Week 5 Flashcards
Requirement for no abritrage with American option
Where Ψ(Snk) is the payoff function or exercise value of the option
Vnk is portfolio at time t with value = to kth value
Adapt binomial pricing algorithm to American options
1) compute the value where you hold as usual
2) compute the value if you exercise at this time
3) set the value to the max of the 2
Consequences of being able to exercise American options early
1) value in any time/state must be >= its European counterpart
2) if in all times/states European option has value greater than intrinsic value, then early exercise is never optimal => American ~ European in this scenario
You can only exchange an expectation with a function if
The function is linear
However, if the function is convex, you can use Jensen’s inequality by replacing the equality with inequality
Show that a European call option is always greater than its intrinsic value
American option Will never be exercised therefore equal to European
Pricing an up and in European call option
We always work backwards in time
Fill out after tree (after barrier crossed)
Fill out before tree (before barrier crossed)
in before tree, for all values above barrier, use after tree values
A stochastic process is a Brownian Motion if
Important properties of Brownian Motion
= ?
0 as Wt is symmetric around 0
Both = 0 as both are symmetrical around 0
How to fill in up and in European call option, key point
Fill in after tree first. Then when filing in before tree (if stock price at that time+price is greater than barrier price, use value from after tree) otherwise price as normal (value is zero if terminal value)
When calculating expectation of Brownian motion Wt =
Sqrt(t) * Z
Where z is a standard gaussian
