· Options are priced on a risk-neutral basis, so a long call (for example) would be paired with a short stock.
· A short-stock position generates interest revenue, which makes the call option more valuable. If interest rates go up, the interest revenue from the short stock position increases, which makes the call worth even more. For put options and dividends, it works in the opposite direction.
· When interest rates are high, the prices of calls are higher and the prices of puts are lower. When rates are high, the option itself is more attractive than the underlying asset. By purchasing an option instead of the underlying asset, an investor saves cash.
· Puts are adversely affected by higher rates because investors lose interest while waiting to sell their underlying assets. This works for all underlying assets except when dealing with bonds or interest rates.
· Interest rate volatility has a huge effect on option prices. When volatility increases, call and put prices both increase because of the increased possibility that a downside or upside event could occur concerning the option. The upside helps call price and has no effect on puts, while downside helps puts with no effect on calls and is especially true when options are out of the money. Downside does begin to matter when options become in the money.
Showing posts with label Options Markets and Contracts. Show all posts
Showing posts with label Options Markets and Contracts. Show all posts
Monday, January 5
Lower bounds of Option Prices and the possibility of early exercise.
Lower bounds for American and European Option Prices
EC0 ≥ Max{0, [S0 – PV (CF,o,T)] – X/(1+r)T}
EP0 ≥ Max {0, X/(1+r)T – {S0-PV(CF,0,T)}
Where: C = Call, S = Strike price, PV = Present value, CF = Cash Flow, r = interest rate, T= Time to expiration of the option
Cash flows for underlying assets are as follows:
· Stocks pay dividends - in formula terms FV (D,O,T) or PV (D,O,T)
· Bonds pay interest - in formula terms FV (CI,O,T) or PV (CI,O,T)
· Currency pays interest
· The underlying price is reduced by the PV of the cash flows of the underlying; therefore, the put-call parity relationship is calculated as:
EC0 +X/(1+r)T = EP0 +{S0 – PV(CF,0,T)}
The Relationship between American and European Options
· At any given point in time, the value of a call option or a put option cannot exceed a particular price.
· A call option can never be worth more than the stock price; therefore, the value of a call option should be lower or equal to the stock price.
· In the case of a put option, the upper bound is the strike price at which the contract has been entered. The value of a put will be lower than or (at most) equal to the strike price of the option. If this condition is violated, an investor can make use of the arbitrage opportunity by writing the option and investing the proceeds at the risk-free rate of interest.
· The lower bounds for an American call option are the same as the lower bounds for a European call option. Prices of American and European options differ mainly in whether they can be exercised before expiration, as is the case with American calls. As such, in most cases American calls and puts will be worth more than European calls and puts and their lower bounds will also differ.
· When the underlying asset does not make cash payments such as dividends or interest payments, the value of an American call option is equal to a European call option.
· When cash payments are involved, the value of an American call option tends to be higher than a European call option.
· American put options are almost always worth more than European put options.
· Cash flows on the underlying security and their effect on the put-call parity and the lowers bounds of options prices
Notes:
· Currency pays interest
· Commodities have carrying costs
EC0 ≥ Max{0, [S0 – PV (CF,o,T)] – X/(1+r)T}
EP0 ≥ Max {0, X/(1+r)T – {S0-PV(CF,0,T)}
Where: C = Call, S = Strike price, PV = Present value, CF = Cash Flow, r = interest rate, T= Time to expiration of the option
Cash flows for underlying assets are as follows:
· Stocks pay dividends - in formula terms FV (D,O,T) or PV (D,O,T)
· Bonds pay interest - in formula terms FV (CI,O,T) or PV (CI,O,T)
· Currency pays interest
· The underlying price is reduced by the PV of the cash flows of the underlying; therefore, the put-call parity relationship is calculated as:
EC0 +X/(1+r)T = EP0 +{S0 – PV(CF,0,T)}
The Relationship between American and European Options
· At any given point in time, the value of a call option or a put option cannot exceed a particular price.
· A call option can never be worth more than the stock price; therefore, the value of a call option should be lower or equal to the stock price.
· In the case of a put option, the upper bound is the strike price at which the contract has been entered. The value of a put will be lower than or (at most) equal to the strike price of the option. If this condition is violated, an investor can make use of the arbitrage opportunity by writing the option and investing the proceeds at the risk-free rate of interest.
· The lower bounds for an American call option are the same as the lower bounds for a European call option. Prices of American and European options differ mainly in whether they can be exercised before expiration, as is the case with American calls. As such, in most cases American calls and puts will be worth more than European calls and puts and their lower bounds will also differ.
· When the underlying asset does not make cash payments such as dividends or interest payments, the value of an American call option is equal to a European call option.
· When cash payments are involved, the value of an American call option tends to be higher than a European call option.
· American put options are almost always worth more than European put options.
· Cash flows on the underlying security and their effect on the put-call parity and the lowers bounds of options prices
Notes:
· Currency pays interest
· Commodities have carrying costs
Protective Put
· aka portfolio insurance or hedged port
· buy a long position in underlying securities + buy a put
· protects the downside risk at a cost
· Appropriate for limiting downside risk when upside potential is still present
· Think stock may go down in near future, want to preserve upside potential
Profit = max(0,X-ST)+ST-S0-P0
Max Profit = ST-S0-P0 (no upside limit)
Max loss = -X+S0+P0
Breakeven price = X+P0
· buy a long position in underlying securities + buy a put
· protects the downside risk at a cost
· Appropriate for limiting downside risk when upside potential is still present
· Think stock may go down in near future, want to preserve upside potential
Profit = max(0,X-ST)+ST-S0-P0
Max Profit = ST-S0-P0 (no upside limit)
Max loss = -X+S0+P0
Breakeven price = X+P0
Covered Call
Friday, January 2
Synthetic strategies
S0=C0 - P0 + Xe-rT
P0=C0 - S0 + Xe-rT
C0= P0+ S0 - Xe-rT
Xe-rT = P0+ S0 - C0
The right hand sied is the synthetic equivalents of the securities of the left.
P0=C0 - S0 + Xe-rT
C0= P0+ S0 - Xe-rT
Xe-rT = P0+ S0 - C0
The right hand sied is the synthetic equivalents of the securities of the left.
Put-Call Parity
· no arbitrage relationship for European-style put & call options that have same characteristics
· a port consisting of a call & a zero coupon bond (earning risk free return) with face value equal to strike = portfolio consisting of a put & a stock
· true only for European options
· the relationship must hold, otherwise arbitrage profit will be produced.
P0+S0=C0+Xe-rT
· a port consisting of a call & a zero coupon bond (earning risk free return) with face value equal to strike = portfolio consisting of a put & a stock
· true only for European options
· the relationship must hold, otherwise arbitrage profit will be produced.
P0+S0=C0+Xe-rT
Put Option
Payoff
PT=max(0,X-ST)
Breakeven at X-P0
Notes:
Both Call & put, payoffs & profits is linear function of ST
Call, infinite payoff -> no upper limit
Put, upper limit to the profit is X-P0
PT=max(0,X-ST)
Breakeven at X-P0
Notes:
Both Call & put, payoffs & profits is linear function of ST
Call, infinite payoff -> no upper limit
Put, upper limit to the profit is X-P0
Call option
Call option gives owner the right, not obligation, buyer or the holder of long position
seller, writer, holder of short position
Payoff
CT=ST-X if ST▷X, CT=0 if ST≦X
=> CT=max(0,ST-X)
Profit = max (0, XT-X)-C0 = CT-Co
If ST◁X+Co, call buyer profit ◁0◁call seller profit
If ST=X+Co, call buyer profit =0=call seller profit
If ST▷X+Co, call buyer profit ▷0▷call seller profit
Breakeven at X+C0, or X+premium
Where:
◁ - smaller than
▷ - greater than
Co - premium
seller, writer, holder of short position
Payoff
CT=ST-X if ST▷X, CT=0 if ST≦X
=> CT=max(0,ST-X)
Profit = max (0, XT-X)-C0 = CT-Co
If ST◁X+Co, call buyer profit ◁0◁call seller profit
If ST=X+Co, call buyer profit =0=call seller profit
If ST▷X+Co, call buyer profit ▷0▷call seller profit
Breakeven at X+C0, or X+premium
Where:
◁ - smaller than
▷ - greater than
Co - premium
How are the Prices of Options are Affected by the Time to Expiration?
The price of an option has two components, the intrinsic value and the time value. The intrinsic value of an option is the difference between the actual price of the underlying security and the strike price of the option. The time value is determined by the remaining lifespan of the option, the volatility and the cost of refinancing the underlying asset (interest rates).
Longer term call, both American and European, typically values more than short term call because there is more time to have an event that can occur to make them go in the money. For American put, same as call.
European puts can only be exercised on the maturity date. If the put were to go into the money, one could exercise that option and receive the funds before the maturity date - that money could then be invested in another asset. Consequently, with the European put an investor has to wait to receive the funds and loses those extra days to earn additional income.
Longer term call, both American and European, typically values more than short term call because there is more time to have an event that can occur to make them go in the money. For American put, same as call.
European puts can only be exercised on the maturity date. If the put were to go into the money, one could exercise that option and receive the funds before the maturity date - that money could then be invested in another asset. Consequently, with the European put an investor has to wait to receive the funds and loses those extra days to earn additional income.
Thursday, January 1
Option Payoffs
Payoffs for interest rate options function are similar to other options. The main difference is that the interest rate options take the days to maturity attached to the agreement into account. Also, the payoff from the option is not made until the end of the number of days attached to the rate. For example, if an interest rate option expires in 60 days and is based on 180-day LIBOR, the holder will not be paid for 180 days.
Interest rate option payoffs
Interest rate call option payoffs are determined by the following formula:
Max {(underlying asset – exercise rate) (days in rate/360) , 0 }
Interest rate put option payoffs are determined by the following formula:
Max {( Exercise rate - underlying asset –) (days in rate/360) , 0 }
Intrinsic Value - Intrinsic value in options is the in-the-money portion of the option's premium. It is the value that any given option would have if it were exercised today.
Intrinsic value of a call = Min {0, CP –X}
Where CP - stock's current price (CP), X - option's strike price (X)
Intrinsic value of a put = Min {0, X-CP}
Time Value - The time value is any value of an option other than its intrinsic value, basically as the risk premium. Fundamentally, time value is related to a stock's beta or volatility. If the market does not expect the stock to move much (if it has a low beta), then the option's time value will be relatively low. Conversely, the option's time value will be high if the stock is expected to fluctuate significantly.
Interest rate option payoffs
Interest rate call option payoffs are determined by the following formula:
Max {(underlying asset – exercise rate) (days in rate/360) , 0 }
Interest rate put option payoffs are determined by the following formula:
Max {( Exercise rate - underlying asset –) (days in rate/360) , 0 }
Intrinsic Value - Intrinsic value in options is the in-the-money portion of the option's premium. It is the value that any given option would have if it were exercised today.
Intrinsic value of a call = Min {0, CP –X}
Where CP - stock's current price (CP), X - option's strike price (X)
Intrinsic value of a put = Min {0, X-CP}
Time Value - The time value is any value of an option other than its intrinsic value, basically as the risk premium. Fundamentally, time value is related to a stock's beta or volatility. If the market does not expect the stock to move much (if it has a low beta), then the option's time value will be relatively low. Conversely, the option's time value will be high if the stock is expected to fluctuate significantly.
| Option | Min. Value | Max. Value |
| European call | ct ³ Max{0, St-X/(1+RFR)T-t} | St |
| American call | Ct ³ Max{St-X/(1+RFR)T-t} | St |
| European put | pt ³ Max{0, X/(1+RFR) T-t - St} | X/(1+RFR) T-t |
| American put | Pt ³ Max{0, X - St} | X |
Interest Rate Options vs. FRAs
Forward rate agreement (FRA)
· an agreement between two parties to exchange a fixed interest payment for a floating interest payment.
· FRAs are OTC derivatives - forward contracts in which one party (which is referred to as the borrow or buyer) pays a fixed interest rate, and another party receives a floating interest rate equal to a reference rate (the underlying rate). The receiver is also referred to as the lender or seller.
· The payments are calculated over a notional amount over a certain period and netted - in other words, only the differential is paid on the termination date.
Interest rate options
· Give buyers the right, but not the obligation, to synthetically pay (in the case of a cap) or receive (in the case of a floor) a predetermined interest rate (the strike price) over an agreed period.
Similarities
· Both Interest rate options and FRAs have interest rates as their underlyer.
· Both use put or call formats.
· Both use a notional amount to define the size of the trade.
· Neither requires an exchange of principal.
Differences
· An FRA is a commitment to make one interest rate payment and receive another one at a future date while an option is the right to make one interest rate payment and receive another one.
· Interest rate options have exercise rate or strike rate instead of an exercise price like an FRA.
· an agreement between two parties to exchange a fixed interest payment for a floating interest payment.
· FRAs are OTC derivatives - forward contracts in which one party (which is referred to as the borrow or buyer) pays a fixed interest rate, and another party receives a floating interest rate equal to a reference rate (the underlying rate). The receiver is also referred to as the lender or seller.
· The payments are calculated over a notional amount over a certain period and netted - in other words, only the differential is paid on the termination date.
Interest rate options
· Give buyers the right, but not the obligation, to synthetically pay (in the case of a cap) or receive (in the case of a floor) a predetermined interest rate (the strike price) over an agreed period.
Similarities
· Both Interest rate options and FRAs have interest rates as their underlyer.
· Both use put or call formats.
· Both use a notional amount to define the size of the trade.
· Neither requires an exchange of principal.
Differences
· An FRA is a commitment to make one interest rate payment and receive another one at a future date while an option is the right to make one interest rate payment and receive another one.
· Interest rate options have exercise rate or strike rate instead of an exercise price like an FRA.
Types of options in terms of the underlying instruments
Financial Options:
Financial options have financial assets, such as an interest rate or a currency, as their underlying assets. There are several types of financial options:
Stock Option
Also known as equity options, these are a privileges sold by one party to another. Stock options give the buyer the right, but not the obligation, to buy (call) or sell (put) a stock at an agreed-upon price during a certain period of time or on a specific date.
Index Option
A call or put option on a financial index, such as the Nasdaq or S&P 500. Investors trading index options are essentially betting on the overall movement of the stock market as represented by a basket of stocks.
Bond Option
An option contract in which the underlying asset is a bond. Other than the different characteristics of the underlying assets, there is no significant difference between stock and bond options. Just as with other options, a bond option allows investors to hedge the risk of their bond portfolios or speculate on the direction of bond prices with limited risk.
A buyer of a bond call option is expecting a decline in interest rates and an increase in bond prices. The buyer of a put bond option is expecting an increase in interest rates and a decrease in bond prices.
Interest Rate Option
Option in which the underlying asset is related to the change in an interest rate. In general, the call buyer of an interest rate option expects interest rates will go up, while the put buyer hopes rates will go down.
Currency Option
A contract that grants the holder the right, but not the obligation, to buy or sell currency at a specified price during a specified period of time. Investors can hedge against foreign currency risk by purchasing a currency option put or call.
Options on Futures
Like other options, an option on a futures contract is the right but not the obligation, to buy or sell a particular futures contract at a specific price on or before a certain expiration date. A call option gives the holder (buyer) the right to buy (go long) a futures contract at a specific price on or before an expiration date. The holder of a put option has the right to sell (go short) a futures contract at a specific price on or before the expiration date.
Commodity Options
These are options in which the underlying asset is a commodity such as wheat, gold, oil and soybeans.
Financial options have financial assets, such as an interest rate or a currency, as their underlying assets. There are several types of financial options:
Stock Option
Also known as equity options, these are a privileges sold by one party to another. Stock options give the buyer the right, but not the obligation, to buy (call) or sell (put) a stock at an agreed-upon price during a certain period of time or on a specific date.
Index Option
A call or put option on a financial index, such as the Nasdaq or S&P 500. Investors trading index options are essentially betting on the overall movement of the stock market as represented by a basket of stocks.
Bond Option
An option contract in which the underlying asset is a bond. Other than the different characteristics of the underlying assets, there is no significant difference between stock and bond options. Just as with other options, a bond option allows investors to hedge the risk of their bond portfolios or speculate on the direction of bond prices with limited risk.
A buyer of a bond call option is expecting a decline in interest rates and an increase in bond prices. The buyer of a put bond option is expecting an increase in interest rates and a decrease in bond prices.
Interest Rate Option
Option in which the underlying asset is related to the change in an interest rate. In general, the call buyer of an interest rate option expects interest rates will go up, while the put buyer hopes rates will go down.
Currency Option
A contract that grants the holder the right, but not the obligation, to buy or sell currency at a specified price during a specified period of time. Investors can hedge against foreign currency risk by purchasing a currency option put or call.
Options on Futures
Like other options, an option on a futures contract is the right but not the obligation, to buy or sell a particular futures contract at a specific price on or before a certain expiration date. A call option gives the holder (buyer) the right to buy (go long) a futures contract at a specific price on or before an expiration date. The holder of a put option has the right to sell (go short) a futures contract at a specific price on or before the expiration date.
Commodity Options
These are options in which the underlying asset is a commodity such as wheat, gold, oil and soybeans.
Basic Options Concept
Option buyer has the right to buy or to sell the underlying asset but no obligation to do so.
Option seller (writer) has no right, but an obligation to complete the trade required by the option contract.
Call option: Gives its holder the right to purchase an asset at the strike price.
Put option: Gives its holder the right to sell an asset at the strike price.
American vs. European:
· American-style options can be exercised at any time up to the expiration date.
· European-style options can only be exercised on the expiration date.
· Since American-style gives more flexibility, it can be more expensive than European-style in some (but not all) cases.
· Both options have the same value on the expiration date.
· Most of the options throughout the world are A. options
Moneyness
The concept of moneyness describes whether an option is in-, out-, at-, or in-the-money by examining the position of strike vs. existing market price of the option's underlying security.
Call option is in-the-money if current asset price is greater than strike price, and out-of-themoney if current asset price is smaller than strike price.
Put option is in-the-money if current asset price is smaller than strike price, and out-of-the-money if current asset price is greater than strike price. If current asset price equals strike price, both are at the money
Over the Counter Options
Many derivative instruments such as forwards, swaps and most exotic derivatives are traded OTC.
· unregulated
· Dealers offer to take either a long or short position in option and then hedge that risk with transactions in other options derivatives.
· Buyer faces credit risk because there is no clearing house and no guarantee that the seller will perform
· Price, exercise price, time to expiration, identification of the underlying, settlement or delivery terms, size of contract, etc. are customized
Exchange-Traded Options
· traded on a regulated exchange
· All terms are standardized except price.
· The exchange establishes expiration date and expiration prices as well as minimum price quotation unit.
· The exchange also establishes whether the option is American or European, its contract size and whether settlement is in cash or in the underlying security.
· Usually trade in lots in which 100 shares of stock = 1 option
· The most active options are the ones that trade at the money
· Can be bought and sold with ease and holder decides whether or not to exercise. When options are in the money or at the money they are typically exercised.
· Most have to deliver the underlying security.
· Regulated at the federal level
Option seller (writer) has no right, but an obligation to complete the trade required by the option contract.
Call option: Gives its holder the right to purchase an asset at the strike price.
Put option: Gives its holder the right to sell an asset at the strike price.
American vs. European:
· American-style options can be exercised at any time up to the expiration date.
· European-style options can only be exercised on the expiration date.
· Since American-style gives more flexibility, it can be more expensive than European-style in some (but not all) cases.
· Both options have the same value on the expiration date.
· Most of the options throughout the world are A. options
Moneyness
The concept of moneyness describes whether an option is in-, out-, at-, or in-the-money by examining the position of strike vs. existing market price of the option's underlying security.
Call option is in-the-money if current asset price is greater than strike price, and out-of-themoney if current asset price is smaller than strike price.
Put option is in-the-money if current asset price is smaller than strike price, and out-of-the-money if current asset price is greater than strike price. If current asset price equals strike price, both are at the money
Over the Counter Options
Many derivative instruments such as forwards, swaps and most exotic derivatives are traded OTC.
· unregulated
· Dealers offer to take either a long or short position in option and then hedge that risk with transactions in other options derivatives.
· Buyer faces credit risk because there is no clearing house and no guarantee that the seller will perform
· Price, exercise price, time to expiration, identification of the underlying, settlement or delivery terms, size of contract, etc. are customized
Exchange-Traded Options
· traded on a regulated exchange
· All terms are standardized except price.
· The exchange establishes expiration date and expiration prices as well as minimum price quotation unit.
· The exchange also establishes whether the option is American or European, its contract size and whether settlement is in cash or in the underlying security.
· Usually trade in lots in which 100 shares of stock = 1 option
· The most active options are the ones that trade at the money
· Can be bought and sold with ease and holder decides whether or not to exercise. When options are in the money or at the money they are typically exercised.
· Most have to deliver the underlying security.
· Regulated at the federal level
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