FIN 406 Exam 3
The Black-Scholes Model
Co (Po) denotes the time-0 price of a European call (put) option
long/short the stock
Consider both a long- and a short-position on KO established respectively at $50/share at time 0. Then ST presents the time-T payoff of the long position -ST presents the time-T payoff of the short position Potential profit/loss (per share) at time T: (P/L)Long = ST -S0 = ST -$50; (P/L)Short = S0 -ST = $50 - ST
ETF stands for "Exchange-Traded-Fund"
true
contributions of the BS model
"Black, Merton and Scholes LAID THE FOUNDATION FOR THE RAPID GROWTH OF MARKETS FOR DERIVATIVES in the last ten years. Their method has more general applicability, however, and has created new areas of research - inside as well as outside of financial economics. A similar method may be used to value insurance contracts and guarantees, or the flexibility of physical investment projects."- Royal Swedish Academy of Sciences, 10/14/97 Considered by many people to be the most successful model so far in economics and finance
You purchase one Microsoft December $140 put contract for a premium of $5.30. What is your maximum possible profit? Assume each contract is for 100 shares.
$13470
An investor purchases a stock for $38 and a put for $0.50 with a strike price of $35. The investor sells a call for $0.50 with a strike price of $40. What are the maximum profit and loss for this position?
$2 -$3
An investor buys a call at a price of $4.50 with an exercise price of $40. At what stock price will the investor break even on the purchase of the call?
$44.50
case study: the PBS documentary "trillion dollar bet"
- The Black-Scholes Option Pricing Model - The rise and fall of the Long-Term Capital Management (LTCM)
collars
A (long) collar is a portfolio of the underlying stock, a long put, and a short call with the same T = a protective put + a short call = a long put + a covered call Initial cost of this portfolio = P0 + S0 - C0 Payoff at T = ST +PT - CT P/L at T = ST + PT - CT - (S0 + P0 - C0)
straddles
A (long) straddle - is a portfolio of a long call and a long put with the same underlying, the same strike price and the same maturity = a long call + a long put Initial cost of this portfolio = C0 + P0 Payoff at T = CT + PT P/L at T = CT + PT - (C0 + P0)
definition of options
A call (put) option is the right, but not the obligation, to buy (sell) a specific asset at a specific price within a specific period of time Examples - One-month JPM (call or put) options - One-month SPY (ETF) options - Two-month S&P 500 index options
covered calls
A covered call A portfolio of a short call and its underlying stock = a short call + a long stock Initial cost of this portfolio =S0 -C0 The portfolio payoff at T =ST -CT The portfolio P/L at T =ST -CT -(S0 -C0)
covered calls: payoff at T
A covered call = a short call + a long stock Payoff of a covered call at T = -CT + ST A short call is also called a naked call
options spreads
A money (or vertical) spread: long and short equal number of calls (or puts) with the same underlying and maturity but different strikes Consider a call spread - a portfolio of a long call with strike X1 and a short call with strike X2 Initial cost of this portfolio = C0(X1) - C0(X2) Payoff at T = CT(X1) - CT(X2) P/L at T = Payoff at T - the initial cost
protective puts
A protective put - A portfolio of a put and its underlying stock = a long stock + a long put Initial cost of this portfolio = S0 + P0 The portfolio's payoff at T = ST + PT The portfolio's P/L at T = ST + PT - (S0 + P0)
option payoffs and P/L: puts
A put option's payoff at time t Pt = max(X-St,0) A put holder's - Payoff = Pt = max(X-St,0) - P/L = max(X-St,0) - put premium A put writer's - Payoff = -Pt = -max(X-St,0) - P/L = - max(X-St,0) + put premium
digression: function N(d)
A standard function in statistics - Its value depends on the value of variable d - N(d=-∞)= 0, N(d=-1)=0.159, N(d=0)=0.5, N(d=1)=.841, N(d=∞)=1 Tabulated in the textbook (Chap 16) Called NORMSDIST in Excel - Example: To calculate N(0) in Excel, enter "NORMSDIST(0)" and it will return 0.5.
a long straddle: payoff at T
A straddle is a portfolio of a call and a put (written on the same asset) with the same strike ($50 here) and expiration. A straddle is a play on volatility.
the volatility parameter
Among the 6 input parameters (S0, X, T, r, gamma, sigma) in the Black-Scholes model (BSM), ONLY VOLATILITY SIGMA IS NOT DIRECTLY OBSERVABLE and has to be estimated Option prices are VERY SENSITIVE to the underlying return volatility sigma -> How to estimate/forecast volatility is an extremely important issue in option markets
delta and hedge ratio
An option's delta represents THE # OF SHARES of (the underlying) stock needed to hedge ONE OPTION - This is one important reason why option pricing models are valuable to banks - Hedging an option based on its delta is called DELTA HEDGING in trader's parlance An option's delta is also called its HEDGE RATIO One reason for the popularity of the Black- Scholes model: it provides a simple formula for the delta hedge ratio
one application of delta
Analyze the sensitivity of options to the underlying stock price - Given an option's delta, we have Changes in C (P) = delata C (delta P) * Changes in S The higher the magnitude of an option's delta, the more sensitive the option to the underlying
buyers/holders of options
Buy a right to acquire (for calls) or sell (for puts) the asset from the option seller Establish a LONG position in the option During the life of the option, holders can - sell (at any time) it at its market price (thus closing out the long position) or - keep the option and do nothing or - exercise it (on exercisable dates)
digression: trading stocks
Buy a stock -> Establish a long position Sell a stock that you own -> Get rid of a long position Sell short a stock that you don't own -> Establish/Open a short position Buy back a short stock -> Close out a short position
buy/sell options
Buy an option -> Establish/Open a long position Sell an option that you own -> Close out a long position Sell/Write an option that you don't own -> Establish/Open a short position Buy back a written option -> Close out a short position
example 4: black scholes delta
Calculate the Black-Scholes delta in Examples 1 though 3 It follows from Eqs (D1) and (D2) that - The IBM call's delta: delta c = 1*0.496 ≈ 0.5; - The SPX call's delta: delta c = 0.999 * 0.8905 = 0.8896; - The SPX put's delta: delta p = - 0.999 * 0.1095 = -0.1094;
example 5: coke call options
Coke (ticker: KO) is trading at $40/share and a 6-month 45 KO call trading at $5. You have $4,000 to invest. Consider the following two strategies (ignoring commissions): - Strategy 1 (All equity): buy 100 KO shares - Strategy 2 (All option): buy 8 calls Analyze potential returns (over the 6-month period) and risk of each strategy - Options magnify both gains and losses - Options are more risky than the underlying stock
example 7: delta hedging
Consider Example 1 on the IBM call. Determine the hedging portfolio of this IBM call using the Black-Scholes model - Need to determine this call option's delta (see Example 1): delta = N(d1) = 0.496 ≈ 0.5 - Given that the IBM 165 call's delta = 1⁄2, a call writer needs to long 1 share of the stock for every 2 short calls in order to hedge away his delta risk
example 3: jpm call option
Consider a JPM call with a strike of $100 and premium of $5.00 (today). The stock's closing price today is $100.25. Analyze this call option's payoff (cash flow per option on the expiration day T), P/L and potential returns. We have - X (or K) = $100; C0 = $5; S0 = $100.25; Need to compute - CT , P/L (at T), and option returns Clearly, option payoff, P/L and returns all depend on the stock price at T
lessons learned
Danger of high leverage and margin calls Importance of liquidity risk Reduction in diversification due to the increase in correlations across different markets Limitations of models
intrinsic value and time value
Decompose an option premium as follows: Option Premium = I.V. + T.V. Intrinsic Value = Value of an option from an immediate exercise = S-X for in-the-money calls = X-S for in-the-money puts = 0 for ATM and OTM calls/puts Time Value = Option Premium/Value - Intrinsic Value
a portfolio's delta
Delta of a SHORT call (put) = - Delta of a long call (put) Delta of the underlying stock - A long stock's delta = 1; - A short stock's delta = -1 A portfolio's delta = sum of the deltas of assets held in the portfolio
option pricing approaches
Discounted Cash Flow Approach - But what discount rate to use? The Relative Valuation Approach (Black and Scholes, 1973) - Valuing the target asset (options or other assets) relative to a TRADED benchmark asset - The STANDARD APPROACH to the valuation of options and other derivatives - Also applicable to non-derivatives
example 5: portfolio delta
E.g., Suppose a portfolio includes 2 short calls (with a call delta of 0.6) and a long stock. Determine this portfolio's delta The portfolio's delta = 2 * short call delta + 1 * stock's delta= 2 * (-0.6) + 1 * (1) = -0.2
empirical evidence
Empirical evidence so far indicates that the Black-Scholes model UNDER-ESTIMATES the INDEX OPTION prices especially for OTM puts (or ITM calls) - The failure of the model in capturing the left tails (downside risk) - There is a huge literature on how to extend the BSM to fit the option price better The BSM with some minor twist might be good enough for pricing EQUITY CALL OPTIONS with zero dividends
intuition of the black schole model
Eqs. (C1) & (P1) have a PORTFOLIO INTERPRETATION - One of the main insights of the Black-Scholes model An option is equivalent to a particular portfolio of the underlying asset and the risk-free asset -> Relative valuation - Option premium = the value of this portfolio - In other words, the option can be REPLICATED by this portfolio - The replicating portfolio varies over time -> Dynamic replication of options (which is discussed in the PBS documentary "Trillion Dollar Bet")
listed equity options
Equity and ETF options - American style; physically settled Stock index options - European or American; cash settled Maturity - Usually up to 3 years - Expiration date: standardized Option contract size - Standardized
factors affecting option prices
Features of the option contract itself - Exercise style of the option - Strike price (denoted X or K) - Expiration/Maturity (T) Characteristics of the underlying stock - So: time-0 price of the underlying stock - sigma: the underlying stock's return volatility
on the LTCM
Feb 1994: LTCM was founded with money from 80 investors with a market cap of $1.25 billion - The largest hedge fund at that time ranked by the new capital raised Dec 1997: LTCM returned $2.7 billion to investors Early 1998: Firm's equity was about $4 billion August 17, 1998: Russia defaulted on its Treasury debt (of $13.5 billion) Sep 1, 1998: LTCM's equity dropped to $2.3 billion - The firm was facing tremendous pressure on meeting margin calls from its lenders Sep 22, 1998: LTCM's equity dropped to $600 million Sep 23, 1998: A consortium of 13 banks organized by the NY Fed agreed to inject $3.65 billion into LTCM in exchange for 90% of the company Spring 2000: LTCM was closed
black, scholes, and merton
Fischer Black, who received a Ph.D. in applied math, was working at Arthur D. Little at that time Myron Scholes was a junior finance prof at MIT Robert Merton, a junior finance prof at MIT, provided a more rigorous and robust derivation of the Black-Scholes option pricing formula using some high-tech math tool
option payoffs and P/L: calls
Fix notation - X: an option's strike price - St: the underlying asset's price at time t - Ct: a (long) call option's payoff at t The call HOLDER'S - Payoff = Ct = max[(St-X),0] - P/L = max(St-X,0) - call premium The call WRITER'S P/L - Payoff = - Ct = -max[(St-X),0] - P/L = call premium - max(St-X,0)
review of the 6 simple strategies
For illustration, consider the Coca-Cola stock (ticker: KO), a KO call option with maturity T (>0), and a KO put option with maturity T and options. We review the payoff and P/L at T for each of the six simple strategies Suppose the stock is currently trading at $50/share at time 0. Let St denote the stock price at time t We have S0 = $50
why options?
Generating alpha (loosely speaking) Specifically, three often mentioned applications - SPECULATION, INCOME, and PROTECTION 1. Speculating/Investing - E.g., you are very bullish on Tesla and want higher returns than buying (the stock) on margin - Recent bets on tech by Michael Burry (in "Big Short") and Softbank 2. Generating income - E.g., option strategies used by Warren Buffett 3. Hedging (risk management) - E.g., you own an index fund and want to buy some "insurance" to limit the downside risk Information content of options markets - More informed trading in options than in stocks - Volatility indexes (e.g. VIX) Employee stock options Financial assets with embedded options - Callable bonds - Defaultable bonds -> Option to default -> Moody's KMV model of default risk
implied volatility
Given a volatility input, the Black-Scholes option price is likely different from the observed option price - Even if when the volatility input used increases, the Black-Scholes model price also increases (Option) implied volatility - Implied volatility of an option is the PARTICULAR VALUE of the volatility input such that the Black-Scholes model price of the option = the observed option price The higher sigma, the higher the option premium -> The implied volatility > 0.12 The implied volatility can be obtained by using Goal Seek in Excel - The implied volatility is 13.61% in Example 2
historical volatility
How about using the standard deviation of historical returns of the underlying asset? - Such an estimate of volatility is often referred to as HISTORICAL VOLATILITY Empirical evidence: - Historical volatility is not a good estimate for input sigma here - Volatility is not constant either -> the notation of time-varying volatility
option payoffs
How to analyze an option's cash flow? The ultimate source of an option's value comes from the right to exercise For European options, just need to consider the exercise decision at expiration only For American options, the analysis of exercise decision before expiration is nontrivial - However, on expiration day, an American option becomes a European option From now on we focus on the exercise of an option at expiration only
main ideas behind option pricing models
Identify factors that can potentially affect option prices - Focus on the most important ones first - Incorporate additional factors later if necessary Establish a link between the option premium and those factors affecting the premium -> an option pricing model
example 4: the payoff
If the option is not exercised and expires worthless, the writer's payoff is zero If the option is exercised, then the writer receives the strike ($100 here) and pays the stock (worth ST). So - Writer's payoff at T = $100-ST<0 The option premium ($5 here) is collected up front and not taken into account in the calculation of the payoff at expiration The option writer's P/L at T is - Writer's P/L = payoff + option premium
contract size: examples
In Example 1, two contracts of the JPM call - cost 2 * (100*$4.43), and - Give the buyer the right to buy 200 shares of the JPM stock at $98/share when the option is exercised In Example 2, one contract of the SPX put - costs $156.00*100, and - gives the buyer the right to sell 100 units of the index at $3,250/unit (cash settled) when the option is exercised
gearing ratio
In this example, - it costs $40 to control one share of the stock via the equity market - On the other hand, it costs only $5 to "control" one share of the stock via the option market The so-called gearing ratio of an option - Gearing ratio = Stock price / option price In this example, - Gearing(ratio)=S0 /C0 =40/5=8 - A rough measure of leverage The call write bets that the stock price would fall or not increase much but could suffer significant losses if the stock price turns against the writer
volatility indexes on stock indexes
Info about volatility extracted from options markets is often used to forecast (equity) volatility VIX: the CBOE Volatility Index - Considered to be the benchmark of U.S. stock market volatility for near term - Based on the S&P 500 index options CBOE NASDAQ Volatility Index (VXN) - Based on NASDAQ-100 options CBOE DJIA Volatility Index (VXD) - Based on options on the DJIA
A call option with a strike price of $50 on a stock selling at $55 costs $6.50. What are the call option's intrinsic and time values?
Intrinsic value = S0 − X = $55 − $50 = $5.00 Time value = C − Intrinsic value = $6.50 − $5.00 = $1.50
zero sum
It follows from Examples 3 and 4 that - Holder's Payoff + Writer's Payoff = 0 - Holder's P/L + Writer's P/L = 0 In another word, options are a zero- sum game
people interviewed in the film including
Leo Melamed (founder of financial futures) Myron Scholes and Robert Merton (see next page) Merton Miller (Scholes's Ph.D. advisor at Chicago) Zvi Bodie (co-author of our textbook and a former Ph.D. student of Robert Merton) Roger Lowenstein (author of a best-selling book on LTCM)
option moneyness
Let St denote the underlying stock price at time t, and X denote the strike. - If St > X, a call (put) option is said to be in the money (out of the money) at time t - If St = X, a call/put option is said to be at the money at time t - IfSt < X, a call (put) option is said to be out of the money (in the money) at time t Notice that an option's moneyness can change due to changes in St
simple option strategies
Long calls - Bullish on the underlying Long put - Bearish on the underlying Writing calls - Bearish on the underlying - Short calls versus long puts Writing puts - Bullish on the underlying - Short puts versus long calls - E.g., Warren Buffett has written put options
contract size for options
Minimal transaction size is ONE (option) contract The standard contract size is - 100 shares for equity and ETF options - 100 units for index options However, the option price/premium is quoted on a per share/unit basis - In Example 1, the call premium $4.43 is the premium for one option; one contract costs $4.43*100
digression: payoff diagram
One convenient and intuitive way to analyze an option's payoff (especially the payoff of a portfolio of options and the underlying asset) is to use the option payoff diagram Option payoff diagram: a plot of an option's payoff vs. the underlying asset's price at expiration - To generate such a diagram, select a set of possible values of the underlying asset, calculate the option payoff for each value of the underlying asset, and then graph OTM (out of the money) Region: Call expires worthless ITM (in the money) region: exercise
option's delta
One way to gauge the risk of an option is to look at its sensitivity to changes in the underlying An option's delta is such a sensitivity measure Option delta = change in the option price / change in the underlying price
exercise of options: a summary
Option EXERCISE STYLE - Determining when the owner of an option can exercise the option - European (style) options - American (style) options - Non-standard (exotic) styles Settlement type - Physical settlement - Physically settled options - Cash settlement
black-scholes delta
Option deltas calculated using the Black-Scholes model (D1) (D2)
HW 7 #13
Put A must be written on the lower-priced stock. Otherwise, given the lower volatility of stock A, put A would sell for less than put B. Put B must be written on the stock with lower price. This would explain its higher value. Call B. Despite the higher price of stock B, call B is cheaper than call A. This can be explained by a lower time to maturity. Call B. This would explain its higher price. Not enough information. The call with the lower exercise price sells for more than the call with the higher exercise price. The values given are consistent with either stock having higher volatility.
benefits of option pricing models
Quantifying each factor's impact on the option premium - If the stock price increases by $1, how much the price of a call (put) option increases (drops)? - Similarly, what's the impact of the strike price on the call or put option premium? Quantifying the impact of the above factors JOINTLY - E.g., what will be the impact on a call premium if both S and sigma move?
buy and sell high
Recall from Example 1 - On 10/30/2020, the JPM call premium is $4.43 and the JPM stock price is $98.04. On 11/3/2020, the JPM call premium: $8.15 and the JPM stock price: $103.81 Calculate the call option and stock returns from 10/30 to 11/3 - The option return = (8.15-4.43)/4.43 = 83.97% - The stock return = (103.81-98.04)/98.04 = 5.89%
example 6: portfolio delta
Refer to Example 1. Consider a portfolio of 2 short IBM 165 calls and 1 long IBM stock. Calculate the portfolio delta - Recall that the call's delta is 0.5. - Verify this portfolio's delta risk is zero!! - Adding one IBM share to a short position in two IBM 165 calls would eliminate the delta risk of the entire portfolio - This is an example of the so-called DELTA HEDGING In general, we can hedge a short call by taking a long position in the stock. But is the key question is - How many shares do we need to be long?
sellers/writers of options
Sell a right and must fulfill his/her obligations when necessary Establish a SHORT position During the life of the option, can - buy it back (thus cancelling or closing out his/her short position) - keep the option -> do nothing unless when it is exercised; fulfill his/her obligations when the option is exercised
6 simple strategies
Six simple strategies (building blocks) - Each is based on a SINGLE instrument - Long/short the underlying (e.g. stock) - long/short a call - long/short a put Any option strategy can be constructed using a PORTFOLIO of these six simple ones
inputs of the black-shole model
So: time-0 price of the underlying asset X: strike T: time to expiration r: risk-free rate sigma: the underlying asset's return volatility delta: annual dividend yield of the underlying asset ln(): the Natural log function N(): the Standard Normal Distribution function
notation
St: the (underlying) stock price at time t X (or K): an option's strike price T: an option's expiration day T-t: option's time-to-expiration at time t Ct: the time-t price of a call option Pt: the time-t price of a put option r: the risk-free rate sigma: the underlying stock's return volatility delta: the dividend yield of the stock
notation
St: the stock price at time t X (or K): an option's strike price T: an option's expiration day Ct: the time-t price of a call option - C0: the initial price/premium of a call - CT: a call's price/premium (per share) at T, often referred to as the (option) payoff at T Pt: the time-t price of a put option P/L: profit & loss
complex options strategies
Strategies using both options and the underlying - Protective puts - Covered calls - Collars Strategies using option portfolios - Straddles - spreads
options spreads: an example
Suppose the KO stock is trading at $50/share. You are bullish on the stock but does not think the stock will go higher than $60 within the next 3 months. Consider a 3-month KO call spread that consists of - a long (3-month) 55 call on KO and - a short (3-month) 60 call on KO - Here, the strikes: X1 = $55, X2 = $60
why option pricing?
Suppose you are interested in buying SPX put options as insurance - Is the put option too expensive or cheap? - If too expensive, then what? - If too cheap, then what? How to estimate the value of an OTC option? - E.g., employee stock options - Exotic options on a bank's trading book How to hedge the risk of options, say, a short call?
collars: payoff at T
Suppose you own (long) Coca-Cola, whose current price St = $50 per share. - Buy a put with X = $40 -> Buy protection - Write a call with X = $60 -> To offset the put premium Put together, you have constructed the following portfolio (a long collar): - A long stock - A long put with strike $40 - A short call with strike $60
example 1 (JPM Options)
The JPM stock (underlying all JPM options) is also referred to as the UNDERLYING STOCK (of JPM options) At any given time, hundreds of different JPM options are WRITTEN on the same JPM stock and traded in the options market - JPM call options with different maturities and strike prices - JPM put options with different maturities and strike prices Exercising an option is essentially an exchange between cash (the strike price) and the underlying asset Holder of the JPM call (put) can exercise the option anytime before it expires to buy (sell) the JPM stock at the strike price ($98/share) from (to) the writer of this call (put) JPM options are PHYSICALLY SETTLED Exercise style of these JPM options: - AMERICAN (by market convention)
example 2 (SPX Options)
The S&P 500 index options are referred to as SPX options - SPX options are among the most liquid stock option contracts in the world Exercise style of these SPX options - SPX options are EUROPEAN - Holder of this call (put) can exercise the option ONLY ON the expiration date to buy (sell) the CASH VALUE of one unit of the index at the strike price ($3250/unit), the strike price, from (to) the writer of this call (put) - Namely, SPX options are CASH SETTLED
Payoff of Calls at Expiration
The call holder - exercises the option only if ST (the inflow) > X (the outflow); - otherwise, simply lets the option expire worthless Then a (long) call option's payoff at T is - CT = ST - X, if ST > X (and option exercised); - CT = 0, otherwise (and option expired). Equivalently, CT = max[(ST -X),0] - Holder's P/L = Payoff - Call Premium = CT - C0
intuition of option delta
The option delta is the change in the option price for a $1 increase in the underlying price - A measure of the DOLLAR risk (not return risk) - This risk is often called the DELTA RISK (of options) The higher the MAGNITUDE of an option's delta, the more sensitive the option to the underlying
black and scholes (1973)
The original Black-Scholes article was actually rejected for publication by several top academic journals and finally published at the Journal of Political Economy in 1973 due to the help of Merton Miller Scholes and Merton won the 1997 Nobel Prize in Economics (Black passed away in 1995)
a protective put: payoff at T
The put acts like an insurance here and is called the protective put "Insured" stock = a long put + a long stock. The minimal payoff of the "insured" stock = the strike of the put
elements of option contracts
The specific asset called the UNDERLYING ASSET of the option The specific price is called the STRIKE (or EXERCISE) price (of the option contract) When the owner of an option exercises his right to buy/sell the underlying asset, the option is said to be exercised The right expires on the EXPIRATION DAY The market price of an option is termed the OPTION PREMIUM
the film is about
The story behind the well-known Black- Scholes option pricing model/formula - "The Formula that Shook the World" (PBS, Feb 2000) The rise and fall of LTCM, a well known hedge fund co-founded by Myron Scholes, Robert Merton and a few other big names
Which one of the following statements about the value of a call option at expiration is false? A short position in a call option will result in a loss if the stock price exceeds the exercise price. The value of a long position equals zero or the stock price minus the exercise price, whichever is higher. The value of a long position equals zero or the exercise price minus the stock price, whichever is higher. A short position in a call option has a zero value for all stock prices equal to or less than the exercise price.
The value of a long position equals zero or the exercise price minus the stock price, whichever is higher. - This is the description of the payoff to a put, not a call.
applications of option pricing models
They can be used to estimate the value of an option - Identify under- and over-valued options using the model as a benchmark - Estimate the value of OTC options More importantly, option pricing models help us better understand how to HEDGE the risk of options
payoff formula
To summarize, the call option's payoff is - CT =ST-$100, if (ST-$100)>0; - CT=0 if (ST-$100)≤0. Importantly, the payoff depends only on the strike (X) and the stock price at T (ST) The call option payoff increases in the underlying (price) and decreases in the strike price There is a simple formula for the call option's payoff at maturity T - CT = max[(ST - $100),0], or in general - CT = max[(ST - X),0] As a result, - P/L = CT - C0 = max(ST-$100,0) - $5 - Option return = (P/L) / C0
trading options
Trading an option is like trading a stock - Both buyers and sellers pay commissions - Many brokers now charge zero commissions Except in some rare cases, listed options are issued by investors or dealers - Employee stock options are not listed
how to estimate delta
Use historical prices of an option and its underlying asset - For instance, examine how much the JPM stock price changes and how much the price of a JPM call option changes over the same period time - E.g., if the JPM stock price and a JPM call option premium increased by $3 and $2 yesterday, respectively, then a rough estimate of this call option's delta is = $2/$3 ≈ 0.67 - Such methods are intuitive but based entirely on historical data Use an option pricing model (a better method) - The main advantage of option pricing models - The benchmark model: the Black-Scholes model
speculating using puts
When an investor is bearish on a particular stock, she/he can speculate by - shorting the stock or - buying a put option written on the same stock The investor can also buy a put option to hedge her/his long position in the underlying - Discussed later when we cover option strategies
speculating using calls
When an investor is bullish on a particular stock, she/he can - Long the stock (even with margins) or - Long a call option written on the same stock What are the differences between these two strategies? - Risk and return - Leverage in options Which strategy is "better"?
sources of the option premium
Why some options are more expensive or cheaper than some others? Option MONEYNESS - One way to classify options, depending their expensiveness or cheapness The decomposition of the option premium - Intrinsic value of an option - Time value of an option
You are a portfolio manager who uses options positions to customize the risk profile of your clients. In each case, what strategy is best given your client's objective? a. Performance to date: Up 16%. Client objective: Earn at least 15%. Your forecast: Good chance of major market movements, either up or down, between now and end of year. Long straddle. Long bullish spread. Short straddle. b. Performance to date: Up 16%.Client objective: Earn at least 15%.Your forecast: Good chance of a major market decline between now and end of year. Long put options. Short call options. Long call options.
a. Long straddle. A long straddle produces gains if prices move up or down and limited losses if prices do not move. A short straddle produces significant losses if prices move significantly up or down. A bullish spread produces limited gains if prices move up. b. Long put options. Long put positions gain when stock prices fall and produce very limited losses if prices instead rise. Short calls also gain when stock prices fall but create losses if prices instead rise. The long call position will not protect the portfolio if prices fall.
You establish a straddle on Fincorp using September call and put options with a strike price of $80. The call premium is $7.00 and the put premium is $8.50. a. What is the most you can lose on this position? b. What will be your profit or loss if Fincorp is selling for $88 in September? c-1. What is the Break-even price for lower bound? c-2. What is the Break-even price for upper bound?
a. Maximum loss happens when the stock price is the same to the strike price upon expiration. Both the call and the put expire worthless, and the investor's outlay for the purchase of both options is lost: $7.00 + $8.50 = $15.50 b. loss of $7.50 c1. $64.50 c2. $95.50
exercise of calls at maturity
buyer will exercise once stock price raises above strike price anything below, will not exercise
exercise of puts at maturity
buyer will make money once stock price falls below strike price anything above, will not exercise
You write a call option with X = $50 and buy a call with X = $60. The options are on the same stock and have the same expiration date. One of the calls sells for $3; the other sells for $9. This strategy is referred to as a call spread (see notes lecs20-27-part2). Note: You should know that between these two call options, the call with the higher strike price is cheaper. As such, you receive $6 (= $9 - $3) initially by using this strategy. See also the solution posted on Canvas. c) What stock price will be the break-even point for this strategy? d) Is ths investor bullish or bearish on the stock?
c) $56 - Breakeven occurs when the payoff offsets the initial proceeds of $6, which occurs at a stock price of S= $56. At S= $56, the payoff of the short call with X=$50 is -(S-X) = -$6 and the payoff of the long call with X=$60 is zero. d) bearish
collars: payoff diagram
corporate insiders can sell their holdings by buying a collar from an investment bank
ETF options and Index options are settled in the same way:
false - ETF options settle like stocks, index options are "cash settled"
ETF options cannot be exercized early meaning they are "European Style" options
false - cash-settled index options
When an ETF option is settled, 60% of the gains is taxed as long-term gains, and 40% of the gains are treated as short-term gains:
false - when an index option settles
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would most likely experience a slight downturn and suggested delta-hedging the BIC portfolio. As predicted, the U.S. equity markets did indeed fall, but BIC's portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected. Which of the following best explains a delta-neutral portfolio? The return on a delta-neutral portfolio is hedged against: small price changes in the underlying asset. small price decreases in the underlying asset. all price changes in the underlying asset.
small price changes in the underlying asset. A delta-neutral portfolio is perfectly hedged against small price changes in the underlying asset. This is true both for price increases and decreases. That is, the portfolio value will not change significantly if the asset price changes by a small amount. However, large changes in the underlying asset will cause the hedge to become imperfect. This means that overall portfolio value can change by a significant amount if the price change in the underlying asset is large.
Prior to the financial crisis in 2008, Warren Buffett made massive derivative bets tied to world stock indices These positions were all... If these market indices performed well (increased in value) Warren Buffet would... Warren Buffet was investing...
true long-term positions make money bullishly
