留学生代考 L4 Optional Reading: Why are American Options RARELY early exercised? Econ – cscodehelp代写

L4 Optional Reading: Why are American Options RARELY early exercised? Economics of Finance
School of Economics, UNSW

Binomial Model

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Binomial states: s ∈ {u, d}, where u : stock price goes up, d : stock price goes down, e.g., u = 1.3, d = 0.6.
Atomic security prices: {p, q}, where p → u, q → d.
Interest rate: r
Arbitrage free implies:
p+q=1; 1+r
p=1+r−d; (1+r)(u−d)
u−1−r (1+r)(u−d)

Option: strike price X, stock price S;
Intrinsic value (IV): the value of exercising immediately; Time value (TV): the value of holding the option until next period.
American option holder solves max{IV,TV}, i.e.,
􏰑 IV > TV : early excercise; IV 􏰒TV : hold

Call Option: IV = S − X Case I: dS 􏰓 X :
TV = p(uS−X)+q(dS−X) = (up+dq)S−(p+q)X
= S− X >IV; 1+r
CaseII:dS (1−p−q)X>0;
Sum up, TV > IV for either Case I or Case II. Early exercise is never preferable for call option.

1+r Early exercise is preferable;
Put option: IV = X − S
CaseI:uS􏰒X,orS􏰒Xu :
TV = p(X−uS)+q(X−dS)
= (up+dq)S−(p+q)X = X −SX,orS>Xu :TV=q(X−dS).
TV−IV = q(X−dS)−(X−S) = (1−dq)S−(1−q)X
= upS−(1−q)X; LetS ̃=(1−q)X =􏰂1+r(u−d)􏰃X >X,then
up 1+r−d u u 􏰑􏰓IV, ifS􏰓S ̃
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