Chapter 21 - Option Valuation

Question 1

what is the assumption of the binomial option pricing model

Answer 1

That at the end of each period, the stock has 2 outcomes.

Question 2

What is the important takeaway from Black and Scholes regaridng hte binomial option pricing model

Answer 2

Option payoffs can be replicated using a portfolio of risk free bonds and underlying stock

Question 3

What is the driver behind being able to replicate option payoffs using portfolio of bonds and stock?

Answer 3

If we can create a portfolio that has the same payoff wit hthe same level of risk and all that, they must be worth the same.

We can use this to udnerstand what the price of the option should be based on what the price of the replicated portfolio is

Question 4

Elaborate on a binomial tree

Answer 4

A binomial tree summarize a lot of important infomration, and is the basic building block of the binomial option pricing model.

It shows the root node as the stock price. Then it splits into the two possible states at the end of the time period. At this point, we have a new stock price, and we can calculate the corresponding call and put value.

Question 5

what do we call the 2 states in a binomial tree?

Answer 5

Up state and down state. whether it is up or down is completley determined by the stock price movement, nto by call/put.

Question 6

Given a binomial tree like the one on the image, what do we do to price the option?

Answer 6

We make use of the fact that we know that option’s value at the end states. Then we need to construct the replicating portfolio of bonds and underlying stock.

The number of shares we buy of the underlying is called ∆.
We assume that the bonds has value TODAY of $1, so that it has a face value of 1xrate.
We denote the amount of cash invested in the bonds as “B”.

So, we have two states with corresponding option values. we need to figure out what portfolio of “B” and “∆” gives the same payoff as the two states.

In the up state, we have the call value of 10. Therefore, we can construct an equation that tell us what we need the bond investment and stock investment to be to create the payoff of 10:
60∆ + 1.06B = 10

For the down state:
40∆ + 1.06B = 0

We solve like system of equations.

When we have the delta and B, we need to figure out what this portfolio is worth.
∆ = 0.5
B = -18.8679

We are long 0.5 shares of stock, but we are short -18… in the bonds, meaning we have borrowed money to do it.

Then we find out what this portfolio is worth.
The stock position is given by 50x0.5 = 25

The bond position is worth what the price is.

Call option value = 25 - 18.8.. = 6.13

Question 7

Elaborate on a negative investment in risk free bonds

Answer 7

just to be clear, shorting (borrowing) the risk free rate without doing anything with it is a completely fucked up thing to do, unless if we use it as a way to replicate exactly the payoffs of a certain security (call option for instance).

it is purely a way to create the payoff.

Question 8

elaborate on how we go from delta and bond ivnestment to finding the option price

Answer 8

We essentially want to figure out what it cost us to purchase the new replicating portfolio. therefore, we use the CURRENT stock price (root node) along with the bond ivnesment.

stockPrice x ∆ + B

Question 9

There are 2 extemely important outcomes of the single period binomial option pricing model

Answer 9

1) although the payoffs from the option and the replicating portfolio are not the same in general, they are the same in exactly the two points of interest, the ones we defined as the up state and down state.

2) The option value is completely independent of probability of movement. This is possible because of the clearly defined states and using the law of one price.

Question 10

Generalize the single period binomial option pricing model

Answer 10

Two states, two outcomes, two known option values at given outcomes.

We need to figure out how much investment in shares (∆) and in the bond (B) we need.

Setting up the two equations:

S_u∆ + rf B = C_u

S_d∆ + rfB = C_d

Solve for ∆ first:

S_u∆ - S_d∆ = C_u - C_d
which we found from subtracting 2 from 1.

∆(S_u - S_d) =) C_u - C_d

∆ = (c_u - c_d) / (s_u - s_d)

And B:
B = (c_d - s_d ∆)/(1+rf)

Then finally, we find the value of the entire option:

C = S∆ + B

Question 11

what happens to the binomial option pricing model if we use a put instead of call?

Answer 11

Very little. same logic, same formulas. The only thing that will change is that we’ll see a negative ∆ and a positive B, while with the calls, we had positive ∆ and negative B. This is not surprising, seeing as the put option require a return the increase wiht negative movement, we need to short the stock to replicate this motion.

Question 12

Main weakness of the single period binomial option pricing model?

Answer 12

Single period and only 2 states

Question 13

if we were to replicate the option in a multi period model, what would we need to do+

Answer 13

Use a dynamic traidng strategy. The ∆ and B change depending on the outcomes. We need to adjust to these levels.

Question 14

Is it actually a weakness that the binomial model is single perido and 2 states only?

Answer 14

No, because it can be generalized to an arbitrary amount of accuracy.

Question 15

Can we use black scholes on american options?

Answer 15

If the stock does not pay dividends, then yes. We know that if the stock aint paying divs witihn the expiraiton date, then the european and american option are worth the same.

If dividends are payed, then it will find the value of european stock.

NB: Calls, not puts. American puts have some cases where early exercise is benefiical

Question 16

Does black scholes apply to puts?

Answer 16

No, it applies to calls.

however, we can use put-call parity to find the put value, given teh call value

Question 17

If the stock pays dividends, what do we do?

Answer 17

We can still value it like a european option. However, we need to make a substitution:

S* = S - PV(div)

We essentially create a new security that is worth the same as S, but without the dividends.
S and S star is the stock prices.

Question 18

Only unobservable parameter in the Black Scholes model

Answer 18

Volatility

Question 19

Since volatility is not observable, how do we use the model?

Answer 19

We need to estimate it somehow. There are basically 2 ways to do this:
1) Use historical data/historical volatility. this can be good, but is not always very good.

2) Use current market prices of the options, and solve backwards for the volatility.

When using method 2, the volatility is called implied volatility.

Question 20

Can the BS model be solved for volatility?

Answer 20

No, it is a transcendental equation, which means that we cannot do this analytically. We therefore have to use trial and error with a numerical tool

Question 21

What are OTM options essentially priced on?

Answer 21

The potential of them ending up in the money. This is why the IV is so important here. A larger volatility means that there is higher potential for the option to expire ITM.

Question 22

What is the black schole sformula for a non diuvidend paying call option?

How does it relate to the replicating portfolio?

Answer 22

C = S N(d1) - PV(K) N(d2)

Recall replicating portfolio:
C = S∆ + B

In this case, S is S
N(d1) = ∆
-PV(K) N(d2) = B

Since N(d1) and N(d2) refer to the cumulative normal distribution, they have lower limit of 0 and upper limit of 1. This means that the ∆ can only be in the interval [0, 1].

Question 23

What is the interpretation of ∆?

Answer 23

Change in option price based on change in stock.

because ∆ is always less than 1, the option’s value can never change more than the underlying.

Question 24

For a call option, what can we say about “B”?

Answer 24

IT will be somewhere between 0 and -K, where K is the strike price of the option.

Recall that B is the initial investment in the bond.

Question 25

Recall what we use the strike price for in the binomial option pricing model

Answer 25

Extract the intrinsic value at each possible final outcome/state.

Question 26

what types of options does the BS model apply to?

Answer 26

European calls. If the stock does not pay dividends, we know that it doesnt matter if it is american or not, as it is worth the same (sell never exercise).

Therefore, BS apply to non-dividend american stocks as well.

We can use it for european puts as well

Question 27

Why can we not use the BS model to evaluate price of American put options that has no diviendes?

Answer 27

Recall from chapter 20 that there are cases where early exercise of American put options are beneficial. Therefore, there are limitations in the usage of BS for such cases.

The entire point here is whether early exercise is beneficial or not. The first thing I need to think of whne regarding the American options are the early exercise benefits (if any).

Question 28

Elaborate on why American puts are not easily valued using BS even if the stock does not pay dividends

Answer 28

Considering the fact that puts are limited in upside because the down-movement is limited, we can actually reach a point where it is not possible to gain much more intrinsic value. When this happens, the time value of the option is reduced significantly, because the additional time serve no benefit. I suppose this time value can even be negative, highlighting the fact that you loose money by not exercising early. I suppose this could happen if the interest rate is high, so that you could exercise the option, which means that the counterpart must give you the intrinsic value, and you can then use this cash in risk free securities rather than holding them in the option contract without benefit. the other case is of course to sell the contract, but if the time value is negative, it is better to exercise it

Question 29

how do we use BS on european puts with no div?

Answer 29

Put call parity + BS for Call, use substitution in the put-call parity.

Question 30

how do we handle dividedns?

Answer 30

We perform substitution S* = S - PV(div)

the theory that the book try to explain here is that if the stock pays dividends, and we (using European stock options) cannot access these dividends, which means that the stock price is not correctly measured in terms of option valuation. Therefore, we seek to remove this, and only this, very specific piece of the future expected cash flow from the valuation. This is why we do the substitution. It also mean that if the stock pay 2 dividends in this option expiration period, we’d remove the present value of both.

Question 31

what is so special about dividends that is fixed size proportionally to the stokc price?

Answer 31

We dont have to do the tedious substitution that introduce new terms. We just add a factor to S, which means that we can keep the original blakc scholes terms, but with a new factor appended on S

Question 32

given BS model equation for calls, how to relate it to the replicating portfolio

Answer 32

BS: C = SN(d1) - PV(K)N(d2)

C = S∆ + B

So,
∆ = N(d1)
B = - PV(K)N(d2)

Question 33

elaborate on ∆ and B in the replicating portfolio of BS model

Answer 33

∆ = N(d1). Since this is cumulative distibution, we know that ∆ is between 0 and 1, strictly.

B = - PV(K) N(d2)

Again, the cum dist is a factor of between 0 and 1. If 0, then 0. If 1, then B = -PV(K).
Present value of strike price can never be greater than K, and never be smaller than 0.
So, we are talking about a range of [-K, 0].

Question 34

Interpretation of ∆

Answer 34

Change in option value from a $1 change in the underlying

Question 35

Interpretation of B in BS

Answer 35

it is in the range [-K, 0].

Notice how for calls, we are always taking a long position in the stock, but a short position in the bond. In other words, we are borrowing money, and investing in the stock.

A leveraged position is always more risky than a non-leveraged. This implies that the call options are more risky than the underlying stock, and therefore have higher returns and higher beta.

Question 36

what happens if the world consists of only risk neutral investors?

Answer 36

The concept of risk compensation would not exist. We would get a world where everyone would base decision on the greatest expecgted return. Risk neutrality implies that the investor dont give a fuck whether the investment is risky or not. he only cares about the expected return.

Question 37

elaborate on the two state risk-neutral model

Answer 37

Assume we have a stock and all investors are risk neutral.

Assume the stock has two states: up or down. There is a probability associated with these states. The assumption is that if we know the probabilities, then we could compute the expected value, and use this to price the option.

Say the stock price is currently 50.
Up state is at 60.
Down state is at 40.

The risk free interest rate is 6%.

the expected value of the stock price would be equal to:

(p60 + (1-p)40)/1.06

Due to the law of one price, we know that this is equal to the stock price, which is 50.

Solveing for p gives us p = 0.65

Using this, we can compute the expeced option payoff:

(0.65x10 + 0.35x0)/1.06 = 6.5/1.06 = 6.13

This happen to be the exact same result as we got from the regular two state single period binomial option pricing model. using that model earlier, we made no assumption of risk neutrality.

This extraordinary outcome tells us that the model, both binommial and BS, work with any assumption regarding investor risk preference.

Question 38

Recall the important conclusion(s) from the risk neutral two state model

Answer 38

We end up having a model that does not depend on risk preference, probability, expected return. The models give the same value regardless.

Question 39

what happens to the expected return of secuirities if everyone is risk neutral?

Answer 39

the expected return of all securities would be equal to the risk free interest rate.

if we are all going after the largest expected return, it will drive the price of the security up, which will decrease the expected return as the capital gain rate decrease from the higher stock price, all the way until the expected return is exactly equal to the risk free rate

Question 40

Elaborate on risk neutral probabilities

Answer 40

So, if we have the case of stock price at 50, and two states at 60 and 40: If we made no assumption regarding the risk preferences of investors, we’d set up the formula:

50 = (60p + 40(1-p))/discountFactor

However, the challenge is:
1) We don’t know the discount factor. The discount factor depends on the risk of the security, right?
2) If we know the discount factor, we solve for p, and p will be the probability of the up state.

The goal would be to solve for the probability so that we can say something about the likelihood of reaching any state.
But since this depends on the discounting rate, we have to take investor preferences to risk into account. Since investors are generally risk averse, at varying levels, all we can say is that investors generally want compensation for additional risk.

However, in order to continue the analysis, we make the assumption that everyone is risk neutral. If everyone is risk neutral, they would not give a fuck about the risk, and always chase the highest expected return investment. Due to supply and demand, this will increase the prices of those investments, and reducing the price of the low expected return investments, until the capital gain rate of those investments are such that the expected return is equal to the risk free return. At this point, we can safely say that each investment will have the expected return of risk free rate, and since we know this, we also know the discounting rate.

in a risk neutral world, the two state case can be discounted with this risk free rate. But what does the result now tell us? It would tell us the probability required for the states in order to achieve an expected return equal to the risk free rate.
If we had no discounting, which is equivalent to required return of 0%, we’d get 50=60p + 40(1-p) which solves for p=0.5. This then yield 0.510+0.5(-10)=0. Which means, expected return of 0%.
So when we use the risk neutral assumption, we get the probability necessary to make the expected return of the two states equal to the required return of the risk free interest rate. Therefore, it is not representative of the real probabilities of the different states. I suppose that this is important because when we make the simplifying assumption of risk neutrality, we must not make the mistake of assuming that the probabilities actually represent the probability of achieving either state.

The probabilities are therefore called risk neutral probabilties.

The only reason for the risk neutral probabilities are because of the simplifying assumption we make regarding risk neutrality.

Answer 41

we know that there can only exist one fair price for the option. And this price is driven by the fact that if the current market price is not equal to this fair price, investors could earn an arbitrage by for instance acquiring a replicating portfolio of stocks and bonds. This will make sure that the current market price converge towards the fair price, by the principle of law of one price.

And then the insight is that the replicating portfolio has no requirement of probabilities. It has only requirement of providing the same payoff for the various states. Ideally, the replicating portfolio, through a dynamic trading strategy, always ensures that the payoff is exactly equal to that of the option.

This is why the principle/assumpiton of risk neutrality is legal/valid. We are essentially finding the expected value of the option, given that there is only one fair price.

Question 42

Does an option’s price/value depend on probabilities?

Answer 42

No, it only depends on the future payoffs. If we can replicate the payoffs, we’re good.

Recall that it is about finding a fair market price. If we happen to know insight, then this represent something that the market does not know. If the market knows, the stock price will adjust and make the case different. Probabilities should not matter. Volatilities does, however. in the binomial model, the up and down state is best created if the volatility of the stock is taken into consideration.

Question 43

What is the simplest way of computing the risk of an option?

Answer 43

Compute the beta of the replicating portfolio

Question 44

elaborate on the option beta

Answer 44

Weighted average of the delta beta and the bond beta.

Question 45

what can we say about the beta of an option (call) in relation to the stock underlying?

Answer 45

Since the bond is riskless, the beta of the option is a multiple of the delta portion:

beta = S∆/(S∆+B) x betaStock

What can we say about the multiple?
S∆+B is the call option’s value. This value is positive.
However, since B is always in the range of [-K, 0], the enumerator is always larger than, or equal to, the denominator. because of this, the beta of the option is always larger than, or equal to, the beta of the underlying.

Of course, this means that the option is more risky than the stock.

Question 46

What can we say about the beta of a put option in relation to the underlying stock?

Answer 46

We know that the S∆ is negative and the bond B is positive.
This will reverse the beta effect. This is not surprising, considering a put is a hedge against the stock’s movement.

Question 47

Define leverage ratio

Answer 47

Leverage ratio is related to the option, and refer to hte weight/factor/multiple we use to multiply by the stock’s beta. It essentialyl tells us how risky the option is in regards to the market.

Question 48

elaborate on risk neutral probabiltiies and all that

Answer 48

There are a couple of extremely important points here:
1) The current stock price reflect the current belief in regards to expected value. Meaning, if investors believed that the true probability of up-state move was 99%, and down was 1%, the current stock price should reflect this. either by placing itself closer to the up state etc

2) The current stock price, the current price levels of the up state and down state, and the risk free interest rate are all known in advance when we compute the option price. This allows us to make use of the principle of law of one price.

3) In a risk neutral world, each security is expected to earn return equal to the risk free interest rate. This is due to how everyone would always chase the highest expected value, driving up prices until the capital gain rate would offset the expected cash flows so that the total expected return is equal to the risk free rate.

4) The future value/payoff of the option needs to be discounted. It is difficult to know what to use to discount if we are not in the risk neutral world. Therefore, it is such a benefit of being there.

So, here is what happens:

We start by assuming risk neutral world. Then we replicate the portfolio using stock and bond investment. This price is well known and defined, and is the same in both worlds. it is the same in both worlds because the cost of simply related to the stock price, and the stock price does not change between the worlds. the stock price doesnt change, only the future expected/required return.

We use the value/price of the replicating portfolio as the price of the option. Now, we are done if we have a single-period binomial tree case. However, if not, we need to continue.

we use the price of the option to compute the probabilties of up state and down state in the risk neutral world. These are called “risk neutral probabilities”. These are no real probabilities, and I dont even know what they represent besides for the fact that they make the discounted expected payoff in the risk neutral world (of the option) equal to the price of the replicating portfolio.

Given these mathematical probability constructs, we can use them to compute the expected payoff, and therefore the price, of the option when multiple periods are concerned. we take the final (leaf) value of the option in the binomial tree, representing the payoff at this specific state, and multiply it by the risk neutral probabilities along the path towards the root node. We also need to discount appropriately using the risk free rate. then we do this for all leaves, finding the discounted expected payoff, which represent what we would be willing to pay for it.

Crucial point: the probabiltieis are not true, and therefore it is misleading to say that they are expected payoff.

however, we know that the final result is correct. Why? Because the replicating portfolio cost the same in both worlds. The replicating portfolio produce the same cash flow in both worlds, and this cash flow is equal to the option’s cash flow in either state. Therefore, the option must cost the same in both worlds.

A common question is what would happen if everyone knew the true probabilities of the up and down states. Theoretically, we would see increased demand for both the option and the stock, driving the prices up for both securities. However, we must notice what this actually do to our case. Now, the stock price is different. We actually assume that the stock price currently reflect the “static” equilibrium price reflecting the current informaiton.

Question 49

what is nice about risk neutrality

Answer 49

Each time we want to value a real option, we can use this theory on risk neutrality, and end up figuring out a way to price the option that also discounts using the risk free rate. This is very nice, as we don’t have to try to figure the cost of capital of theoretical decisions