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计算代写-FIN 524B

时间：2021-03-31

FIN 524B Derivatives

Exotic Options I: Barrier Options

Professor Linda M. Schilling

Olin Business School

March 30, 2021

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 1 / 12

Generalization: Binomial Trees

Recall from last lecture

The time zero value of an option in an n-step binomial tree (without

path-dependence) is given as

f0 = e

−rn∆t

n∑

k=0

(

n

k

)

qk(1− q)n−kf(n, k) (1)

where

each time step describes a time period ∆t, and the option has

maturity n∆t = T .

f(n, k) denotes the value of the option after n time steps and

exactly k ≤ n up movements (final nodes).

q = e

r∆t−d

u−d is the risk-neutral probability of an up-jump, where r is

the annual risk-free rate, u is the factor by which the underlying

stock jumps up and d is the factor by which the stock jumps down(

n

k

)

= n!k!(n−k)!

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 2 / 12

Binomial Coefficients

Binomial coefficients form the Pascal’s triangle

⇒ Trick to count paths for large trees

From first to second layer: 1-step tree

From first to third layer: 2-step tree

From first to fourth layer: 3-step tree

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 3 / 12

Application: Large Binomial Tree

Exercise Consider a time horizon of one year T = 1, split into 4 time

periods n = 4 each with length 3 months ∆t = 3/12.

Assume the current stock price in the market equals S0 = 50. The

risk-free rate equals r = 0.05. The stock increases or drops by 10% in

each period, i.e. jumps up or down by factors u = 1.1, d = 0.9 at every

time step. The stock pays no dividends. You are interested in a

European Call option on the stock with strike K = 50 and maturity

T = 1 year

(a) Calculate the risk-neutral probability of an up-jump of the stock.

b) Calculate the values of the underlying stock and then the values of

the European Call with strike K = 50 in all final nodes.

c) Price a European Call option on the stock with strike K = 50 and

maturity T = 1 year in the given binomial tree.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 4 / 12

Application: Large Binomial Tree

Solution

(a) Calculate the risk-neutral probability of an up-jump of the stock.

For each jump the stock makes, the risk-neutral probability of an

up-jump equals

q =

e0.05×3/12 − 0.9

1.1− 0.9 = 0.5629

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 5 / 12

Application: Large Binomial Tree

Solution continued

b) Calculate the value of a European Call option on the stock with

strike K = 50 and maturity T = 1 year in all final nodes.

The stock jumps n = 4 times. Therefore Suuuu = S0u

4 = 73.205,

Suuud = S0u

3d = 59.895, Suudd = S0u

2d2 = 49.005,

Suddd = S0ud

3 = 40.095, and Sdddd = 32.805

The value of the Call option is given by fET = max(ST −K, 0).

Denote by f(4, k) the value of the Call option after 4 jumps when

exactly k up jumps have occured. [e.g. since there is no

path-dependence, f(4, 1) = fuddd = fdudd = fdddu, similarly

f(4, 2) = fuudd = fudud = fduud, i.e. for the final option value only

the number of up jumps matters, not the sequence in which they

occur.]

The option values in the final nodes are therefore

f(4, 4) = fuuuu = 23.205, f(4, 3) = fuuud = 9.895,

f(4, 2) = fuudd = 0, f(4, 1) = fuddd = 0, and f(4, 0) = fdddd = 0.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 6 / 12

Application: Large Binomial Tree

Solution continued

c) Price the European Call on the stock with strike K = 50 and

maturity T = 1 year in the given binomial tree.

Using the binomial formula, we calculate the fair time zero price of the

option as

fC0 = e

−0.05×4×3/12

4∑

k=0

(

4

k

)

qk(1− q)4−kf(4, k)

= e−0.05×4×3/12

(

q4f(4, 4) + 4q3(1− q)f(4, 3) + 0 + 0 + 0)

= e−0.05×4×3/12

(

q4 23.205 + 4q3(1− q) 9.895)

= 5.1511

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 7 / 12

Barrier Options

Definition: Down-and-in Call (Knock-in Call): A Down-and-in

Call option with a barrier B, strike K and maturity T on an

underlying stock St transforms to a European Call option with strike

K and maturity T if the underlying stock crosses or hits the barrier at

least once until maturity T . If the barrier is not hit until maturity,

then the option payoff is zero.

Definition: Down-and-out Call (Knock-out Call): The

down-and-out Call option with a barrier B, strike K and maturity T

on an underlying stock St pays off like a European Call option if the

underlying does not cross or hit the barrier at any point in time until

the expiration of the option. If the underlying hits or crosses the

barrier, the option value becomes zero.

Barrier options are ’path-dependent’. The precise value

evolution of the underlying stock matters.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 8 / 12

Barrier Options

The value of the Down-and-out Call option (DOC) with strike K,

barrier B, maturity T = n∆t in an n-step Binomial tree is given as

fDOC0 = e

−rn∆t

n∑

k=0

(

# paths to node (n,k) that

do not hit/cross barrier B

)

×qk(1−q)n−kf(n, k)

(2)

where

here f(n, k) = fC(n, k) = max(S0u

kdn−k −K, 0) since we are

considering a Call option

and q = e

rδt−d

u−d is the risk-neutral probability of an up-jump of the

underlying stock.

For a Down-and-out Put option: In (2) you would only

exchange the final option values to

fP (n, k) = max(K − S0ukdn−k, 0). As with the DO-Call option,

you will count paths that do not hit the barrier and lead to nodes

at which the option is in the money.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 9 / 12

Barrier Options

Down-and-out Call (DOC)

S0

f

u S0

d S0

4

4

fuuuu=f(4,4)

fdddd=f(4,0)

u d S03

fuuud=f(4,3)

u d S02

fuudd=f(4,2)

2

ud S0

fuddd=f(4,1)

3

B

every path to node (4,1)

and (4,0) must cross B

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 10 / 12

Barrier Options

The value of the Down-and-in Call option (DIC) with strike K,

barrier B, maturity T = n∆t in an n-step Binomial tree is given as

fDIC0 = e

−rn∆t

n∑

k=0

(

# paths to node (n,k) that

do hit/cross barrier B

)

×qk(1−q)n−kf(n, k)

(3)

where

f(n, k) = fC(n, k) = max(S0u

kdn−k −K, 0)

For a Down-and-in Put option: In (3) you would only exchange

the final option values to fP (n, k) = max(K − S0ukdn−k, 0). As

with the DI-Call option, you will count paths that do hit the

barrier and lead to nodes at which the option is in the money.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 11 / 12

Barrier Options

Exercise: Consider a time horizon of one year T = 1, split into 4 time

periods n = 4 each with length 3 months ∆t = 3/12.

Assume the current stock price in the market equals S0 = 50. The

risk-free rate equals r = 0.05. The stock jumps up or down by factors

u = 1.1, d = 0.9 at each time step. The stock pays no dividends.

Consider a Down-and-out Call with a strike K = 40 and barrier

B = 48 with maturity T = 1 in this 4-step binomial tree.

(a) Price the Down-and-out Call (DOC).

(b) Price the equivalent Down-and-in Call (DIC)

(c) Compare to the price of the European Call option with the same

strike.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 12 / 12

学霸联盟

Exotic Options I: Barrier Options

Professor Linda M. Schilling

Olin Business School

March 30, 2021

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 1 / 12

Generalization: Binomial Trees

Recall from last lecture

The time zero value of an option in an n-step binomial tree (without

path-dependence) is given as

f0 = e

−rn∆t

n∑

k=0

(

n

k

)

qk(1− q)n−kf(n, k) (1)

where

each time step describes a time period ∆t, and the option has

maturity n∆t = T .

f(n, k) denotes the value of the option after n time steps and

exactly k ≤ n up movements (final nodes).

q = e

r∆t−d

u−d is the risk-neutral probability of an up-jump, where r is

the annual risk-free rate, u is the factor by which the underlying

stock jumps up and d is the factor by which the stock jumps down(

n

k

)

= n!k!(n−k)!

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 2 / 12

Binomial Coefficients

Binomial coefficients form the Pascal’s triangle

⇒ Trick to count paths for large trees

From first to second layer: 1-step tree

From first to third layer: 2-step tree

From first to fourth layer: 3-step tree

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 3 / 12

Application: Large Binomial Tree

Exercise Consider a time horizon of one year T = 1, split into 4 time

periods n = 4 each with length 3 months ∆t = 3/12.

Assume the current stock price in the market equals S0 = 50. The

risk-free rate equals r = 0.05. The stock increases or drops by 10% in

each period, i.e. jumps up or down by factors u = 1.1, d = 0.9 at every

time step. The stock pays no dividends. You are interested in a

European Call option on the stock with strike K = 50 and maturity

T = 1 year

(a) Calculate the risk-neutral probability of an up-jump of the stock.

b) Calculate the values of the underlying stock and then the values of

the European Call with strike K = 50 in all final nodes.

c) Price a European Call option on the stock with strike K = 50 and

maturity T = 1 year in the given binomial tree.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 4 / 12

Application: Large Binomial Tree

Solution

(a) Calculate the risk-neutral probability of an up-jump of the stock.

For each jump the stock makes, the risk-neutral probability of an

up-jump equals

q =

e0.05×3/12 − 0.9

1.1− 0.9 = 0.5629

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 5 / 12

Application: Large Binomial Tree

Solution continued

b) Calculate the value of a European Call option on the stock with

strike K = 50 and maturity T = 1 year in all final nodes.

The stock jumps n = 4 times. Therefore Suuuu = S0u

4 = 73.205,

Suuud = S0u

3d = 59.895, Suudd = S0u

2d2 = 49.005,

Suddd = S0ud

3 = 40.095, and Sdddd = 32.805

The value of the Call option is given by fET = max(ST −K, 0).

Denote by f(4, k) the value of the Call option after 4 jumps when

exactly k up jumps have occured. [e.g. since there is no

path-dependence, f(4, 1) = fuddd = fdudd = fdddu, similarly

f(4, 2) = fuudd = fudud = fduud, i.e. for the final option value only

the number of up jumps matters, not the sequence in which they

occur.]

The option values in the final nodes are therefore

f(4, 4) = fuuuu = 23.205, f(4, 3) = fuuud = 9.895,

f(4, 2) = fuudd = 0, f(4, 1) = fuddd = 0, and f(4, 0) = fdddd = 0.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 6 / 12

Application: Large Binomial Tree

Solution continued

c) Price the European Call on the stock with strike K = 50 and

maturity T = 1 year in the given binomial tree.

Using the binomial formula, we calculate the fair time zero price of the

option as

fC0 = e

−0.05×4×3/12

4∑

k=0

(

4

k

)

qk(1− q)4−kf(4, k)

= e−0.05×4×3/12

(

q4f(4, 4) + 4q3(1− q)f(4, 3) + 0 + 0 + 0)

= e−0.05×4×3/12

(

q4 23.205 + 4q3(1− q) 9.895)

= 5.1511

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 7 / 12

Barrier Options

Definition: Down-and-in Call (Knock-in Call): A Down-and-in

Call option with a barrier B, strike K and maturity T on an

underlying stock St transforms to a European Call option with strike

K and maturity T if the underlying stock crosses or hits the barrier at

least once until maturity T . If the barrier is not hit until maturity,

then the option payoff is zero.

Definition: Down-and-out Call (Knock-out Call): The

down-and-out Call option with a barrier B, strike K and maturity T

on an underlying stock St pays off like a European Call option if the

underlying does not cross or hit the barrier at any point in time until

the expiration of the option. If the underlying hits or crosses the

barrier, the option value becomes zero.

Barrier options are ’path-dependent’. The precise value

evolution of the underlying stock matters.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 8 / 12

Barrier Options

The value of the Down-and-out Call option (DOC) with strike K,

barrier B, maturity T = n∆t in an n-step Binomial tree is given as

fDOC0 = e

−rn∆t

n∑

k=0

(

# paths to node (n,k) that

do not hit/cross barrier B

)

×qk(1−q)n−kf(n, k)

(2)

where

here f(n, k) = fC(n, k) = max(S0u

kdn−k −K, 0) since we are

considering a Call option

and q = e

rδt−d

u−d is the risk-neutral probability of an up-jump of the

underlying stock.

For a Down-and-out Put option: In (2) you would only

exchange the final option values to

fP (n, k) = max(K − S0ukdn−k, 0). As with the DO-Call option,

you will count paths that do not hit the barrier and lead to nodes

at which the option is in the money.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 9 / 12

Barrier Options

Down-and-out Call (DOC)

S0

f

u S0

d S0

4

4

fuuuu=f(4,4)

fdddd=f(4,0)

u d S03

fuuud=f(4,3)

u d S02

fuudd=f(4,2)

2

ud S0

fuddd=f(4,1)

3

B

every path to node (4,1)

and (4,0) must cross B

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 10 / 12

Barrier Options

The value of the Down-and-in Call option (DIC) with strike K,

barrier B, maturity T = n∆t in an n-step Binomial tree is given as

fDIC0 = e

−rn∆t

n∑

k=0

(

# paths to node (n,k) that

do hit/cross barrier B

)

×qk(1−q)n−kf(n, k)

(3)

where

f(n, k) = fC(n, k) = max(S0u

kdn−k −K, 0)

For a Down-and-in Put option: In (3) you would only exchange

the final option values to fP (n, k) = max(K − S0ukdn−k, 0). As

with the DI-Call option, you will count paths that do hit the

barrier and lead to nodes at which the option is in the money.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 11 / 12

Barrier Options

Exercise: Consider a time horizon of one year T = 1, split into 4 time

periods n = 4 each with length 3 months ∆t = 3/12.

Assume the current stock price in the market equals S0 = 50. The

risk-free rate equals r = 0.05. The stock jumps up or down by factors

u = 1.1, d = 0.9 at each time step. The stock pays no dividends.

Consider a Down-and-out Call with a strike K = 40 and barrier

B = 48 with maturity T = 1 in this 4-step binomial tree.

(a) Price the Down-and-out Call (DOC).

(b) Price the equivalent Down-and-in Call (DIC)

(c) Compare to the price of the European Call option with the same

strike.

Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 12 / 12

学霸联盟