Assignment 10

!"# P A R T V I Options, Futures, and Other Derivatives

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Two-State Option Pricing A complete understanding of commonly used option-valuation formulas is difficult with- out a substantial mathematics background. Nevertheless, we can develop valuable insight into option valuation by considering a simple special case. Assume that a stock price can take only two possible values at option expiration: The stock will either increase to a given higher price or decrease to a given lower price. Although this may seem an extreme sim- plification, it provides a useful introduction to more complicated and realistic models. Moreover, it can be extended to describe far more reasonable specifications of stock price behavior. In fact, several major financial firms employ variants of this simple model to value options and securities with option-like features.

Suppose the stock now sells at S0 = $100, and the price will either increase by a factor of u = 1.20 to $120 (u stands for “up”) or fall by a factor of d = .9 to $90 (d stands for “down”) by year-end. A call option might specify an exercise price of $110 and a time to expiration of 1 year. The interest rate is 10%. At year-end, the payoff to the call will be either $0, if the stock price falls, or $10, if the stock price increases to $120.

These possibilities are illustrated by the following value “trees”:

$%" $"

$"" C

&" '"

Stock price Call option value

Now compare the payoff of the call to that of a portfolio consisting of one share of the stock and borrowing of $81.82 at the interest rate of 10%. The payoff of this portfolio also depends on the stock price at year-end:

Value of stock at year-end $!" $#$" ! Repayment of loan with interest !!" !!" Total $ " $ %"

We know the cash outlay to establish the portfolio is $18.18: $100 for the stock, less the $81.82 proceeds from borrowing. Therefore the portfolio’s value tree is

("

$).$)

'"

The payoff of this portfolio is exactly three times that of the call option for either value of the stock price. Because the portfolio replicates the payoff of the three calls, we call it a replicating portfolio. Moreover, because their payoffs are the same, the three calls and the replicating portfolio must have the same value. Therefore,

3C"="$18.18

or each call should sell at C = $6.06. Thus, given the stock price, exercise price, interest rate, and volatility of the stock price (as represented by the spread between the up or down movements), we can derive the fair value for the call option.

21.3 Binomial Option Pricing

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C H A P T E R !" Option Valuation #$#

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This valuation approach relies heavily on the notion of replication. With only two pos- sible end-of-year values of the stock, the payoffs to the levered stock portfolio replicate the payoffs to three call options and, therefore, command the same market price. Replication is behind most option-pricing formulas. For more complex price distributions for stocks, the replication technique is correspondingly more complex, but the principles remain the same.

One way to view the role of replication is to note that, using the numbers assumed for this example, a portfolio made up of one share of stock and three call options written is perfectly hedged. Its year-end value is independent of the ultimate stock price:

Stock value $!" $#$" ! Obligations from % calls written !" !%" Net payoff $!" $ !"

The investor has formed a riskless portfolio, with a payout of $90. Its value must be the present value of $90, or $90/1.10 = $81.82. The value of the portfolio, which equals $100 from the stock held long, minus 3C from the three calls written, should equal $81.82. Hence $100 ! 3C = $81.82, or C = $6.06.

The ability to create a perfect hedge is the key to this argument. The hedge locks in the end-of-year payout, which therefore can be discounted using the risk-free interest rate. To find the value of the option in terms of the value of the stock, we do not need to know either the option’s or the stock’s beta or expected rate of return. When a perfect hedge can be established, the final stock price does not affect the investor’s payoff, so the stock’s risk and return parameters have no bearing.

The hedge ratio of this example is one share of stock to three calls, or one-third. This ratio has an easy interpretation in this context: It is the ratio of the range of the values of the option to those of the stock across the two possible outcomes. The stock, which originally sells for S0 = 100, will be worth either d " $100 = $90 or u " $100 = $120, for a range of $30. If the stock price increases, the call will be worth Cu = $10, whereas if the stock price decreases, the call will be worth Cd = 0, for a range of $10. The ratio of ranges, 10/30, is one-third, which is the hedge ratio we have established.

The hedge ratio equals the ratio of ranges because the option and stock are perfectly correlated in this two-state example. Because they are perfectly correlated, a perfect hedge requires that they be held in a fraction determined only by relative volatility.

We can generalize the hedge ratio for other two-state option problems as

H#=# Cu#!#Cd _________ u S0#!#d S0

where Cu or Cd refers to the call option’s value when the stock goes up or down, respec- tively, and uS0 and dS0 are the stock prices in the two states. The hedge ratio, H, is the ratio of the swings in the possible end-of-period values of the option and the stock. If the investor writes one option and holds H shares of stock, the value of the portfolio will be unaffected by the stock price. In this case, option pricing is easy: Simply set the value of the hedged portfolio equal to the present value of the known payoff.

Using our example, the option-pricing technique would proceed as follows: 1. Given the possible end-of-year stock prices, uS0 = 120 and dS0 = 90, and the

exercise price of 110, calculate that Cu = 10 and Cd = 0. The stock price range is 30, while the option price range is 10.

2. Find that the hedge ratio of 10/30 = 1 ⁄ 3 . 3. Find that a portfolio made up of 1 ⁄ 3 share with one written option would have an

end-of-year value of $30 with certainty.

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C H A P T E R !" Option Valuation #$%

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The midrange value of 104.50 can be attained by two paths: an increase of 10% followed by a decrease of 5%, or a decrease of 5% followed by a 10% increase.

There are now three possible end-of-year values for the stock and three for the option:

Cuu Cu C Cud!=!Cdu Cd Cdd

Using methods similar to those we followed above, we could value Cu from knowledge of Cuu and Cud, then value Cd from knowledge of Cdu and Cdd, and finally value C from knowledge of Cu and Cd. And there is no reason to stop at 6-month intervals. We could next break the year into four 3-month units, or twelve 1-month units, or 365 1-day units, each of which would be posited to have a two-state process. Although the calculations become quite numerous and correspondingly tedious, they are easy to program into a computer, and such computer programs are used widely by participants in the options market.

Suppose that the risk-free interest rate is !% per "-month period and we wish to value a call option with exercise price $##$ on the stock described in the two-period price tree just above. We start by finding the value of Cu!. From this point, the call can rise to an expiration- date value of Cuu = $## (because at this point the stock price is u ! u ! S$ = $#%#) or fall to a final value of Cud = $ (because at this point, the stock price is u ! d ! S$ = $#$&.!$, which is less than the $##$ exercise price). Therefore the hedge ratio at this point is

H"=" Cuu"#"Cud __________ uuS$"#"udS$

"=" $##"#"$ _____________ $#%#"#"#$&.!$ "="

% __

'

Thus, the following portfolio will be worth $%$( at option expiration regardless of the ultimate stock price:

! udS" = $#"$.%" uuS" = $#&#

Buy % shares at price uS$ = $##$ $%$( $%&% Write ' calls at price Cu )))) ))$ )#'' Total $%$( $%$(

The portfolio must have a current market value equal to the present value of $%$(:

%"!"##$"#"'*Cu"="$%$(/#.$!"="$#((.$&+

Solve to find that Cu = $".(,&. Next we find the value of Cd. It is easy to see that this value must be zero. If we reach

this point (corresponding to a stock price of $(!), the stock price at option expiration will be either $#$&.!$ or $($.%!; in either case, the option will expire out of the money. (More formally, we could note that with Cud = Cdd = $, the hedge ratio is zero, and a portfolio of zero shares will replicate the payoff of the call!)

Finally, we solve for C using the values of Cu and Cd!. Concept Check %#.! leads you through the calculations that show the option value to be $&.&'&.

Example 21.1 Binomial Option Pricing

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C H A P T E R !" Option Valuation #""

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4Using this probability, the continuously compounded expected rate of return on the stock is .10. In general, the formula relating the probability of an upward movement to the annual expected rate of return, r, is p!=! exp(r"t )!#!d ___________

u!#!d .

5We adjust the probabilities of up versus down movements using the formula in footnote 4 to make the distribu- tions in Figure 21.5 comparable. In each panel, p is chosen so that the stock’s expected annualized, continuously compounded rate of return is 10%.

Suppose you are using a !-period model to value a "-year option on a stock with volatil- ity (i.e., annualized standard deviation) of $ = .!#. With a time to expiration of T = " year, and three subperiods, you would calculate "t = T/n = "/!, u = exp ($$ !

__ "t $) = exp (.!#$ !

___ "/! $)!=

"."%& and d!= exp (# $$ ! __

"t $) = exp (#.!#$ ! ___

"/! $) = .%'". Given the probability of an up move- ment, you could then work out the probability of any final stock price. For example, suppose the probability that the stock price increases is .((' and the probability that it decreases is .'').' Then the probability of stock prices at the end of the year would be as follows:

Event Possible Paths Probability Final Stock Price

! down movements

ddd #.'')! = #.#%& * (&.'% = "## % #.%'"!

+ down and " up

ddu, dud, udd

! % #.'')+ % #.((' = #.!!# * %'."# = "## % "."%& % #.%'"+

" down and + up

uud, udu, duu

! % #.'') % #.(('+ = #.'"" ""%.%& = "## % "."%&+ % #.%'"

! up movements

uuu #.(('! = #.",# ")%.#& = "## % "."%&!

We plot this probability distribution in Figure +".(, Panel A. Notice that the two middle end-of- period stock prices are, in fact, more likely than either extreme.

Example 21.2 Calibrating u and d to Stock Volatility

In Example +".+, we broke up the year into three subperiods. Let’s now look at the cases of ) and +# subperiods.

Subperiods, n !t = T/n u!=!exp ("! ! ___

!t !) d!=!exp (#"! ! ___

!t !)

! #.!!! exp(#.",!) = "."%& exp(##.",!) = #.%'" ) #."), exp(#."++) = "."!# exp(##."++) = #.%%(

+# #.#(# exp(#.#),) = ".#)& exp(##.#),) = #.&!(

We plot the resulting probability distributions in Panels B and C of Figure +".(.(

Example 21.3 Increasing the Number of Subperiods

Now we can extend Example 21.2 by breaking up the option maturity into ever-shorter subintervals. As we do, the stock price distribution becomes increasingly plausible, as we demonstrate in Example 21.3.

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resembles a lognormal distribution.6 Thus the apparent oversimplification of the two-state model can be overcome by progressively subdividing any period into many subperiods.

At any node, one still can set up a portfolio that is perfectly hedged over the next tiny time interval. Then, at the end of that interval, on reaching the next node, a new hedge ratio can be computed and the portfolio composition could be revised to remain hedged

6Actually, more complex considerations enter here. The limit of this process is lognormal only if we assume also that stock prices move continuously, by which we mean that over small time intervals only small price move- ments can occur. This rules out rare events such as sudden, extreme price moves in response to dramatic infor- mation (like a takeover attempt). For a treatment of this type of “jump process,” see John C. Cox and Stephen A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics 3 (January–March 1976), pp. 145–66; or Robert C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics 3 (January–March 1976), pp. 125–44.

A Risk-Neutral Shortcut W

O R

D S

FR O

M T

H E

S T

R E

E T

We pointed out earlier in the chapter that the binomial model valuation approach is arbitrage-based. We can value the option by replicating it with shares of stock plus borrow- ing. The ability to replicate the option means that its price relative to the stock and the interest rate must be based only on the technology of replication and not on risk prefer- ences. It cannot depend on risk aversion or the capital asset pricing model or any other model of equilibrium risk-return relationships.

This insight—that the pricing model must be independent of risk aversion—leads to a very useful shortcut to valuing options. Imagine a risk-neutral economy, that is, an economy in which all investors are risk-neutral. This hypothetical economy must value options the same as our real one because risk aversion cannot affect the valuation formula.

In a risk-neutral economy, investors would not demand risk premiums and would therefore value all assets by discounting expected payoffs at the risk-free rate of interest. Therefore, a security such as a call option would be valued by discounting

its expected cash flow at the risk-free rate: C! =! “E”(CF) ______ "! +! rf

. We

put the expectation operator E in quotation marks to signify that this is not the true expectation, but the expectation that would prevail in the hypothetical risk-neutral economy. To be consistent, we must calculate this expected cash flow using the rate of return the stock would have in the risk-neutral economy, not using its true rate of return. But if we successfully maintain consistency, the value derived for the hypothetical economy should match the one in our own.

How do we compute the expected cash flow from the option in the risk-neutral economy? Because there are no risk premiums, the stock’s expected rate of return must equal the risk-free rate. Call p the probability that the stock price increases. Then p must be chosen to equate the expected rate of increase of the stock price to the risk-free rate (we ignore dividends here):

“E”(S")!=!p(uS)!+!("!"!p)dS!=!("!+!rf$$)S

This implies that p!=! "!+!rf!"!d ________ u!"!d

. We call p a risk-neutral probability

to distinguish it from the true, or “objective,” probability. To illustrate, in our two-state example at the beginning of Section %".#, we had u = ".%, d = .&, and rf = ."'. Given these

values, p!=! "!+!."'!"!.& __________ ".%!"!.&

!=! % __ #

.

Now let’s see what happens if we use the discounted cash flow formula to value the option in the risk-neutral economy. We continue to use the two-state example from Section %".#. We find the present value of the option payoff using the risk- neutral probability and discount at the risk-free interest rate:

C!=! “E”((CF)

_______ "!+!rf

!=! pCu!+!("!"!p)Cd ______________

"!+!rf !=!

%/#!#!"'!+!"/#!#!' _______________

"."' !=!).')

This answer exactly matches the value we found using our no- arbitrage approach!

We repeat: This is not truly an expected discounted value.

• The numerator is not the true expected cash flow from the option because we use the risk-neutral probability, p, rather than the true probability.

• The denominator is not the proper discount rate for option cash flows because we do not account for the risk.

• In a sense, these two “errors” cancel out. But this is not just luck: We are assured to get the correct result because the no-arbitrage approach implies that risk preferences can- not affect the option value. Therefore, the value computed for the risk-neutral economy must equal the value that we obtain in our economy.

When we move to the more realistic multiperiod model, the calculations are more cumbersome, but the idea is the same. Footnote * shows how to relate p to any expected rate of return and volatility estimate. Simply set the expected rate of return on the stock equal to the risk-free rate, use the resulting probability to work out the expected payoff from the option, discount at the risk-free rate, and you will find the option value. These calculations are actually fairly easy to program in Excel.

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!"# P A R T V I Options, Futures, and Other Derivatives

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Some of the important assumptions underlying the formula are the following: 1. The stock will pay no dividends until after the option expiration date. 2. Both the interest rate, r, and variance rate, !2, of the stock are constant (or in

slightly more general versions of the formula, both are known functions of time— any changes are perfectly predictable).

3. Stock prices are continuous, meaning that sudden extreme jumps such as those in the aftermath of an announcement of a takeover attempt are ruled out.

Variants of the Black-Scholes formula have been developed to deal with many of these limitations.

Second, even within the context of the Black-Scholes model, you must be sure of the accuracy of the parameters used in the formula. Four of these—S0, X, T, and r—are straightforward. The stock price, exercise price, and time to expiration are readily deter- mined. The interest rate used is the money market rate for a maturity equal to that of the option, and the dividend payout is reasonably predictable, at least over short horizons.

The last input, though, the standard deviation of the stock return, is not directly observable. It must be estimated from historical data, from scenario analysis, or from the prices of other options, as we will describe momentarily.

We saw in Chapter 5 that the historical variance of stock market returns can be calcu- lated from n observations as follows:

!2"=" n _____ n"#"1

! t = 1

n (rt"#" ̄ r )

2 _______

n

where ̄ r is the average return over the sample period. The rate of return on day t is defined to be consistent with continuous compounding as rt = ln(St /St # 1). [We note again that the natural logarithm of a ratio is approximately the percentage difference between the numer- ator and denominator so that ln(St /St # 1) is a measure of the rate of return of the stock from time t # 1 to time t.] Historical variance commonly is computed using daily returns over periods of several months. Because the volatility of stock returns must be estimated, however, it is always possible that discrepancies between an option price and its Black- Scholes value are simply artifacts of error in the estimation of the stock’s volatility.

In fact, market participants often give the option-valuation problem a different twist. Rather than calculating a Black-Scholes option value for a given stock’s standard devia- tion, they ask instead: What standard deviation would be necessary for the option price that I observe to be consistent with the Black-Scholes formula? This is called the implied volatility of the option, the volatility level for the stock implied by the option price. Inves- tors can then judge whether they think the actual stock standard deviation exceeds the implied volatility. If it does, the option is considered a good buy; if actual volatility seems greater than the implied volatility, its fair price would exceed the observed price.

Another variation is to compare two options on the same stock with equal expiration dates but different exercise prices. The option with the higher implied volatility would be considered relatively expensive, because a higher standard deviation is required to justify its price. The analyst might consider buying the option with the lower implied volatility and writing the option with the higher implied volatility.

The Black-Scholes valuation formula, as well as the implied volatility, is easily calcu- lated using an Excel spreadsheet like Spreadsheet 21.1. The model inputs are provided in column B, and the outputs are given in column E. The formulas for d1 and d2 are provided in the spreadsheet, and the Excel formula NORMSDIST(d1) or NORM.S.DIST(d1, TRUE) is used to calculate N(d1)."Cell E6 contains the Black-Scholes formula. (The formula in the spreadsheet actually includes an adjustment for dividends, as described in the next section.)

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C H A P T E R !" Option Valuation #!"

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Dividends and Call Option Valuation We noted earlier that the Black-Scholes call option formula applies to stocks that do not pay dividends. When dividends are to be paid before the option expires, we need to adjust the formula. The payment of dividends raises the possibility of early exercise, and for most realistic dividend payout schemes the valuation formula becomes significantly more complex than the Black-Scholes equation.

When stocks pay quarterly dividends, their share prices decline by a discrete amount, roughly equal to the amount of the dividend. In some cases, it will be rational for the call holder to exercise just before the stock goes ex dividend. This introduces uncertainty into “maturity” of the call. Will it be exercised at the ex-dividend date or held until the expira- tion date?12 Variations on the Black-Scholes formula have been developed that can accom- modate dividends, but the resulting valuation formulas are more complex and become rapidly more difficult as the number of possible dividend payments increase.13

In one special case, the dividend adjustment takes a simple form and allows us to use a slight variant of the Black-Scholes formula. Suppose the underlying asset pays a con- tinuous flow of income. This might be a reasonable assumption for options on a stock index, where different stocks in the index pay dividends on different days, so that dividend income arrives in a more or less continuous flow. If the dividend yield, denoted !, is con- stant, one can show that the present value of that dividend flow accruing until the option expiration date is S0 (1 " e"!T).14# In this case, S0 " PV(Div) = S0 e"!T, and we can use the Black-Scholes call option formula on the dividend-paying asset simply by substituting S0 e"!T for S0 in the original formula. This approach is used in Spreadsheet 21.1.

One warning about this practice, however. Even with continuous dividends, it may be rational to exercise the call option early, so strictly speaking, the modified Black-Scholes formula would apply only to European options. As a general rule, American calls on divi- dend paying stocks will be worth more than European ones even if dividends are continuous.

Put Option Valuation We have concentrated so far on call option valuation. We can derive Black-Scholes European put option values from call option values using the put-call parity theorem. To value the put option, we simply calculate the value of the corresponding call option in Equation#21.1 from the Black-Scholes formula, and solve for the put option value (on a non-dividend paying stock) as P#=#C#+#PV(X )#"#S0 =#C#+#X e"rT#"#S0

(21.2)

We calculate the present value of the exercise price using continuous compounding to be consistent with the Black-Scholes formula.

12While the stock price falls by a discrete amount on the ex-dividend date, the option price does not. The dividend is announced in advance and is anticipated by the market. The option price will adjust smoothly over time to reflect the approaching dividend payment. 13An exact formula for American call valuation on dividend-paying stocks has been developed in Richard Roll, “An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends,” Journal of Financial Economics 5 (November 1977). The technique has been discussed and revised in Robert Geske, “A Note on an Analytical Formula for Unprotected American Call Options on Stocks with Known Dividends,” Journal of Financial Economics 7 (December 1979); and Robert E. Whaley, “On the Valuation of American Call Options on Stocks with Known Dividends,” Journal of Financial Economics 9 (June 1981). 14For intuition about this formula, notice that e"!T approximately equals 1 " !T, so the value of the dividend is approximately !TS0.

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!"" P A R T V I Options, Futures, and Other Derivatives

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Sometimes, it is easier to work with a put option valuation formula directly. If we sub- stitute the Black-Scholes formula for a call in Equation 21.2, we obtain the value of a European put option as

P!=!Xe"rT [ 1!"!N(d2) ] " S0 [ 1!"!N(d1) ] (21.3)

Using data from Example !".# (C = $"$.%&,' X = $(),' S& = $"&&,' r = ."&,' # = .)&, and T = .!)), Equation !".$ implies that a European put option on that stock with identical exercise price and time to expiration is worth

$()*e"."&** $ .!)("!"!.)%"#)!"!$"&&("!"!.+++#)!=!$+.$)

Notice that this value is consistent with put-call parity:

P!=!C!+!PV(X)!"!S&!=!"$.%&!+!()*e"."&****$ .!)!"!"&&!=!+.$)

As we noted traders can do, we might then compare this formula value to the actual put price as one step in formulating a trading strategy.

Example 21.5 Black-Scholes Put Valuation

Dividends and Put Option Valuation Equation 21.2 and Equation 21.3 are valid for European puts on non-dividend-paying stocks. As we did for call options, if the underlying asset pays a dividend, we can find European put values by substituting S0 " PV(Div) for S0. Cell E7 in Spreadsheet 21.1 allows for a continuous dividend flow with a dividend yield of %. In that case S0 " PV(Div) = S0e"%T.

However, listed put options on stocks are American options that offer the opportunity of early exercise, and we have seen that the right to exercise puts early can turn out to be valuable. This means that an American put option must be worth more than the cor- responding European option. Therefore, Equation 21.2 and! Equation 21.3 describe only the lower bound on the true value of the American put. However, in many applications the approximation is very accurate.

Hedge Ratios and the Black-Scholes Formula In Chapter 20, we considered two investments in FinCorp stock: 100 shares or 1,000 call options. We saw that the call option position was more sensitive to swings in the stock price than was the all-stock position. To analyze the overall exposure to a stock price more precisely, however, it is necessary to quantify these relative sensitivities. We can summa- rize the overall exposure of portfolios of options with various exercise prices and times to expiration using the hedge ratio, the change in option price for a $1 increase in the stock price. A call option, therefore, has a positive hedge ratio and a put option a negative hedge ratio. The hedge ratio is commonly called the option’s delta.

21.5 Using the Black-Scholes Formula

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!"# P A R T V I Options, Futures, and Other Derivatives

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insurers gambled that the futures price would recover to its expected premium over the stock index and chose to defer sales, they remained underhedged. As the market fell farther, their portfolios experienced substantial losses.

Although most observers at the time believed that the portfolio insurance industry would never recover from the market crash, delta hedging is still alive and well on Wall Street. Dynamic hedges are widely used by large firms to hedge potential losses from options positions. For example, when Microsoft ended its employee stock option program in 2003, its investment banker J.P. Morgan purchased many already-issued options of Microsoft employees, and it was widely expected that Morgan would protect its options position by selling shares in Microsoft using a delta hedging strategy.16

Option Pricing and the Crisis of 2008–2009 Merton17 shows how option pricing models can provide insight into the financial crisis of 2008–2009. The key to understanding his argument is to remember that when banks lend to or buy the debt of firms with limited liability, they implicitly write a put option to the borrower (see Chapter 20, Section 20.5). If the borrower has sufficient assets to pay off the loan when it comes due, it will do so, and the lender will be fully repaid. But if the borrower has insufficient assets, it can declare bankruptcy and discharge its obligations by transferring ownership of the firm to its creditors. The borrower’s ability to satisfy the loan by transferring ownership is equivalent to the right to “sell” itself to the creditor for the face value of the loan. This arrangement is therefore just like a put option on the firm with exercise price equal to the stipulated loan repayment.

Consider the payoff to the lender at loan maturity (time T ) as a function of the value of the borrowing firm, VT, when the loan, with face value L, comes due. If VT ! L, the lender is paid off in full. But if VT < L, the lender gets the firm, which is worth less than the promised payment L.

We can write the payoff in a way that emphasizes the implicit put option:

payoff"=" { L VT ="L"#" { 0 if"VT"!"L L"#"VT if"VT"<"L (21.4) Equation 21.4 shows that the payoff on the loan equals L (when the firm has sufficient assets to pay off the debt), minus the payoff of a put option on the value of the firm (VT) with an exercise price of L. Therefore, we may view risky lending as a combination of safe lending, with a guaranteed payoff of L, combined with a short position in a put option on the borrower.

When firms sell credit default swaps (see Chapter 14, Section 14.5), the implicit put option is even clearer. Here, the CDS seller agrees to make up any losses due to the insol- vency of a bond issuer. If the issuer goes bankrupt, leaving assets of only VT for the credi- tors, the CDS seller is obligated to make up the difference, L # VT. This is in essence a pure put option.

Now think about the exposure of these implicit put writers to changes in the financial health of the underlying firm. The value of a put option on VT appears in Figure 21.13. When the firm is financially strong (i.e., V is far greater than L), the slope of the curve

16To read more about this episode, see"Jathon Sapsford and Ken Brown, “J.P. Morgan Rolls Dice on Microsoft Options,” The Wall Street Journal, July 9, 2003. 17This material is based on a lecture given by Robert Merton at MIT in March 2009. You can find the lecture online"at http://video.mit.edu/watch/observations-on-the-science-of-finance-in-the-practice-of-finance-9449/.

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C H A P T E R !" Option Valuation #$%

out-of-the-money puts would be nearly worthless if stock prices evolve smoothly, because the probability of the stock falling by a large amount (and the put option thereby moving into the money) in a short time would be very small. But a possibility of a sudden large downward jump that could move the puts into the money, as in a market crash, would impart greater value to these options. Thus, the market might price these options as though there is a bigger chance of a large drop in the stock price than would be suggested by the Black- Scholes assumptions. The result of the higher option price is a greater implied volatility.

Interestingly, Rubinstein points out that prior to the 1987 market crash, plots of implied volatility like the one in Figure 21.15 were relatively flat, consistent with the notion that the market was then less attuned to fears of a crash. However, postcrash plots have been consistently downward sloping, exhibiting a shape often called the option smirk. When we use option-pricing models that allow for more general stock price distributions, including crash risk and random changes in volatility, they generate downward-sloping implied vola- tility curves similar to those shown in Figure 21.15.19

19For an extensive discussion of these more general models, see R. L. McDonald, Derivatives Markets, 3rd ed. (Boston: Pearson Education [Addison-Wesley], 2013).

1. Option values may be viewed as the sum of intrinsic value plus time or “volatility” value. The volatility value is the right to choose not to exercise if the stock price moves against the holder. Thus the option holder cannot lose more than the cost of the option regardless of stock price performance.

2. Call options are more valuable when the exercise price is lower, when the stock price is higher, when the interest rate is higher, when the time to expiration is greater, when the stock’s volatility is greater, and when dividends are lower.

3. Call options must sell for at least the stock price less the present value of the exercise price and dividends to be paid before expiration. This implies that a call option on a non-dividend-paying stock may be sold for more than the proceeds from immediate exercise. Thus European calls are worth as much as American calls on stocks that pay no dividends, because the right to exercise the American call early has no value.

4. Options may be valued using a binomial pricing model that assumes the stock price can take on only two values by the end of any short time period. As the number of such periods increases, the binomial model can approximate more realistic stock price distributions. The Black-Scholes for- mula may be seen as a limiting case of the binomial option model as the holding period is divided into progressively smaller subperiods when the interest rate and stock volatility are constant.

5. The Black-Scholes formula applies to options on stocks that pay no dividends. Dividend adjustments may be adequate to price European calls on dividend-paying stocks, but the proper treatment of American calls on dividend-paying stocks requires more complex formulas.

6. It may be optimal to exercise put options early, whether or not the stock pays dividends. Therefore, American puts generally are worth more than European puts.

7. European put values can be derived from the call value and the put-call parity relationship. This technique cannot be applied to American puts for which optimal early exercise is a possibility.

8. The implied volatility of an option is the standard deviation of stock returns consistent with the option’s market price. It can be backed out of an option-pricing model by finding the stock volatility that makes the option’s value equal to its observed price.

9. The hedge ratio (or option delta) is the number of shares of stock required to hedge the price risk involved in writing one option. Hedge ratios are near zero for deep out-of-the-money call options and approach 1.0 for deep in-the-money calls.

SUMMARY

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!"# P A R T V I Options, Futures, and Other Derivatives

Binomial model: u!=!exp (" ! ___

#t ) ;!d!=!exp ($ " ! ___

#t ) ;!p!=! exp (r#t )!$!d ___________ u!$!d

Put-call parity: P!=!C!+!PV(X )!$!S0!+!PV(dividends)

Black-Scholes formula (no dividend case): SN(d1)!$!Xe$rT N(d2)

where d1!=! ln(S/X )!+!(r!+!! "2)T

__________________ " !

__ T ; d2!=!d1!$!" !

__ T

Delta (or hedge ratio): H!=! Change!in!option!value ___________________ Change!in!stock!value

KEY EQUATIONS

10. Although call option deltas are less than 1.0, their elasticities are greater than 1.0. The rate of return on a call (as opposed to the dollar return) responds more than one-for-one with stock returns.

11. Portfolio insurance can be obtained by purchasing a protective put option on an equity position. When the appropriate put is not traded, portfolio insurance entails a dynamic hedge strategy where a fraction of the equity portfolio equal to the desired put option’s delta is sold and placed in risk-free securities.

12. The option delta is used to determine the hedge ratio for options positions. Delta-neutral portfolios are independent of price changes in the underlying asset. Even delta-neutral option portfolios are subject to volatility risk, however.

13. Empirically, implied volatilities derived from the Black-Scholes formula tend to be higher on options with lower exercise prices. This may be evidence that the option prices reflect the possibility of a sudden dramatic decline in stock prices. Such “crashes” are inconsistent with the Black-Scholes assumptions.

intrinsic value time value binomial model Black-Scholes pricing formula implied volatility

KEY TERMS hedge ratio delta option elasticity portfolio insurance dynamic hedging

gamma delta neutral vega

1. We showed in the text that the value of a call option increases with the volatility of the stock. Is this also true of put option values? Use the put-call parity theorem as well as a numerical example to prove your answer.

2. Would you expect a $1 increase in a call option’s exercise price to lead to a decrease in the option’s value of more or less than $1?

3. Is a put option on a high-beta stock worth more than one on a low-beta stock? The stocks have identical firm-specific risk.

4. All else equal, is a call option on a stock with a lot of firm-specific risk worth more than one on a stock with little firm-specific risk? The betas of the two stocks are equal.

5. All else equal, will a call option with a high exercise price have a higher or lower hedge ratio than one with a low exercise price?

6. In each of the following questions, you are asked to compare two options with parameters as given. The risk-free interest rate for all cases should be assumed to be 4%. Assume the stocks on which these options are written pay no dividends. a. Put T X " Price of Option

A !." "! !.#! $$! B !." "! !.#" $$!

PROBLEM SETS

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C H A P T E R !" Option Valuation #$#

Which put option is written on the stock with the lower price? i. A. ii. B. iii. Not enough information.

b. Put T X ! Price of Option

A !." "! !.# $$! B !." "! !.# $$#

Which put option must be written on the stock with the lower price? i. A. ii. B. iii. Not enough information.

c. Call S X ! Price of Option

A "! "! !.#! $$# B "" "! !.#! $$!

Which call option must have the lower time to expiration? i. A. ii. B. iii. Not enough information.

d. Call T X S Price of Option

A !." "! "" $$! B !." "! "" $$#

Which call option is written on the stock with higher volatility? i. A. ii. B. iii. Not enough information.

e. Call T X S Price of Option

A !." "! "" $$! B !." "! "" $ %

Which call option is written on the stock with higher volatility? i. A. ii. B. iii. Not enough information. 7. Reconsider the determination of the hedge ratio in the two-state model (see Section 21.2), where

we showed that one-third share of stock would hedge one option. What would be the hedge ratio for the following exercise prices: (a) 120, (b) 110, (c) 100, (d) 90? (e) What do you conclude about the hedge ratio as the option becomes progressively more in the money?

8. Show that Black-Scholes call option hedge ratios also increase as the stock price increases. Consider a 1-year option with exercise price $50, on a stock with annual standard deviation 20%. The T-bill rate is 3% per year. Find N(d1) for stock prices (a) $45, (b) $50, and (c) $55.

9. We will derive a two-state put option value in this problem. Data: S0 = 100; X = 110; 1 + r = 1.10. The two possibilities for ST are 130 and 80. a. Show that the range of S is 50, whereas that of P is 30 across the two states. What is the hedge

ratio of the put? b. Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this

portfolio? c. What is the present value of the portfolio? d. Given that the stock currently is selling at 100, solve for the value of the put.

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!"# P A R T V I Options, Futures, and Other Derivatives

10. a. Calculate the value of a call option on the stock in Problem 9 with an exercise price of 110.

b. Verify that the put-call parity theorem is satisfied by your answers to Problem 9 and part (a). (Do not use continuous compounding to calculate the present value of X in this example because we are using a two-state model here with discrete periods, not a continuous-time Black-Scholes model.)

11. Use the Black-Scholes formula to find the value of a call option on the following stock:

Time to expiration ! months Standard deviation "#% per year Exercise price $"# Stock price $"# Annual interest rate $% Dividend #

12. Find the Black-Scholes value of a put option on the stock in Problem 11 with the same exercise price and expiration as the call option.

13. Recalculate the value of the call option in Problem 11, successively substituting one of the changes below while keeping the other parameters as in Problem 11: a. Time to expiration = 3 months. b. Standard deviation = 25% per year. c. Exercise price = $55. d. Stock price = $55. e. Interest rate = 5%.

Consider each scenario independently. Confirm that the option value changes in accordance with the prediction of Table 21.1.

14. A call option with X = $50 on a stock currently priced at S = $55 is selling for $10. Using a volatility estimate of ! = .30, you find that N(d1) = .6 and N(d2) = .5. The risk-free inter- est rate is zero. Is the implied volatility based on the option price more or less than .30? Explain.

15. What would be the Excel formula in Spreadsheet 21.1 for the Black-Scholes value of a straddle position?

Use the following case in answering Problems 16 through 21: Mark Washington, CFA, is an ana- lyst 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 experience a downturn of approximately 4% over a 12-month period. However, portfolio performance for BIC was disappointing, lagging its peer group by nearly 10%. Washington has been told to review the options strategy to determine why the hedged portfolio did not perform as expected. 16. Which of the following best explains a delta-neutral portfolio? A delta-neutral portfolio is per-

fectly hedged against: a. Small price changes in the underlying asset. b. Small price decreases in the underlying asset. c. All price changes in the underlying asset.

17. After discussing the concept of a delta-neutral portfolio, Washington determines that he needs to further explain the concept of delta. Washington draws the value of an option as a function of the underlying stock price. Using this diagram, indicate how delta is interpreted. Delta is the: a. Slope in the option price diagram. b. Curvature of the option price graph. c. Level in the option price diagram.

18. Washington considers a put option that has a delta of ".65. If the price of the underlying asset decreases by $6, then what is the best estimate of the change in option price?

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C H A P T E R !" Option Valuation #$%

19. BIC owns 51,750 shares of Smith & Oates. The shares are currently priced at $69. A call option on Smith & Oates with a strike price of $70 is selling at $3.50 and has a delta of .69. What is the number of call options necessary to create a delta-neutral hedge?

20. Return to Problem 19. Will the number of call options written for a delta-neutral hedge increase or decrease if the stock price falls?

21. Which of the following statements regarding the goal of a delta-neutral portfolio is most accurate? One example of a delta-neutral portfolio is to combine a: a. Long position in a stock with a short position in call options so that the value of the portfolio

does not change with changes in the value of the stock. b. Long position in a stock with a short position in a call option so that the value of the

portfolio changes with changes in the value of the stock. c. Long position in a stock with a long position in call options so that the value of the portfolio

does not change with changes in the value of the stock. 22. Should the rate of return of a call option on a long-term Treasury bond be more or less sensitive

to changes in interest rates than is the rate of return of the underlying bond? 23. If the stock price falls and the call price rises, then what has happened to the call option’s

implied volatility? 24. If the time to expiration falls and the put price rises, then what has happened to the put option’s

implied volatility? 25. According to the Black-Scholes formula, what will be the hedge ratio (delta) of a call option as

the stock price becomes infinitely large? Explain briefly. 26. According to the Black-Scholes formula, what will be the hedge ratio (delta) of a put option for

a very small exercise price? 27. The hedge ratio of an at-the-money call option on IBM is .4. The hedge ratio of an at-the-money

put option is !.6. What is the hedge ratio of an at-the-money straddle position on IBM? 28. Consider a 6-month expiration European call option with exercise price $105. The underlying

stock sells for $100 a share and pays no dividends. The risk-free rate is 5%. What is the implied volatility of the option if the option currently sells for $8? Use Spreadsheet 21.1 (available in Connect; link to Chapter 21 material) to answer this question. a. Go to the Data tab of the spreadsheet and select Goal Seek from the What-If menu. The

dialog box will ask you for three pieces of information. In that dialog box, you should set cell E6 to value 8 by changing cell B2. In other words, you ask the spreadsheet to find the value of standard deviation (which appears in cell B2) that forces the value of the option (in cell E6) equal to $8. Then click OK, and you should find that the call is now worth $8, and the entry for standard deviation has been changed to a level consistent with this value. This is the call’s implied standard deviation at a price of $8.

b. What happens to implied volatility if the option is selling at $9? c. Why has implied volatility increased? d. What happens to implied volatility if the option price is unchanged at $8, but the time until

option expiration is lower, say, only 4 months? Why? e. What happens to implied volatility if the option price is unchanged at $8, but the exercise

price is lower, say, only $100? Why? f. What happens to implied volatility if the option price is unchanged at $8, but the stock price

is lower, say, only $98? Why? 29. A collar is established by buying a share of stock for $50, buying a 6-month put option with

exercise price $45, and writing a 6-month call option with exercise price $55. On the basis of the volatility of the stock, you calculate that for a strike price of $45 and expiration of 6 months, N(d1) = .60, whereas for the exercise price of $55, N(d1) = .35. a. What will be the gain or loss on the collar if the stock price increases by $1? b. What happens to the delta of the portfolio if the stock price becomes very large? c. What happens to the delta of the portfolio if the stock price becomes very small?

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!"# P A R T V I Options, Futures, and Other Derivatives

30. These three put options are all written on the same stock. One has a delta of !.9, one a delta of !.5, and one a delta of !.1. Assign deltas to the three puts by filling in this table.

Put X Delta

A !" (a)# B $" (b)# C %" (c)

31. You are very bullish (optimistic) on stock EFG, much more so than the rest of the market. In each question, choose the portfolio strategy that will give you the biggest dollar profit if your bullish forecast turns out to be correct. Explain your answer. a. Choice A: $10,000 invested in calls with X = 50. Choice B: $10,000 invested in EFG stock. b. Choice A: 10 call option contracts (for 100 shares each), with X = 50. Choice B: 1,000 shares of EFG stock.

32. You would like to be holding a protective put position on the stock of XYZ Co. to lock in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year the stock price will increase by 10% or decrease by 10%. The T-bill rate is 5%. Unfortunately, no put options are traded on XYZ Co. a. Suppose the desired put option were traded. How much would it cost to purchase? b. What would have been the cost of the protective put portfolio? c. What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that

would be provided by a protective put with X = 100? Show that the payoff to this portfolio and the cost of establishing the portfolio match those of the desired protective put.

33. Return to Example 21.1. Use the binomial model to value a 1-year European put option with exercise price $110 on the stock in that example. Confirm that your solution for the put price satisfies put-call parity.

34. Suppose that the risk-free interest rate is zero. Would an American put option ever be exercised early? Explain.

35. Let p(S, T, X) denote the value of a European put on a stock selling at S dollars, with time to maturity T, and with exercise price X, and let P(S, T, X) be the value of an American put. a. Evaluate p(0, T, X). b. Evaluate P(0, T, X). c. Evaluate p(S, T, 0). d. Evaluate P(S, T, 0). e. What does your answer to part (b) tell you about the possibility that American puts may be

exercised early? 36. You are attempting to value a call option with an exercise price of $100 and one year to expira-

tion. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and a 50% chance of decreasing to $80. The risk-free rate of interest is 10%. Calculate the call option’s value using the two-state stock price model.

37. Consider an increase in the volatility of the stock in the previous problem. Suppose that if the stock increases in price, it will increase to $130, and that if it falls, it will fall to $70. Show that the value of the call option is now higher than the value derived in the previous problem.

38. Calculate the value of a put option with exercise price $100 using the data in Problem 36. Show that put-call parity is satisfied by your solution.

39. XYZ Corp. will pay a $2 per share dividend in two months. Its stock price currently is $60 per share. A call option on XYZ has an exercise price of $55 and 3-month time to expiration. The risk-free interest rate is .5% per month, and the stock’s volatility (standard deviation) = 7%

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C H A P T E R !" Option Valuation #$"

per month. Find the Black-Scholes value of the option. (Hint: Try defining one “period” as a month, rather than as a year, and think about the net-of-dividend value of each share.)

40. “The beta of a call option on General Electric is greater than the beta of a share of General Electric.” True or false?

41. “The beta of a call option on the S&P 500 index with an exercise price of 1,930 is greater than the beta of a call on the index with an exercise price of 1,940.” True or false?

42. What will happen to the hedge ratio of a convertible bond as the stock price becomes very large?

43. Goldman Sachs believes that market volatility will be 20% annually for the next three years. Three-year at-the-money call and put options on the market index sell at an implied volatility of 22%. What options portfolio can Goldman establish to speculate on its volatility belief with- out taking a bullish or bearish position on the market? Using Goldman’s estimate of volatility, 3-year at-the-money options have N(d1) = .6.

44. You are holding call options on a stock. The stock’s beta is .75, and you are concerned that the stock market is about to fall. The stock is currently selling for $5 and you hold 1 million options on the stock (i.e., you hold 10,000 contracts for 100 shares each). The option delta is .8. How much of the market-index portfolio must you buy or sell to hedge your market exposure?

45. Imagine you are a provider of portfolio insurance. You are establishing a 4-year program. The portfolio you manage is currently worth $100 million, and you hope to provide a minimum return of 0%. The equity portfolio has a standard deviation of 25% per year, and T-bills pay 5% per year. Assume for simplicity that the portfolio pays no dividends (or that all dividends are reinvested). a. How much should be placed in bills? How much in equity? b. What should the manager do if the stock portfolio falls by 3% on the first day of trading?

46. Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike price $90 are selling at an implied volatility of 30%. ExxonMobil stock currently is $90 per share, and the risk-free rate is 4%. a. If you believe the true volatility of the stock is 32%, would you want to buy or sell call

options? b. Now you need to hedge your option position against changes in the stock price. How many

shares of stock will you hold for each option contract purchased or sold? 47. Using the data in Problem 46, suppose that 3-month put options with a strike price of $90 are

selling at an implied volatility of 34%. Construct a delta-neutral portfolio comprising positions in calls and puts that will profit when the option prices come back into alignment.

48. Suppose that JPMorgan Chase sells call options on $1.25 million worth of a stock portfolio with beta = 1.5. The option delta is .8. It wishes to hedge its resultant exposure to a market advance by buying a market-index portfolio. a. How many dollars’ worth of the market-index portfolio should it purchase to hedge its

position? b. Now it decides to use market index puts to hedge its exposure. Should it buy or sell puts?

How many? The index at current prices represents $1,000 worth of stock. 49. Suppose you are attempting to value a 1-year expiration option on a stock with volatility

(i.e., annualized standard deviation) of ! = .40. What would be the appropriate values for u and d if your binomial model is set up using: a. 1 period of 1 year. b. 4 subperiods, each 3 months. c. 12 subperiods, each 1 month.

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!"# P A R T V I Options, Futures, and Other Derivatives

50. You build a binomial model with one period and assert that over the course of a year, the stock price will either rise by a factor of 1.5 or fall by a factor of 2/3. What is your implicit assumption about the volatility of the stock’s rate of return over the next year?

51. Use the put-call parity relationship to demonstrate that an at-the-money call option on a nondividend-paying stock must cost more than an at-the-money put option. Show that the prices of the put and call will be equal if S0 = (1 + r)T.

52. Return to Problem 36. Value the call option using the risk-neutral shortcut described in the box in Section 21.3. Confirm that your answer matches the value you get using the two-state approach.

53. Return to Problem 38. a. What will be the payoff to the put, Pu, if the stock goes up? b. What will be the payoff, Pd, if the stock price falls? c. Value the put option using the risk-neutral shortcut described in the box in Section 21.3. d. Confirm that your answer matches the value you get using the two-state approach.

1. The board of directors of Abco Company is concerned about the downside risk of a $100 mil- lion equity portfolio in its pension plan. The board’s consultant has proposed temporarily (for 1 month) hedging the portfolio with either futures or options. Referring to the following table, the consultant states: a. “The $100 million equity portfolio can be fully protected on the downside by selling

(shorting) 4,000 futures contracts.” b. “The cost of this protection is that the portfolio’s expected rate of return will be zero percent.”

Market, Portfolio, and Contract Data

Equity index level !!."" Equity futures price #""."" Futures contract multiplier $$%" Portfolio beta #.$" Contract expiration (months) &

Critique the accuracy of each of the consultant’s two statements. 2. Michael Weber, CFA, is analyzing several aspects of option valuation, including the determinants

of the value of an option, the characteristics of various models used to value options, and the potential for divergence of calculated option values from observed market prices. a. What is the expected effect on the value of a call option on common stock if the volatility of

the underlying stock price decreases? If the time to expiration of the option increases? b. Using the Black-Scholes option-pricing model and an estimate of stock return volatility,

Weber calculates the price of a 3-month call option and notices the option’s calculated value is different from its market price. With respect to Weber’s use of the Black-Scholes option- pricing model,

i. Discuss why the calculated value of an out-of-the-money European option may differ from its market price.

ii. Discuss why the calculated value of an American option may differ from its market price. 3. Joel Franklin is a portfolio manager responsible for derivatives. Franklin observes an American-

style option and a European-style option with the same strike price, expiration, and underly- ing stock. Franklin believes that the European-style option will have a higher premium than the American-style option. a. Critique Franklin’s belief that the European-style option will have a higher premium. Franklin

is asked to value a 1-year European-style call option for Abaco Ltd. common stock, which last traded at $43.00. He has collected the information in the following table.

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C H A P T E R !" Option Valuation #$%

Closing stock price $!".## Call and put option exercise price !$.## %-year put option price !.## %-year Treasury bill rate $.$#% Time to expiration One year

b. Calculate, using put-call parity and the information provided in the table, the European-style call option value.

c. State the effect, if any, of each of the following three variables on the value of a call option. (No calculations required.)

i. An increase in short-term interest rate. ii. An increase in stock price volatility. iii. A decrease in time to option expiration. 4. A stock index is currently trading at 50. Paul Tripp, CFA, wants to value 2-year index options

using the binomial model. The stock will either increase in value by 20% or fall in value by 20%. The annual risk-free interest rate is 6%. No dividends are paid on any of the underlying securities in the index. a. Construct a two-period binomial tree for the value of the stock index. b. Calculate the value of a European call option on the index with an exercise price of 60. c. Calculate the value of a European put option on the index with an exercise price of 60. d. Confirm that your solutions for the values of the call and the put satisfy put-call parity.

5. Ken Webster manages a $400 million equity portfolio benchmarked to the S&P 500 index. Webster believes the market is overvalued when measured by several traditional fundamental/ economic indicators. He is concerned about potential losses but recognizes that the S&P 500 index could nevertheless move above its current 1,766 level.

Webster is considering the following option collar strategy: ! Protection for the portfolio can be attained by purchasing an S&P 500 index put with a strike

price of 1,760. ! The put can be approximately financed by selling one 1,800 strike-price call for every put

purchased. ! Because the combined delta of the call and put positions (see following table) is greater than

"1 (i.e., ".44 " 30 = ".77), the options will not lose more than the underlying portfolio will gain if the market advances.

The information in the following table describes the two options used to create the collar.

Characteristics ",&'' Call ",#(' Put

Option price $"!.%# $"&.&# Option implied volatility &&% &!% Option’s delta #."# "#.!!

Notes: • Ignore transaction costs. • S&P $## historical "#-day volatility = &"%. • Time to option expiration = "# days.

a. Describe the potential returns of the combined portfolio (the underlying portfolio plus the option collar) if after 30 days the S&P 500 index has:

i. Risen approximately 5% to 1,854. ii. Remained at 1,766 (no change). iii. Declined by approximately 5% to 1,682. (No calculations are necessary.)

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!"" P A R T V I Options, Futures, and Other Derivatives

b. Discuss the effect on the hedge ratio (delta) of each option as the S&P 500 approaches the level of!each of the potential outcomes listed in part (a).

c. Evaluate the pricing of each of the following in relation to the volatility data provided: i. The put. ii. The call.

E#INVESTMENTS EXERCISES $. Use information from finance.yahoo.com to answer the following questions.

a. What is Coke’s current price?

b. Now enter the ticker “KO” (for Coca-Cola) and find the AnalystOpinion tab. What is the mean $%-month target price for Coke? Based on this forecast, would at-the-money calls or puts have the higher expected profit?

c. What is the spread between the high and low target stock prices, expressed as a percent- age of Coke’s current stock price? How (qualitatively) should the spread be related to the price at which Coke options trade?

d. Calculate the implied volatility of the call option closest to the money with time to expira- tion of about three months. You can use Spreadsheet %$.$ (available in Connect) to calcu- late implied volatility using the Goal Seek command.

e. Now repeat the exercise for Pepsi (ticker: PEP). What would you expect to be the relation- ship between the high versus low target price spread and the implied volatility of the two companies? Are your expectations consistent with actual option prices?

f. Suppose you believe that the volatility of KO is going to increase from currently antic- ipated levels. Would its call options be overpriced or underpriced? What about its put options?

g. Could you take positions in both puts and calls on KO in such a manner as to speculate on your volatility beliefs without taking a stance on whether the stock price is going to increase or decrease? Would you buy or write each type of option?

h. How would your relative positions in puts and calls be related to the delta of each option?

%. Calculating implied volatility can be difficult if you don’t have a spreadsheet handy. Fortunately, many tools are available on the Web to perform the calculation; for example, www.numa.com contains option calculators that also compute implied volatility.

Using daily price data (available from finance.yahoo.com), calculate the annualized stan- dard deviation of the daily percentage change in a stock price. Try calculating standard devia- tion using historical data covering (a) &' days, (b) $%' days, and (c) $(' days. For the same stock, use the numa Web site to find the implied volatility. The input for the risk-free rate may be found at www.bloomberg.com/markets/rates/index.html.Option price data can be retrieved from www.cboe.com.

Which sample period for calculating historical standard deviation seems most correlated with implied volatility?

SOLUTIONS TO CONCEPT CHECKS 1. To understand the impact of higher volatility, consider the same scenarios as for the call. The

low-volatility scenario yields a lower expected payoff. High volatility Stock price $!" # $$" # $%" # $&" # $'"

Put payoff $$" # $!" # $##" # $##" # $##" Low volatility Stock price $$" # $$' # $%" # $%' # $&"

Put payoff $!" # $##' # $##" # $##" # $##"

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!"" P A R T V I Options, Futures, and Other Derivatives

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The value S0 ! X is sometimes called the intrinsic value of in-the-money call options because it gives the payoff that could be obtained by immediate exercise. Intrinsic value is set equal to zero for out-of-the-money or at-the-money options. The difference between the actual call price and the intrinsic value is commonly called the time value of the option.

“Time value” is unfortunate terminology because it may confuse the option’s time value with the time value of money. Time value in the options context refers simply to the dif- ference between the option’s price and the value it would have if it were expiring imme- diately. It is the part of the option’s value that may be attributed to the fact that it still has positive time to expiration.

Most of an option’s time value typically is a type of “volatility value.” Because the option holder can choose not to exercise, the payoff cannot be worse than zero. Even if a call option is out of the money now, it still will sell for a positive price because it offers the potential for a profit if the stock price increases, while imposing no risk of additional loss should the stock price fall. The volatility value lies in the value of the right not to exercise if that action would be unprofitable. The option to exercise, as opposed to the obligation to exercise, provides insurance against poor stock price performance.

As the stock price increases substantially, it becomes likely that the call option will be exercised by expiration. Ultimately, with exercise all but assured, the volatility value becomes minimal. As the stock price gets ever larger, the option value approaches the “adjusted” intrinsic value, the stock price minus the present value of the exercise price, S0 ! PV(X).

Why should this be? If you are virtually certain the option will be exercised and the stock purchased for X dollars, it is as though you own the stock already. The stock certifi- cate, with a value today of S0, might as well be sitting in your safe-deposit box now, as it will be there shortly. You just haven’t paid for it yet. The present value of your obligation is the present value of X, so the net value of the call option is S0 ! PV(X).1

Figure 21.1 illustrates the call option valuation function. The value curve shows that 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 adjusted intrinsic value. In the midrange case, where the option is approxi- mately at the money, the option curve diverges from the straight lines corresponding to adjusted intrinsic value. This is because although exercise today would have a negligible (or negative) payoff, the volatility value of the option is quite high in this region.

The call always increases in value with the stock price. The slope is greatest, however, when the option is deep in the money. In this case, exercise is all but assured, and the option increases in price one-for-one with the stock price.

Determinants of Option Values We can identify at least six factors that should affect the value of a call option: the stock price, the exercise price, the volatility of the stock price, the time to expiration, the interest rate, and the dividend rate of the stock. The call option should increase in value with the stock price and decrease in value with the exercise price because the payoff to a call, if exercised, equals ST ! X. The expected payoff increases with the difference S0 ! X.

1This discussion presumes that the stock pays no dividends until after option expiration. If the stock does pay dividends before expiration, then there is a reason you would care about getting the stock now rather than at expiration—getting it now entitles you to the interim dividend payments. In this case, the adjusted intrinsic value of the option must subtract the value of the dividends the stock will pay out before the call is exercised. Adjusted intrinsic value would more generally be defined as S0 ! PV(X) ! PV(D), where D is the dividend to be paid before option expiration.

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C H A P T E R !" Option Valuation #$%

2. If This Variable Increases . . . The Value of a Put Option

S Decreases

X Increases

! Increases T Increases

*

rf Decreases

Dividend payouts Increases

*For American puts, increase in time to expiration must increase value. One can always choose to exercise early if this is optimal; the longer expiration date simply expands the range of alternatives open to the option holder which must make the option more valuable. For a European put, where early exercise is not allowed, longer time to expira- tion can have an indeterminate effect. Longer expiration increases volatility value because the final stock price is more uncertain, but it reduces the present value of the exercise price that will be received if the put is exercised. The net effect on put value can be positive or negative.

3. The parity relationship assumes that all options are held until expiration and that there are no cash flows until expiration. These assumptions are valid only in the special case of European options on non-dividend-paying stocks. If the stock pays no dividends, the American and European calls are equally valuable, whereas the American put is worth more than the European put. Therefore, although the parity theorem for European options states that

P"="C"#"S0"+"PV(X )

in fact, P will be greater than this value if the put is American. 4. Because the option now is underpriced, we want to reverse our previous strategy.

! ! Cash Flow in " Year for Each Possible Stock Price

& Initial Cash Flow S = '( S = "!(

Buy ! options #"#.$% % !% Short-sell " share; repay in " year "%% #&% #"'% Lend $(!.$% at "%% interest rate #(!.$% &".($ )&".($ TOTAL % ".($ ".($

The riskless cash flow in 1 year per option is $1.85/3 = $.6167, and the present value is $.6167/1.10 = $.56, precisely the amount by which the option is underpriced.

5. a. Cu # Cd = $6.984 # 0 b. uS0 # dS0 = $110 # $95 = $15 c. 6.984/15 = .4656 d. Value in Next Period as

Function of Stock Price

Action Today (time () dS( = $'% uS( = $""(

Buy %.*#$# shares at price S% = $"%% $**.'!' $$".'"# Write " call at price C% ++++ +++++++% ##.&(* TOTAL $**.'!' $**.'!'

The portfolio must have a market value equal to the present value of $44.232. e. $44.232/1.05 = $42.126 f. .4656 $ $100 # C0 = $42.126 C0 = $46.56 # $42.126 = $4.434

6. When %t shrinks, there should be lower possible dispersion in the stock price by the end of the subperiod because each shorter subperiod offers less time in which new information can move stock prices. However, as the time interval shrinks, there will be a correspondingly greater number of these subperiods until option expiration. Thus, total volatility over the remaining life

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!"# P A R T V I Options, Futures, and Other Derivatives

of the option will be unaffected. In fact, take another look at Figure 21.5. There, despite the fact that u and d each get closer to 1 as the number of subintervals increases and the length of each subinterval falls, the total volatility of the stock return until option expiration is unaffected.

7. Because ! = .6, !2 = .36.

d1"=" ln(100/95)"+"(.10"+".36/2).25 ________________________

.6 ! ___

.25 "=".4043

d2"="d1"#".6 ! ___

.25 "=".1043

Using Table 21.2 and interpolation, or from a spreadsheet function:

N(d1)"=".6570 N(d2)"=".5415

C"="100"$".6570"#"95e#.10 $ .25"$".5415"="15.53

8. Implied volatility exceeds .2783. Given a standard deviation of .2783, the option value is $7. A higher volatility is needed to justify an $8 price. Using Spreadsheet 21.1 and Goal Seek, you can confirm that implied volatility at an option price of $8 is .3138.

9. A $1 increase in stock price is a percentage increase of 1/122 = .82%. The put option price will fall by (.4 $ $1) = $.40, a percentage decrease of $.40/$4 = 10%. Elasticity is #10/.82 = #12.2.

10. The delta for a call option is N(d1), which is positive, and in this case is .547. Therefore, for every 10 option contracts purchased, you would need to short 547 shares of stock.

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C H A P T E R !" Option Valuation #$%

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Several quantitative models of option pricing have been devised, and we will examine some of them later in this chapter. All models, however, rely on simplifying assumptions. You might wonder which properties of option values are truly general and which depend on the particular simplifications. To start with, we will consider some of the more impor- tant general properties of option prices. Some of these properties have important implica- tions for the effect of stock dividends on option values and the possible profitability of early exercise of an American option.

Restrictions on the Value of a Call Option The most obvious restriction on the value of a call option is that its value cannot be nega- tive. Because the option need not be exercised, it cannot impose any liability on its holder; moreover, as long as there is any possibility that at some point the option can be exercised profitably, it will command a positive price. Its payoff is zero at worst, and possibly posi- tive, so it has some positive value.

We can place another lower bound on the value of a call option. Suppose that the stock will pay a dividend of D dollars just before the option expiration date, denoted by T (where today is time 0). Now compare two portfolios, one consisting of a call option on one share of stock and the other a leveraged equity position consisting of that share and borrowing of (X + D)/(1 + rf)T dollars. The loan repayment is X + D dollars, due on the expiration date of the option. For example, for a one-year maturity option with exercise price $70, dividends to be paid of $5, and an effective annual interest of 5%, you would purchase one share of stock and borrow $75/1.05 = $71.43. In one year, when the loan matures, the payment due is $75.!At that time, the payoff to the leveraged equity position is given by the following table (where ST denotes the stock price at the option expiration date).

In General Our Numbers

Stock value ST + D ST + ! ""Payback of loan "(X + D) "#!

Total ST " X ST " #$

Notice that the payoff to the stock is the ex-dividend stock value plus dividends received. Whether the total payoff to the stock-plus-borrowing position is positive or negative depends on whether ST exceeds X. The net cash outlay required to establish this leveraged equity position is S0 " $71.43, or, more generally, S0 " (X + D)/(1 + rf)T, that is, the current price of the stock, S0, less the initial cash inflow from the borrowing position.

The payoff to the call option will be ST " X if the option expires in the money and zero otherwise. Thus the option payoff is equal to the leveraged equity payoff when that payoff is positive and greater when the leveraged equity position has a negative payoff. Because the option’s payoff is always greater than or equal to that of the leveraged equity position, its price must exceed the cost of establishing that position.

Therefore, the value of the call must be greater than S0 " (X + D)/(1 + rf)T, or, more generally,

C!#!S0!"!PV(X )!"!PV(D) where PV(X) denotes the present value of the exercise price and PV(D) is the present value of the dividends the stock will pay at the option’s expiration. More generally, we can inter- pret PV(D) as the present value of any and all dividends to be paid prior to the option expi- ration date. Because we know already that the value of a call option must be nonnegative, we may conclude that C is greater than the maximum of either 0 or S0 " PV(X) " PV(D).

21.2 Restrictions on Option Values

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