STA 2503 / MMF 1928 Project - 1 American Options

Suppose that an asset price process S = (Stk )k∈{0,1,...,N} (with tk = k ∆t and ∆t = TN

, for a fixed N)

are given by the stochastic dynamics

Stk = Stk 1 er ∆t+σ√∆t k ,

where k are iid rv with k ∈ {+1, ;1} and

P(k = ±1) = 12 1 ± (µ  r) ) 12σ2 σ √∆t .

Here, r ≥ 0 and σ > 0 are constants.

Moreover, let B = (Btk )k∈{0,1,...,N} denote the bank account with Bt = e

r t

.

1. Let X(N) denote the random variable X(N)

:= log(ST /S0). Prove that

X(N) d

−→

N→∞

(µ  12σ2) T + σ√

T Z, and Z P∼ N (0, 1)

2. Derive the probabilities Q(k = ±1) and QS(k = ±1), as well as the Q and QS distribution of ST

in the limit as N → ∞.

[Recall that Q refers to the martingale measure induced by using the bank account B as a numeraire,

and QS

refers to the martingale measure induced by using the asset S as a numeraire.]

3. In this part, you will evaluate an American option. Assume that T = 1, S0 = 10, µ = 5%, σ = 20%,

and the risk-free rate r = 2%. Use N = 5000.

(a) Implement a binomial tree to value the American put option with strike K = 10.

i. Plot the exercise boundary as a function of t.

ii. Show how the various plots vary as volatility and risk-free rate change.

(b) Assume you have purchased the American option with the base set of parameters.

i. Simulate 10,000 sample paths of the asset and obtain a kernel density estimate of (i) profit

and loss you will receive, and (ii) the distribution of the time at which you exercise the

option. Explore how the various model parameters effect these distributions.

ii. Suppose that the realized volatility is σ = 10%, 15%, 20%, 25%, 30%, but you were able

to purchase the option with a volatility of σ = 20% and you use the σ = 20% exercise

boundary in your trading strategy. Explore how the distributions of profit and loss and

exercise time vary in this case.