That is
35 S0 2c0 B0 34.65 1 0.01
Then
40 34.65 c0 2.695 2
This is the investor should pay $2.695
for this stock option.
Analysis of the Example
V0 Q VT E B0 BT
.
Theorem 3.1
Under the probability measure Q, an
option's discounted price is its expectation on the expiration date. i.e.
One-Period & Two-State
One-period: assets are traded at t=0 &
t=T only, hence the term one period. Two-state: at t=T the risky asset S has u d ST & ST , two possible values (states): with their probabilities satisfying
Δ- Hedging Definition
Definition:
for a given option V, trade Δ shares of the underlying asset S in the opposite direction, so that the portfolio
V S
Definitions
the probability measure Q defined by d u
qu ProbQ ST ST ud u d qd ProbQ ST ST ud ,
is called by risk-neutral measure. The option price given under the riskneutral measure is called the risk-neutral price.
u d 0 Prop ST ST , Prop ST ST 1 u d Prop ST ST Prop ST ST 1
One-Period & Two-State Model
The model is the simplest model.
ST E BT
Q
1 u d qu ST qd ST B0 1 d u S0 S0u S0 d B0 u d ud B0
Risk-Neutral World
Under the probability measure Q, the

(for strike price K, expired time T)
Analysis of the Model
S t - Stock Price, is a stochastic variable
S0
S S0u
u T
Up, with probability p Down, with probability 1-p
① the idea of hedging: it is possible to
construct an investment portfolio with S and c such that it is risk-free. ② The option price thus determined (c_0=$2.695) has nothing to do with any individual investor's expectation on the future stock price.
Let U be a certain risky asset, and B a risk-free asset, then U t / Bt is called
the discounted price (also known as the relative price) of the risky asset U at time t.
From the discussion above,
V0
Q
1

E (VT ),
Q
where E (VT ) denotes the expectation of the random variable VT under the probability measure Q.
Definition of Discounted Price
Consider a market consisting of two
assets: a risky S and a risk-free B If: risky asset S t and risk free asset Bt
known S0 , B0, when t=0, S u S u , T 0 u d. t=T, 2 possibilities ST : d ST S 0 d , Option Price at t=0?
so that
VT ST 0
Analysis of Δ- Hedging cont.

VT , ST are random variables, when t=T,
V S0u (V0 S0 ),
u T
both of them have 2 possible values
Example cont.1
cT ( ST K ) payoff =
STu $45, cT (45 40) $5
S0 $40
d ST $35, cT (35 40) $0
Consider a portfolio
S 2c
Example cont.2
expected return of a risky asset S at t=T is the same as the return of a risk-free bond. A financial market possessing this property is called a Risk-Neutral World In a risk-neutral world, no investor demands any compensation for risks, and the expected return of any security is the risk-free interest rate.
Chapter 3
Binomial Tree Methods ------ Discrete Models of Option Pricing
An Example
S $45
u T
S0 $40
d ST $35
Question: When t=0, buying a call option of
Define a new Probability Measure
qu ProbQ ST S
u T
Байду номын сангаасud ,
qu qd 1.
d
u qd ProbQ ST S ud
d T
Obviously 0 q , q 1, u d
Solution of Premium
is risk-free.
Analysis of Δ- Hedging
risk free asset BT B0 , 1 rT If Π is risk free, then, on t=T,
T VT ST
is risk free. i.e.
T 0
deposit of B=35/(1+0.01) after 1 month 35 1 VT ( B) (1 12%) 35 VT () 1 0.01 12
By arbitrage-free principle
V0 ( B) V0 ().
Example cont.4
When t=T,
45 2*5 35, if S , VT () $35 35 2*0 35, if S .
has fixed value $35, no matter S is
up or down

Example cont.3
If risk free interest r =12%, a bank

STd S0 d
u T
V ( S 0u K )
V0
where
VTd ( S0 d K )
Vt is a stochastic variable.
Question & Analysis
If known VT ( ST )
at t=T, how to find out V0 when t=0? Assume the risky asset to be a stock. Since the stock option price is a random variable, the seller of the option is faced with a risk in selling it. However, the seller can manage the risk by buying certain shares (denoted asΔ) of the stocks to hedge the risk in the option. This is the idea!