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3.1 Option pricing for dummies
[ SLIDE
forecasting 
no arbitrage 
uncertainty 
eliminate risk 
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Because investors have different opinions about the value of every
security that is traded in open markets, the spot price does not
evolve smoothly with time and
cannot be predicted with any certainty.
Nevertheless, sect.2.1.1 showed that
possible realizations can be simulated
by adding price increments that are typical of volatilities from the
past.
This chapter examines increasingly sophisticated models to forecast
market prices, using them first to estimate the average terminal
payoff from financial derivatives, and later to calculate what is a
fair price for an option before it expires.
For a qualitative understanding, you can think of an option as a form
of insurance covering a financial risk without having any obligation:
clearly, an insurance gets increasingly valuable when the market becomes
risky and volatile... but how large should this risk premium be?
To show here with a simple example how the price of options can be
calculated, imagine a vanilla put expiring in 3 months (
years)
with a strike price
, for an underlying currently valued at
EUR and risk free interest rate of 3% (
).
In a first study, consider the case without drift nor volatility
(
) for which the forecast price simply remains constant at
EUR: the terminal payoff from the put option (2.1.3#eq.1)
can then easily be calculated three months into the future as suggested
by (3.1#fig.1) and yields
EUR:
Figure 3.1#fig.1:
Sketch showing an evolution of the underlying price in absence of drift
and volatility: the forecast remains constant
EUR, which makes
it easy to calculate the terminal payoff from a put option with a strike
of
EUR and 3 months to expiry.

A general noarbitrage argument
states that without taking any risk, a security has to grow exactly at
the risk free interest (spot) rate:
if the security grew more quickly, investors could make a risk less
profit by borrowing money at the spot rate, buy large amounts of that
security and sell it later for a higher price; on the contrary, if
the security grew more slowly, investors could make a risk less profit
by selling short large amounts of that security and reinvest the
proceeds for the higher yield of the spot rate.
This shows that a risk less investment always grow exactly at the
risk free interest rate.
In absence of volatility, the terminal payoff from the option can
therefore be discounted back in time using the risk free rate during
the entire lifetime of the option, which yields the present value of
the option
.
In a second study, repeat the calculation using what appears to be a
simplistic model of the uncertainty, where the forecast price can
take only two distinct values.^{44}Starting from the initial price
EUR, the sketch in (3.1#fig.2)
shows how, after three months, the price of underlying can either move
up to
or down to
EUR. The corresponding terminal option
payoff are easily calculated and yield
and
.
Figure 3.1#fig.2:
Sketch showing two possible realizations of the bond market, with an
initial price at
that can either to move up
or down
by a normal increment
before the put option expires
after
year with a strike at
.

Dealing with an uncertain outcome, the general strategy is to
eliminate the risk by settingup a perfectly hedged portfolio
that combines an (a priori unknown) amount
of the underlying
security with a (negatively correlated) derivative.
The initial value of the portfolio
evolves
to new values until the expiry date, depending whether the underlying
moves up or down.
Choosing exactly the right hedging factor
, it is however
possible to force both outcomes to be equal, which in effect makes the
investment risk less

(3.1#eq.1) 
Indeed, the forecasted value of the portfolio in both cases becomes

(3.1#eq.2) 
Repeating the general noarbitrage argument,
this a portfolio earns exactly the risk free interest rate and can be
discounted back during the entire lifetime of the option
.
This has to be equal to the initial value of the portfolio:

(3.1#eq.3) 
The example illustrates how the price of an option can be calculated
before the expiry date, assuming a normal distribution of the price
increments
that are typical for bonds.
The study also shows that the price can be dramatically different
when accounting for the uncertain evolution of the underlying.
The next section takes these ideas further and shows how the price
levels can be chosen to match the volatility observed in real markets.
SYLLABUS Previous: 3 FORECASTING WITH UNCERTAINTY
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Next: 3.2 Simple valuation model
