FIN 524B Derivatives
Exotic Options I: Barrier Options
Professor Linda M. Schilling
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  