Options

Question 1

what is the principal question regarding options

Answer 1

how do we value the right to back away?

Question 2

Recall put call parity

Answer 2

Call - Put = PV(Forward price - strike price)

Question 3

what is the intuition behind put call parity equation

Answer 3

Buying a call and selling/writing a put at the same strike price creates a synthetic forward contract, and if the strike is equal to the forward price the position MUST have zero price to avoid arbitrage.

Question 4

what happens if we establish a synthetic forward contract at a strike price different to the forward price

Answer 4

There is a benefit in either party. This benefit is therefore balanced by offsetting the price of initiating the position (premium).

Question 5

what do we need to know about parity and option category?

Answer 5

Generally fails for american options. It is a european concept, but has use cases with american options as well.

Question 6

extend put call parity formula to include discrete dividends on a stock

Answer 6

The RHS is PV(forward price - strike)

which is:

PV(F_{0,T} - K) = e^(-rt) F_{0,T} - e^(-rt)K

First part is actually the prepaid forward price. Second part is just the value of investing the strike price worth into bonds of no risk.
Dividends only make an impact on the prepaid forward. We know that the prepaid forward price when discrete dividends are considered, must account for the fact that dividends are not recevied. Therefore, we get:

= S_0 - PV(div) - e^(-rt) K

So, the put call parity equation becomes

Call - Put = S_0 - PV(div) - e^(-rt) K

Question 7

what is put call parity formula for index options?

Answer 7

Now there is continuous dividends, so we do not subtract the present value of these, but instead discount:

Call - Put = S_0 e^(-∂t) - e^(-rt) K

Question 8

why does the price of an ATM call exceed the price of the ATM put?

Answer 8

Buying the call and selling the put creates a synthetic forward long position. But the strike is below the forward price.

If call == put price, we could do the following:
Buy the cheaper synthetic forward long position.
Short the actual forward position.
At expiration, we acquire the asset from the synthetic long position. Since this was cheap, it cost us say 3 usd.
Also at expiration, we sell the assert (short position) that we acquired from the long position, but this one was relatively speaking more expensive. Say 3.3 usd. The outcome is that we have made 0.3 usd completely risk free with no net investment. We never had any exposure to the risk of the asset, since our obligation was to buy it and sell it. Since the prices were locked in advance, it was a risk free investment for us.

This example shows what happens if the strike price is not equal to the forward price, AND the call premium is equal to put premium.

In order to offset the advantage of lower or higher strike prices relative to the forward price, the price of the options will change. For instance, if the strike price is lower than the forward price, this favors the long position, since it makes it more likely that we can acquire the asset for the cheap price COMPARED to its spot price. Opposite applies as well.
Therefore, the CALL PRICE is more expensive relative to the PUT PRICE when the strike price is below the forward price.

Question 9

elaborate on what happens with premiums when the strike price is higher than the forward price

Answer 9

Assume that premiums are the same.

The synehtetic forward acquire asset at a less favorable price than the forward contract. As a result, assuming that premiums are the same, we can short sell the synthetic forward (and receive the expensive cash) while also going long a regular forward. We pay less, receive more, and just transfer shares. NO net investment (zero net investment). No risk.

Recall that shorting a synthetic forward implies that we write the call and buy the put. Thus, we make money should the asset decline in value.

The high strike price favors the shorted synthetic forward. It is relatively speaking MORE likely to see the assets being sold for less than the strike price. Because of this, the premium of the put option must be larger than the premium of the call option. Otherwise, there is arbitrage opportunity.
There is nothing wrong with having the synthetic forward short position with a high strik price, but if we have this then the premium for the put will be more significant

Question 10

the book present an alternative reasoning for why ATM calls always exceed ATM puts in premium. Elaborate on it (NO DIVIDENDS)

Answer 10

If we buy a synthetic forward vs outright purchase, we end up wit hthe shares/asset regardless. Payoff is the same in other words.

However, one case defer payment, while the other does not. This is an advantage for the option holders. This is an advantage because they can earn interest on it in the mean time, while the outright purchaser does not.

As a result, the differnece in premiums reflect the interest one can earn from holding the synthetic forward.

Specifically, the difference in premiums must equate the interest on the strike price.

AND WHY MUST DOES IMPLY THAT CALLS ATM ARE WORTH MORE THAN PUTS ATM?

Because we receive the premium of the put, and pay for the call premium. The position must cost something, if not the strike price interest advantage is not accounted for. As a result, the calls must cost more than the puts benefit us.

Question 11

what would be required for a call option to have a negative premium?

Answer 11

It must force the user to exercise. This is not how options work, and therefore the call premium is always non negative.

Question 12

what is the maximum possible premium for a call?

Answer 12

Stock price itself. This is because best case scenario is that we end up owning the stock.

Question 13

what is the maximum worth of a put option?

Answer 13

Strike price

Question 14

when might we want to exercise a call option early?

Does it also apply to puts?

Answer 14

Depends on dividends. If the stock doesnt pay dividends, there are no situations that benefit early exercise.

Doesnt apply to puts. There are certain scenarios where exercising puts early makes sense.

Question 15

Demonstrate why exercising calls early are never advantageous for non-dividend paying stocks

Answer 15

Say we consider some time step T-t. if we keep, the call has value equal to Call price.

If we exercise, we get intrinsic value, which is the difference between the strike price and the spot price at T-t time left. Or more specifically, we get Spot - Strike.

AS a result, early exercise is beneficial whenever S-K > C.

So the question is, can this be?

From put call parity, we have that

C - P = S - e^(-rt) K

C = P + S - e^(-rt)K + K - K

C = P + (S - K) + K(1 - e^(-rt))

We recognize (S-K) as the intrinsic value we get from the early exercise.
Then we also have the non-negative value of the put option.
Then we also have the non negative effect of strike price interest, which we label as the time value of money.

We refer to the put option as insurance.

Therefore, we view the call option as a combinatoin of intrinsic value, insurance, and time value of money. Insurance and time value are always positive. As a result, the call is always worth more than the intrinsic value from early exercise.

Question 16

what two effects are related to early exercise

Answer 16

1) We throw away the insruance from the put protection

2) we accelerate payment of the stock,

Question 17

can we have parity with dividends?

Answer 17

yes

Question 18

recall the parity with discrete dividends

Answer 18

Question 19

what determines the early exercise condition for american options with discrete diviends (call options)?

Answer 19

If we are at time T-t, and the strike price less the time t present value of the strike price is greater than the time t presnet value of the dividend, then early exercise is not beneifcial.

So, this is bascially the relation:

K - PV(K) > PV(div)

The difference “K - PV(K)” is the interest we get by holding.

NB: If the inequality is violated, it does not tell us that we should exercise early. It only tell us that we cannot rule out early exercise if it is violated.

So, we can here only say for certain when we will not exercise early. We cannot say that we will exercise.

Recall why this is

Question 20

The book say that the following inequality:

K - PV(K) > PV(div)

makes sure that early exercise is never done when it holds. However, if vioalted ,the book sya that we cant guarantee that early exercise is the way to go. Why is that?

Answer 20

It is difficult to account for the put protectoin benefits, the time value benefits, and the intrinsic value.

Recall that this is about finding analytical proofs of conditions for otpimal early exercise, and is not really related to binomial model etc.

Question 21

elaborate on early exercise for puts

Answer 21

It can be optimal to exercise puts early.

Consider a put. If we do not exercise early, we get K at expiration, which today is worth PV(K).
if we instead exercise NOW, we get K NOW.
So, we are looking for scenarios where K > PV(K).
In general, this is the case, since interest rates are typically positive.

So, since puts sort of benefit from early exercise, it is a little opposite in regards to the calls.

Question 22

what do we receive when we early exercise an optin?

Answer 22

depends n the type.

if call, we receive asset. Asset pay dividend.

If put, we receive strike price.

Question 23

are options always worth more with more time to maturity?

Answer 23

not necessarily. For american options, yes.

But european, no. Many cases can make it beneficial to have less time.

Question 24

what is K - PV(K)?

Answer 24

Represent the amount of cash we earn from interest on the strike price by deferring payment.

If we wait, we need K at the term. This amounts to having PV(K) now.
If we exercise early, we therefore do not get the difference in interest. We’d pay K today, which is worth more than K later, assuming interest is positive.

Question 25

recall the early exercise condition for calls with dividend

Answer 25

Early exercise is never optimal if the inequality K - PV(K) > PV(div) If the interest earned is greater than the dividend, there is simply no reason

Question 26

why not include interest you can earn on the dividend as well? like this: K - PV(K) > PV(div) + interest (div)

Answer 26

because when we take present value, we auto-adjust to a specific point in time. Doing this allows us to make a simple straighforward comparison.

Question 27

consider a put. if the stock goes to 0, and there is a year left, what is the put worth now vs at maturity?

Answer 27

now: K later: PV(K) K > PV(K), favor of early exercise

Question 28

foundational assumption of the binomial model?

Answer 28

Price can move up and down in fixed sizes only. Hence, the name binomial.

Question 29

what can we say about the portfolio of borrwoing some cash and buying long some portion?

Answer 29

It is a synthetic call in the context of binomial model.

Question 30

what can we say about expected return of an option in the context of binomial model?

Answer 30

Since the synthetic call includes borrowing to fund a part of it, it is a leveraged position. therefore, there is higher risk and we should expect higher return.

Question 31

what is u and what is d

Answer 31

u = 1+rate of capital gain on the stock if UP d = 1 + rate of capital loss on the stock if DOWN

Question 32

what is the no arbitrage condition for the system-of-equations approach to finding option price?

Answer 32

u > e^(r-∂)t > d COnsider 0 div. if it is violated, say u <= e^(rt), then we'd be better of investing in risk free bonds. We could short the stock, invest in bonds.

Question 33

how can we interpret delta?

Answer 33

sensitivity of the option in regards to the underlying asset

Question 34

consider a synthetic option. elaborate on what happens initially, and what happens at expiration

Answer 34

This is for CALLS: We invest in delta shares. Pay now, sell later. We sell at the different price. We borrow money. At expiration, we repay it WITH interest.

Question 35

graphically, how does the synthetic call look compared to the regular call?

Answer 35

illustrates the fact that the synthetic call is only accurate at the specific points

Question 36

mathematically, how do we change the synthetic call "line"?

Answer 36

delta represent slope, B represetn the position.

Question 37

why doesnt binomial model need probabilities?

Answer 37

the payoff is perfectly replicated regardless of outcome. Our only goal is to price the option correctly. And to do this, it is sufficient with a replication of the payoff. If payoff is the same, we know that the actual option and the synthetic option must have the same price.

Question 38

formula for risk neutral probabilities

Answer 38

p = (e^(r-∂)t - d)/(u - d)

Question 39

how do we "find" risk neutral probabilities?

Answer 39

we can take the original expression for the option price, and re-write it. C = e^(-rh) [p Cu + (1-p) Cd] one can verify that p is indeed a probability-like value. It sums to 1, and both are positive. and the expression is an expected value, which is then discounted using risk free rate. Hence the name "risk neutral pricing".

Question 40

can we use riks neutral pricing to price forward and prepaid forward?

Answer 40

Yes. We swap the price of the option at up and down moves with the Su and Sd. We should get this: e^(-∂h) [p Su + (1-p) Sd] = S e^(-rh) RHS is the prepaid forward price for some period, given the stock price up and down the next period. If we omit the discounting at RHS, we get the forward price instead. This means that we can create a binomial tree of stock prices, and use this tree to create binomial trees of forward and prepaid forward prices.

Question 41

formulas for up and down u and d

Answer 41

Question 42

summarize the important properties of continuously compounded returns

Answer 42

the log function (of return) computes continuously compounded returns The expo function computes prices from continuously compounded returns. continuously compounded returns are additive.

Question 43

why is it important that cont comp returns are additive?

Answer 43

if we have say daily returns of a year, we can sum these returns and we know that we then get the annual return

Question 44

what is the variance of a sum of cont comp returns?

Answer 44

Var(r_annual) = var(∑r_i)[year] assuming the returns are uncorrelated, we get that the variance of a sum is the sum of variances, and we get the relaiton: sigma^2 = 12 sigma^2_monthly etc so if we want the volatility annual, based on monthly, we need to take sqrt(12)sigma.

Question 45

why do we start binomial option pricing at the terminal state?

Answer 45

This is the only time we know the true payoff given the various scenarios.

Question 46

why do we bother with icluding early exercise in the binomial modeling? continuous dividends make this never happen?

Answer 46

for put options, it can happen. therefore we must include it.

Question 47

elaborate on the economics considerations regarding early exercise of options

Answer 47

1) The holder receives stock, therefore receive future dividend 2) interest cost, since the buyer pay the strike price before expiration 3) loses the implicit put insurance

Question 48

case: consider a call option. strike = 100. Current stock price = 200. Interest = 5%. Dividend = 5%. Should we exercise early?

Answer 48

The dividend is much greater than the interest. This would make a case for early exercise. However, we cannot know this for sure. We do not know the value of the insurance. Waiting will make the owner of the call option sure that he wont sustain big losses. In general, if volatility is 0, the insurance has no value. We generalize this to saya that higher volatility give higher insurance value, which will make a greater case towards not exercising early.

Question 49

for an infinitely lived call option, when should we exercise?

Answer 49

Assuming volatility is 0, we should wait until the dividend times stock price exceeds the interest times strike price. so, if: ∂ S > rK, we exercise. The ratio, S > (r/∂)K determine when to exercise. NB: This is with zero volatility.

Question 50

elaborate on early exercise wiht puts

Answer 50

early exercise means that we are giving up dividends, but receiving strike price earlier.

Question 51

when modeling with binomial tree, we never explicitly account for early exercise for calls and puts by considering the dividends (continuous). Why is that?

Answer 51

we make the assumption that those dividends are auto reinvested. Therefore, we can adjust the amount of shares we buy instead. Makes our life easier, since the model is exactly the same with or without the continuously compounded diviends. For discrete dividends that are payed out, however, the case is more intricate.

Question 52

what happens in the risk neutral world scenario?

Answer 52

no one would require risk premium. Therefore, everyone would chase the highest expected return. Doing this eventually make each asset yield the risk free rate.

Question 53

what is the first theorem of asset pricing

Answer 53

the theorem is about establishing the fact that if it is not possible to perform arbitrage, then the prices of various assets must behave in a certain way. This is a two way street, since we can flip it and say that if the assets are not priced/behaving in a certain way, we can exploit it in arbitrage

Answer 54

we observe a call price in the market, and we also "happen to know" the possible future states that the stock price can be in. By using the binomial model, we arrive at a true price. This price is apparantly the price that ensures no arbitrage. it is also complete because binomial model use replicating portfolio. Since the two states are fully accounted for, and we have 2 assets in our replication, the solution is unique. Therefore, the price is unique. THerefore, for this specific price, there is only a single pair of probabilities that give us this price under the risk neutral rate.

Answer 55

The pricing map must be linear. This means that if we add two assets together, their price together must equal the sum of individual prices. For instance, two portfolios that are equal, by one being an option and another being replciating portfolio, price must be the same. if this was not the case, arbitrage is present. Some assets sell at inconsistent prices. For instance, the stock can be undervalued compared to the call. this creates arbitrage opportunity. Any linear function can be written like: π = ∑q_i x_i, where q is weight and x is payoff.

Question 56

elaborate on why the fundamental theorem of asset pricing holds

Answer 56

If no arbitrage, then we must have consistent pricing. This means that portfolios that are the same in payoffs must cost the same. it also means that the sum of individual prices of assets in a portfolio must be equal to a portfolio that holds the same assets together. This allows us to establish a linear pricing rule. π = ∑q_i x_i The price is a weighted sum of the payoffs given by the assets. The weight is unknown, but I believe we generally know the payoffs. for instance, assuming binomial model, the payoffs are clearly defined, though the weight is not. becasue of the linear pricing rule, we can adjust the weights to make them probabilities. We can do this by considering the risk free asset. This asset will give a payoff of (1+r) given that we invest 1. Meaning, with a price of 1, payoff is 1+r. This gives us: 1 = (1+r)∑q_i This then gives us: 1/(1+r) = ∑q_i However, if we define "p_i" as "p_i = q_i (1+r)", we get: 1 = ∑p_i, which now is a probability, granted that all values are non-negative. Now we can write: price = π = 1/(1+r) ∑p_i x_i, where p_i is the adjusted probabilties, which means that we are now using the differnet measure Q. As a result, we can write: price = 1/(1+r) E^(Q)[payoff] Due to the rules of mathematical expectation, we can move 1/(1+r) to the inside of the E or keep it at the outside, outcome is the same, since the random variable is the payoff. However, E^Q only works with probabilities that makes sure that the expected payoff is risk free rate. This is because we normalized the weights by using the risk free rate. I suppose we could have normalized using a differnet asset, but obviously this is mroe difficult because it requires careful analysis of risk etc. In my understanding, the following equation: price = E^(Q) [payoff / (1+r)] is a martingale. This is because the expected payoff is computed using risk free return, which we then discount immediately. So the price is sat so that it is equal to the expected payoff. When we do this, there is no expected advantage. The game is fair, which is because all prices are consistent. This does not mean it is impossible to make money in the market using the derivatives. It simply means that we cant do so without risk.

Question 57

why do we care about random walk

Answer 57

Random walk provides a foundation for looking at prices of stocks and other assets

Question 58

elaborate on the basics of a random walk

Answer 58

Coin flip example. flip a coin, +1 or -1 reward depending on outcome. Cumulative sum (fortune) is recorded. It turns out that the more times we flip the coin, the further away from the starting fortune we end up at. This is because the expected value is 0, but we always get something else. When we get some value, we change our starting position. We start at 0. Next state is either +1 or -1. Say we get +1. The next state is either 0 or +2. Thus, we have changed the outcomes compared to before. we still expect to see the same number of up and down movements, but we do not expect them to occur in a perfect oscillation.

Question 59

elaborate on random walk as a change process

Answer 59

We can define the process in terms of change: Z_{n} - Z_{n-1} = Y_n, where Y_n is the coin flip or whatever value we look it at the n'th iteration.

Question 60

elaborate on the random walk model's shortcomings in regards to asset price modeling

Answer 60

It allows negative prices. The magnitude of a movement needs to be scaled with time. this is related to the volatility. The stock should have positive return on average.

Question 61

relate binomial model to random walk

Answer 61

The binomial model is a model that accounts for the shortcomings of the random walk model. The binomial model assumes that continuously compounded returns are a random walk with drift. We knwo that the binomial model price movements like this: S_{t+h} = S_t e^((r-∂)h +- sqrt(h) sigma) Taking logs, we obain: log(S_{t+h}/S_t) = (r-∂)h +- sigma sqrt(h) Key thing is that this RHS epxression has one CERTAIN part and one UNCERTAIN part. The certain part creates the drift. The uncertain part creates the variation. Therefore the binomial model represent random walk with continuously copounded returns with drift

Question 62

why does the binomail model account for the shortcomings of the random walk model

Answer 62

No negative prices since we multiply the current stock price by some factor that is never negative. It grows towrds 0, but never beyond. Volatility is scaled correctly, because we use sqrt(h) sigma. This makes sure that our movements correctly represent the time in relation to the volatility of the stock/asset. We know that assets that are risky require a return. This is the drift term, and is accounted for in the binomial model with the certain drift term.

Question 63

why does the binomial tree approximate a log normal distribution?

Answer 63

Log normal distribution is built on the assumption that continuously compounded returns are normally distributed. The definition of log normal distribution is: If x is normally distributed, then y = e^x is log normally distributed. So, if we have that contniuously compounded returns are normally distributed, then the price movemnt is log normal. The return itself. the return itself is S_{t+h} = S_t e^(x) where x = (r-∂)h +- sigma sqrt(h). It is easy to see why this is normally distributed. if we add a variable that is 1 or -1 to the sigma part, we get: x = (r-∂)h + Y sigma sqrt(h). Y represent random variable, normally distributed, then we have scaled it so that the movement represent movement that is consistent with the voaltility of the stock, and we have shifted it to account for the drift. This is a simple trnasformation. It preserves the normal distribution, it just change the location adn the variability of the distribution. Thus, x is still normally distributed. AS a result, e^x is log normal. therefore, we have binomial mdoel approximate the log normal distribution. Formally, in the limit, the sum of the small binomial steps approximate the normal variable.

Question 64

what is the probability of reaching a specific node in the binomial tree?

Answer 64

If the binomial tree has n periods, it is given by: p^{n-i} (1-p)^{i} (n! / ((n-i)! i!))

Question 64

do we say that returns or prices are log normally distributed?

Answer 64

Prices For the returns, we use normal distribution along with continuously compounded returns

Question 64

book mentio ndiffernet trees. We have the regualr one. then we have cox-ross-rubinstein which omits drift term. how do we deal with the alteration of prices that result from these differences?

Answer 64

In both cases, the uncertain term with the volatility dominates the drift term when dt is less than 1. as a result, in the limit, these methods yield the same result. Therefore, there is no difference. The book do mention it as a problem with cox-ross-rubinstein, in that it is possible for drift to be larger than the other part. however, they also say that in practice, dt is likely to be small, so this never happens. The change occurs at dt=1.

Question 65

what is the issue with modelign discrete dividends using the regular "subtraction" method?

Answer 65

The binomail tree does not recombine

Question 66

elaborate on the correct way of modeling discrete dividends in binomial tree

Answer 66

We start by recognizing that if a stock pay a dividend, we can view the stock as a sum of two components: 1) Dividend 2) Prepaid forward price The prepaid forward price is the present value of hte ex dividend price. Doing this allows us to place all uncertainty to the volatility, and no uncertainty in the dividend. The issue with the other method is that we mix these together. If we know that a stock will pay a dividend at time t < T_d, then the stock price for all t