LECTURE 8 Flashcards
(23 cards)
binomial option pricing model =
a model for pricing options based on the assumption that a stock’s return can only undertake 2 values within a period
What does this lecture try to determine?
we previously talked about ask and bid prices for options - are these fair prices?
Replicating portfolio
stock + risk free bond that has the same payoff in each period as an option written on the same stock.
BY LOOP:
PV of replicating portfolio = PV of option
Do we need probabilities of states for binomial?
NO
delta (number of shares) formula
Delta = Cu - Cd / Su - Sd
B (value of bonds) formula
B = (Cd - Sd*delta) / 1 + Rf
Price of call:
C = delta*S + B
What does delta represent and what are range of values?
Delta < 1 implies Cu - Cd < Su - Sd
hedge ratio
In which period can we set value of option = intrinsic value?
In the final period only - when option expires
Dynamic trading strategy:
delta and B must be adjusted at the end of every period as prices change in order to replicate the option’s payoff
Limitation of binomial
In reality, > 2 possible prices in each period
What does B-S do to periods? What’s the distribution?
Shorten them to they’re infinitesimally small such that within each period only 2 prices still.
Binomial –> normal distribution
In B-S PV(K) =
present value of a risk-free zero-coupon bond that pays K on maturity
N(d) =
cumulative normal distribution - th probability a normally distributed variable is < d.
5 B-S inputs
- Stock price.
- Strike price
- Exercise date T
- Volatile of stock returns
- risk free rate
What kind of options is BS for?
european - can use for American too and find reaonsable estimate.
What do we use to get from Call BS to Put BS?
use put call parity: P = C - S + PV(K)
How do we adapt BS when we have divided paying stocks?
Replace S with Sx = S - PV(div)
or Sx = S / (1 + q) if stock pays compounded dividend yield until expiration.
Risk neutral assumption. Why do we make this assumption?
Assume all investors are risk neutral
Allows price estimate of rWACC which must = risk-free rate.
What are risk-neutral probabilities?
If we do not know rWACC, adjust expected cash flows for risk i.e. apply probabilities then discount @ risk-free rate.
NOT actual probabilities - they show how actual p would have to be adjusted to keep the stock price the same in a risk-neutral world.
3 other names for risk-neutral probabilities?
state contingent prices, state prices, martingale prices
How do we find risk neutral probabilities?
Set stock’s expected return / current price equal to risk-free rate
