Returning to the earlier example, there are 3 key points:
Why should theoretical price be the market price? This is an important question. If not then there is risk-free money to be made. If C < 50p buy it and hedge to make pro…t. If C > 50p sell it and hedge, make a guaranteed pro…t. Supply and demand should make this price converge to 50p.
How do I know to sell 1/2 the stock for hedging (and not another value)? means the amount of stock sold for hedging purposes. The right choice for hedging means that the value of the portfolio does not depend on the direction of the stock. Earlier we had
1

101 =

99
1 0
=
=
101 99

1
2

Note it is purely a coincident in this example that delta has the same value as the option. Note
=

V+
S+

V
S

=

Range of option payo¤s
Range of stock prices

This model is discrete time, discrete stock. When we go to continuous time continuous stock delta will become @V :
@S
How does this change if interest rates are non-zero? Everything is as before but we now have a discount factor.
Consider the earlier example but with r = 10% over one day, i.e.
1
1
=
1 + rt
1 + 0:1=252

0:9996

Now discount tomorrow’ value to get to todays s V

0:5

100 =
0:5 99
V = 0:51963

0:9996

So the portfolio value today must be the Present Value of the portfolio value tomorrow.
Consider a portfolio asset price Si :

; long an option and short

assets. Vi denotes the option value corresponding to

S+

V+

S0

V0
S
0

V
= V0

S0

At time T there are two possible outcomes:

+

Choosing

so

=

+

= V
= V+

S
S+

gives
=

V+
S+
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V
S

This choice of
i.e.

makes

risk-free, so no-arbitrage suggests that the return on erT =

=

+

= V

S =V

V (S +

=

S )
S+
S V+
S

V S+
S+

=

equal the risk-free rate,

V+ V
S
S+ S
S (V + V )
S

Hence we can write erT (V0

V S+ S V +
S+ S
V+ V
V S+ S V +
S0
=
+ erT
S+ S
S+ S erT S0 S
S + erT S0
=
V + + erT
V
S+ S
S+ S
= qV + + (1 q) V

S0 ) = erT V0

and …nally we can write rT V0 = e

qV + + (1

where we de…ne

q) V

erT S0 S
S+ S

q= with 0 < q < 1:
If compounding is discrete q =

((1+rT )S0
S+ S

S

)

rT ) (qV + + (1

and V0 = (1

q) V ) :

q is a risk-neutral probability which come about from insistance on no arbitrage. It has nothing to do with the real probabilities of S + and S occurring. Pricing a call using the real probability, p; will probably make you a pro…t, but a loss is also possible. Pricing an option using the risk neutral probability, q, you will certainly make neither a pro…t nor a loss.
Another way to understand this model is in terms of replication. The option can be replicated using only cash/bonds and the underlying asset.
At time t = 0 construct a portfolio with

bonds (which grow at the risk-free rate) and
0

=

+

assets

S0

and as before option values corresponding to S are V : At expiry T the two possible outcomes are

+

= erT +
= erT +

S
S+

To replicate the option we must have
=V ; hence V = erT +
V + = erT +

S
S+
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+

=V+ subtracting =
Also

=e

rT

V S+
S+

V +S
S

V+
S+

V
S

: This can be obtained from S + V

S V +:

Then no arbitrage implies that
0

=

+

S0 = V0

which can be rearranged to give
V0 = e

rT

= e

rT

erT S0 S
S+ S qV + + (1

S + erT S0
V++
V
S+ S

!

q) V

and thus we have replicated the option’ value by trading in bonds/cash and the underlying asset. This s approach is equivalent to delta-hedging the option.

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Pricing in the risk neutral world: Returning to our initial example, if the stock is correctly priced today what would be the probabilities of it rising or falling q =?

101

100
1

q =?

99

We calculate these from q 101 + (1

q)

99 = 100

which gives q = 0:5: Now use this to calculate the option q = 0:5

1

V
1

q = 0:5 0

and taking the risk neutral expectation gives V = 0:5.
In the presence of interest rates, perform the same calculation but allow for present valuing when equating values at di¤erent times. Same example as before but r = 10%; so risk neutral probabilities are calculated from 0:9996

(q

101 + (1

q)

99) = 100 q = 0:51984

0:51984)

0 = 0:51984

Now calculate the option value using expectations
0:51984

1 + (1

and present valuing gives
0:9996

0:51984 = 0:51963

and this is the earlier option value.

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