Options v2 (extension of chapter 11 options) Flashcards
Elaborate on the economic factors governing whether you will early exercise an option or not.
We have 3 factors:
1) Dividends
2) Interest rate return
3) Protective position
If we exercise a call early, we may receive dividends, but we give away the option to earn risk free return. At the same time we would be exposed to the downside of holding the asset.
As a result, it really is dividends vs the rest.
For puts, we receive these dividends, and we have a protection. Therefore, the only benefit of early exercise comes as a result of potentially receiving interest on the strike price that we receive for the asset.
elaborate on what happens if we consider volatility of stock price movement to be zero
No volatility means movement with certainty. In such case, there would be no value of the protection that options otherwise would carry.
Give the formula for option price using risk neutral probabilities
C = e^(-rh)[pCu + (1-p)Cd]
In other words, the price of the option is equal to the present value of future expected? payoff. Expected in the sense of risk neutrality.
Note: doesnt matter that it is not the actual expected payoff or not. All that matters is no arbitrage.
can we use discounted cash flow method to price options?
Yes
do we usually use discounted cash flow method to price options?
No. there is simply no reason to do it. It is more intricate than using risk neutral proabilities.
If we were to price an option based on discounted cash flow method, how would we do it?
we’d use the true probability distribution of the returns of the stock, and find the expected return, and then discount this using an appropriate rate.
An appropriate rate would be something like the required return from CAPM.
However, this is easy on paper, but difficult in practice.
First, we dont know the true probability distribution.
Second, we dont necessarily know the rate to discount with.
So, we’d try to find out what the true probability is, by using the same formula as the risk neutral probability use, but now we use a rate differnet from the risk free rate.
Here is the important case: we cannot use the stock’s alpha to discount, because the option is a leveraged position and does not follow that rate. instead, we use the weighted average rate of the replicating portfolio.
However, solving the expression for the option price, it turns out that it is equivalent with the risk neutral one, which removes any reason to use this method in practice.
define a log normal random variable
A random variable is said to be log-normal if when we take ln- of it, we get a normally distributed variable.
elaborate on the overall workings on the log-normality and normality of the binomial model
First, we need to recall that if we have a random variable that is normally distributed, and we use it as the exponent to base euler, we get a log-normal distributed random variable.
Likewise, if we have a random variable X that when subject to the function “ln(x)” becomes a normally distributed variable, X was originally a log-normal distributed random variable.
to build the binomial tree, we start by considering “sigma sqrt(h)”. If we consider making h very small and increasing the number of periods correspondingly, this constitutes a random walk of normally distributed variable. The expectation is 0, but the variance is dependent on other things. this would just simply give us a normally distributed variable with mean 0.
However, our returns shouldnt have mean 0, they should have mean equal to some fixed value. In our case, it is equal to (r-∂)h. The way we make sure this applies, is by adjusting the probabilities of up vs down moves so that the expectation is equal to this certain term.
There is no stochasticity here, so the mean, regardless of up or down move, is equal to the certain term. There is no variance in the certain term, so the variance remains as it was before.
Since the mean has changed, we know the location of the normal distribution has changed. A simple mean shift does not skew the distribution, but only change the position of the mean. As a result, we still have a perfect bell shaped curve that represent the exponent-movements. Since these are normally distributed, they follow the 68-95-99 rule in regards to standard deviations etc.
Since the exponents are normally distributed random variables, the e^(exponent) is log normal. Therefore, the prices are log-normal. This means that when we look at the distribution of the prices we get at the final layer of the binomial tree, the prices are log-normally distributed. This carry benefits, such as no negative prices and upward return bias.
A point of confusion may arise from the surroundings of probability change, and how it affects the distributions. but, what happens is this: When we add the certain term in the exponent, we shift the mean of the distribution (normal distribution location shift). but if we keep the probabilities at say 0.5, we would not get the same expected returns as before. This follow from the fact that we did not change the probabiltiies, but we changed the mean.
We ultimately want to keep the return equal to the risk free rate, so we adjust the probabilities accordingly.
CRUCIAL POINT: If we were to simulate many stock price moves, and we did not change the probabilities, we’d end up at expected exponent value of n(r-∂)h. This will likely not match the risk free rate return that we know we need to ensure no arbitrage. By adjusting the probabilities, we end up at an exponent value that corresponds with the no arbitrage return.
To clarify, the values of the probabilities do not alter the distribuiton shape. Neither does the certain term. It only change where the mean is located. With the correct certain term, and the correct probabilities, the mean of the normal distribution will be the mean that gives us an expected return equal to the risk free rate.
recall why we are actually allowed to use the stock price movements to find the risk neutral probabilities which we then plug into the no aribtrage formula for option price
We can use the sock price movements to find the risk neutral probabilities because the movement in the stock price is ultimately directly tied to C_u and C_p. The probability of achieving either state MUST be the same for both the stock and the option.
what are some issues with modelling stock prices as random walks?
1) We may get negative stock prices
2) The magnitude of the stock price movement each period should be relative to the number of periods
3) Stocks require positive return, otherwise no one would hold. Therefore, a simple random walk does not include this compensation.
how do we cope with the random walk issues?
in the binomial model, this works out because we use stock movements that makes the stock price movements follow log normal distribution and all that.
We avoid negative prices.
Price movements scale with the number of periods.
We get a constant positive growth.
What is the underlying assumption for the usage of log-normal distribution in the binomial model?
Continuously compounded returns are normally distributed.
What does it even mean that “Continuously compounded returns are normally distributed.”?
it mean that if we look at some time period, and take many samples from the same stock where each sample is basically the ln(S_t/S_(t-1)), each covering the same time duration, we expect the measure to be normally distributed
why do we have a framework for accurately pricing options using binomial model and normally distributed returns that become log normal etc, but not for stocks?
the only reason why we have such a sophisticated method for options and DERIVATIVES, but not for stocks in general, is because we can replicate the option by a portfolio in stock+bond. The very moment we do this, we no longer care about the stock price, we care that the option payoff is the same as the replicating portfolio at every possible outcome. Therefore, the modeling of stock prices in the binomial tree is more about getting the ratios right, rather than predicting the actual real life distribution of stock returns.
To clarify, we still care about the stock prices, but it is secondary to the other things.
The CORE IDEA is that derivatives can be replicated, which allow us to attack the problem from the angle of no-arbitrage.
what are the more pressing assumptions regarding the binomial model=?
constant volatility
Independent returns
are there alternative ways to construct binomial trees?
Yes. A known one is the Cox-Ross-Rubinstein variant
elaborate on the requirement for constructing binomial trees
An acceptable tree must match the standard deviation of the continously compounded returns on the asset, and must generate an appropriate distribution as the length of the binomial period approach 0.
elaborate on the Cox-Ross-Rubinstein way of constructing the binomial tree
u=e^(sigma sqrt(h))
d=e^(- sigma sqrt(h))
what is the time t forward price for delivery at (t+h) for a stock when dividend is payed in the mean time?
F_{t, t+h} = e^(rh)S_t - D
When discrete dividends are included, how do we find up and down moves in the binomial tree?
We base the up and down moves on the ex-dividend price, which is S_t e^(rh) - D
it is basically the same as before. we need to add the movement term about the constant growth rate:
S_t (e^(rh)-D)e^(+-sigma sqrt(h))
how do we do the option portfolio replication when dividends are imminent?
discuss some problems with the discrete dividend binomial tree
The tree does not recombine after a discrete dividend.
Also, since the dividend is a fixed size, and is being subtracted, we may model a negative stock price. therefore, we actually not another way that is conceptually sound to model discrete dividends.
reflect on why we subtract the dividend like we do, and in what periods it is relevant
because we want to subtract the value of the dividend to highlight the likely drop in stock price that follow from the dividend. When we use a binomial tree for pricing options, this is the way it should work because we are ultimately interested in stock prices at expiration, and these prices follow the drops that we see from ex-dividend dates.
It is all about making the stock prices reflect the drop that follow in the stock.
so in regards to the construction on the binomial tree, we do not consider the advantage of exercising early as a result of the dividends, but we are interested in how the stock prices move. When we later use this tree to compute the option prices, we can, at each node, consider the benefit of early exercise by comparing the regular expected payoff vs the immediate benefit of exercising.
when we model the tree with discrete dividends as described, we automatically include all the “penalties” of not receiving the dividends, right, Assuming the option is European, this way of modeling it would make sure that the final prices reflect the penalty. Then it is up to us to identify that there are penalties included, and check for early exercise at each node.
what is the better way of modelling discrete dividends in the binomial model?
An approach using pre paid forwards.
The idea is that if we know a stock will pay a dividend for certain, we can view it as a combination of two things:
1) The dividend
2) The present value of the ex-dividend stock price
The present value of the ex-dividend stock price is the same as the pre-paid forward price. Therefore we can find the stock price by summing the pre-paid forward (that doesnt include the dividend) and the actual dividend, after taking present value on it:
S_t = F^P_{t, t+h} + De^(-rh)
then we assume that all uncertainty lies in the forward price, and that the dividend is given with 100% certainty. This will entail that the stock price movement only relates to the prepaid forward, and not the dividend. The OBSERVED stock price is then:
S_ud = F^P_{t}e^(rh+-sigma sqrt(h)) + De^(-rh)
Finally, we need to note what volatility is in this case: since we are using the forward to price, the volatility is related to the forward price. Therefore, we need to perform an ad-hoc adjustment to it:
sigma_f = sigma_s (S/F^P)
