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Summary SV Investment and Portfolio Theory 2
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H4, 17 to 20, 24, 26. Chapters are ranked on in which week they were discussed
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H17. Option Markets: Introduction.Options: right (not obligation) to buy (call) or sell (put) the underlying security before or at a
certain time for a certain price (strike/exercise price).
The three major reasons for the use of derivatives (securities whose prices are determined by
the price of other securities) are: to reduce risk, to change the nature of one’s financial
exposure and to reduce transaction costs.
Buyer = Holder = Long position = right to buy.
Seller = Writer = Short position = obligation to sell.
Stock price = ST Exercise price = X
Call options.
The holder of a call option is not required to exercise the
option. He will choose to exercise only if the market value
of the underlying asset exceeds the exercise price.
The net profit of the call is the value of the option minus
the price originally paid to purchase it. The purchase price
of the option is called the premium. It represents the
compensation the purchaser of the call must pay for the
right to exercise the option if exercise becomes profitable.
Sellers of call options (writers), receive premium income
now as payment against the possibility they will be required
at some later date to deliver the asset in return for an exercise
price less than the market value of the asset.
Value at expiration (Proceeds) = ST – X.
Profit = final value (proceeds) – price of the call option.
Payoff to Call Holder: Payoff to Call Writer:
If ST > X = ST – X If ST >X = –(ST - X) = X - ST
If ST ≤ X = 0 If ST ≤ X = 0
Profit to Call Holder: Profit to Call Writer:
Payoff – purchase price Payoff + Premium
Put options.
A put option gives its holder the right to sell an asset for a
specified exercise or strike price on or before some expiration
date. It entitles the owner to sell for example his IBM stock to
the put writer.
Payoff to Put Holder: Payoff to Put Writer:
If ST ≥ X = 0 If ST ≥X = 0
If ST < X = X – ST If ST <X = -(X– ST) = ST – X
Profit to Put Holder: Profit to Put Writer:
Payoff – premium Payoff + Premium
- In the money: when exercising the option would be profitable.
So for a Call this means if ST > X, and for a Put if ST < X.
- Out of the money: when exercising the option would not be
profitable. For a Call when ST ≤ X, and for a Put ST ≥ X.
- At the money: when the exercise price and asset price are
equal. So when ST = X.
,When we compare prices of call options with the same expiration date but different exercise
prices, we see that the value of a call is lower when the exercise price is higher. This makes
sense, because the right to purchase a share at a lower exercise price is more valuable than the
right to purchase at a higher price. Conversely, put options are worth more when the exercise
price is higher. You would rather have the right to sell a share for $135 than for $130, and this
is reflected in the prices of the puts.
There are different types of options:
• American: allows it holder to exercise the option at any time on or before the
expiration or maturity date.
• European: the option can only be exercised on the expiration or maturity date.
• Exotic: customized options that would have been highly unusual a few years ago.
- Barrier options: payoff depends on whether the underlying asset price crosses
trough some barrier.
- Asian options: payoff depends on the average price of the underlying asset during
at least some portion of the lifetime of the option.
- Lookback options: payoff depends on the minimum or maximum price of the
underlying asset during the lifetime of the option.
Bullish strategy: when you purchase a call option (or write a put). That is, the calls provide
profits when stock prices increase.
Bearish strategy: when you purchase a put option (or write a call). That is, the puts provide
profits when stock prices decrease.
Because option values depend on the price of the underlying
stock, purchase of options may be viewed as a substitute for
direct purchase or sale of a stock. Why might an option
strategy be preferable to direct stock transactions?
Answer: an option offers leverage. Calls are a levered
investment on the stock. Their values respond more than
proportionately to changes in the stock value. The figure
illustrates this point. The slope of the all-option portfolio is
far steeper than that of the all-stock portfolio, reflecting its
greater proportional sensitivity to the value of the underlying
asset. The leverage factor is the reason investors (illegally)
exploiting inside information commonly choose options as
their investment vehicle.
Protective put.
You use this as an insurance against stock price declines and you therefore invest in stock and
purchase a put option on the stock. Whatever happens to the stock price, you are guaranteed a
payoff at least equal to the put option’s exercise price because
the put gives you the right to sell your shares for that price.
Postion: long the stock and long the put. The profit is negative
and equal to the cost of the put if ST is below X. The protective
put limits losses.
Pay off: ST ≤ X ST > X
Long Stock ST ST
Long Put X– ST 0
Total profit X ST
,Covered Call.
A covered call position is the purchase of a share of stock with simultaneous sale of a call on
that stock. The call is ‘covered’ because the potential obligation to deliver the stock is covered
by the stock held in the portfolio. The value of a covered call position at the expiration of the
call equals the stock value minus the value of the call. The call value is subtracted because the
covered call position involves writing a call to another
investor who may exercise it at your expense. Regardless
of how high the stock price rises, the covered call writer’s
maximum payoff will be the strike price + the premium
received for the call.
Position: long the stock and write a call.
Pay off: ST ≤ X ST > X
Long Stock ST ST
Short Call 0 X - ST
Total profit ST X
! Straddle: is established by buying both a long call
and a long put on a stock, each with the same exercise price,
C, and the same expiration date T. Straddles are useful
strategies for investors who believe a stock will move a lot in
price but are uncertain about the direction of the move. The
straddle position will do well regardless of whether the stock
price makes extreme upward or downward moves from X.
The worst-case scenario for a straddle is no movement in the
stock price. Straddle positions, therefore, are bets on volatility. Conversely, investors who
write straddles – selling both a call and a put – must believe the stock is less volatile. They
accept the option premiums now, hoping the stock price will not change much before option
expiration. A short straddle is illustrated as an ‘upside-down long straddle’.
! Collar: an options strategy that brackets the value of a portfolio between two bounds
(buy a put, sell a call, different strike prices, buy stock). A collar would be appropriate for an
investor who has a target wealth goal in mind but is unwilling to risk losses beyond a certain
level.
! Spread: is a combination of two or more call options or put options on the same asset
with differing exercise prices or times to expiration. Some options are bought, whereas other
are sold, or written. A money spread (vertical spread) involves the purchase of one option and
the simultaneous sale of another with different exercise prices but same maturity. A time
spread refers to the sale and purchase of options with differing maturities but the same
exercise prices. There are two types of spreads.
1. Bull (Call) Spread. This involves a
money spread in which one call option is bought at an
exercise price of X1, whereas another call with identical
expiration date, but higher exercise price X2, is written.
The payoff to this position will be the difference in the
value of the call held and the value of the call written.
There are now three instead of two profit outcomes to
distinguish. It is called a bullish spread because the payoff either increases or is unaffected by
stock price increases. Holders of bullish spreads benefit from stock price increases.
A bull
put spread involves being short a put option and long another put option for same expiration
but with a lower strike. The short put generates income, whereas the long put's main purpose
,is to offset assignment risk. Because of the relationship between the two strike prices, the
investor will always be paid a premium (credit) when initiating
this position.
2. Bear (Put) Spread. This is a type of money
spread. It consists of buying one put in hopes of profiting from
a decline in the underlying stock, and writing another put with
the same expiration, but with a lower strike price, as a way to
offset some of the cost. Thus, the bear put spread pays off if the
price of the underlying asset depreciates.
A bear call spread is also a type of money spread. It contains
two calls with the same expiration but different strikes. The strike price of the short call is
below the strike of the long call, which means this strategy will always generate a net cash
inflow at the outset. The short call's main purpose is to generate income, whereas the long call
simply helps limit the risk from possible assignment.
! Strangle.
The long strangle involves going long (buying) both a call option and a put
option of the same underlying security. The options
expire at the same time have different strike prices. The
owner of a long strangle makes a profit if the underlying
price moves far enough away from the current price,
either above or below. Thus, an investor may take a long
strangle position if he thinks the underlying security is
highly volatile, but does not know which direction it is
going to move. This position is a limited risk, since the
most a purchaser may lose is the cost of both options. At
the same time, there is unlimited profit potential.
Put-Call Parity.
For European put & call options with the same X & T on the same underlying asset, we have:
𝑷 + 𝑺 = 𝑪 + 𝑷𝑽(𝑿)
This relation must hold regardless of your view about the security and the options prices. That
is, you may believe that some securities are overpriced, underpriced, or correctly priced but
the put-call parity must hold true in all cases. If the parity relation is ever violated, an
arbitrage opportunity arises. The parity only holds for European options (for American
options bounds can be derived).
Arbitrage opportunity: if the left-hand side of the equation (put & stock) is less than the right-
hand side (call & present value of the exercise price), it means that the put & stock strategy is
less expensive and thus, you will acquire the low cost alternative and sell the high cost
alternative:
, H18. Option Valuation.
Intrinsic value: payoff that could be made if the option was immediately exercised. It is set
equal to zero for out-of-the money or at-they-money options and it cannot be negative.
– Call: stock price ‐ exercise price (S0 – X)
– Put: exercise price ‐ stock price (X – S0)
Time value: the difference between the actual option price and the intrinsic value.
‘Adjusted intrinsic value’: Stock price – PV(X).
Call valuation: When the stock price is very low, the
option is nearly worthless, because there is almost no
chance that it will be exercised. When the stock price
is very high, the option value approaches the adjusted
intrinsic value.
Six determinants of option values:
1. Stock price. Call: If ST↑, this gives a higher
payoff and thus a higher option value.
Put: If ST↑, this gives a lower payoff since this
is calculated by X – S, and thus a lower option
value.
2. Exercise price. Call: If XT↑, this reduces the option value, because the payoff from
(S – X) will be lower.
Put: If XT↑, this increases the option value because the payoff from (X – S) will be
higher.
3. Volatility of stock price. Volatility influences stock prices and it is the main driver of
‘optionality’. The expected payoff for an option under high-volatility is greater than
under low-volatility, because the average option payoff is greater in a high-volatility
scenario.
4. Time to expiration. Longer time to expiration increases the value op an option. For
more distant expiration dates, there is more time for unpredictable future events to
affect prices, and the range of likely stock prices increases.
5. Interest rate. Call option values are higher when interest rates rise because higher
interest rates reduce the present value of the exercise price.
Put option values are lower when interest rates rise, because it reduces the present
value of the exercise price and thus the difference between (X – S).
6. Dividend rate on stock. Call: a higher dividend rate reduces the value of a call option
because when there is a dividend payment, the price of the stock will decrease by the
amount of the payment. In other words, the drag on stock price appreciation decreases
the potential payoff from the call option, thereby lowering the call value.
Put: a higher dividend rate increases the value of a put option.
Put options get more
expensive due to the fact that stock price always drop by the dividend amount after ex-
dividend date. So if S↓, this means the payoff (X – S) increases.
Restrictions on option values.
• The value cannot be negative. Because the option need not be exercised, it cannot
impose any liability on its holder.
• The value cannot exceed the stock value. You’ll never pay more for it than it’s price!
• The value of a call must be greater than the value of levered equity.
𝐶 ≥ 𝑆! − 𝑃𝑉 𝑋 − 𝑃𝑉(𝐷)
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