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FIN 524B Derivatives
创建日期：2024-04-07 16:14:15
标签： FIN 524B
Derivatives

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FIN 524B Derivatives


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.
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