Define an optinon and in and out of the money
-An option is a finalncial derivative contract hat gives its buyer the right, but not the obligation, to buy or sell an asset at a predetermined price (called the exercise/strike price) on or before a specified date (called the expiration date).
If it is profitable to exercise the option at the current price,–> “in the money option”
If it is unprofitable to exercise the option at the currentprice -> “out of the money option”
Define a call option, (european and american call) + value
-Call option gives the holder the right, not obligation, to buy the underlyng asser at a specified strike price on or ebfore the experation date.
European option refers to ptions that can only be executed on a specific day whereas american options refer to options that can be exercised on or before the experiation date
-Ct = Max(0, St - X), profit when St - X - a >0
show call option graphically
show put option graphically
Define a put option + value
-Put option contracts give the holder the right, not obligation, to sell an asset as a specified strike price on or before the experiation date.
-Value of put option Pt with asset price S and strike price X is given by:
-Pt = max(0, X - St)
-The put is worth the difference between current asset price and strike price, profit depends on whether Pt > a, premium.
Main purpose of options for investtors and how are options used to hedge risk briefly
Options allow investors to transfer risk—one party pays a premium to reduce risk, while another takes on that risk.
-Options can protect against interest rate, currency, or stock price movements.
describe protective puts (option strategies)
-investors buy/own stock and hedge against price drops by buying a put option.
-Can be seen as insurance against a stock with the price of insurance being the premium.
-Approrpaite when expect a short term devline and want to stay invested or tax reasons stop you from selling
describe spreads option (strategie) + bullish call spread option
-Refers to the purchasing of one option and selling another, typically same type with different strike prices (money spread) or different experiations (time spreda)
y aims to limit risk and cost and profit off price moves/mispricing.
-Bullish call spread refers to when you expect stock to go up but not by a large amount. are used to profit from a moderate rise in a stock’s price while limiting both risk and cost.
-Bullish call spread: buy call option at low strike price and sell call option at high strike price, same experiation darte.
describe covered call options (option strategies)
-Refers to buying stock/owning and selling a call option against it.
-Goal is to generate additoanl income from premiums(a) if you expect price to rise moderatly.May miss out on gains greater than the strike price if prices rise.
-If experation price st rises above strike price, buyer of call will rationally exerecise call and you’re the seller required to sell shares at strike price.
-If experation price st falls below X strike price, seller/you keep premium and buyer doesnt execute.
define short and long straddle and when both appropriaete
-Long straddle refers to buying a call and a put with same strike price and expperation date. Pay both premiums upfront.
-Long straddle approapite when expected share prices to be volatile
-Short straddle refers to selling a call and a put with same strike price and experation date. Recieve both premiums upfront.
-Suitable whe expect share prices to remain in a trading range. Expect low voitilty.
Calculate payoff/net value of a long straddle
-Refers to buying a call and put optino at same price and expery.
-Calculate put payoff = Max(0, X- St)
-Calcluate call payoff: = max(0,St-X)
-Total payoff = put + call payoff
-net value = Total payoff - total premiums paid
Calculate payoff/et value of a short straddle
-Refers to selling a call and a put option at same strike price x and expiry date.
-Calculate call payoff: = - max(0,St-X)
-Calculate Put payoff= - max (0, X-st)
-negative beacuse paying out option
x strike price, st experiation price.
-Total payoff = put payoff + call payoff (negative)
-Net value = total payoff + total premiums gained from selling contracs
Calculate net payoff/value of a protective put
-buy/own stock and buy put option
Find stock payoff = Stock price at experiation St - initial stock price (may be bought at strike price)
-Find the put payoff: = Max(0, X - St)
x is strike price, st stock price at experiation.
-Total payoff = put payoff + stock payoff
-Net value = Total payoff - put premium
Calculate payoff/net value of a covered call
Refers to buying/owning stock and SELLING a call option against it.
-Find stock payoff: Price at experation St - inital price (maybe bought at strike price)
-Find call payoff = as sold its negative or 0
= - MAX(0, St-X)
-Total payoff = stock payoff+ call payoff
-Net value = total payoff (stock payoff + call payoff) + premium recived from selling option
Calculate payoff/net value of a spread option (Bull call spread)
-Refers to buying a call at a lower strike price and selling a call at a higher strike price.
-Long call payoff (call bought) = max(0, St - X), premium paid
-Short call ( call sold) payoff =
- Max (0, St-X ),
will recived premium, negative payoff
-Total payoff = long call payoff (bought) + short call payoff (sold)
-Value = total payoff - premoum paid for long call + premoium recived for short call
Formula for the put call parity relationship with continuous and discrete compounding
Continous compounding of X
Ct + Xe^-rT = Pt + So
-Discrete compounding of X
Ct + X / (1+r)^T = Pt + So
Ct/Pt - price/value of call and put option, X strike price, r is the interest rate/risk free rate, So is the current stock price
Describe the put call parity relationship and what used for
-refers to relationship between european call option and put option prices of the same underlying stock with the same price and experation date.
-if relationship doesnt hold –> mispricing exists (should have fair prices relative to the other, market prices being different)
- mispricing creates arbitrage opportunities for investors to earn profits.
-USED FOR:
-Calculaying fair option prices based on the price of other type of option.
-Check for arbitrage opportunties.
Put call parity relationship formula involving dividends
-Discrete
-Ct + X / (1 + r)^T = Pt + So - D / (1+r)^T
Continous
-Ct + Xe^-rT = Pt + So - De^-rT
Limitations with Black-Scholes model
Underestimates extreme price movements: Large price changes occur more frequently in real markets than the model predicts.
Assumes stock (volatility) and the risk-free rate remain constant, unrealistic.
Assumes stock prices follow a lognormal distribution, may be unrealistic.
-Despite limitations, model still is still a fundamental model in options trading.
Assumptions of Black-Scholes model
The underlying asset pays no dividends or interest during the option’s life.
The option is European.
Rf is constant over the option’s life.
-No transaciton costs/fees in market.
-The price of the underlying asset is lognormally distributed.
Price moves are continuous .
Define Black-Scholes model + what used for
Black and Scholes (1973) developed a formula to evaluate European Options, used to calculate fair price of options based on multiple variables
used for:
stimates current value of a call option by calculating expected payoff at experation, adjusted for risk and discounting to pv.
To price options accurately.
To help traders and investors make better investmentl decisions.
Explain why lower bound of european call options is the lower bound write out on paper
-Ct > St - Xe^-r(T-t)
-Xe^-r(T-t) is the present value of the strike price.
-Assume Ct < lower bound, Investor contructs portfolio at time t < T by buying a ‘cheap’ call option priced Ct, short sell the stock, recivee st and invest Xe^-r(T-t) into a risk free asset.
-Initial value of constructed portfolio:
Vt = St - Ct -Xe^-r(T-t)
-At expeiry T:
-Case 1: if X < St, call option bough is exercised. Short sold srock must be repurchased at St, RFA grows to X and call pays out.
Final payoff V = St - X + X - St = 0
-Case 2: option not exercised X > St, call worth 0, rfa grows to X and short sold stock repurchased at St. Payoff X - St > 0
Even if call bought exercised or not, the portfolio always has a (weakly) positive value. Lower bound prevents risk free profit (arbitrage) by ensuring portfolio cant be built for free which gives guairenteed return.
Explain why upper bound is the upper bound using example with payoffs
-Upper bound Ct < St (stock price at time t)
-Assume the Ct (call price) costs more than the stock, a rational investor could sell the overpriced call option for Ct, use the proceeds to buy the stock for St and can make risk free profit/sure gain before experation of at least Ct - St > 0.
-At expeiry T, if the call is exercised seller recieves strike price X, if not exercised keep the stock worth St therefore min profit is MIN(X,St) > 0
-Known as a money pump situation and violates the no arbitrage principle.
Describe intrinsic and time value of options
-Intrinsic value refers to the value of the option if exercised now.
-Call = Max(0,St - X), Put = Max(0, X-St)
-Time value refers to the extra value of an option/extra paid because there is still time for it to become profitable (in the money)
TimeValue=OptionPremium (cost to buy option)−IntrinsicValue
