Reading 58 & 59 LOS's Flashcards
LOS 58a: Explain how the concepts of arbitrage, replication, and risk neutrality are used in pricing derivatives.
Also explain some limits to arbitrage
The price or value of a financial asset is determined as the present value of expected future price + any benefits (dividends, coupons) - any costs (storage, transaction), discounted at a rate appropriate for the risk assumed (the risk-free rate with some risk premium). This fundamental value is compared to market price to see if there is profit potential.
Abitrage helps this price be found by trading an unequally hedged portfolio. Since the derivative is based on the underlying, if you hold both the actual asset and the derivative of that asset, in theory you should have a perfectly hedged portfolio.
- If the hedged portfolio generates returns in excess of the risk free, arbitrageurs borrow at the risk free and go long the hedged
- If the hedged portfolio generates returns less than the risk free, abitrageurs will short the hedge and invest in the risk free.
This trading will be done until the perfect hedged portfolio exists, returning only the risk free rate.
Replication refers to the exercise of creating an asset or a portfolio from another asset, portfolio, and/or derivative. Starting with what we just learned:
- Asset + Derivative = Risk-free asset
- Asset- Risk-free asset = -Derivative
- Derivative- Risk-free asset = - Asset
Risk Neutrality is the idea that derivatives can be combined with assets to produce the risk-free asset. So when discounting derivatives we use only the risk-free rate with no premium attached. This is because traders of derivatives are considered risk neutral. In fact derivative pricing is sometimes called risk-neutral pricing
Limits to Abitrage:
- significant transaction costs
- transactions require a lot of capital that the arbitrageur may not have access to
- transaction may requiring shorting an asset that is difficult to short
LOS 58b: Distinguish between value and price of forward and futures contracts
Price as it relates to forwards, futures, and swaps refers to the fixed price, agreed upon at the contract, at which the underlying transaction will take place in the future.
Value fluctuates in response to a change in current market price of the underlying ( ex. value increases for the long position as value of the underlying increases in current market price)
The takeaway here is that while value and price can be compared in the equity market, in the derivative market they are not comparable
LOS 58c: Explain how the value and price of a forward contract are determined at expiration, during the life of the contract, and at initiation.
LOS 58d: describe monetary and nonmonetary benefits and costs associated with holding the underlying asset, and explain how they affect the value and price of a forward contract.
The forward price
- At initiation is calculated as the current spot price of the asset times 1 + the risk free rate raised to however many holding periods
- At initiation with costs and/or benefits- Consider the costs and benefits and the periods in which they are incurred. Discount them by the risk free rate raised to the power of the period received. Then simply add/subtract from current spot price and multiply by the risk-free raised to the holding period
- During Life and Expiration the price will remain the same as it was at initiation
Valuing a Forward contract:
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At initiation the value should be 0, meaning both the long and short positions receive no immediate value. If this were not the case abitrage would be available:
- Cash and Carry arbitrage can be executed when the forward price is too high compared to the risk free rate. They can borrow at the risk free, purchase the underlying asset at the spot price, and then short the asset to sell at the high price in the future. When they sell the asset, they can pay back their loan (lesser value) and receive profit.
- Reverse Cash and Carry can be executed when the forward price is too low. You can short the underlying asset, and invest the proceeds at the risk-free. Then take a long position to buy the asset back at the low future price. You will be able to buy the asset back at a lesser value than the money you receive from investing.
- During the life For the long position it is the current spot price of the underlying plus/minus any benefits/costs compounded at the risk free rate minus the forward price discounted by the risk free rate raised to the current period. For the short position just flip the terms around the minus sign
- At expiration the value for the long is the current spot price minus the forward price and for the short it is the forward price minus the current spot
At expiration, the contract can be delivered in 2 ways:
- Physical delivery- the long position buys the asset at the forward price, and then can sell it to see profit/loss
- Cash settlement the difference between the price of the asset and the forward price is paid to the long
Example Sally is long on GOOG for $600 and Steve is short. At expiration GOOG is $800. Without considering time value for sake of simplicity, a physical delivery would be Sally giving Steve $600 and receiving the stock, which she can then sell for $800 to realize profit. A cash settlement would have Steve pay Sally $200, and he would keep the stock
Los 58e: Define a forward rate agreement and describe its uses
A FRA is a forward contract where the underlying is an interest rate (usually LIBOR). The long party benefits when interest rates rise above the FRA rate, while the short party benefits when interest rates fall below the FRA. EXAMPLE:
- Dealer (short) quotes a rate of 4% on an FRA and the end user (long) agrees.
- Expiration is in 90 days
- the notional amount is $5 million
- The underlying interest rate is the 180 LIBOR time deposit.
- In 90 days the 180 LIBOR is at 5%. That 5% will be paid in 90 more days.
- For the contract, the long receives a payment from the short since rates when up.
- Payment is calculated as such:
$5,000,000 x (( 0.05- 0.04) (180/360)) = $47,600
1 + 0.05 (180/360)
This formula at is base is :
((Underlying rate at expiration - Forward contract rate)(days in underlying rate/360)) /
1 + underlying rate (days in underlying rate/360)
If a company needs to borrow $ 1 million in 30 days for 120 days, and we are given the 30 LIBOR and the 150 LIBOR, we can find the rate of the FRA in the following steps:
- 1st unannualize the rates by multiplying them by (30/360) and (150/360) repectively.
- Then from here we know that (1+ 30LIBOR)(1+ 120LIBOR) = (1+ 150LIBOR)
- simple arithmatic will get us to ((1+150LIBOR)/(1+30LIBOR)) - 1 = 120LIBOR
- This rate will be the FRA agreement for a contract that begins in 30 days and expires in 120.
Takeaway— The FRA rate is really just a forward rate derived from the term structure of interest rates, even though the underlying is not an asset
LOS 58f: Explain why forward and futures prices differ
List Characteristics of Futures contracts
Future contract characteristics:
- Standardization- They are traded on exchange. The exchange specifies expiration date, size of contract, and all other terms besides that of the price. This makes future contracts acceptable to a wider variety of users
- Clearinghouse- every futures exchange has a clearinghouse which takes the opposite side of every tade. This comforts traders as they know they will be able to exit their positions, without having to worry about counterparty default
- Futures margins- futures contracts will have initial margin and maintenance margins. When the maintenance margin is broken, the investor receives a margin call and must replenish funds to the initial margin or have their position closed out.
- Marking-to-Market- is a process of adjusting the balance in an investor’s futures account to reflect change in the value of the futures position since the last mark to market. Most exchanges use daily.
- Price limits- some futures contracts use price limits to restrict the change in settlement price of a contract from one day to the next. If it breaks the celing or floor its called limit up or limit down and at that point the price is said to be locked limit
Forward vs Future prices
- if underlying asset prices are positively correlated with interest rates, any gains from mark-to-market can be reinvested at higher rates, while loses can be financed at lower rates. This idea causes future prices to be higher than forward prices
- If there is negative correlation then the opposite will be true
- If interest rates are constant, forwards and futures will have the same price.
In general the derivatives industry makes no distinction between futures and forward price
LOS 58g: Explain how swap contracts are similar to but different from a series of forward contracts.
LOS 58h: Distinguish between the value and price of swaps
A swap is an agreement to exchange a series of cash flows at periodic settlement dates over a certain period of time, known as the tenor of the swap. The simplest kind is a plain-vanilla swap. In this case a company has taken a loan from a bank at a floating rate but wishes to pay a fixed rate. They enter into a swap with a counterparty, where they agree to pay a fixed rate to the swap dealer in return for a floating rate. The interest payments are calculated off a notional principal, but the contract only has to deal with the rates. So the company (pay-fixed side) pays a fixed rate and receives from the swap dealer (pay-floating side) a floating rate.
Net fixed rate payment= [Swap fixed rate - (LIBOR + spread)] x (no. of days/360) x notional principal
Swaps are pretty much a combination of FRAs. For example a plain-vanilla interest rate swap, is a combination of FRAs, where one FRA expires on each settlement date over the tenor of the swap, and the FRA rate equals the swap fixed rate.
- On a given reset date, if LIBOR is greater than the swap rate, the fixed-rate payer would receive the interest savings (difference between floating and swap fixed rate)
- If LIBOR is less than the swap rate, then the fixed-rate payer will pay the interest savings.
Swaps can also be compared to issuing a fixed rate bond (on which fixed payments must be made) and purchasing a floating rate bond ( on which floating payments will be received). The same payouts work as above.
LOS 58i: Explain how the value of a European option is determined at expiration
The value of a European call option at expiration is as follows:
- the holder of the option has the right, not obligation to buy at the exercise price, while the writer of the option has the obligation to sell at the exercise price, if the holder calls.
- There is normally an option premium, that is a price the holder pays the writer for the option.
- If the spot price of the asset is above the exercise price at expiration, the holder will have a profit of the spot minus (the exercise + the option premium)
- If the spot price of the asset is below the exercise price at expiration, the holder will not call and thus lose the option premium
The value of a European put option at expiration is as follows:
- the holder of the option has the right, not obligatoin, to sell an asset at a certain exercise price, while the writer of the option has the obligation to buy the asset at the exercise price if the holder puts
- Again there will be an option premium
- The payouts will be the opposite of the call. If the spot if above the exercise, the option will not be used, and the holder will lose the option premium
- If the spot is below the exercise, the holder will receive profits of the exercise minus (the spot price plus the option premium)
LOS 58J; Explain the exercise value, time value, and moneyness of an option
The exercise value is the price at which the option can take place. The holder of the option has the right to buy/sell the asset at the exercise price.
Moneyness refers to whether an option is in-the-money or out-of-the-money
- For calls:
- In the money when the spot price is greater than the exercise price
- At-the-money when the spot and exercise price are equal
- Out-of-the-money when spot price is less than the exercise price
- For puts:
- In the money when the spot price is less than the exercise price
- At-the-money when the spot and exercise price are equal
- Out-of-the-money when spot price is greater than the exercise price
Time value- at any point in time before expiration, an option is worth as much as its exercise value. However on top of its exercise value, there is also an element of speculative value, the idea that the underlying will move in a beneficial way. This speculative value is known as the time value of an option and it increases with volatility of the underlying, but decreases as the option nears expiration ( known as time value decay)
LOS 58k: Identify the factors that determine the value of an option and explain how each factor that affects the value of an option
Value of the underlying
- the value of a call option is directly related to the value of the underlying
- the value of a put option is inversely related to the value of the underlying
Note the value of the underlying serves as an upper bound on the price of a call option. It would not make sense to pay more for the right to buy an asset, than what the asset actually costs.
Exercise Price
- Call option values and exercise prices are inversely related
- Put option values and exercise prices are directly related
Note that the exercise price helps form an upper bound for the European option. The best outcome is for the stock price to fall to 0, giving the holder of the put the exercise price as profit
Risk-Free Rate of Interest
- the value of a call option is directly related to the risk-free rate
- the value of a put option is inversely related to the risk-free rate
Time to Expiration
- The value of a European call is directly related to time to expiration
- The value of a European put can be either directly or inversely related to the time to expiration. The idea is that with a put the person is waiting on cash inflow from the sale of the good. If they have to wait a long time and the risk-free rate is high, the lower the present value of the cash inflow will be. Typically the direct effect holds, but the inverse effect can prevail if a longer time to expiration comes with a higher risk-free rate and if the option is deep in-the-money
Volatility of the Underlying
- both call and put prices are directly related to volatility of the underlying
Payments from the underlying and the cost of carry
- Benefits such as dividend payments, reduce the value of the underlying. Thus benefits are negative for calls and positive for puts
- Costs are the opposite. Costs are positive for calls and negative for puts
Lowest Prices of Calls and Puts
- the lowest value of a European call is either the greater of 1) 0 or 2) the value of the underlying minus the present value of the exercise price
- the lowest value of a European put is either the greater of 1) 0 or 2) the present value of the exercise price minus the value of the underlying
LOS 58l: Explain put-call parity for European options
Supposed we have two portfolios- A and B and they are comprised of:
- A has a 1) European call option on a stock and 2) a zero-coupond riskless bond that pays X at maturity ( this portfolio is known as a fiduciary call)
- B has a 1) European put option on a stock and 2) a share of the same stock ( this is known as a protective put)
The call, put, and bond have the same maturity and the same value X. The underlying on the call and the put is the stock
If the stock price is greater than exercise price:
- The call option is exercised for a payoff of Stock price - X, and the bond’s face value is paid X, therefore making payout of portfolio A, the stock price
- The put option is not exercised as it is out-of-the-money giving it 0 value, but portfolio B still has the stock, so its payout is the stock price
- Here we notice that the payouts are the same
If the stock price is less than the exercise price:
- The call option is not exercised because it is out-of-the-money but the principal on the bond X is still receive, for a payout of X for port A
- The put option is exercised for a payoff of X- Stock price, and the portfolio has the stock, so payoff for portfolio B is X aswell
*
LOS 58m: explain put-call forward parity for European options
This is very similar to the put-call parity, with the fiduciary call (call option plus zero-coupon bond) and a protective put ( put option plus underlying stock), but instead of holding the underlying stock with the protective put, we will instead 1) take a long position on a forward contract on the underlying stock and 2) hold a risk-free bond that has a face value equal to the forward price ( the fudiciary call is not a part of this). This lead to the protective put having a cost of the future value of the bond discounted at the risk free rate (determines prices paid for it) plus the put option premium.
At expiration if the stock price is greater than the exercise price, the pay off will equal the price of the stock:
- the long forward will be worth Stock price - Forward price
- The zero coupon bond will be worth the forward price
- the option will expire out-of-the-money
At expiration if the stock price is less than the exercise price, the pay off will equal the exercise price on the put:
- The long forward will be worth Stock - Forward price
- the zero coupon bond will be worth the forwad price
- the put option will be worth The exercise price - stock price
LOS 58n: Explain how the value of an option is determined using a one-period binomial model
LOS 58o: Explain under which circumstances the values of European and American options differ
American options have every characteristic of European options, plus they come with the added flexibility that they can be exercised at any time prior to expiration. Since American options can be exercised at anytime, the minimum value they should take is the exercise price. If it were selling for less, someone could buy it and immediately exercise it, earning risk-free arbitrage.
American calls offer more flexibility than European calls and therefore should not trade for a lower price than them. This will lead the American calls to always be worth more than there exercise price in the market
For example consider a deep-in-the-money call. You might think that if the investor didn’t want the call anymore, the best option woud be to simply exercise it. Actually the better option would be to sell it, as investors would be willing to pay more than the exercise price. The reason behind this is that there is limited downside to the option ( the premium paid to buy it), while there is unlimited upside.
In regards to puts, it would be most beneficial to exercise a put when a company is close or near to bankruptcy, as this would max profits. Since the American put can be called when needed, it is worth more than the European put which has to wait to the exercise date.
LOS 59a: Determine the value at expiration, the profit, maximum profit, maximum loss, breakeven underlying price at expiration, and payoff graph of the strategies of buying and selling calls and puts and determine the potential outcomes for investors using these strategies.
CALL EXAMPLE Lets take for example a share of TKS, that has an exercise option of $100, a option price of $5, and at expiration, TKS stock trades at $115. The option is in the money once it hits $100, but the investor doesn’t break even until the cost of the option is covered, when the stock hits $105.
Call holder’s perspective:
- If option expires out-of-the-money, the max loss is $5
- Breakeven occurrs at $105
- As the price rises, the profits are unlimited
- The holder will exercise the option if there is a positive payoff between the stock price and the exercise price
Call writer’s perspective:
- When the option expires out of the money, the write makes a profit of $5
- The breakeven for the writer occurrs at the same point, $105
- The writer’s losses are unlimited.
When the option is exercised at $115, the holder makes a profit of $10 while the writer losses $10. The sum of the buyers and writers profits will always be zero, as options are a zero sum game.
PUT EXAMPLE Once again we will work with TKS, the put having an exercise value of $100, costing $5, but this time the stock price is $80 at expiration.
Put holder’s Perspective:
- If the put expires out-of-the-money, the max loss is the cost of the option, $5
- Breakeven occurrs at $95 ($100-5)
- The holders profits are maximized if the stock price falls to zero, making her max profits $95.
- The option will be exercised if there is a positive payoff between stock and exercse price
Put Writer’s perspective:
- When the put option expires out-of-the-money, the writer makes a profit of $5
- The writers breakeven is the same at $95
- The max loss occurrs if the stock price falls to $0, making the loss $95
When the option is exercised at stock price of $80, the holder makes profits of $15, while the writer losses $15
2 TAKEAWAYS
- Call option holders and put option writers benefit when underlying asset prices increase. Call option buyers believe that the underlying asset is undervalues
- Put option holders and call option writiers benefit when the underlying asset price decreases. Put option buyers believe that the underlying asset is overvalued
LOS 59b: Determine the value at expiration, profit, max profit, max loss, breakeven underlying price at expiration, and payoff graph of a covered call strategy and a protective put strategy, and explain the risk management application of each strategy.
Covered Call strategy this involves owning the stock and writing a call option ( usually out of the money) on the stock. A covered call is written when an investor believes that the stock price will not increase in the near future. That way they can supplement the return on the stock, with the premium from the option. If they are wrong and the price does increase
Lets supposed that we write a call on a stock that we already own. We earn $8 from writing a call on the stock that currently trades at $100. The exercise price of the call is $110
- If the stock price is below $110 at expiration, the option won’t be exercised, we will continue to hold the stock and make $8 profit from the option
- We breakeven at a stock price of $92, the loss from the value of our stock equals the profit from the option
- Between the price of $100 and $110, our profit increases from $8 to $18
- Once the option becomes in the money, we have reached our max profit, $8 from the option, and $10 from the stock appreciation
- The max loss we could bare is $92, if the stock falls from $100 to $0, the only upside would be the $8 earned from selling the option
Protective Put Strategy this is a hedging strategy that protects a portfolio from falling in value below a particular level. Its constructed by owning a stock and purchasing a put option on the stock.
Lets supposed that we own a stock that is worth $50 currently and we want to protect ourselves against a decline, while not limiting our participation in any upside of the stock. We can accomplish this by purchasing a put on the stock with a strike of $50. Assume the option is available for $2.
- If the stock price exceeds the exercise price at option expiration, the put expires worthless, and we suffer a loss of $2 for the option
- IF the stock price falls below $50, we will exercise the put. The reduction in the value of our stock is offset by the payoff from the option
- Between the prices of $50 and $52, our losses will vary between $0 and $2.
- Once we cross the breakeven of $52, profits are made
