call put parity
Binary or digital options have a payoff at expiry that is discontinuous in the underlying asset price.
For example, a binary call pays some quantity (for example, 1 euro) at maturity time, if the asset price is then greater than the exercise price K (for example K=100).
Why would you invest in a binary call? Assume you think that the asset price will rise by expiry.
To finish above the strike price then we can choose to buy either a vanilla call or a binary call.
The vanilla call has the best upside potential, growing linearly with S beyond the strike.
The binary call, however, can never pay off more
than 1 euro (in our example).
If you expect the underlying to rise dramatically then it may be best to buy the vanilla call.
If you believe that the asset rise will be less dramatic then
buy the binary call.
A binary put pays some quantity (for example, 1 euro) at
maturity time, if the asset price is then below the exercise
price K (for example, K=100). The payoff is given by a
function of the type
The binary put would be bought by someone expecting a modest fall in the asset price.
Notice that the sum of the payoffs of the binary call and the binary put in our example is equal to 1. Then
price(binary call)+price(binary put)=e-r(T-t)
A payoff that is similar to a binary option can be made up with vanilla calls.
This is a first example of a portfolio of options or an option
strategy.
Suppose I buy one call option with a strike (K1
) of 100 and write
another with a strike of (K2
) 120 and with the same expiration.
This payoff is zero below 100, 20 above 120 and linear in between. It is continuous, unlike the binary call, but has a payoff that is similar.
This strategy is called a bull spread (or a call spread) because
it benefits from a bull, i.e. rising, market.
Suppose I buy one put option with a strike (K2
) of 120 and
write another with a strike of (K1
) 100 and with the same
expiration.
This payoff is zero above 120, 20 below 100 and linear in between. It is continuous, unlike the binary put, but has a payoff that is similar.
This strategy is called a bear spread (or a put spread)
because it benefits from a bear, i.e. falling, market.
Exotics
exotic options are not standard ones-
knock-out options: go out of existence if the asset price reaches the barrier:
– down and out: has to fall to reach the barrier
– up and out: has to rise to reach the barrier
knock in options: come into existence if the asset price reaches the barrier:
– down and in: has to reach the barrier
– up and in: has to rise to reach the barrier
example: knock in: spot eurusd is trading at 1.1150. will eurusd trade below 1.08 within 2 months, and will it be volatile?
yes. buy a .08 put option for expiring 2 months with a knock in at 1.13. the option will only be triggered once eurusd touches 1.13 at any time up to expiry
binomial trees
We are going to present a very simple and popular model for
the random behavior of an asset
This simple model will allow us to start valuing options
Later we’ll be seeing a more sophisticated model, but the
ideas we first encounter with this model will be seen over and
over again. These are the fundamental concepts of hedging
and no arbitrage
The binomial model is very important because it shows that you
don’t need a simple formula for everything.
Often in real life a contract may contain features that make
analytic solution very hard or impossible. Some of these features may be only a minor modification of some other easy price
contract, but even minor changes in a contract can have
significant effects on the value and especially on the solution of
the method.
The classic example is an American put. Early exercise may
seem to be a small change in a contract but the difference
between the values of a European and an American put can be
large and certainly there is no simple closed-form solution for the
American option and its value must be found numerically.
We have to take into account that the binomial model does
not fit real market data. As a model of stock price
behavior it is poor. The binomial model says that the stock
can either go up by a known amount or down by a known
amount, there are but two possible stock prices
‘tomorrow.’ This is clearly unrealistic.
But, before working with more sophisticated methods,
binomial trees will allow us to understand several
fundamental concepts.
We will have a stock, and a call option on that stock expiring
tomorrow.
• The stock can either rise or fall by a known amount
between today and tomorrow.
• Interest rates are zero.
Which of the two prices is realized tomorrow is completely
random.
There is a certain probability of the stock rising and one minus
that probability of the stock falling. In this example the probability
of a rise to 101 is 0.6, so that the probability of falling to 99 is 0.4.
Now let’s introduce the call option on the stock. This call
option has a strike of $100 and expires tomorrow.
If the stock price rises to 101, what will then be the option’s
payoff?
It is just 101 − 100 = 1.
And if the stock falls to 99 tomorrow, what is then the
payoff? The answer is zero, the option has expired out of
the money
If the stock rises the option is worth 1, and if it falls it is
worth 0. There is a 0.6 probability of getting 1 and a 0.4
probability of getting zero. Interest rates are zero. . .
What is the option worth today?
No, the answer is not 0.6. If that is what we tend to think,
based on calculating simple expectations, but this is the
wrong answer.
The correct answer is. . .
1/2.
Why?
To see how this can be the only correct answer we must
first construct a portfolio consisting of one option and
short 1/2 of the underlying stock.
If the stock rises to 101 then this portfolio is worth 1 − 1/2 ×
101; the one being from the option payoff and the −1/2× 101
being from a short position in the stock (now worth 101).
If the stock falls to 99 then this portfolio is worth 0 − 1/2× 99;
the zero being from the option payoff and the −1/2× 99 being
from a short position in the stock (now worth 99).
In either case, tomorrow, at expiration, the portfolio takes the
value−99/2 and that is regardless of whether the stock rises or
falls.
If the portfolio is worth −99/2 tomorrow, and interest
rates are zero, how much is this portfolio worth today?
It must also be worth −99/2 today.
This is an example of no arbitrage: There are two ways
to ensure that we have −99/2 tomorrow.
1. Buy one option and sell one half of the stock.
2. Put the money under the mattress.
Both of these ‘portfolios’ must be worth the same today.
Therefore, using ‘?’ as in the figure to represent the
unknown option value
? − 1/2× 100 = the option value − 1/2× 100 = −1/2× 99
and so
? = the option value = 1/2.
The value of an option does not depend on the probability of the
stock rising or falling. This is equivalent to saying that the stock
growth rate is irrelevant for option pricing.
This is because we have hedged the option with the stock. We
do not care whether the stock rises or falls. We do care
about the stock price range, however. The stock volatility is
very important in the valuation of options.
Three questions follow from the above simple argument:
• Why should this ‘theoretical price’ be the ‘market price’?
• How did I know to sell ½ of the stock for hedging?
• How does this change if interest rates are non-zero?
Introduce a symbol. Use to denote the quantity of stock
that must be sold for hedging.
We start off with one option, -Δ of the stock, giving a
portfolio value of
?- Δ × 100.
Tomorrow the portfolio is worth
1 -Δ × 101
if the stock rises, or
0 −Δ × 99
if it falls.
The key step is the next one; make these two
equal to each other:
1 − Δ × 101 = 0 − Δ × 99.
Therefore
(101 − 99)Δ = 1
Δ = 0.5.
Another example: Stock price is 100, can rise to 103
or fall to 98.
Value a call option with a strike price of 100. Interest
rates are zero.
Again use to denote the quantity of stock that must
be sold for hedging.
The portfolio value is
? − Δ × 100.
How did I know to sell ½ of the stock for hedging?
22
Tomorrow the portfolio is worth either3 − Δ 103 or 0 − Δ 98.
So we must make
3 − Δ 103 = 0 −Δ 98.
That is,
Δ = (3 − 0)/(103 − 98)= 3/5= 0.6.
The portfolio value tomorrow is then
−0.6 × 98.
With zero interest rate, the portfolio value today must equal the
risk-free portfolio value tomorrow:
? − 0.6 × 100 = −0.6 × 98.
Therefore the option value is 1.2.
Delta hedging means choosing such that the portfolio value does
not depend on the direction of the stock.
When we generalize this (using symbols instead of numbers later
on) we will find that
Δ=(Range of option payoffs)/(Range of stock prices).
We can think of as the sensitivity of the option to changes in
the stock.
Example: Same as first example, but now r = 0.1.
The discount factor for going back one day is
e
-0.1/252= 0.9996.
The portfolio value today must be the present value of the
portfolio value tomorrow
? − 0.5 × 100 = −0.5 × 99 × 0.9996.
So that
? = 0.51963.
Earlier, we tried to trick you into pricing the option by
looking at the expected payoff. Suppose, for the sake of
argument, that we had been successful in this. We would
then have asked you what was the expected stock price
tomorrow; forget the option.
The expected stock value tomorrow is
0.6 × 101 + 0.4 × 99 = 100.2.
In an expectation sense, the stock itself seems incorrectly
priced. Shouldn’t it be valued at 100.2 today? Well, we
already kind of know that expectations aren’t the way to
price options, so this is also not the way to price stock. But
we can go further than that, and make some positive
statements.
We ought to pay less than the future expected value because
the stock is risky. We want a positive expected return to
compensate for the risk.
We can plot the stock (and all investments) on a risk/return
diagram.