Chapter 11 - Options Flashcards
what can we consider options as
The option to back away from an unfavorable position. Futures, forwards and all that is an obligation. With options, we dont have this. But naturally, this flexibility has a price. So most of options is about trying to figure out how to price the advantage of being exposed to a certain upside potential, while not having the downside risk.
elaborate on the most important relationship between option prices
Put call parity is essentially about relating the price of a call to the price of a put (with the same strike) by relating them to other financial instruments.
We recognize that a call option has the same “payoff” at expiration as the combination of a long stock and put option.
Because the payoff is the same, we know the cost must be the same. If not, there would be an arbitrage opportunity.
It is temping to say that the price of the call must equal the price of the stock+put, but this doesn’t account for the fact that options work a certain way. A put is a right to sell shares. Since we already have the shares in the long stock+put position, we have all we need. But in the case of the call option, just simply buying the call doesn’t mean that we are “done”. If we want to exercise the call, we only have the right to buy for a certain fixed price. Therefore, we must pay the fixed price. however, since this price is to be paid in the future, we say that the cost is PresentValue(StrikePrice).
Now we can create the equality:
C + PV(K) = P + S
C - P = S - PV(K)
The interpretation of the last line is that if the strike price is very large, it will make the Put more expensive, and the call less expensive, and vice versa.
There is another way to build the relationship between calls and puts. The LHS of the line above can be interpreted as buying a call and selling a put. When we buy a put, we buy the right to acquire an asset, which we will naturally exercise if the asset value is above strike price. When we sell a put, we take on the obligation to buy an asset, should the price of it drop below the strike. As a result, regardless of the asset value, we will end up acquiring the asset for the strike price (assuming strike price is the same for the call and put).
Therefore, this acts as a forward, and we call it synthetic forward.
As a result, and a direct consequence of no arbitrage, we can establish the relationship that the cost of buying a call, less the benefit of selling the put, must be equal to the advantage of buying the asset later, rather than now. We pay the forward price, which we treat as the present value of the forward price, but by paying later we get payed time value of money, which is worth (in today) PV(K), where k is strike price.
The idea here is that the regular forward contract gives us the asset for the forward price. If we could buy a call, sell a put, and use PV(K) to make sure can pay for it, it could be used in arbitrage cases against the actual forward contract.
The whole case is that acquiring assets for a fixed price, which both of these methods do (although the forward price is/can be different from the strike price), they must cost the same. IF they do not cost the same, we can trade arbitrage (same assets, different price). And therefore we can establish the equality relationship:
C-P=PV(forwardprice - K)
what is the more sophisticated put call parity equation
C(K, T) - P(K, T) = e^(-rT)(F_{0,T} - K)
are put call parity referring to american options?
It is actually referring to european
elaborate on put call parity by considering the point where strike equals forward price and go from there
When the strike price is equal to forward price, we have a case where we are ultimately creatign a synthetic forward position to acquire asset. We are looking to acquire it for the forward price, which is usually just the psot price compounded by the risk free rate.
The idea is that doing this with strike equal to forward price is EXACTLY the same as a regular forward contract. Therefore, we expect nothing to be paid to us or us to pay out as a result of it. and since we do it synthetically using options, this means that whatever we pay for the call and whatever we get payed for the put, must equal each other out. This creates a position that is identical to the regular forward.
Think about why this is.
if it was not this way, then it would open up the possibility to make risk free profit. For instance, if the put pay us more than the call cost us, we’d make profit on the synthetic position. This means that we could create it, and at the same time short the forward contract to pay for it.
Same applies for hte other way around.
Now, if we move the strike price away from the forward price, it will shift the balance. For instnace, if we move the strike price below the forward price, it means that the call option is more likely to expire ITM. This is an advantage to us, because we bought the call. However, this is also a disadvantage for the put holders.
And from teh equation, we see that:
P - C = e^(-rT)(F_{0,T} - K)
… a decrease in K makes RHS larger. This makes LHS larger. Both P and C will be adjusted, with P growing smaller and C growing larger.
what can we say about the PV(F_{0,T}) term for options on stocks with dividends
e^(-rT)(F_{0,T}) = S_0 - PV(div)
what is the formula for put call parity when considering options on sticks that pay dividends
C = P + S_0 - PV(div) + e^(-rT)K
what is the formula for put call parity whne considering options on indices that have continuously compounded dividends
recall that dividends for indices are continuouslly re-invested, which will result in a new number of shares rather than cash.
If we are payed ∂ anually, we get:
e^(∂T) new number of shares.
This typically measn that if we invest S_0e^(-∂T) we have a tailed position that gets us 1 share at the end of the year.
For the options pricing, we arrive at:
C = P + S_0 e^(-∂T) - e^(-rT)K
what can we say about hte difference between the call and put?
Why is the price of a call for ATM always larger than price of PUT AMT?
It is equal to the itnerest rate of holding cash until expiration. If we lock in 50 bucks, we have to pay interest on this.
This is why the call is more expensive. We have to pay interest on the fact that we are deferring payment.
elaborate on the statement that calls are more worth because upside is ulimited while this is not the case for puts
while true, the statement is false. If calls are more expensive, it is because of the time value of money.
for the ATM options, this is easy to understand. We could pay now, or later. Since we purchase option, we pay later. And therefore, we pay the time value of money.
what can we say about american vs european options
since the american options have more flexibility, they can never be worth less than the their european counterpart.
can european call options be negative?
no, because they need not be exercised.t
what is the maximum value of a european call option?
The stock price itself. The worst case scenario is that we imply pay for the stock.
elaborate on early exercise
American call options on non-dividend paying stocks should never be exercised early becasue of the time value of money.
explian this notation:
C(S, Q, T-t)
the call price when considering some asset S that we are giving away in exchange for asset Q when there is T-t time left.
This is a converntion that is abstract, but represent two assets that are exchanged. Typically for financial options, S is the stock price while Q is the strike price. Thus, it represent how we receive S while giving up Q.
For call options, the payoff at the point in time when there is 0 time left:
C(S_T, Q_T, 0) = maximum(S_T - Q_T, 0)
Opposiute for the put.
what is the most general way of stating the put call parity equaiton that we use
C(S_t, Q_t, T-t) - P(S_t, Q_t, T-t) = F^P_{t, T}(S) - F^P_{t,T}(Q)
We are buying a call, which entails us receiving S_t in exchange for Q_t. we sell a put, which entail us buying Q_t in exchange for S_t.
The prepaid forward price of the assets represent their values when deferring payment, including all kinds of dividends etc.
discuss the distinction between calls and puts
Calls and puts are determined on what we choose to label as the asset we buy, and the asset we sell.
If we buy a call regular call, we exchange our cash to receive stock. but from the perspective of someone who is looking to purchase cash, this option is a put. Why? because the derivative increase in value if the strike asset is worth more than the regular asset.
For us, the option is worth more if the regular asset is increasing in value relative to the strike asset.
But if we look from the perspective of the strike asset, it is the other way around.
For instance, if the regular asset is a house, and the strike asset is a car: usually, we’d consider this a call, and say that if the house is worth more than the car, we earn a profit.
But, if we consider it the other way around, and we want to acquire the car for the house, we will make money if the value of the house drop relative to the car. So it is sort of determined on what we label as the strike asset.
give the proof of why early exercise on non-dividend paying american call option is never beneficial
we use the original parity equaiton:
This is for european options.
C(S_t, K, T-t) - P(S_t, K, T-t) = PV(F_{t,T}) - PV(K)
re-arranging we get:
C(S_t, K, T-t) = P(S_t, K, T-t) + S_t - e^(-r(T-t))K
C(S_t, K, T-t) = P(S_t, K, T-t) + S_t - K + K(1 - e^(-r(T-t)))
Now we can isolate S_t - K. We basically have that the value of the put, which can never be negative, plus K(1-e^(-r(ttt))), which can never be negative since exponent is negative, we can conclude that the CALL is always greater than the (S_t - K), which represent the exercise value.
Then we also know that the american value must be greater than the european. Therefore, the american call option is never good to exercise early either.
in the result of the proof of why early exervise is not beneficial, we decompose the price of a call option into 3 parts. Elaborate on these
We have the exervise value, S_t - K.
Then we have the implicit put protection that protects us from downside movement.
Thirdly, we have the time value of money. This is listed as K(1-e^(-r(T-t)))
Therefore, by exercising early, we throw away the protection AND we accelerate the payment by neglecting that we have time value.
what can we say is the short conclusion regarding the early exercise of american call options?
We should never see them sell for less than S_t - K
give the formula for put call paritty in the case of dividnds.
Can we say that early exercise is stil never beneficial?
We end up with:
C(S_t, K, T-t) = P(S_t, K, T-t) + S_t - PV(div) - PV(K),
where the present values are in regards to time t, of course.
The case now is that there may be cases where the dividends are greater than the interest we’d get from deferring payment. The thing is that we have to exercise if we want the dividends, so simply selling the option further is no bueno as the new buyer would have to exercise as well if he want the dividends.
Recall what we did for the no dividend case. We had only positive terms, except for (S_t - K). It is easy to conclude that the price of the call therefore would always be larger than this difference, which makes is never beneficial to exercise early.
However, now we do have a negative component in the dividends.
We can conclude by saying that if early exercise is optimal as a result of the dividnesxd, we should wait as long as possible, which is the day before ex-dividend date to do so.
elaborate on early exercise for puts
THe case for puts is different than for options. Suppose the firm’s share goes to 0.
if we do not exercise, we will end up with PV(K), taken relative to timestep t.
However, if we exercise immediately, we get K. Why would we want this?
because if the interest rate is positive, we would get K > PV(K).
can parity give a condition where we will early exercise for puts?
no. The case is that we can not say whether we should or shouldnt early exercise.
For calls, the case is simple as we could rule out early exercise. For puts, this is not the case. We cannot say that we should early exercise. We can only say that parity cannot rule it out.
generally speaking, what can we say is a neceassry condition for early exercise?
the necessary condition is that we want something now, ratehr than later. For stock returns without dividneds, there is no reason to have this wish. This is why calls never should be exercised early.
BUT: If there are dividends, we may want them more than we want the risk free return. In such case, there may be a case to exercise early.
is it always the case that adding time to an option will increase its value?
generally, but not always.
For american options, this is always the case for both calls and outs. This is because both calls and puts have the ability to be exercised early. And therefore, the additional time is simply more flexibility.
For european options, it is more difficult.
Consider a liquidating dividend. If you cant exercise prior to the dividend, you end up with a worthless option.
For european puts there is the case with the bankrupt company. If bankrupt, we’d have to wait before we receive the strike price. It is obviously beneficial to receive it sooner.
why is the binomial option pricing model called binomial?
It conists of a series of steps where each step assume that the price can only move up or down. this is a binary choice.
why is the binomial model so simple?
The assumption of binary stock price movement
OG name for the binomial option pricnig model
Cox-Ross-Rubinstein pricing model
there is a slight problem with the usage of buying shares as far as the binomial model is concerned. Elaborate
Buying fractional shares. Instead, we need to multiply by an order of 10 to get non-fractional shares
elaborate on the rationale behind the binomial option pricing model
We consider an investment in stock, where some of it is used with borrowed money. The trick is to create a portfolio of stock and bond so that the binomial outcomes for the option and the replicating portfolio is the same.
We basically start by saying that we buy X amount of shares, and borrow Y amount of cash. The cost of this position is given as XPriceOfStock - Y
Then at expiration we sell the shares and repay the debt (with interest).
Another word for the replicating portfolio
Synthetic call/put
What do we call the “thing” we use to argue that the price of the option must equal the price of hte synthetic option?
The law of one price
Can we say anything about the expected return of the option?
We borrow money to fund a position in the stock. Since stocks has positive risk premium, we increase the risk with the borrowed amount, we have a leveraged position. Recall why it is leveraged: The possible returns are amplified in relation to the original investment amount by the usage of leverage. As a reuslt, we should expect a higher return
If we consifder continuous dividend, how many shares do we get if we invest in time t, and we consider a duration to t+h?
we get the multiplier of e^(∂h) shares. If we invested in 1 share, we now have e^(∂h). if we invested in e^(-∂h) shares, we know have 1 share.
what is the value of the replicating portfolio at time h?
Consists of 2 parts: Stock and bond:
S_h∆e^(∂h) + Be^(rh)
The stock price is now S_h, which we multiply by the number of shares we bought. We also use the second share multiplier.
Note that ∆ and B are negative/positive depends on whether it is a call or put.
give the equations that we solve to find price
we have 2 equations that we solve as a system:
S_u∆e^(∂h) + Be^(rh) = C_u
S_d∆e^(∂h) + Be^(rh) = C_d
Then we solve for ∆ and B. First we subtract 2 from 1:
S_u∆e^(∂h) - S_d∆e^(∂h) = C_u - C_d
∆(S_u - S_d) = e^(-∂h) (C_u - C_d)
∆ = e^(-∂h) (C_u - C_d) / (S_u - S_d)
B is really ugly
what can we say about “u” and “d”?
We must have the following relationship:
u > e^((r-∂)h) > d
Suppose there are no dividends. The middle would be e^rh.
If u is less than e^rh, it means that we get more cash by holding risk free asset than from up movement in the stock. If this was the case, we should short the stock and invest in risk free.
the same would apply if “d” was larger than e^(rh). then we’d borrow all we could to buy the stock, because the stock would guarantee to deliver so that we could re-pay the debt.
interpretation of ∆
sensitivity of the option. It provides a direct relation between a one-point movement in the underlying and its impact on the option.
If we are only interested in the price of the option, what can we do?
The formula doesnt actually require us to compute ∆ and B, so we can solve it directly.
give the formula for risk neutral probabilities
give the rationale behind risk neutral proibabilities
Assume everyone is risk neutral.
this would cause everyone to chase the highest expected profit.
this would cause eveyr price to balance returns out so that every expected return is equal to the risk free rate.
because everything is equal to the risk free rate, we know what the expected return for our option is.
we try to find probabilities that make the return equal to the risk free return.
pSu + (1-p)Sd = e^(r-∂)t
given the risk neutral probabilities, how to find the call price?
how do we determine u and d in the binomial tree?
recall that “h” is the length of the binomial period in years. sigma is the annual volatility.
given two stock prices, how do we find the continuously compounded returns in the interval between them?
we use the ln function:
If we have the continously compounded return and the stock price, how do we find the next period stock price?
we use the exponential function.
what is a very useful property of continuously compounded returns?
They are additive.
elaborate on how we relate volatility in differnet time frames
Var(r_annual) = Var(∑r_periodic)
IF WE ASSUME THE RETURNS TO BE UNCORRELATED, the variance is the sum of individual variances. Recall also that this only works for contniuously compounded rates. Otherwise we dont have the euler notation and all that.
Var(r_annual) = Var + Var + …
Var(r_annual) = Periods x Var(r_periodic)
for instance:
Var(r_annual) = 12 x Var(r_monthly)
then we can get the volatility:
volatility(r_annual) = volatility(r_monthly)x sqrt(12)
This also means:
volatility(monthly) = volatility(r_annual) / sqrt(12)
key result of the volatility in various time frames
Volatility scales with the square root of time
assume there is certainty in the future stock price. What can we say about the movements u and d in the binomial tree?
Must be equal to the forward price.
then add uncertainty. what will our prices in binomial tree be?
we multiply the forward price by a factor that either has a compounding effect or a discounting effect.
F_{t, t+h}e^(+-(sigma sqrt(h)))
This is the same as saying:
1) for up movement, we set the price equal to the forward price multiplied by the compounded (continuous) effect of volatility.
2) same but discount
if h=1:
Therefore, we are basically saying that if the volatility is X%, then in this period the movement is restricted to be either the forward price compounded by continuois return of X%, or discounted by X%.
This is the same as saying that we expect the stock to increase with APR equal to X% etc.
Ultimately, we end up with: uS = F_{t,t-h} e^(sigma sqrt(h)) and negative for dS.
what does zero volatility mean?
It means that the prices are known in advance. It does not mean that there is not movement.
what parameter is the most difficult in the binomial model?
Volatility. It is difficult because we cannot directly observe it. we have to estiamte it.
define historical volatility
historical volatility is the result we get when we try to estimate volatility using previous stock returns. We compute the standard deviation of previously continuously compounded historical returns.
should we base volatility on total returns, or only capital gain?
the payoffs for both american and european options depends on the ex dividend price. so the calculation should exclude dividends.
there is a problem with the binomial problem, that we solve easily. What is this problem?
Doesnt account for early exercise.
We can do this by checking each node against the value of exercising at that point, and simply setting the value of the option at that point to be the maximum of holding and exercising early.
what are the economic factors that play a role in determining early exercise
1) We receive the stock, and therefore earn dividend
2) we pay the strike price early, which means that we are giving up interest. I suppose if the interest rate is expected to drop, we may benefit from doing this instead.
3) we lose the insurance provided by the position. For instance, a call option that is exercised can have large losses in the period between the exercise date and the would-be-maturity date. If we decide to not exercise, the loss is limited to 0 (or the premium).
CASE:
call option with strike 100, 1 year maturity, interest rate 5%, continuous dividends of 5%, and the stock is curently trading at 200. What should we do?
This case has dividends, which is a benefit if we exercise early, but it also has the disadvantage of giving away the strike price interest.
I suppose there is the time value as well, but this is not covered.
If we exercise now, the dividend we’d get is: 200 x e^(5%)-200 = 10.25 ish.
The strike price interest is 100 x e^(0.05) - 100 = 5.127
Balancing these two against each other would indicate that the advantage is in exercising early.
in the previous case, we decided to early exercise the option because the dividends were greater than the interest received on the stirke price. is this the whole story?
No. Recall that there were 3 big factors we need to consider when we consider early exercise:
1) Strike price interest
2) Dividends
3) The insurance provided by the option
The insurance is valuable. the longer the duration, the more valuable it will be.
what can we say when volatility is zero
if volatility is zero, the insurance abilities of the options will have no value.
In such cases, it is simple to create a deciding rule on when we should exercise the option. it is simply saying that we exercise if dividend yield multiplied by share price is greater than the itnerest multiplied by the strike price.
compare calls vs puts in regards to early exercise factors
same, but we reverse the dividend and interest advantage. The insurance is still valid.
But if you hold a put, you currently receive dividends and can trade it for interest.
if we use the true probability distribution to find the expected payoff of an option, we run into an issue. explain the issue
We end up not knowing how to discount the result we get. This is because using the alpha is wrong because alpha is the risk measure of the stock, not the option. the option is a leveraged position in the stock, whcih means that it needs to be discounted more heavily.
What we need to do, is compute the weighted average rate of the components of the option, which is the bond position and the stock position. The weighted average gives us the discount rate.
However, it turns out that doing this actually give us the same price for the option as the risk neutral way. therefore, there is no reason to do it more complicated.
is risk neutrality the reasoning or the interpretation?
Interpretation. We arrive at the formulas in an other way
can we use discounted cash flows with options to price them?
Yes. but in practice, this is not done because there is simply no reason to do it.
tt
Q_h/(Q_h+Q_l) can be regarded as a weight that tell us how favorable the high state is compared to the weighted average of the high state and low state. since high state is typically less desirable with averse risk, we can expect p* to be lower than its counterpart (1-p*)
elaborate on the entire explanation/intuition of risk neutral probabilitites
we start by saying that “we want a relationship between two assets that make sure there is no arbitrage opportunities”.
In general, we establish an abstract and simple relationship of “asset price today is equal to the discounted expected value of the asset price later”. The issue with this is that it doesn’t really tell us anything. If we used the required return to discount and to find the expected value of the future asset price, which would be correct, we would just receive a circular argument of saying that the current asset price is equal to the present value of the future asset price. The goal was to establish a price that make sure no arbitrage exist, and so far we haven’t really said anything.
But, if we introduce the idea of a forward, we can start saying something. For instance, if we know that the expected asset price in the future would be equal to the forward price of this asset, we are closer to establishing a direct link between the asset, and the forward on this asset. If successful, this is all we need in order to ensure that those two assets, or asset and instrument, are priced in a way that makes sure that we cannot use them together to create a risk free position with no initial investment.
So we want to say that “Asset price now equals the present value of the forward price of this asset”. And since we know that the expected value of the forward price is greater than the current price by a factor of e^(rT) where r is the risk free rate, the discounting is simple to do.
However, we still have not solved the issue of actually making the expected asset price in the future be equal to the forward price of the asset. Since it is expected value, we can model it using ∑p_i state_i, and we know that this must equal the forward price.
This leaves us with 2 options: 1) Change the states, which means changing the outcomes. 2) Change the probabilities. In regards to changing the states, this doesn’t make much sense to me. If we have continuous price movement, we have states that perfectly create a continuous range of possible future values. We can’t change those. Therefor,e we must change the probabilities.
What we now know, is that we can establish a direct connection between the price of an asset, and the price of its forward contract, by using a simple discounting expected value, BUT we need to adjust the probabilities.
What these probabilities are, is not immediately obvious. In the case of a binomial model, we can solve directly for them and receive a value for them. This value is actually all we need to use the concept, but it doesn’t provide much interpretation in itself.
Since we are able to find the new probabilities in the binomial case, we can price assets in a way that ensure no arbitrage. This is very valuable, because it doesn’t rely on anything other than observing a single price, for instance the forward price, and determining what the actual asset price should we, and make a simple decision based on this.
The reason why this is usually referred to as risk neutral probabilities, is that by mere chance (or as a non-intended outcome) the new probabilities that create the desired connection between the assets, are exactly those probabilities that WOULD BE USED if all investors were risk neutral.
So if we try to relate this abstraction to something like options, here is my understanding:
We have the option which we wish to price. Then we have the forward price of the option. Although the forward price of the option is not observable, we can observe it by considering what happens at expiration. At expiration, we know the payoffs. This payoff will be our forward price, meaning: If we only had a single state with certainty, the payoff at this stage+state would be the forward price of what the option price is. Since we have 2 states at each stage, we compute expected value using the probabilities that achieve the desired result, which we know is the adjusted probabilities. (check this: If we did not use the new probabilities, we would not be able to say that the option price at expiration is the forward price etc). So we are allowed to take the expected value using the adjusted probabilities, and at each stage backwards in the binomial tree, we’d simply consider each stage a forward price (expected forward price) and therefore be able to ultimately discount using the risk free rate.
so we end up calling it risk neutral pricing because this is a sort of intuitive way of explaining the ultimate behavior of the system, but the actual explanation is that we relate the asset to the forward price of said asset, and by doing so we achieve behavior that would be the same as in a risk neutral world.
The only reason why this approach is actually viable, is because we are able to compute the new probabilities when there are only two states. If we had more states, we would not be able to do this. And if this is the case, we’d still have the same issue of figuring out what the actual probabilities should be. These probabilities would incorporate things like risk aversion and true probability, and this is extremely difficult to compute. The advantage is that by assuming the only possible future states, we have enough observed parameters to compute the binary case new probabilities.
OR: to be clear, black scholes solves the issue of more states. The key is that we need a way of computing the probabilities that doesnt require us knowing the risk preferneces.
we always have the possibility going long the asset and short the forward on the asset. This creates a scenario where we will earn risk free profit if the asset price is less than the forward price discounted using the risk free rate.
If we short the asset and long the forward, we make money if the asset is currently higher than the discounted forward price.
From this alone, we can establish the following:
current price must equal the discounted forward price.
the actual explanation of risk neutrality
we know that the only price of an asset that prevent arbitrage is the price that corresponds to the forward price, so that it is not possible to enter a position in the stock and the opposite in the forward, and make risk free profit based on it. Since the risk free profit rate is the risk free rate, we know that this rate determine the time value of money. In other words, if we invest X today, we get Xe^(rT) in T time.
This relates to the forward contract because if the asset price currently is X, the forward price must be Xe^(rT), otherwise there are arbitrage opportunities.
This is the relationship we need to enforce.
Price today = e^(-rT) ForwardPrice.
This relationship is actually finished. If we have the forward price, we automatically know the no-arbitrage price of the asset.
However, to make it more relatable, we find an expression for the forward price. As I see it, the keyword is “expression for the forward price”.
if we know that the asset price can only be in one of two states at time T, we know that there is a probability distribution that makes the expected value be equal to the forward price of the asset. this has nothing to do with expected returns at all. It has nothing to do with the states or the probabilities. We only care about the fact that we can represent the forward price as a function of the possible states and some probability distribution.
The reason why we dont care about the possible states, is that we are only interested in creating an expression on the shape of Price = e^(-rT)forwardPrice.
when we insert the expression for the forward price, we get:
price = e^(-rT)∑p_i s_i,
I suppose it is “ok” to use a stochastic process because asset returns are not certain, and should never be expected to be. However, their averages are typically known to a certain extend. And there is always a way to assign probabilities to states that create expected return equal to the risk free rate.
Since we actually dont really care about what ∑p_i s_i is, but only care that it should be equal to the forward price, we are free to choose p_i’s whatever we need in order to make the expected value be equal to the forward price.
The benefit is that the p_i’s are relatively easy to compute compared to the true probabilities. But that is not that relevant, because we were never interested in computing the true probabilities nor the true returns.
we only want an expression for the forward price.
There are obvious advantages to having a simple expression for the forward price. In some cases, we might rely on the observed value on an exchange. But in many cases, there is no option to do this. Therefore, the mathematical expression is great.
Sum up: We have defiend the relationship we wish to create that ensure no arbitrage. This is :
Price = e^(-rT)ForwardPrice.
to make it applicable, we need a more usable expression for the forward price.
price = e^(-rT) ∑p_i s_i
GIVEN the fact that p_i’s are easy to compute (which they obviously are in the case of only 2 states and binomial model), we now have an expression for what the price should be that ensure no arbitrage.
the model doesn’t give a fuck about what p_i’s are. Although they can be interpreted, all that matters is that we can compute them.
walk through pricing a call using the no arbitrage framework, and tie it to risk neutral probabilities
for pricing an option, say a call:
call = e^(-rT)ForwardCallPrice
Let us assume we have only a single-period binomial model.
We start at the back, where we can consider the payoff states independently.
the up state has a payoff, and this payoff represent a lump of cash that we’d be willing to discount using “r” and pay that discounted amount for. So, if there was only this payoff with certainty, the forward price would be this payoff, and the price of the call would simply be e^(-rT)payoff.
But, we need to consider the other state as well.
The downstate has a different payoff.
We take these two payoffs, and we run them through a probability distribution so that the expected value is equal to the risk free rate. to do this, we’d need the actual value of the underlying. So, we take Sup + Sd(1-p) = e^(rT), which gives us the ability to compute the probability p, and thereby also 1-p. With these probabilities, the forward price is simply payoff1p+payoff2(1-p). With this value, we can discount it to get the call price.
we are ultimately looking for a way to establish an expression for the forward prices. the discounting is trivial, and the no arbitrage connectoin is trivial as well, but the work lies in determining an expression for the forward price. and for options, the forwadrd price can be determined by considering what we’d pay for a certain payoff some time earlier.
what is actually the put call parity equation saying
it is based on the idea of a synthetic forward.
buy a call and sell a put to create a position that has the same properties as the regular forward (as long as strike equals forward price).
If the position does not equate forward price with strike prices, then we have a slight imbalance. For instance, if we place the strike lower than the forward price:
Since we fix a price (for the underlying asset) lower than the forward price, if it didnt cost anything to establish the position, we could short the actual forward, but long the synthetic forward. Since the synthetic forward is cheaper for us in regards to the price we end up paying, we acquire the assets from the synthetic forward, and then deliver the assets to the counterparty to the actual forward because he was willing to pay more for them. This obviously is arbitrage, so we assume such opportunities does not exist for long.
If the strike price doesnt match the forward price, there is an advantage at expiration to one party.
The benefit is quantified as:
(forward price - strike). This is an advantage in the hands of the long synthetic forward position. Therefore, we know that the cost of acquiring the position initially must offset this.
This gives rise to the equation:
C - P = PV(Forward price - Strike)
elaborate on the alternative way of deriving the put call parity formula
We consider a long position in stock, and use a protective put on it. This creates a piecewise linear payoff curve. Linear as the stock typically is above the strike price, 0 for spot prices below strike.
A call option has the payoff curve as the long stock+put.
so they must cost the same.
StockPrice + Put = Call
But we also need to consider the cost of exercising. For the put, we already have the stock to give away, so there is no additional cost. But for the call, we need a sum equal to the strike price. Therefore, we invest PV(K) now, to get K at expiration.
This gives us:
S+P=C+PV(K)
Alternatively:
C - P = S - PV(K)
How do we account for dividends in the put call parity formula? both accounts
for both the case of discrete and continuous dividends, we simply subtract the value of the dividend from the cost of the forward price.
This is because we naturally want to pay less for the forward when we miss out on something.
what can we say about the prices of ATM call vs put
The difference is due to the interest on the strike price.
The call require us to have the strike price ready, while the put doesnt.
We interpret this as the benefit of avoiding early payment. IF we are to pay later, we will basically pay the time value of money. And this is the interest on the strike price. By acquiring a long synthetic forward contract, we defer payment, which is an advantage.
Since we have bought a call and sold a put, we need the strike regardless. We invest the strike price in risk free asset, and receive risk free rate. Since we get this, it must cost sometihng, and this is the difference in the premiums.
can ATM call sell for less than ATM puts?
Yes, if dividends are involved.
for calls with dividends, when will we for certain never exercise early?
if the interest on the strike price is greater than the dividends, htere is no reason to exercsie rearly
elaborate on early exercise on puts
by exercising early, we receive K. there are certain cases where it is beneficial to receive K early. Recall that we receive K and pay spot.
For instancei, if the firm is bankrupt, we’ll receive K regardless. However, we’d like to get it early because then we can invest it. If not, it will just depreciate in value.
elaborate on the value of american and european options as we change time to maturity.
For american, more than is always more valuable.
For european, we may miss liquidating dividends etc. Not good. Therefore, there are cases where the european option can lose value with longer TTM.
