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Derivation and Comparative Statics of the

Black-Scholes Call and Put Option Pricing Formulas James R. Garven∗

First Version: October 2004Second Version: February 2006Current Version: March 2009

ABSTRACT

This paper provides an alternative derivation of the Black-Scholes call and put option pricing formulas using an integration rather than differential equations approach. The economic andmathematical structure of these formulas is discussed, and comparative statics are derived.

The purposes of this paper are: (1) to provide a concise overview of the economic

and mathematical assumptions needed to derive the Black-Scholes (1973) call and put optionpricing formulas; (2) to provide an alternative derivation and comparative statics analysis of these formulas.

This paper is organized as follows. In the next section, we discuss the economic andmathematical structure of the Black-Scholes model. In the third section, we provide analternative derivation of the Black-Scholes option pricing formulas based upon anintegration rather than differential equations approach. The advantage of the integration

approach lies in its simplicity, as only basic integral calculus is required. The integrationapproach also makes the link between the economics and mathematics of the call and putoption formulas more transparent for most readers. This paper concludes with comparativestatics analyses of these formulas.

ECONOMIC AND M ATHEMATICAL S TRUCTURE OF THE BLACK -SCHOLES MODEL

The most important economic insight contained in Black and Scholes’ seminal paperis the notion that an investor can create a riskless portfolio by dynamically hedging a long (short) position in the underlying asset with a short (long) position in a European call option.In order to prevent arbitrage, the expected return on such a portfolio must be the risklessrate of interest. Cox and Ross (1976) note that since the creation of such a hedge portfolio



a call option written against that asset will trade for the same price in a risk neutral economy as it would in a risk averse or risk loving economy. Consequently, options can be priced as if

they are traded in risk neutral economies.

Black and Scholes assume that stock prices change continuously according to theGeometric Brownian Motion equation; i.e.,

.dS Sdt Sdz σ = + (1)

Equation (1) is a stochastic differential equation because it contains the Wiener process dz =

dt ε , where ε is a standard normal random variable. dS corresponds to the stock price

change per dt time unit, S is the current stock price, μ is the expected return, and

σ represents volatility. The Geometric Brownian Motion equation is commonly referred toas an exponential stochastic differential equation because its solution is an exponential

function; specifically, t S = μ σ σ − +2( /2) ,t t z Se where t S represents the stock price t periods from

now. Since t z is normally distributed, it follows that t S is lognormally distributed.

't S s mean is ,t

t S Se μ μ = and its variance is

22 2 2 ( 1).t

t t S S e e μ σ σ = −

Ito’s Lemma provides a rule for finding the differential of a function of one or more variables in which at least one of the variables follows a stochastic differential equation. Atany given point in time, the value of the call option ( C ) depends upon the value of theunderlying asset; i.e., C = C ( S , t ).1 Since the value of the call option is a function of the stock price which evolves over time according to equation (1), Ito’s Lemma justifies the use of a Taylor-series-like expansion for the differential dC :

22

2

1

2

C C C dC dt dS dS

t S S

∂ ∂ ∂= + +

∂ ∂ ∂. (2)

Since 2 2 2 ,dS S dt σ = substituting for 2dS in equation (2) yields equation (3):

dC =2

2 2

2

1

2

C C C dt dS S dt

t S S σ

∂ ∂ ∂+ +∂ ∂ ∂

. (3)

Suppose we wish to construct a hedge portfolio consisting of one long call option

position worth C ( S , t ) and a short position in some quantity Δt of the underlying asset worth



dV = dC - Δt d S =22 2

21 .2

t C C C S dt dS t S S

σ ⎛ ⎞∂ ∂ ∂⎛ ⎞+ + − Δ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠(4)

Note that there are stochastic as well as deterministic terms on the right hand side of equation (4). The deterministic terms are represented by the first product, whereas thestochastic terms are represented by the second product. However, by setting t Δ equal to

,C

S

∂

∂ the second product equals zero and we have:

dV =2

2 2

2

1.

2

C C S dt

t S σ

⎛ ⎞∂ ∂+⎜ ⎟∂ ∂⎝ ⎠

(5)

Since this is a perfectly hedged portfolio, it has no risk. In order to prevent arbitrage, the

hedge portfolio must earn the riskless rate of interest r ; i.e.,

dV =rVdt . (6)

We will assume that t

C

S

∂Δ =

∂, so V = C ( S , t ) -

C S

S

∂

∂. Substituting this into the right hand

side of equation (6) and equating the result with the right hand side of equation (5), weobtain:

22 2

2

1.

2

C C C r C S dt S dt

S t S σ

⎛ ⎞∂ ∂ ∂⎛ ⎞− = +⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ (7)

Dividing both sides of equation (7) by dt and rearranging results in the Black-Scholes partialdifferential equation:

22 2

2

1

2

C C C S rS rC

t S S σ

∂ ∂ ∂+ + −∂ ∂ ∂

= 0. (8)

Equation (8) shows that dynamic hedging causes the valuation relationship between a calloption and its underlying asset to be risk neutral. In other words, a relative pricing



price at expiration.

Black and Scholes transform their version of equation (8) into the heat transferequation of physics,2 which allows them to (quite literally) employ a “textbook” solutionprocedure taken from Churchill’s (1963) classic introduction to Fourier series and theirapplications to boundary value problems in partial differential equations of engineering andphysics. This solution procedure results in the following equation for the value of aEuropean call option:

1 2( ) ( ),

rt

C SN d Xe N d

−= − (9)

where2

1

ln( / ) ( .5 )S X r t d

t

σ

σ

+ += ;

2 1 ;d d t σ = − 2σ = variance of underlying asset’s rate of return; and

N ( z ) = standard normal distribution function evaluated at z .If it is not possible to form a riskless hedge portfolio, then other more restrictive

assumptions are needed in order to obtain a risk neutral valuation relationship between anoption and its underlying asset. Rubinstein (1976) has shown that the Black-Scholes optionpricing formulas also obtain under the following set of assumptions:

1) The conditions for aggregation are met so that securities are priced as if all investorshave the same characteristics as a representative investor;

2) The utility function of the representative investor exhibits constant relative risk aversion (CRRA);

3) The return on the underlying asset and the return on aggregate wealth are bivariatelognormally distributed;

4)

The return on the underlying asset follows a stationary random walk through time;and5) The riskless rate of interest is constant through time.

Collectively, assumptions 1, 2, and 3 constitute sufficient conditions for valuing the option as if both it and the underlying asset are traded in a risk neutral economy. Assumptions 4 and 5



variance of the underlying asset will be proportional to time.3

DERIVATION OF THE BLACK -SCHOLES OPTION PRICING FORMULAS

Rubinstein (1987) notes that the Black-Scholes call option pricing formula can bederived by integration if one assumes that the probability distribution for the underlying asset is lognormal. Fortunately, as noted earlier, this assumption is implied by the GeometricBrownian Motion equation given by equation (1). Therefore, our next task is to derive theBlack-Scholes call option pricing formula by integration. Once we obtain the Black-Scholescall option pricing formula, the put option pricing formula follows directly from the put-callparity theorem.

The value today ( C ) of a European call option that pays C t = Max [ S t – X , 0] at date t is given by the following equation:

( ) ( )[ ,0] ,t t C V C V Max S X = = − (10)

where V represents the valuation operator. Having established in equation (8) that a risk neutral valuation relationship exists between the call option and its underlying asset, itfollows that we can price the option as if investors are risk neutral. Therefore, the valuationoperator V determines the current price of a call option by discounting the risk neutral

expected value of the option’s payoff at expiration ( ( )ˆt E C ) at the riskless rate of interest, as

indicated in equation (11):

( ) ˆˆ ( ) ( ) ,rt rt t t t t

X C e E C e S X h S dS

∞− −= = −∫ (11)

where ˆ( )t h S represents the risk neutral lognormal density function of t S .

We begin our analysis by computing the expected value of :t C

( ) [ ( ,0)] ( ) ( ) ,t t t t t X

E C E Max S X S X h S dS ∞

= − = −∫ (12)

where ( )t h S represents 'st S lognormal density function.4 To evaluate this integral, we



rewrite it as the difference between two integrals:

[ ]( ) ( ) ( ) ( ) 1 ( ) ,rt t t t t t t X t

X X E C S h S dS X h S dS E S Xe H X

∞ ∞−= − = − −∫ ∫ (13)

where

( )t E S = the partial expected value of t S , truncated from below at X ;5

( )H X = the probability that t S X ≤ .

By rewriting the terminal stock price t S as the product of the current stock price S and the t -

period lognormally distributed price ratio / ,t S S ( )/ ,t t S S S S = equation (13) can be

rewritten as

/ /

/

( ) ( / ) ( / ) / ( / ) /

( / ) [1 ( / )],

t t t t t t X S X S

X S t

E C S S S g S S dS S X g S S dS S

SE S S X G X S

∞ ∞= −

= − −

∫ ∫ (14)

where( / )t g S S = lognormal density function of S t /S ;6

/ ( / ) X S t E S S = the partial expected value of S t /S , truncated from below at X /S ; and

( / )G X S = the probability that S t /S ≤ X /S .

Next, we evaluate the right-hand side of equation (14) by considering its two integralsseparately. First, consider the integral that corresponds to the partial expected value of the

terminal stock price, / ( / ) X S t SE S S . Let / ,kt

t S S e = where k is the rate of return on the

underlying asset per unit of time. Since /t S S is lognormally distributed, ( )ln /t S S kt = is

normally distributed with density f ( kt ), mean kt and variance 2k t σ .7 Furthermore, g ( S t /S )

= ( S/S t ) f ( kt ).8 Also, since / ,kt

t S S e = / .kt

t dS S e tdk= Consequently, ( / ) ( / ) /t t t S S g S S dS S

= ( ) .kt e f kt tdk Thus,

rt . The ˆ( )t h S density function is “risk neutral” in the sense that its location parameter t μ is replaced by 2( .5 )r t σ − .

5 The partial expected value of X for values of X ranging from y to z is ( ) ( ) .z

z yE X Xf X dX = ∫ If we replace the y and z



2 2{.5[( ) / ]}

/ 2ln( / ) ln( / )

1( / ) ( ) .

2

kt kt kt t t

X S t X S X S

SE S S S e f kt tdk S e e tdkt

μ σ

πσ

∞ ∞ − −= =∫ ∫ (15)

Next, we simplify equation (15)’s integrand by adding the terms in the two exponents,

multiplying and dividing the resulting expression by 2.5 t e σ − , and rearranging:

2 2 2 2 2 2

2 2 2 4 4 2

2 2 4 2 2

2 2 2 2

{.5[( ) / ]} {.5 [( 2 2 )/ ]}

{.5 [( 2 2 )/ ]}

{.5 [(( ) 2 )/ ]}

( .5 ) {.5[( ( ) ) / ]}.

kt kt t t t k k k

t k k k

t k

t kt t t

e e e

e

e

e e

μ σ μ μ σ σ

μ μ σ σ σ σ

μ σ σ μσ σ

μ σ μ σ σ

− − − − + −

− − + − + −

− − − − −

+ − − +

=

=

=

=

(16)

In equation (16), the term2( .5 )t e μ σ + = E( t S /S ), the mean of the t -period lognormally

distributed price ratio t S /S . Therefore, we can rewrite equation (15) with this term

appearing outside the integral:

2 2 2{.5[( ( ) ) / ]}/ 2 ln( / )

1( / ) ( / )

2

kt t t

X S t t X S

SE S S SE S S e tdkt

μ σ σ

πσ

∞ − − += ∫

2 2 2{.5[( ( ) ) / ]}

2 ln( / )

1( ) .

2

kt t t

t X S

E S e tdkt

μ σ σ

πσ

∞ − − += ∫ (17)

Next, we let y = 2[ ( ) ]/kt t t σ σ − + , which implies that kt = 2( )t t y μ σ σ + + and

tdk = .t dy σ With this change in variables, [ln( X /S) - 2( ) ]/t t μ σ σ + = 1δ − becomes the

lower limit of integration. Thus equation (17) may be rewritten as:

2

1

1

1

.5/

1

( / ) ( ) [ / 2 ]

( ) ( ) ( ) ( )

( ) ( ),

y

X S t t

t t

t

SE S S E S e dy

E S n y dy E S n y dy

E S N

δ

δ

δ

π

δ

∞ −

−

∞

− −∞

=

= =

=

∫

∫ ∫ (18)

where n ( y ) is the standard normal density function evaluated at y and 1( ) N δ is the standard

l di ib i f i l d δ



2t σ . Furthermore, this change in variable implies that ( ) f kt tdk must be substituted in place

of ( / ) /t t g S S dS S . Therefore,

2 2

/ ln( / )

{.5[( ) / ]}

2 ln( / )

( / ) / ( )

1.

2

t t X S X S

kt t t

X S

X g S S dS S X f kt tdk

X e tdkt

μ σ

πσ

∞ ∞

∞ − −

=

=

∫ ∫

∫ (19)

Next, we let z = [ ]/kt t t μ σ − , which implies that kt t tz μ σ = + and tdk = t dz σ .

With this change in variables, the lower limit of integration becomes[ln( / ) ]/ X S t t μ σ − =

-( 1δ - t σ ) = 2 .δ − Substituting these results into the right-hand side of equation (19) yields

equation (20):

2

2

2

.5

/

2

( / ) / [ / 2 ]

( )

( ),

z

t t X S X g S S dS S X e dz

X n z dz

XN

δ

δ

π

δ

∞ ∞ −

−

−∞

=

=

=

∫ ∫ ∫ (20)

where n ( z ) is the standard normal density function evaluated at z and 2( ) N δ is the standard

normal distribution function evaluated at z = 2 .δ Substituting the right-hand sides of

equations (18) and (20) into the right-hand side of equation (14), we obtain

1 1( ) ( ) ( ) ( ).t t E C E S N XN t δ δ σ = − − (21)

Since equation (11) defines the price of a call option as the discounted, risk neutral

expected value of the option’s payoff at expiration; i.e., ( )ˆ ,rt t C e E C −= our next task is to

compute the risk neutral expected value of the stock price on the expiration date ( ˆ ( )t E S )and the risk neutral value for 1δ (which we will refer to as 1d ). In footnote (3), we noted

that 2( .5 )t rt μ σ + = in a risk neutral economy. Since2( .5 )( ) t

t E S Se μ σ += , this implies that

ˆ ( ) .rt

t E S Se = In the expression for 1 ,δ the term 2( )t σ + appears. In a risk neutral2 2 2



Now that we have the Black-Scholes pricing formula for a European call option, theput option pricing formula follows directly from the put-call parity theorem. Suppose two

portfolios exist, one consisting of a European call option and a riskless discount bond, andthe other consisting of a European put option and a share of stock against which bothoptions are written. The call and put both have exercise price X and t periods to expiration,and the riskless bond pays off X dollars at date t . Then these portfolios’ date t payoffs areidentical, since the payoff on the first portfolio is ( ,0 )t Max S X − + X = ( , )t Max S X and the

payoff on the second portfolio is ( ,0)t Max X S − + S t = ( , ).t Max S X Consequently, the

current value of these portfolios must also be the same; otherwise there would be a riskless

arbitrage opportunity. Therefore, the price of a European put option, P , can be determinedas follows:

[ ] [ ]1 2

2 1

2 1

( ) ( )

1 ( ) 1 ( )

( ) ( ).

rt

rt rt

rt

rt

P C Xe S

SN d Xe N d Xe S

Xe N d S N d

Xe N d SN d

−

− −

−

−

= + −

= − + −

= − − −

= − − −

(23)

COMPARATIVE S TATICS OF THE BLACK -SCHOLES C ALL AND PUT EQUATIONS

As indicated by equations (22) and (23), the prices of European call and put options

depend upon five parameters: S , X , t , r , and σ . Next, we provide comparative statics

analysis of call and put option prices.

Relationship between the call and put option prices and the price of the underlying asset

Consider the relationship between the price of the call option and the price of the

underlying asset, ∂C/∂S :

∂C/∂S = N ( 1d ) +S ( ∂ N ( 1d )/∂ 1d )( ∂ 1d /∂S ) - X rt e − ( ∂ N ( 2d )/∂ 2d )( ∂ 2d /∂S )

= N ( 1d ) + Sn ( 1d )( ∂ 1d /∂S ) - X rt e − n ( 2d )( ∂ 2d /∂S ) (24)

Since 2d = 1d - t σ , ∂ 2d /∂S = ∂ 1d /∂S and n ( 2d ) = n ( 1d - t σ ). Substituting these



Recall that 1d = [ln( S / X ) + 2( .5 ) ]/ .r t t σ σ + Solving for S , we find that S =2

1 ( .5 )d t r t

Xe

σ σ − +

. Substituting this expression for S into equation (25)’s bracketed term yields

∂C/∂S = N ( 1d ) + ( ∂ 1d /∂S )1

2π [ X

21.5d e −

21 ( .5 )d t r t e σ σ − + - X

.5 21.5( )rt d t e e σ − − − ])]

= N ( 1d ) + ( ∂ 1d /∂S )1

2π [ X (

2 21 1( .5 ) .5r t d t d e σ σ − + + − -

2 21 1( .5 ) .5r t d t d e σ σ − + + − )] (26)

Since the bracketed term on the right hand side of equation (26) equals zero, ∂C/∂S = N ( 1d ) > 0; i.e., the price of a call option is positively related to the price of its underlying

asset. Furthermore, from our earlier analysis of the Black-Scholes partial differentialequation, we know that a dynamically hedged portfolio consisting of one long call positionper /t C S Δ = ∂ ∂ short shares of the underlying asset is riskless (cf. equation (5)). Since

∂C/∂S = N ( 1d ), N ( 1d ) is commonly referred to as the option’s delta , and the dynamic

hedging strategy described here is often referred to as delta hedging .

Intuitively, since the put option provides its holder with the right to sell rather thanbuy, one would expect a negative relationship between the price of a put option and theprice of its underlying asset.

∂P/∂S = - N (- 1d ) + X rt e − ( ∂ N (- 2d )/∂ 2d )( ∂ 2d /∂S ) - S ( ∂ N (- 1d )/∂ 1d )( ∂ 1d /∂S )

= - N (- 1d ) - X rt e − n (- 2d )( ∂ 1d /∂S ) + Sn (- 1d )( ∂ 1d /∂S )

= - N (- 1d ) + ( ∂ 1d /∂S )[ Sn (- 1d ) - X rt e − n ( t σ - 1d )] (27)

Rearranging the bracketed term in equation (27), we find that Sn (- 1d ) - X rt e − n ( t σ - 1d ) =

Sn ( 1d ) - X rt e − n ( 1d - t σ ). From our analysis of equations (25)-(26), we know that this term

is equal to zero. Therefore, ∂P/∂S = - N (- 1d ) < 0. Thus our intuition is confirmed; i.e., the

price of a put option is negatively related to the price of its underlying asset.

Relationship between call and put option prices and the exercise price



= - rt e − N ( 2d ) +Sn ( 1d )( ∂ 1d /∂ X ) - X rt e − n ( 2d )( ∂ 2d /∂ X ) (28)

Substituting 2d = 1d - t σ , ∂ 2d /∂ X = ∂ 1d /∂ X and n ( 2d ) = n ( 1d - t σ ) into (28) gives us

∂C/∂ X = - rt e − N ( 2d ) + ( ∂ 1d /∂ X )[ Sn ( 1d ) - X rt e − n ( 1d - t σ )]. (29)

Since [ Sn ( 1d ) - X rt e − n ( 1d - t σ )] = 0, it follows that ∂C/∂ X = - rt e − N ( d 2 ) < 0; i.e., the price

of a call option is negatively related to its exercise price.

Intuition suggests that since the price of a call option is negatively related to itsexercise price, one would expect to find that a positive relationship exists between the priceof a put option and its exercise price, which we confirm in equation (30):

∂P/∂ X = rt e − N (- 2d ) + X rt e − ( ∂ N (- 2d )/∂ 2d )( ∂ 2d /∂ X ) - S ( ∂ N (- 1d )/∂ 1d )( ∂ 1d /∂ X )

= rt e − N (- 2d ) - X rt e − n (- 2d )( ∂ 1d /∂ X ) + Sn (- 1d )( ∂ 1d /∂ X )

= rt e − N (- 2d ) + ( ∂ 1d /∂ X )[ Sn (- 1d ) - X rt e − n ( t σ - 1d )] (30)

Since [ Sn (- 1d ) - X rt e − n ( t σ - 1d )] = 0, it follows that ∂P/∂ X = rt e − N (- 2d ) > 0; i.e., the price

of a put option is positively related to its exercise price.

Relationship between call and put option prices and the interest rate

Consider next the relationship between the price of the call option and the rate of

interest, ∂C/∂r :

∂C/∂r = tX rt

e −

N ( d 2 ) +S ( ∂ N ( 1d )/∂ 1d )( ∂ 1d /∂r ) - X rt

e −

( ∂ N ( 2d )/∂ 2d )( ∂ 2d /∂r )

= tX rt e − N ( 2d ) + Sn ( 1d )( ∂ 1d /∂r ) - X rt e − n ( 2d )( ∂ 1d /∂r )

= tX rt e − N( 2d ) + (∂ 1d /∂r) [Sn( 1d ) - X rt e − n( 1d - tσ )]



Intuition suggests that since the price of a call option is positively related to the rate of interest, one would expect to find that a negative relationship exists between the price of a

put option and the rate of interest, which we confirm in equation (32):

∂P/∂r = -tX rt e − N (- 2d ) + X rt e − ( ∂ N (- 2d )/∂ 2d )( ∂ 2d /∂r ) - S ( ∂ N (- 1d )/∂ 1d )( ∂ 1d /∂r )

= -tX rt e − N (- 2d ) - X rt e − n (- 2d )( ∂ 1d /∂r ) + Sn (- 1d )( ∂ 1d /∂r )

= -tX rt

e −

N (- 2d ) + ( ∂ 1d /∂r )[ Sn (- 1d ) - X rt

e −

n ( t σ - 1d )]

= -tX rt e − N (- 2d ). (32)

Since the right-hand side of Equation (32) is negative, this implies that the price of a putoption is negatively related to the rate of interest.

Relationship between call and put option prices and the time to expiration

Consider next the relationship between the price of the call option and the time to

expiration, ∂C/∂t :

∂C/∂t = rX rt e − N ( 2d ) + S ( ∂ N ( 1d )/∂ 1d )( ∂ 1d /∂t ) - X rt e − ( ∂ N ( 2d )/∂ 2d )( ∂ 2d /∂t ). (33)

Substituting X =2

1 ( .5 )d t r t Se σ σ − + + into equation (33) and simplifying further, we obtain

∂C/∂t = rX rt e − N ( 2d ) + S [ n ( 1d )( ∂ 1d /∂t ) -2

1 ( .5 )d t r t rt e σ σ − + + − n ( 1d - t σ )( ∂ 2d /∂t )]

=2 2 21 1 1- .5 ( .5 ) .5( )

2 1 2

1( ) [( / ) ( / ) ]

2

rt d d t r t rt d t rXe N d S d t e d t e σ σ σ

π

− − + + − − −+ ∂ ∂ − ∂ ∂

=2 2 2 21 1 1 1- .5 .5 .5 .5

2 1 2

1( ) [( / ) ( / ) ]

2

rt d d t d t rt rt t t d rXe N d S d t e d t e σ σ σ σ

π

− − + + − + − −+ ∂ ∂ − ∂ ∂

= - ( ) ( )[( / ) ( / )]rt X N d S d d t d t+ ∂ ∂ ∂ ∂



call options are worth more than otherwise identical shorter lived European call options.

Next, we consider the relationship between the price of a put option and the time toexpiration:

∂P/∂t = -rX rt e − N (- 2d ) + X rt e − ( ∂ N (- 2d )/∂ 2d )( ∂ 2d /∂t ) - S ( ∂ N (- 1d )/∂ 1d )( ∂ 1d /∂t ) (35)

Substituting X =2

1 ( .5 )d t r t Se σ σ − + + into equation (35) and simplifying further, we obtain

∂P/∂t = -2

1 ( .5 )-2 1 2 1 1( ) ( )( / ) ( )( / )d t r t rt rt rXe N d Se n t d d t Sn d d t σ σ

σ − + + −− + − ∂ ∂ − − ∂ ∂

= -2 2

1 1( .5 ) .5( )-2 2 1 1( ) ( / ) ( )( / )

2

d t r t rt t d rt S rXe N d e d t Sn d d t σ σ σ

π

− + + − − −− + ∂ ∂ − − ∂ ∂

= -

-

2 1 2 1( ) ( )[( / ) ( / )]

rt

rXe N d Sn d d t d t − + ∂ ∂ − ∂ ∂

= -2 1

.5( ) + ( ) 0

?rt krXe N d Sn d

t

σ − − <> (36)

It is not possible to sign ∂P/∂t , because the first term is negative and the second term ispositive. Therefore, the relationship between put value and time to expiration is

ambiguous.9

Relationship between call and put option prices and volatility of the underlying asset

Finally, consider the relationship between the price of the call option and the volatility of the underlying asset, /C σ ∂ ∂ :

9 The “theta” for an option represents the rate at which its value changes with the passage of time, given a fixed time to

expiration. Therefore, the first term on the left-hand side of equation (8) corresponds to the theta for a call option

which expires in T periods, where T represents a fixed unit of time. Thus theta for a call option, ,C Θ is equal to



-1 1 2 2

1 2

( ) ( )/ - .rt N d d N d d

C S Xe d d

σ σ σ

∂ ∂ ∂ ∂∂ ∂ =

∂ ∂ ∂ ∂(37)

Substituting 22

2

( )( )

N d n d

d

∂=

∂= 1( )kn d t σ − , 2 1d d

t σ σ

∂ ∂= −

∂ ∂, and X =

21 ( .5 )d t r t Se σ σ − + + into

equation (37) and simplifying further, we obtain

∂C/∂σ = S [ n ( 1d )( ∂ 1d /∂σ ) -2

1 ( .5 )d t r t rt e σ σ − + + − n ( 1d - t σ )( ∂ 2d /∂σ )]

=( )

22

112

1

.5.5( .5 ) -1 1

2 2

d t d d t r t rt e d e d

S Se e t

σ

σ σ

σ σ π π

− −−− + +∂ ∂⎛ ⎞− −⎜ ⎟

∂ ∂⎝ ⎠(38)

Since( )

22 21 1 1

.5 .5 .5d t rt t rt d t d e e σ σ σ − + + − − − −= and

21.5

1( )2

d e n d

π

−

= , equation (38) can be rewritten as

21.5

1 11

1 11 1 1

/ ( )2

( ) ( ) ( )

d d e d C Sn d S t

d d Sn d Sn d Sn d t

σ σ σ π

σ σ

−∂ ∂⎛ ⎞∂ ∂ = − −⎜ ⎟∂ ∂⎝ ⎠∂ ∂

= − +∂ ∂

1( ) .Sn d t =

(39)

As equation (39) indicates, the price of a call option is positively related to the volatility of the underlying asset. This comparative static relationship is commonly referred to as theoption’s vega.

Next, we consider the relationship between the price of a put option and the volatility of the underlying asset, /P σ ∂ ∂ :

- 2 2 1 1

2 1

( ) ( )/ rt N d d N d d

P Xe S d d

σ σ σ

∂ − ∂ ∂ − ∂∂ ∂ = −

∂ ∂ ∂ ∂

2 1t d d∂ ∂



∂P/∂σ =( )

221 1.5 .5

1 11[ ( ) ( )]

2

d t rt t rt t d d e d

S n d t

σ σ σ

σ σ π

− + + − − −∂ ∂

− − −

∂ ∂

1 11( )[ ( )]

d d Sn d t

σ σ

∂ ∂= − − −

∂ ∂

1( ) .Sn d t = − (41)

As equation (41) indicates, the price of a put option is positively related to the volatility of the underlying asset. Table 1 summarizes the comparative statics of the Black-Scholes call

and put option pricing formulas.



R EFERENCES

Black, F., and M. Scholes (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81:637-659.

Brennan M. J., 1979, “The Pricing of Contingent Claims in Discrete Time Models”, Journal of Finance , Vol. 34, No. 1 (March), pp. 53-68.

Churchill, R. V. (1963), Fourier Series and Boundary Value Problems , 2nd edition, New York:

McGraw-Hill.

Cox, J. C., and S. A. Ross (1976), “The Valuation of Options for Alternative StochasticProcesses,” Journal of Financial Economics 3:145-166.

Rubinstein, M. (1976), “The Valuation of Uncertain Income Streams and the Pricing of Options,” Bell Journal of Economics 7:407-425.

Rubinstein, M. (1987), “Derivative Assets Analysis,” The Journal of Economic Perspectives,1(Autumn): 73-93.

Stapleton, R. C. and M. G. Subrahmanyam, 1984, “The Valuation of Multivariate ContingentClaims in Discrete Time Models”, Journal of Finance , Vol. 39, No. 1 (March), pp. 207-228.

Wilmott, P. (2001), Paul Wilmott Introduces Quantitative Finance , Chichester, England: John Wiley & Sons Ltd.

Winkler, R. L., G. M. Roodman and R. R. Britney (1972), “The Determination of PartialMoments,” Management Science 19(3):290-296.



17

Table 1. Comparative Statics of the Black-Scholes Model

Derivative Call Option Put Option

C

S

∂

∂and

P

S

∂

∂

( delta )

C

S

∂

∂= N ( 1d ) > 0

P

S

∂

∂= - N ( - 1d ) < 0

C

X

∂

∂

andP

X

∂

∂

C

X

∂

∂

= - -rt e N ( 2d ) < 0 P

X

∂

∂

= -rt e N ( - 2d ) > 0

C

r

∂

∂and

P

r

∂

∂

( rho )

C

r

∂

∂= tX -rt e N ( 2d ) > 0

P

r

∂

∂= -tX -rt e N ( - 2d ) < 0

C

t

∂

∂and

P

t

∂

∂

C

t

∂

∂= -

2 1

.5( ) ( )rt rXe N d Sn d

t

σ + > 0 P

t

∂

∂= -

2 1

.5( ) + ( ) 0

?rt krXe N d Sn d

t

σ − − <>

theta -C

t

∂∂

= -2 1

.5( ) ( )rt rXe N d Sn d

t

σ − − < 0 -

P

t

∂

∂= -

2 1

.5( ) ( ) 0

?rt krXe N d Sn d

t

σ − − <>

C

σ

∂

∂and

P

σ

∂

∂

( vega )

1( ) 0C

Sn d t σ

∂= >

∂ 1( ) 0

P Sn d t

σ

∂= − >

∂

garven options

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