Derivativos - Problemas 9 e 10

Vinicius Perez Marques

May 2021

1 Introduction
A useful and very popular technique for pricing an option involves constructing a binomial tree.
This is a diagram representing different possible paths that might be followed by the stock price
over the life of an option. The underlying assumption is that the stock price follows a random walk.
In each time step, it has a certain probability of moving up by a certain percentage amount and a
certain probability of moving down by a certain percentage amount.

2 A One-Step Binomial Model and a No-Arbitrage Theo-

rem
To price an option using s binomial tree, the only assumption needed is that arbitrage oppor-
tunities do not exist. We set up a portfolio of the stock and the option in such a way that there is
no uncertainty about the value of the portfolio at the end of a given period. We then argue that,
because the portfolio has no risk, the return it earns must equal the risk-free interest rate. This
enables us to work out the cost of setting up the portfolio and therefore the option’s price.
Consider a portfolio consisting of a long position in ∆ shares of the stock and a short position in
one call option. We calculate the value of ∆ that makes the portfolio riskless; the portfolio is riskless
if the value of ∆ is chosen so that the final value of the portfolio is the same for both alternatives.
Riskless portfolios must, in the absence of arbitrage opportunities, earn the risk-free rate of
interest.

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 We can generalize the no-arbitrage argument just presented by considering a stock whose price
is S0 and an option on the stock (or any derivative dependent on the stock) whose current price is
f . We suppose that the option lasts for time T and that during the life of the option the stock price
can either move up from S0 to a new level, S0 u, where u > 1, or down from S0 to a new level, S0 d,
where d < 1. The percentage increase in the stock price when there is an up movement is u − 1; the
percentage decrease when there is a down movement is 1 − d. If the stock price moves up to S0 u,
we suppose that the payoff from the option is fu ; if the stock price moves down to S0 d, we suppose
the payoff from the option is fd .
As before, we imagine a portfolio consisting of a long position in ∆ shares and a short position
in one option. We calculate the value of ∆ that makes the portfolio riskless. If there is an up
movement in the stock price, the value of the portfolio at the end of the life of the option is

S0 u − fu

If there is a down movement in the stock price, the value becomes

S 0 d − fd

The two are equal when

S0 u − fu = S0 d − fd

Or
fu − fd
∆= (1)
S0 u − S0 d
In this case, the portfolio is riskless and, for there to be no arbitrage opportunities, it must earn
the risk-free interest rate.
If we denote the risk-free interest rate by r, the present value of the portfolio is

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 (S0 u∆ − fu )e−rT

The cost of setting up the portfolio is

S0 ∆ − f

It follows that

S0 ∆ − f = (S0 u∆ − fu )e−rt

or

f = S0 ∆(1 − ue−rT ) + fu e−rT

Substituting from equation (1) for ∆, we obtain

 
fu −fd
S0 = S0 u−S0 d (1 − ue−rT ) + fu e−rT

or
fu (1−de−rT )+fd (ue−rT −1)
f= u−d

or
f = e−rT [pfu + (1 − p)fd ] (2)

where
erT − d
p= (3)
u−d
Equations (2) and (3) enable an option to be priced when stock price movements are given by a
one-step binomial tree. The only assumption needed for the equation is that there are no arbitrage
opportunities in the market.

2.1 Irrelevance of the Stock’s Expected Return

The option pricing formula in equation (2) does not involve the probabilities of the stock price
moving up or down. It is natural to assume that, as the probability of an upward movement in the
stock price increases, the value of a call option on the stock increases and the value of a put option
on the stock decreases. This is not the case.
The key reason is that we are not valuing the option in absolute terms. We are calculating
its value in terms of the price of the underlying stock. The probabilities of future up or down
movements are already incorporated into the stock price: we do not need to take them into account
again when valuing the option in terms of the stock price.

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3 Risk Neutral Valuation
Risk-neutral valuation states that, when valuing a derivative, we can make the assumption that
investors are risk-neutral. This assumption means investors do not increase the expected return
they require from an investment to compensate for increased risk.
The world we live in is, of course, not a risk-neutral world. The higher the risks investors take,
the higher the expected returns they require. However, it turns out that assuming a risk-neutral
world gives us the right option price for the world we live in, as well as for a risk-neutral world.
Should not a person’s risk preferences affect how they are priced? The answer is that, when we
are pricing an option in terms of the price of the underlying stock, risk preferences are unimportant.
As investors become more riskaverse, stock prices decline, but the formulas relating option prices
to stock prices remain the same.
A risk-neutral world has two features that simplify the pricing of derivatives:

1. The expected return on a stock (or any other investment) is the risk-free rate.

2. The discount rate used for the expected payoff on an option (or any other instrument) is the
risk-free rate.

Returning to equation (2), the parameter p should be interpreted as the probability of an up

movement in a risk-neutral world, so that 1 − p is the probability of a down movement in this world.
We assume u > erT , so that 0 < p < 1. The expression

pfu + (1 − p)fd

is the expected future payoff from the option in a risk-neutral world and equation (2) states that
the value of the option today is its expected future payoff in a risk-neutral world discounted at the
risk-free rate. This is an application of risk-neutral valuation.
To prove the validity of our interpretation of p, we note that, when p is the probability of an up
movement, the expected stock price E(ST ) at time T is given by

E(ST ) = pS0 u + (1 − p)S0 d → E(ST ) = pS0 (u − d) + S0 d

Substituting from equation (3) for p gives

E(ST ) = S0 erT (4)

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This shows that the stock price grows, on average, at the risk-free rate when p is the probability of
an up movement. In other words, the stock price behaves exactly as we would expect it to behave
in a risk-neutral world when p is the probability of an up movement.

4 Two-Step Binomial Trees

The objective of the analysis is to calculate the option price at the initial node of the tree. This
can be done by repeatedly applying the principles established earlier.
The stock price is initially S0 . During each time step, it either moves up to u times its initial
value or moves down to d times its initial value. The notation for the value of the option is shown
on the tree. (For example, after two up movements the value of the option is fuu .) We suppose
that the risk-free interest rate is r and the length of the time step is ∆t years.
Because the length of a time step is now ∆t rather than T , equations (2) and (3) become

f = e−r∆t [pfu + (1 − p)fd ] (5)

er∆t − d
p= (6)
u−d
Repeated application of equation (5) gives

fu = e−r∆t [pfuu + (1 − p)fud ] (7)

fd = e−r∆t [pfud + (1 − p)fdd ] (8)

f = e−r∆t [pfu + (1 − p)fd ] (9)

Substituting from equations (7) and (8) into (9), we get

f = e−2r∆t [p2 fuu + 2(1 − p)fud + (1 − p)2 fdd ] (10)

The variables p2 , 2p(1 − p) and (1 − p)2 are the probabilities that the upper, middle, and lower
final nodes will be reached. The option price is equal to its expected payoff in a risk-neutral world
discounted at the risk-free interest rate.

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 As we add more steps to the binomial tree, the risk-neutral valuation principle continues to
hold. The option price is always equal to its expected payoff in a riskneutral world discounted at
the risk-free interest rate.

5 Delta
The delta (∆) of a stock option is the ratio of the change in the price of the stock option to the
change in the price of the underlying stock. It is the number of units of the stock we should hold for
each option shorted in order to create a riskless portfolio. It is the same as the ∆ introduced earlier
in this chapter. The construction of a riskless portfolio is sometimes referred to as delta hedging.
The delta of a call option is positive, whereas the delta of a put option is negative.

6 Matching Volatility With u and d

The three parameters necessary to construct a binomial tree with time step ∆t are u, d, and p.
Once u and d have been specified, p must be chosen so that the expected return is the risk-free rate
r. We have already shown that
er∆t − d
p= (11)
u−d
The parameters u and d should be chosen to match volatility. The volatility of stock (or any other
asset), σ, is defined so that the standard deviation of its return in a short period of time ∆t is
√
σ ∆t. Equivalently the variance of the return in time ∆t is σ 2 ∆t. During a time step of length

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∆t, there is a probability p that the stock will provide a return of u − 1 and a probability 1 − p
that it will provide a return of d − 1. It follows that volatility is matched if

p(u − 1)2 + (1 − p)(d − 1)2 − [p(u − 1) + (1 − p)(d − 1)]2 = σ 2 ∆t (12)

Substituting for p from equation (11), this simplifies to

er∆t (u + d) − ud − e2r∆t = σ 2 ∆t (13)

When terms in ∆t2 and higher powers of ∆t are ignored, a solution to equation (13) is
√ √
u = eσ ∆t
and d = e−σ ∆t

7 The Binomial Tree Formulas

The analysis in the previous section shows that, when the length of the time step on a binomial
tree is ∆t, we should match volatility by setting
√
u = eσ ∆t
(14)

and
√
d = e−σ ∆t
(15)

Also, from equation (6),

a−d
p= (16)
u−d
where
a = er∆t (17)

8 Increasing the Number of Steps

As the number of time steps is increased (so that ∆t becomes smaller), the binomial tree
model makes the same assumptions about stock price behavior as the Black– Scholes–Merton
model. When the binomial tree is used to price a European option, the price converges to the
Black–Scholes–Merton price, as expected, as the number of time steps is increased.

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9 Options on Other Assets

9.1 Options on Stocks Paying a Continuous Dividend Yield

Consider a stock paying a known dividend yield at rate q. The total return from dividends and
capital gains in a risk-neutral world is r. The dividends provide a return of q. Capital gains must
therefore provide a return of r − q. If the stock starts at S0 , its expected value after one time step
of length ∆t must be S0 e(r−q)∆t . This means that

pS0 u + (1 − p)S0 d = S0 e(r−q)∆t

so that

e(r−q)∆t−d
p= u−d

√
As in the case of options on non-dividend-paying stocks, we match volatility by setting u = eσ ∆t
√
and d = e−σ ∆t
. This means that we can use equations (14) to (17), except that we set a = e(r−q)∆t
instead of a = er∆t .