P.<symbol><month code><year><strike price> for puts.

The strike price is 2-5 digits.

Example: C.SPZ081500 = December 2008 1500 call on the S&P 500 futures contract.

An alternate format is C.<symbol>_<month code><year>.<strike price> for calls and with P. for puts.

C.SP_U8.1500 = September 2008 1500 call on the S&P 500 futures contract. On Options windows, you can enter the symbol only.

For at the money for the nearby month, type C. or P., the symbol, and ?.

For at the money for some other month, type C. or P., the symbol, the month, the year, and ? and then press CTRL+ENTER.

For strikes for the most active month, type C. or P. and the symbol and ? and then press

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Options User Guide

Opening Options Applications

Click the Options button on the main toolbar, and then click the name of the options window you want to open:

This button provides access to all options windows without having to display the button for each window.

If the Options button is not displayed, click the More button, and then click Options. You can also add individual options windows to the toolbar:

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CQG API and Options

CQGs API supports efficient access to options strike properties through the use of the CQGInstrumentsGroup interface.

With one request to CQG servers, your application can subscribe to all strikes in any given contract month or a range of months.

Data subscription levels can also be configured to optimize instrument resolution for strike properties and market data, allowing for the delivery of critical information without unnecessary overhead.

CQG also offers access to common real-time values for all subscribed options strikes: Greeks, theoretical values and implied volatilities.

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Options User Guide

Greeks and Volatility Definitions

As you work with options in CQG IC, it’s helpful to understand how implied and average volatility are calculated and how the Greeks are defined.

Delta

Delta shows the change in the price of a derivative to the change in the price of the underlying assets. Sometimes delta is known as the “hedge ratio,” as delta indicates how much of the underlying asset needs to be bought or sold to hedge the option. Traders take advantage of delta by creating delta hedging, delta spreads, and delta neutral.

Delta values are positive numbers less than or equal to 100. They represent the ratio of the change in the theoretical value over the change in the underlying price.

Values:

Out of the money = close to 0 At the money = close to +0.5 In the money = close to +1 Calls = positive

Puts = negative

Delta values for the out-of-the-money series move closer to 0 as expiration nears. Likewise, more in-the-money options have deltas close to 1 as expiration approaches.

For example: If the underlying S&P 500 contract stands at 134020, with a delta of 52.73, and a theoretical value of 2600.5, and the underlying price increases to 134220, while the delta rises to 54.02, the theoretical value increases to 2707.

The calculations are: 134220 – 134020 = 200 (52.73 + 54.02)/2 = 106.750 53.375 *2 = 106.750

The deltas from one underlying price to the next are interpolated. 106.750 + 2600.5 = 2707.25 new theoretical value

Gamma

Gamma is the amount the delta changes when the underlying price changes by one tick. Gamma is greatest for at-the-money options. Gamma increases as the option moves closer to expiration. Traders try to limit gamma risk because short gamma positions create a potential for losses.

For example: If the delta of an S&P future was 91.80, the gamma was .01 and the price of an S&P future increased from 1340.80 to 1340.90 i.e., a one-tick increase, the delta would increase to 91.81.

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Theta

Theta represents the loss in theoretical value in one day, if all other factors are constant. In other words, it attempts to isolate the time decay factor.

For example: Assume the amount showing the Value column was 2725.1, with 15 days until expiration and a theta value of 92.053. You would expect to see the amount in the Value column decrease approximately 92 dollars the following day. A more precise definition of the amount of the time value lost is an average of the Thetas on the dates under consideration. So, if the theta on the following day was 95.201, the decrease in theoretical value would be: (92.053 + 95.201)/2 = 93.6

Vega

Vega is the amount that the theoretical value changes when the volatility changes by 1 point. For example: Assume a June Corn contract had a vega of 1.421, a volatility of 25.90, and a theoretical value of 45.4. If the volatility were to increase to 26.90, the vega says that the theoretical value would increase by 1.4 dollars to 46.8, provided the other factors affecting options prices remained constant.

The display also indicates the days until expiration, as well as the volatility and interest rate assumptions underlying the data.

Rho

Rho is the change in option price to a unit change in interest rates. When the interest rate increases, the call option price increases also and put option price falls.

For example: Assume the starting call value is 4.2012, the interest rate r is 5% and zero-coupon rate b is 2%. Rho(r)(per 1%)= 0.1243, and Rho(b)(per 1%)=0.1328, If r rises to 6% and b stays at 5%, the call value is 4.3255. If r stays at 5% and b rises to 3%, the call value is 4.334.

Implied Volatility

The implied volatility calculated from an options display represents the volatility that, if entered into a theoretical pricing model, would produce a theoretical value equal to the market price of the option. Unlike the Historical Volatility study, the Implied Volatility calculation depends on

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Options User Guide

Standard Options Pricing Models

Options pricing models describe mathematically how a set of input parameters – typically underlying price, strike price, time to expiration, interest rate, and volatility – combine to determine a theoretical value of an option.

CQG offers seven basic option models that serve as the framework for valuing options: Black, Black-Scholes, Bourtov, Cox-Ross-Rubinstein, Garman-Kohlhagen, Merton, and Whaley.

Term Definition

TheoV option theoretic value

sigma, σ volatility of the relative price change of the underlying stock price ImpV implied volatility

Greeks Partial derivatives of the option price to a small movement in the underlying variables. Main greeks are delta, gamma, theta, vega, rho.

Delta, ∆ delta is the first derivative of the option price by underlying price Gamma, Γ gamma is the second derivative of the option price by underlying price Vega vega is the first derivative of the option price by volatility

Theta, Θ theta is the first derivative of the option price by time to expiration Rho, ρ rho is the first derivative of the option price by interest rate

N(x) the cumulative normal distribution function

∫

∞ − −

⋅

=

x z

dz

e

x

N

2 2

2

1

)

(

π

n(x) normal distribution function

2 2

2

1

)

(

x

e

x

n

−

⋅

=

π

, 2 2

2

1

)

(

x

e

x

n

−

⋅

−

=

′

π

S underlying price

X strike price of option r risk-free interest rate

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Term Definition

σ volatility of the relative price change of the underlying instrument b the cost-of-carry rate of holding the underlying security

For further reading, we suggest:

• The Complete Guide to Option Pricing Formulas. ISBN 0071389970.

• Options, Futures, and Other Derivatives. ISBN 0132164949.

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Options User Guide

Black Model

In 1976, Fisher Black developed a modification to the Black-Scholes model designed to price options on futures more precisely. The model assumes that futures can be treated the same way as securities, providing a continuous dividend yield equal to the risk-free interest rate. The model provides a good correction to the original model concerning options on futures. However, it still carries the restrictions of the Black-Scholes evaluation.

Notation

Theoretical value of a call Theoretical value of a put Underlying price

Strike price Interest rate

Time to expiration in years Volatility

Cumulative normal density function The theoretical values for calls and puts are:

Where:

Note: Although similar, this definition of is different from the one used in the Black-Scholes model.

An alternative form for is:

C

P

U

E

r

t

ν

)

(

x

N

)

(

)

(

h

Ee

N

h

t

N

Ue

C

=

−rt

−

−rt

−

ν

)

(

)

(

h

Ee

N

t

h

N

Ue

P

=

−

−rt

−

+

−rt

ν

−

2

)

/

ln(

t

E

U

h

ν

+

=

h

t

E

U

h

ν

2

)

/

ln(

2

+

2

=

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Generalized Black-Scholes (Black-Scholes extended) Model

The generalized Black-Scholes model can be used to price European options on stocks without dividends [Black and Scholes (1973) model], stocks paying a continuous dividend yield [Merton (1973) model], options on futures [Black (1976) model], and currency options [Garman and Kohlhagen (1983) model].

TheoV

Call

)

(

)

(

C

( ) ₁ ₂ GBS

S

e

N

d

X

e

N

d

c

=

⋅

b−r⋅T

⋅

−

⋅

−r⋅T

⋅

Put

)

(

)

(

P

_GBS

X

e

N

d

₂

S

e

( )

N

d

₁

p

=

⋅

−r⋅T

⋅

−

⋅

b−r⋅T

⋅

−

where

(

)

T

b

X

S

d

⋅

+

=

σ

/

2

)

/

ln(

2 1

T

d

₂

=

₁

−

σ

⋅

N(x) – the cumulative normal distribution function; S – underlying price;

X – strike price of option; r – risk-free interest rate; T – time to expiration in years;

σ – volatility of the relative price change of the underlying stock price. b – the cost-of-carry rate of holding the underlying security.

b = r gives the Black and Scholes (1973) stock option model.

b = r – q gives the Merton (1973) stock option model with continuous dividend yield q. b = 0 gives the Black (1976) futures option model.

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Options User Guide

Gamma

Gamma is identical for put and call options.

T

S

e

d

n

b rT

⋅

=

Γ

−

σ

) ( 1

)

(

where 2 2

2

1

)

(

x

e

x

n

−

⋅

=

π

- normal distribution function.

Vega

Vega is identical for put and call options.

T

d

n

e

S

Vega

=

⋅

b−rT

(

1

)

⋅

) (

Theta

Call

)

(

)

(

)

(

2

)

(

2 1 ) ( 1 ) (

d

N

X

r

d

N

e

S

r

b

T

d

n

e

S

b rT rT T r b − − −

⋅

+

⋅

−

+

⋅

=

Θ

σ

Put

)

(

)

(

)

(

2

)

(

2 1 ) ( 1 ) (

d

N

X

r

d

N

e

S

r

b

T

d

n

e

S

b rT rT T r b

−

⋅

−

⋅

−

⋅

=

Θ

−

σ

− −

Rho

Call







=

⋅

−

<>

⋅

=

−

0

),

(

2

b

when

c

T

b

when

d

N

e

X

T

rT

ρ

where c – call TheoV Put







=

⋅

−

<>

−

⋅

−

=

−

0

),

(

2

b

when

p

T

b

when

d

N

e

X

T

rT

ρ

where p – put TheoV

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Implied volatility

To find implied volatility the following equations should be solved for the value of sigma: Call

)

(

)

(

₁ ₂ ) (

d

N

e

X

d

N

e

S

c

=

⋅

b−r⋅T

⋅

−

⋅

−r⋅T

⋅

Put

)

(

)

(

1 ) ( 2

S

e

N

d

N

e

X

p

=

⋅

−r⋅T

⋅

−

⋅

b−r⋅T

⋅

−

where

(

)

T

b

X

S

d

⋅

−

+

=

σ

/

2

)

/

ln(

2 1

T

d

₂

=

₁

−

σ

⋅

This equation has no closed form solution, which means the equation must be numerically solved to find σ.

Bourtov’s Model

Bourtov’s model is based on the Black-Scholes model. It defines a special method to calculate volatility, which is an input parameter of the Pricing Model Calculator.

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Options User Guide

Cox-Ross-Rubinstein Model

The Cox-Ross-Rubinstein binomial model can be used to price European and American options on stocks without dividends, stocks and stock indexes paying a continuous dividend yield, futures, and currency options.

TheoV

The main binomial model assumption is the underlying price can either increase by a fixed amount u with probability p, or decrease by a fixed amount d with probability 1-p. So the underlying price at each node is set equal to

j

i

d

u

S

⋅

i

⋅

j−i

,

=

0

,

1

,...,

where S – underlying price;

u, d – up and down jump sizes that underlying price can take at each time step.

Option pricing is done by working backwards, starting at the terminal date. Here we know all the possible values of the underlying price. For each of these, we calculate the payoffs from the derivative, and find what the set of possible derivative prices is one period before. Given these, we can find the option one period before this again, and so on. Working ones way down to the root of the tree, the option price is found as the derivative price in the first node.

Call At expiration date:

n

i

X

d

u

S

f

i,n

=

max(

⋅

i

⋅

n i

−

,

0

),

=

0

,

1

,...,

− where n – number of time steps.

At each previous step: European exercise

[

1, 1 , 1

]

, + +

(

1

)

+ ∆ ⋅ −

⋅

+

−

⋅

=

i j ij t r j i

e

p

f

p

f

American exercise

[

]

(

1, 1 , 1

)

,

max

,

+ +

(

1

)

+ ∆ ⋅ − −

−

⋅

+

−

⋅

=

i j i j t r i j i j i

S

u

d

X

e

p

f

p

f

where

−

=

∆t

e

u

σ price up movement size;

−

=

− ∆

u

e

d

σ t

1

/

price down movement size;

n

T

t

=

/

∆

– size of each time step;

−

=

⋅∆

d

u

d

e

p

t b up movement probability; b – the cost-of-carry, defined as:

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b = r – q to price European and American options on stocks and stock indexes paying a continuous dividend yield q;

b = 0 to price European and American options on futures;

b = r – rf to price European and American currency options (rf – risk-free rate of the foreign currency). Put At expiration date:

n

i

d

u

S

X

p

i,n

=

max(

−

⋅

i

⋅

n i

,

0

),

=

0

,

1

,...,

− At each previous step:

European exercise

[

1, 1 , 1

]

, + +

(

1

)

+ ∆ ⋅ −

⋅

+

−

⋅

=

i j ij t r j i

e

p

f

p

f

American exercise

[

]

(

1, 1 , 1

)

,

max

,

+ +

(

1

)

+ ∆ ⋅ − −

⋅

+

−

⋅

−

=

i j i j t r i j i j i

X

S

u

d

e

p

f

p

f

Delta

Given the

f

_i,_j values calculated for the price, Delta approximation is

d

S

u

S

f

S

f

⋅

−

⋅

−

=

∆

=

∆

1,1 1,0

Gamma

Gamma approximation is

(

)

(

)

[

]

[

(

)

(

)

]

(

2 2

)

2 0 , 2 1 , 2 2 1 , 2 2 , 2 2 2

5

.

0

S

u

S

d

S

d

u

S

f

d

u

S

u

S

f

S

f

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

=

∂

=

γ

Theta

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Options User Guide

Garman-Kohlhagen Model

This model, developed to evaluate currency options, considers foreign currencies analogous to a stock providing a known dividend yield. The owner of foreign currency receives a “dividend yield” equal to the risk-free interest rate available in that foreign currency. The model assumes price follows the same stochastic process presumed in the Black-Scholes model.

This model is used to evaluate options written on currencies. The interest rate of the native currency is used as the default, but you can set the foreign interest rate in Model preferences. This model corrects the difference between native and foreign interest rates. However, as a modification of Black-Scholes model, it possesses all its limitations.

Notation

Strike price Interest rate

Interest rate in the foreign country Time to expiration in years

Volatility

The European call price is given by:

Where:

The European put price is given by:

C

P

U

E

r

f

r

t

ν

)

(

)

(

h

Ee

N

h

t

N

Ue

C

=

−rft

−

−rt

−

ν

t

r

E

U

h

f

ν

/

2

)

(

)

/

ln(

+

−

+

2

=

)

(

)

(

h

Ee

N

t

h

N

Ue

P

=

−

−rft

−

+

−rt

ν

−

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Merton Model

In 1973, Merton produced a model with a non-constant interest rate. He assumed that interest rates follow a special type of random process.

By taking into consideration the dynamic process of interest rate determination, and the correlation between the underlying price and the options price, this model provides an improvement over the Black-Scholes model. This model is generally used to value European options written on stocks.

Notation

Strike price

Time to expiration in years

Cumulative normal density function Volatility

Volatility of an interest rate contract Interest rate

Correlation between the underlying and interest rate contracts The theoretical values for European calls and puts are:

Where:

C

P

U

E

t

)

(

x

N

ν

p

ν

)

(

t

R

ρ

)

(

)

(

)

(

h

B

t

EN

h

t

UN

C

=

−

ϑ

)

(

)

(

)

(

h

B

t

EN

t

h

UN

P

=

−

+

ϑ

−

t

B

X

U

h

ϑ

(

)

/

2

)

(

ln

)

/

ln(

−

+

=

t

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Options User Guide

Whaley Model

The quadratic approximation method by Baron-Adesi and Whaley (1987) can be used to price American options.

TheoV

Call











≥

−

<

⋅

+

=

* * * 2 2

)

/

(

)

,

(

)

(

)

,

(

S

when

X

S

when

S

A

b

r

T

X

S

C

stoks

on

options

r

b

when

b

r

T

X

S

C

c

_GBS q GBS

σ

where

b – the cost-of-carry rate;

b = r to price options on stocks.

b = r – q to price options on stocks and stock indexes paying a continuous dividend yield q b = 0 to price options on futures.

b = r – rf to price currency options (rf – risk-free rate of the foreign currency). CGBS – the generalized Black-Scholes call TheoV expression;

(

)

[

1

( ) ₁

(

*

)

]

2 * 2

e

N

d

S

q

S

A

=

−

b−rT

⋅

T

b

X

S

d

σ

⋅

+

=

ln(

/

)

(

/

2

)

(

2 1

2

/

4

)

1

(

)

1

(

2 2

K

M

N

q

=

−

+

−

+

⋅

2

/

2

r

σ

M

=

2

/

2

b

σ

N

=

rT

e

K

=

1

−

S*_{– the critical commodity price for the call option that satisfies}

(

)

[

1

(

)

]

)

,

(

( ) ₁ * 2 * * *

S

d

N

e

q

S

b

r

T

X

S

C

X

S

−

=

_GBS

σ

+

−

b−rT

⋅

The last equation should be numerically solved to find S*_. Put









≤

−

>

⋅

+

=

* * * * * * 1 1

)

/

(

)

,

(

S

when

S

X

S

when

S

A

b

r

T

X

S

P

p

q GBS

σ

(26)

where

PGBS – the generalized Black-Scholes put TheoV expression;

(

)

[

1

( ) ₁

(

**

)

]

1 * * 1

e

N

d

S

q

S

A

=

−

b−rT

⋅

−

2

/

4

)

1

(

)

1

(

2 1

K

M

N

q

=

−

+

⋅

S**_{– the critical commodity price for the put option that satisfies}

(

)

[

1

(

)

]

)

,

(

( ) ₁ ** 1 * * * * * *

S

d

N

e

q

S

b

r

T

X

S

P

S

X

−

=

_GBS

σ

−

b−rT

⋅

−

The last equation should be numerically solved to find S**_.

Delta

Call











≥

<

⋅

+

∆

=

∆

=

∆

− * * * 1 2 2

1

)

/(

)

,

(

)

(

)

,

(

2 2

S

when

S

when

S

q

A

b

r

T

X

S

stoks

on

options

r

b

when

b

r

T

X

S

q q GBS GBS

σ

where

∆GBS - the generalized Black-Scholes call ∆ expression.

Put









≤

−

>

⋅

+

∆

=

∆

− * * * * * * 1 1 1

1

)

/(

)

,

(

1 1

S

when

S

when

S

q

A

b

r

T

X

S

q q GBS

σ

where

(27)

Options User Guide

Gamma

Call











≥

<

⋅

−

⋅

+

Γ

=

Γ

=

Γ

− * * * 2 2 2 2

0

)

/(

)

1

(

)

,

(

)

(

)

,

(

2 2

S

when

S

when

S

q

A

b

r

T

X

S

stoks

on

options

r

b

when

b

r

T

X

S

q q GBS GBS

σ

Put









≤

>

⋅

−

⋅

+

Γ

=

Γ

− * * * * * * 2 1 1 1

0

)

/(

)

1

(

)

,

(

1 1

S

when

S

when

S

q

A

b

r

T

X

S

q q GBS

σ

Vega

Call











≥

<

=

* *

0

)

(

)

,

(

S

when

S

when

ation

differenti

Numerical

stoks

on

options

r

b

when

b

r

T

X

S

Vega

GBS

σ

Put









≤

>

=

* * * *

0

when

S

when

ation

differenti

Numerical

Vega

Theta

Call











≥

<

=

Θ

=

Θ

* *

0

)

(

)

,

(

S

when

S

when

ation

differenti

Numerical

stoks

on

options

r

b

when

b

r

T

X

S

GBS

σ

where

ΘGBS - the generalized Black-Scholes call Θ expression. Put









≤

>

=

Θ

_*_* * *

0

when

S

when

ation

differenti

Numerical

where

(28)

Rho

Call











≥

<

=

* *

0

)

(

)

,

(

S

when

S

when

ation

differenti

Numerical

stoks

on

options

r

b

when

b

r

T

X

S

GBS

σ

ρ

where

ρ_GBS - the generalized Black-Scholes call ρ expression. Put









≤

>

=

* * * *

0

when

S

when

ation

differenti

Numerical

ρ

where

ρGBS - the generalized Black-Scholes put ρ expression.

Implied volatility

System numerically finds implied volatility.

Implied volatility can’t be calculated for call option if option value is less than (underlying price - strike).

Implied volatility can’t be calculated for put option if option value is less than (strike - underlying).

(29)

Options User Guide

Exotic Option Models

For further reading, we suggest:

• The Complete Guide to Option Pricing Formulas. ISBN 0071389970.

• Barrier Options, Binary/Digital Options, and Lookback Options at www.global-derivatives.com.

Standard (Vanilla) Barrier

There are two kinds of the barrier options:

• In = Paid for today but first come into existence if the underlying price hits the barrier H before expiration.

• Out = Similar to standard options except that the option is knocked out or becomes worthless if the underlying price hits the barrier before expiration.

TheoV

In Barriers Down-and-in call c(X>=H) = C + E η = 1, φ = 1 c(X<H) = A – B + D + E η = 1, φ = 1 Up-and-in call c(X>=H) = A + E η = -1, φ = 1 c(X<H) = B – C + D + E η = -1, φ = 1 Down-and-in put p(X>=H) = B – C + D + E η = 1, φ = -1 p(X<H) = A + E η = 1, φ = -1 Up-and-in put p(X>=H) = A – B + D + E η = -1, φ = -1 p(X<H) = C + E η = -1, φ = -1 Out Barriers Down-and-out call c(X>=H) = A – C + F η = 1, φ = 1 c(X<H) = B – D + F η = 1, φ = 1

(30)

Up-and-out call c(X>=H) = F η = -1, φ = 1 c(X<H) = A – B + C – D + F η = -1, φ = 1 Down-and-out put p(X>=H) = A – B + C – D + F η = 1, φ = -1 p(X<H) = F η = 1, φ = -1 Up-and-out put p(X>=H) = B – D + F η = -1, φ = -1 p(X<H) = A – C + F η = -1, φ = -1 where

(

x

)

X

e

N

(

x

T

)

N

e

S

A

=

φ

⋅

b−rT

⋅

φ

⋅

1

−

φ

⋅

−rT

⋅

φ

⋅

1

−

φ

⋅

σ

⋅

) (

(

x

)

X

e

N

(

x

T

)

N

e

S

B

=

φ

⋅

b−rT

⋅

φ

⋅

2

−

φ

⋅

−rT

⋅

φ

⋅

2

−

φ

⋅

σ

⋅

) (

(

y

)

X

e

H

S

N

(

y

T

)

N

S

H

e

S

C

=

φ

⋅

b−rT

⋅

µ+

⋅

η

⋅

−

φ

⋅

−rT

⋅

µ

⋅

η

⋅

1

−

η

⋅

σ

⋅

2 1 ) 1 ( 2 ) (

)

/

(

)

/

(

y

)

X

e

H

S

N

(

y

T

)

N

S

H

e

S

D

=

φ

⋅

b−rT

⋅

µ+

⋅

η

⋅

−

φ

⋅

−rT

⋅

µ

⋅

η

⋅

2

−

η

⋅

σ

⋅

2 2 ) 1 ( 2 ) (

)

/

(

)

/

(

)

(

)

[

N

x

T

H

S

N

y

T

]

e

K

E

=

⋅

−rT

⋅

η

⋅

₂

−

η

⋅

σ

⋅

−

(

/

)

2µ

⋅

η

⋅

₂

−

η

⋅

σ

⋅

( )

(

)

[

H

S

N

z

H

S

N

z

T

]

K

F

=

(

/

)

µ+λ

η

⋅

+

(

/

)

µ−λ

η

⋅

−

2

⋅

η

⋅

λ

⋅

σ

⋅

T

X

S

x

=

+

µ

⋅

σ

⋅

σ

(

1

)

/

ln(

1

T

H

S

x

=

+

µ

⋅

σ

⋅

σ

(

1

)

/

ln(

2

(

)

(

H

S

X

)

T

y

=

⋅

+

µ

⋅

σ

⋅

σ

(

1

)

/

ln

2

(31)

Options User Guide 2 2

2

σ

µ

λ

=

+

r

K – possible cash rebate, b – the cost-of-carry.

b = r – rf to price currency options (rf – risk-free rate of the foreign currency).

Delta, Gamma, Vega, Theta, Rho

The system uses the numerical differentiation to calculate the Greeks.

Implied volatility

(32)

Asset-or-Nothing Binary

At expiry, the asset-or-nothing call option pays 0 if S <= X and S if S > X. Similarly, a put option pays 0 if S >=X and S if S < X.

TheoV

Call

)

(

) (

d

N

e

S

c

=

⋅

b−r⋅T

⋅

Put

)

(

) (

d

N

e

S

p

=

⋅

b−r⋅T

⋅

−

where 2

ln( /

)

2

S X

b

T

d

T

σ





+



+



⋅





=

⋅

b – the cost-of-carry.

b = r – rf to price currency options (rf – risk-free rate of the foreign currency).

Delta

( ) ( )

( )

( ( )

)

(

)

( (

)

b r T call b r T put

n d

e

N d

T

n

d

e

N

d

T

σ

− −

∆

=

+

−

∆ =

− −

Gamma

(33)

Options User Guide

Vega

( ) 2 ( ) 2

ln( /

)

( )

2

ln( /

)

(

)

2

b r T call b r T put

T

S X

bT

V

Se

n d

T

S X

bT

V

Se

n

d

T

σ

− −



+



=



−









+



= −

−



−







Theta

(

)





































+

−

+

⋅

−

⋅

=

Θ

−

2

/

ln

2

)

(

)

(

)

(

2 ) (

σ

T

b

X

S

T

d

n

d

N

r

b

e

S

b rT call

(

)





































+

−

⋅

−

⋅

=

Θ

−

2

/

ln

2

)

(

)

(

)

(

2 ) (

σ

T

b

X

S

T

d

n

d

N

r

b

e

S

b rT put

Rho

( ) ( )

( )

0

(

)

0

( )

0

(

)

0

b r T call b r T put rT call rT put

Se

n d T

b

T

Se

n

d T

b

T

STe

N d

b

STe

N

d

b

ρ

σ

ρ

σ

ρ

− − − −

=

≠

−

= −

≠

= −

=

= −

−

=

Implied volatility

To find implied volatility the following equations should be solved for the value of sigma: Call

)

(

) (

d

N

e

S

c

=

⋅

b−r⋅T

⋅

Put

)

(

) (

d

N

e

S

p

=

⋅

b−r⋅T

⋅

−

(34)

Floating Strike Lookback

The Lookback models are used to price European lookback options on stocks without dividends, stocks and stock indexes paying a continuous dividend yield and currency options.

A floating strike lookback call gives the holder of the option the right to buy the underlying security at the lowest price observed, Smin, in the life of the option. Similarly, a floating strike lookback put gives the option holder the right to sell the underlying security at the highest price observed, Smax, in the option’s lifetime.

TheoV

Call













−

⋅

−













−

+

⋅













⋅

+

⋅

−

⋅

=

− − − −

)

(

2

)

(

)

(

1 1 2 min 2 2 min 1 ) ( 2

a

N

e

T

b

a

N

S

b

e

S

a

N

e

S

a

N

e

S

c

bT b rT rT T r b

σ

σ where b – the cost-of-carry;

b = r to price options on stocks;

b = r – q to price options on stocks and stock indexes paying a continuous dividend yield q;

b = r – rf to price currency options (rf – risk-free rate of the foreign currency);

T

b

S

a

σ

⋅

+

=

ln(

/

min

)

(

2

/

2

)

1

T

a

2

=

1

−

σ

Put













⋅

+











 −

⋅













−

⋅

+

−

⋅

−

⋅

=

− − − −

)

(

2

)

(

)

(

₁ ₁ 2 max 2 1 ) ( 2 max 2

b

N

e

T

b

N

S

b

e

S

b

N

e

S

b

N

e

S

p

bT b rT T r b rT

σ

σ where

(35)

Options User Guide

Delta, Gamma, Vega, Theta, Rho

Implied volatility

(36)

Fixed Strike Lookback

In a fixed strike lookback call, the strike is fixed in advance, and at expiry the option pays out the maximum of the difference between the highest observed price, Smax, in the option lifetime and the strike X, and 0. Similarly, a put at expiry pays out the maximum observed price, Smin, and 0.

TheoV

Call when X > Smax













⋅

+











 −

⋅













−

⋅

+

⋅

−

⋅

=

− − − −

)

(

2

)

(

)

(

₁ ₁ 2 2 2 1 ) ( 2

d

N

e

T

b

d

N

X

S

b

e

S

d

N

e

X

d

N

e

S

c

bT b rT rT T r b

σ

σ where b – the cost-of-carry;

b = r to price options on stocks;

b = r – q to price options on stocks and stock indexes paying a continuous dividend yield q;

b = r – rf to price currency options (rf – risk-free rate of the foreign currency);

T

b

X

S

d

σ

⋅

+

=

ln(

/

)

(

2

/

2

)

1

T

d

2

=

1

−

σ

when X <= Smax           ⋅ +       − ⋅       − ⋅ + ⋅ ⋅ − ⋅ ⋅ + − = − − − − − ) ( 2 2 ) ( ) ( ) ( 1 1 2 max 2 2 max 1 ) ( max 2 e N e T b e N S S b e S e N e S e N e S X S e c bT b rT rT T r b rT

σ

σ where

T

b

S

e

=

ln(

/

)

+

(

+

σ

/

2

)

⋅

2 max

(37)

Options User Guide           − ⋅ −      ₋ ₊ ⋅       ⋅ + − ⋅ ⋅ + − ⋅ ⋅ − − = − − − − − 2 ₍ ₎ 2 ) ( ) ( ) ( ₁ ₁ 2 max 2 2 min 1 ) ( min 2 f N e T b f N S S b e S f N e S f N e S S X e p bT b rT rT T r b rT σ σ σ where

T

b

S

f

σ

⋅

+

=

ln(

/

min

)

(

2

/

2

)

1

T

f

2

=

1

−

σ

By defining the following variables all four formulas can be combined into one:

z

- option type adjustment,







−

=

option

put

1

,

option

call

1

z

observed,

extreme

price

−

S







=

option;

put

a

g

calculatin

if

,

option,

call

a

g

calculatin

if

min max,

S

,

limit

price

−

L

S







>

<

=

otherwise;

,

puts,

for

or

calls

for

if

,

X

S

X

S

L

Now the formulas transform into:

(

)





































−

⋅

−

⋅

+

⋅

−

⋅

=













⋅

+























 −

⋅













−

⋅

+

⋅

−

⋅

+

−

⋅

=

− − − − − − −

σ

T

b

d

N(z

S/S

)

d

N(z

e

b

σ

S

)

d

N(z

S

X

S

e

z

)

d

N(z

e

T

σ

b

d

z

N

S

b

σ

e

S

z

)

d

N(z

e

S

z

)

d

N(z

e

S

z

X)

(S

e

z

TheoV

σ b L bT L L rT bT σ b rT rT r)T (b L rT L

2

1 2 1 2 2 1 1 2 max 2 2 1 2 2

Delta, Gamma, Vega, Theta, Rho

Implied volatility

(38)

Interest-Rate Option Models

For further reading, we suggest The Complete Guide to Option Pricing Formulas. ISBN 0071389970.

The Vasicek Model

The Vasicek (1977) model is a yield-based one-factor equilibrium model. The model allows closed-form solutions for European options on zero-coupon bonds.

TheoV

Call

)

(

)

(

h

X

P

T

N

h

p

N

P

L

c

=

⋅

τ

⋅

−

⋅

−

σ

Put

)

(

)

(

h

L

P

N

h

N

P

X

p

=

⋅

T

⋅

−

+

σ

p

−

⋅

τ

⋅

−

where

L – bond principal (i.e. face value),

τ – bond time to maturity,

)

(

T

P

_T

=

_,

)

(

τ

P

=

_,

P(T)-the price at time zero of a zero-coupon bond that pays $1 at time T,

2

ln

1

p T p

P

X

P

L

h

σ

τ





+









⋅

=

d

p

=

σ

⋅

σ

(

) (

e

)

e

d

aT T a

1

( ) 2 − − −

−

=

τ

(39)

Options User Guide

a – the speed of the mean reversion, b – the mean reversion level.

Delta

Since, Delta is the option value sensitivity to small movements in the underlying price then Call

(

)

(

)

( )

T p p p p T

P

h

n

P

L

h

N

X

h

n

X

P

c

⋅

−

⋅

−

⋅

−

⋅

=

∂

=

∆

σ

1

_τ

1

Put

(

)

(

)

( )

T p p p p T

P

h

n

P

L

h

N

X

h

n

X

P

p

⋅

−

+

−

⋅

+

⋅

−

⋅

=

∂

=

∆

σ

1

_τ

1

Gamma

( )

(

)

T p p p T p T

P

h

n

X

h

P

h

n

P

L

P

2

1

2

1

σ

τ



+

⋅

−

⋅











−

⋅

=

∂

∆

∂

=

Γ

Vega

System uses the numerical differentiation to calculate Vega.

Theta

System uses the numerical differentiation to calculate Theta.

Rho

Since the price at time zero of a zero-coupon bond that pays $1 at time t is r t B

e

t

A

t

P

(

)

=

(

)

⋅

− ()⋅ _then T T T

B

P

′

=

−

⋅

τ τ τ

B

P

′

=

−

⋅

p T

B

h

σ

τ

−

=

′

where

)

(

τ

B

=

_,

)

(

T

B

_T

=

Call

(40)

( )

(

)

(

)

p T p T p T T p T T

B

h

n

P

X

h

N

B

P

X

h

N

B

P

L

B

h

n

P

L

r

c

σ

ρ

τ τ τ τ τ

−

⋅

−

⋅

−

⋅

+

⋅

−

⋅

=

∂

=

Put

( )

(

)

(

)

p T p T p T T p T

B

h

n

P

X

h

N

B

P

X

B

h

n

P

L

h

N

B

P

L

r

p

σ

ρ

τ τ τ τ τ

−

⋅

−

⋅

−

+

−

⋅

−

⋅

+

−

⋅

=

∂

=

Implied volatility

(41)

Options User Guide

The Hull and White Model

The Hull and White (1990) model is a yield-based no-arbitrage model. This is extension of the Vasicek model. The model allows closed-form solutions for European options on zero-coupon bonds.

TheoV

Call

)

(

)

(

h

X

P

T

N

h

p

N

P

L

c

=

⋅

τ

⋅

−

⋅

−

σ

Put

)

(

)

(

h

L

P

N

h

N

P

X

p

=

⋅

T

⋅

−

+

σ

p

−

⋅

τ

⋅

−

Where

τ – bond time maturity,

)

(

T

P

_T

=

_,

)

(

τ

P

=

_,

P(T) - the price at time zero of a zero-coupon bond that pays $1 at time T,

2

)

(

)

(

ln

1

p p

P

T

X

P

L

h

τ

σ





+









⋅

=

(

) (

)

a

e

a

aT T a p

2

1

2 ) ( − − −

−

=

σ

τ

σ

a – the speed of the mean reversion.

Unlike Vasicek model, PT and Pτ are input parameters.

Delta

Call

(

)

(

)

( )

T p p p p T

P

h

n

P

L

h

N

X

h

n

X

P

c

⋅

−

⋅

−

⋅

−

⋅

=

∂

=

∆

σ

1

_τ

1

Put

(

)

(

)

( )

T p p p p T

P

h

n

P

L

h

N

X

h

n

X

P

p

⋅

−

+

−

⋅

+

⋅

−

⋅

=

∂

=

∆

σ

1

_τ

1

(42)

Gamma

( )

(

)

T p p p T p T

P

h

n

X

h

P

h

n

P

L

P

2

1

2

1

σ

τ



+

⋅

−

⋅











−

⋅

=

∂

∆

∂

=

Γ

Vega

Because

n

( ) ( )

x

=

n

−

x

( )

(

)

σ

τ

h

n

P

X

h

n

P

L

p

c

Vega

p



+

⋅

_T

⋅

−

_p

⋅











−

⋅

=

∂

=

∂

=

Theta

Call













+













−

⋅

−

⋅

−

⋅

+













+













−

+

⋅

=

∂

=

Θ

p p p T p T p p

r

g

h

n

P

X

h

N

P

r

X

r

g

h

n

P

L

T

c

σ

τ

'

2

1

)

(

)

(

'

2

1

)

(













⋅

=

X

P

L

g

T p τ

σ

ln

1

2

( )

P

T

r

=

−

1

ln

Put         +      ₋ ₊ ⋅ − ⋅ ⋅ +         +      ₋ ₋ ⋅ + − ⋅ ⋅ − + − ⋅ ⋅ ⋅ − = ∂ ∂ = Θ p p p p p T p T r g h n P L r g h n P X h N P r X T p

σ

τ ' 2 1 ) ( ' 2 1 ) ( ) ( where

(

)

(

)

(

)













−

⋅

−

⋅

+

−

=

− − − − − − − aT T a aT aT T a p

e

a

e

a

e

2 ) ( 2 2 ) (

1

2

1

2

1

'

τ τ

σ

(43)

Options User Guide

Rho

Since, the price at time zero of a zero-coupon bond that pays $1 at time t is r t

e

t

P

(

)

=

−⋅ _then T T

T

P

′

=

−

⋅

τ τ

τ

P

′

=

−

⋅

p

T

h

σ

τ

−

=

′

Call

( )

(

)

(

)

p p T p T p T

T

h

n

P

X

h

N

T

P

X

h

N

P

L

T

h

n

P

L

r

c

σ

τ

σ

τ

σ

τ

ρ

=

⋅

_τ

⋅

−

⋅

_τ

⋅

+

⋅

−

⋅

−

⋅

−

∂

=

Put

( )

(

)

(

)

p p T p T p

T

h

n

P

X

h

N

T

P

X

T

h

n

P

L

h

N

P

L

r

p

σ

τ

σ

τ

ρ

=

⋅

_τ

⋅

−

+

⋅

_τ

⋅

−

⋅

−

+

−

⋅

−

⋅

−

∂

=

Implied volatility

(44)

The Ho and Lee Model

Ho and Lee (1986) model is the no-arbitrage model. The model allows closed-form solutions for European options on zero-coupon bonds.

TheoV

Call

)

(

)

(

h

X

P

T

N

h

p

N

P

L

c

=

⋅

τ

⋅

−

⋅

−

σ

Put

)

(

)

(

h

L

P

N

h

N

P

X

p

=

⋅

_T

⋅

−

+

σ

_p

−

⋅

_τ

⋅

−

Where

τ – bond time maturity,

)

(

T

P

_T

=

_,

)

(

τ

P

=

_,

P(T) - the price at time zero of a zero-coupon bond that pays $1 at time T,

2

)

(

)

(

ln

1

p p

P

T

X

P

L

h

τ

σ





+









⋅

=

(

T

)

T

p

=

σ

τ

−

⋅

σ

The distinctions from Vasicek model are - PT and Pτ are input parameters, - σp expression is different.

Delta

(45)

Options User Guide

Gamma

( )

(

)













−

⋅

−

⋅

+













+

⋅

=

∂

∆

∂

=

Γ

p T p p p T p T

P

h

n

X

P

h

n

P

L

P

τ

σ

1

2

Vega

Because

n

( ) ( )

x

=

n

−

x

( )

(

)

σ

τ

h

n

P

X

h

n

P

L

p

c

Vega

p



+

⋅

_T

⋅

−

_p

⋅











−

⋅

=

∂

=

∂

=

Theta

Call













+













−

⋅

−

⋅

−

⋅

+













+













−

+

⋅

=

∂

=

Θ

p p p T p T p p

r

g

h

n

P

X

h

N

P

r

X

r

g

h

n

P

L

T

c

σ

τ

'

2

1

)

(

)

(

'

2

1

)

(













⋅

=

X

P

L

g

T p τ

σ

ln

1

2

( )

P

T

r

=

−

1

ln

Put

�

CQG Integrated Client Options User Guide. November 14, 2012 Version 13.5

CQG Integrated

About this Document

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