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The Greeks: Derivatives of Option PricesAn Undergraduate Introduction to Financial Mathematics

J. Robert Buchanan

2010

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Sensitivity Analysis

In calculus a derivative gives you a measure of the rate ofchange of a dependent variable as an independentvariable is changed.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Sensitivity Analysis

In calculus a derivative gives you a measure of the rate ofchange of a dependent variable as an independentvariable is changed.

In the world a finance an option is an example of aderivative, any financial instrument whose value is derivedfrom that of an underlying security.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Sensitivity Analysis

In calculus a derivative gives you a measure of the rate ofchange of a dependent variable as an independentvariable is changed.

In the world a finance an option is an example of aderivative, any financial instrument whose value is derivedfrom that of an underlying security.

Here we will calculate the partial derivatives (in the senseof calculus) of option value formulas. These partialderivatives will allow us to determine how sensitive thevalues of options are to changes in independent variablesand parameters.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Black-Scholes Option Pricing Formulas

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t

C(S, t) = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)

P(S, t) = Ke−r(T−t)φ(σ√

T − t − w) − Sφ(−w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Cumulative Distribution Function

The function φ(w) is the cumulative distribution function

φ(w) =1√2π

∫ w

−∞e−x2/2 dx

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Cumulative Distribution Function

The function φ(w) is the cumulative distribution function

φ(w) =1√2π

∫ w

−∞e−x2/2 dx

which by the Fundamental Theorem of Calculus has derivative

φ′(w) =1√2π

e−w2/2.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Partial Derivatives of w

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Partial Derivatives of w

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t∂w∂t

=1

2σ√

T − t

(

ln(S/K )

T − t− r − σ2

2

)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Partial Derivatives of w

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t∂w∂t

=1

2σ√

T − t

(

ln(S/K )

T − t− r − σ2

2

)

∂w∂S

=1

σS√

T − t

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Partial Derivatives of w

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t∂w∂t

=1

2σ√

T − t

(

ln(S/K )

T − t− r − σ2

2

)

∂w∂S

=1

σS√

T − t∂w∂r

=

√T − tσ

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Partial Derivatives of w

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t∂w∂t

=1

2σ√

T − t

(

ln(S/K )

T − t− r − σ2

2

)

∂w∂S

=1

σS√

T − t∂w∂r

=

√T − tσ

∂w∂σ

=√

T − t − wσ

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Theta Θ (1 of 3)

Theta Θ is the partial derivative with respect to time t .

Time is the only independent variable we are certain willchange before expiry. It is also the only deterministicindependent variable.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Theta Θ (2 of 3)

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂t

= Sφ′(w)∂w∂t

− rKe−r(T−t)φ(w − σ√

T − t)

− Ke−r(T−t)φ′(w − σ√

T − t)[

∂w∂t

+σ

2√

T − t

]

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Theta Θ (2 of 3)

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂t

= Sφ′(w)∂w∂t

− rKe−r(T−t)φ(w − σ√

T − t)

− Ke−r(T−t)φ′(w − σ√

T − t)[

∂w∂t

+σ

2√

T − t

]

=Se−w2/2 − Ke−r(T−t)−(w−σ

√T−t)2/2

2σ√

2π(T − t)

(

ln(S/K )

T − t− r − σ2/2

)

− Ke−r(T−t)

(

rφ(w − σ√

T − t) +σe−(w−σ

√T−t)2/2

2√

2π(T − t)

)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Theta Θ (3 of 3)

For a European Put

∂P∂t

=Se−w2/2

2σ√

2π(T − t)

(

ln(S/K )

T − t− r − σ2/2

)

+ Kre−r(T−t)φ(σ√

T − t − w)

− Ke−r(T−t)−(w−σ√

T−t)2/2

2σ√

2π(T − t)

(

ln(S/K )

T − t− r + σ2/2

)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (1 of 2)

∆ was involved in the derivation of the Black-Scholes PDE andis defined to be the partial derivative with respect to the price ofthe security.

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂S

= φ(w) +(

Sφ′(w) − Ke−r(T−t)φ′(w − σ√

T − t)) ∂w

∂S

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (1 of 2)

∆ was involved in the derivation of the Black-Scholes PDE andis defined to be the partial derivative with respect to the price ofthe security.

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂S

= φ(w) +(

Sφ′(w) − Ke−r(T−t)φ′(w − σ√

T − t)) ∂w

∂S= φ(w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (1 of 2)

∆ was involved in the derivation of the Black-Scholes PDE andis defined to be the partial derivative with respect to the price ofthe security.

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂S

= φ(w) +(

Sφ′(w) − Ke−r(T−t)φ′(w − σ√

T − t)) ∂w

∂S= φ(w)

Question: what is the range of Delta for a European calloption?

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (2 of 2)

Recall the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (2 of 2)

Recall the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂S(P + S) =

∂

∂S(C + Ke−r(T−t))

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (2 of 2)

Recall the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂S(P + S) =

∂

∂S(C + Ke−r(T−t))

∂P∂S

+ 1 =∂C∂S

∂P∂S

= φ(w) − 1

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Delta ∆ (2 of 2)

Recall the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂S(P + S) =

∂

∂S(C + Ke−r(T−t))

∂P∂S

+ 1 =∂C∂S

∂P∂S

= φ(w) − 1

Question: what is the range of Delta for a European putoption?

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (1 of 2)

Example

The current price of a stock is $77 and its volatility is 35% peryear. The risk-free interest rate is 3.25% per year. A portfolio isconstructed consisting of one six-month European call optionwith a strike price of $80 and the cash obtained from shorting ∆shares of the stock. The portfolio’s value is non-random. Whatis ∆?

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

The assumption the portfolio’s value is non-random is theassumption

(∆)S − C =

(

∂C∂S

)

S − C = 0

made in deriving the Black-Scholes equation.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

The assumption the portfolio’s value is non-random is theassumption

(∆)S − C =

(

∂C∂S

)

S − C = 0

made in deriving the Black-Scholes equation.

S = 77 σ = 0.35 T =612

r = 0.0325 K = 80 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t∂C∂S

= ∆ = φ(w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

The assumption the portfolio’s value is non-random is theassumption

(∆)S − C =

(

∂C∂S

)

S − C = 0

made in deriving the Black-Scholes equation.

S = 77 σ = 0.35 T =612

r = 0.0325 K = 80 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.0349666

∂C∂S

= ∆ = φ(w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

The assumption the portfolio’s value is non-random is theassumption

(∆)S − C =

(

∂C∂S

)

S − C = 0

made in deriving the Black-Scholes equation.

S = 77 σ = 0.35 T =612

r = 0.0325 K = 80 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.0349666

∂C∂S

= ∆ = φ(w) ≈ 0.513947

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Gamma Γ

Gamma is the second partial derivative with respect to S, thus

Γ =∂

∂Sφ(w)

= φ′(w)∂w∂S

∂2P∂S2 =

∂2C∂S2 =

e−w2/2

σS√

2π(T − t).

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Gamma vs. S(0)

50 100 150 200SH0L

0.005

0.010

0.015

G

K = 100, σ = 0.25 T = 1 r = 0.0325

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Gamma vs. K and T

60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

K

T

S(0) = 100, σ = 0.25 r = 0.0325

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Relationships Between ∆, Θ, and Γ

Remember the Black-Scholes PDE:

rF = Ft + rSFS +12

σ2S2FSS

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Relationships Between ∆, Θ, and Γ

Remember the Black-Scholes PDE:

rF = Ft + rSFS +12

σ2S2FSS

Since Ft = Θ, ∆ = FS , and FSS = Γ then the Black-Scholesequation can be thought of as

rF = Θ + rS∆ +12

σ2S2Γ.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Vega V (1 of 2)

Vega is the partial derivative with respect to volatility σ.

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂σ

= Sφ′(w)∂w∂σ

− Ke−r(T−t)φ′(w − σ√

T − t)(

∂w∂σ

−√

T − t)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Vega V (1 of 2)

Vega is the partial derivative with respect to volatility σ.

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂σ

= Sφ′(w)∂w∂σ

− Ke−r(T−t)φ′(w − σ√

T − t)(

∂w∂σ

−√

T − t)

=S√

T − t√2π

e−w2/2

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Vega V (2 of 2)

According to the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Vega V (2 of 2)

According to the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂σ(P + S) =

∂

∂σ(C + Ke−r(T−t))

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Vega V (2 of 2)

According to the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂σ(P + S) =

∂

∂σ(C + Ke−r(T−t))

∂P∂σ

=∂C∂σ

∂P∂σ

=S√

T − t√2π

e−w2/2

Remark: vega is identical for puts and calls.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (1 of 2)

Example

Consider a three-month European put option on a stock whosecurrent value is $200 and whose volatility is 30%. The optionhas a strike price of $195 and the risk-free interest rate is6.25%.

1 Find the vega of the option.2 If the volatility of the stock increases to 31%, approximate

the change in the value of the put.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t

V =S√

T − t√2π

e−w2/2

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.347952

V =S√

T − t√2π

e−w2/2

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.347952

V =S√

T − t√2π

e−w2/2 ≈ 37.5509

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.347952

V =S√

T − t√2π

e−w2/2 ≈ 37.5509

Using the linear approximation,

dP = Vdσ = (37.5509)(0.01) = 0.375509.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Rho ρ (1 of 2)

Rho is the partial derivative with respect to the risk-free interestrate r .

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂r

= Sφ′(w)∂w∂r

+ K (T − t)e−r(T−t)φ(w − σ√

T − t)

− Ke−r(T−t)φ′(w − σ√

T − t)∂w∂r

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Rho ρ (1 of 2)

Rho is the partial derivative with respect to the risk-free interestrate r .

C = Sφ(w) − Ke−r(T−t)φ(w − σ√

T − t)∂C∂r

= Sφ′(w)∂w∂r

+ K (T − t)e−r(T−t)φ(w − σ√

T − t)

− Ke−r(T−t)φ′(w − σ√

T − t)∂w∂r

= K (T − t)e−r(T−t)φ(w − σ√

T − t)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Rho ρ (2 of 2)

Starting with the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Rho ρ (2 of 2)

Starting with the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂r(P + S) =

∂

∂r(C + Ke−r(T−t))

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Rho ρ (2 of 2)

Starting with the Put-Call Parity formula:

P + S = C + Ke−r(T−t)

∂

∂r(P + S) =

∂

∂r(C + Ke−r(T−t))

∂P∂r

=∂C∂r

− K (T − t)e−r(T−t)

∂P∂r

= −K (T − t)e−r(T−t)φ(σ√

T − t − w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (1 of 2)

Example

Consider a three-month European put option on a stock whosecurrent value is $200 and whose volatility is 30%. The optionhas a strike price of $195 and the risk-free interest rate is6.25%.

1 Find the rho of the option.2 If the interest rate increases to 7.00%, approximate the

change in the value of the put.

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t

ρ = −K (T − t)e−r(T−t)φ(σ√

T − t − w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.347952

ρ = −K (T − t)e−r(T−t)φ(σ√

T − t − w)

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.347952

ρ = −K (T − t)e−r(T−t)φ(σ√

T − t − w) ≈ −20.2315

J. Robert Buchanan The Greeks: Derivatives of Option Prices

Example (2 of 2)

S = 200 σ = 0.30 T = 3/12

K = 195 r = 0.0625 t = 0

Using these values

w =ln(S/K ) + (r + σ2/2)(T − t)

σ√

T − t≈ 0.347952

ρ = −K (T − t)e−r(T−t)φ(σ√

T − t − w) ≈ −20.2315

Using the linear approximation,

dP = ρdr = (−20.2315)(0.0075) = −0.151737.

J. Robert Buchanan The Greeks: Derivatives of Option Prices