COMPOUND OPTIONS AND STOCK OPTIONS

7.3 COMPOUND OPTIONS AND STOCK OPTIONS

Stocks are assets that represent equity shares issued by individual firms. They have various forms, granting various powers to stockholders. In general, stock- holders are entitled to dividend payments made by the firm and to the right to vote at the firm’s assembly. Stocks are also a claim to the value of the firm that they share with bondholders. For example, if the firm defaults on its interest pay- ments, bondholders can force the firm into bankruptcy to recover the loans. A stockholder, a junior claimant in this case, has generally nothing left to claim. Hence, a bondholder has the right to sell the company at a given threshold or,

OPTIONS AND PRACTICE

equivalently, the bondholder holds a put on the value of the firm that the stock- holder must hold short. Hence, a stock can be viewed as a claim or option on the value of the firm that is shared with bondholders. In practice, managers are often given stock options on their firm so they may align their welfare with those of the shareholders. The rationale of such compensation is that a manager whose income is heavily dependent on an upward move of the firm’s stock price will

be more likely to pursue an aggressive policy leading to a stock price rise as his payoff is a convex increasing function of the stock price. The shareholders will, of course, benefit from such a rise while it assumes some risk due to the call (stock) option’s limited liability granted to the manager. This case illustrates some of the economic limits of risk-neutral pricing, which presumes that risk can be elim- inated by trading it away. Further, this supposes the existence of another party willing to take the risk for no extra compensation. This can happen only if markets are perfectly liquid or there exists another investor willing to take on the exact opposite risk. Risk-neutrality presupposes therefore that there is always such an exact opposite. In reality, as is the case for executives’ options, the strategy is set up so that the risk is not shifted away. For most applications, risk-neutrality may

be used comfortably. But, the more out of the money options are, the less risk can

be transferred and, thus, the more speculators are needed to take this risk. This means that in crash times or other extreme events, risk-neutral pricing tends to break down.

With these limitations in mind, we can apply risk-neutral pricing to value options or compound options (options on a stock option or some other underlying asset). Define a stock option (a claim) on the value of the firm (its stock price). To do so, say that a firm has N shares whose price is S and let the firm’s debt

be expressed by a pure discount bond B with maturity T . Initially, the value of the firm V can be written as V = NS + B. Assuming risk-neutral pricing, the

stockprice (using an annual risk-free discount rate) over one and two periods is:

S=

˜S(1) =

E ∗ ˜S(2)

(1 + R f )

(1 + R f ) 2

For a binomial process, shown in Figure 7.1, we have: ˜S(1) = (S h , S d ) and ˜S(2) = (S hh , S hd S dd ). By the same token, we compute recursively the value of

the compound (stock) option by:

with ˜ C c (1) = C ˜ c h ,˜ C c d C c (2) = C ˜ c hh ,˜ C hd c ,˜ C c dd

C c 2 c 2 hh c = Max [0, h V − B]; C dd = Max [0, d V − B]; C hd = Max [0, h dV − B] and therefore:

 p ∗2

(Max[0, h

2 V − B])+ 

  2(1 − p ∗ )p ∗ (Max [0, h dV − B]) +  1+R f (1 − p ∗ ) 2 Max(0, d 2 V − B)

COMPOUND OPTIONS AND STOCK OPTIONS

hV 2 hV

V hdV

dV 2 dV

S hh = max 0, ( hV − B

S hd = max 0, ( hdV − B )

dd = max 0, ( dV − B

Figure 7.1 Compound option.

Here the risk neutral probability is:

1+R f −d

h−d

Note that this model differs from the simple plain vanilla model treated earlier,

since in this case, S h d

is V = NP, a portion is invested in a risk-free asset and the other in a risky asset, similarly to the previous binomial case. For example, say that u = 1.3

while d = 0.8 and the risk free rate is R f = 0.1. Thus the risk-neutral probability is:

0.36 [Max (0, 1.69V − B)] + =

0.48 [Max (0, 0.8V − B)] + 

0.16 [Max (0, 0.64V − B)] Now, if bondholders have a claim on 40 % of the firm value, we have:

C c =V

1.1 [0.36 (1.29) + 0.48 (0.4) + 0.16 (0.24)] = (0.57421)V

OPTIONS AND PRACTICE

Problem

What are the effects of an increase of 5 % on bondholders’ share of the firm on the option’s price?

Problem

High-tech firms (and in particular start-ups) often offer their employees stock options instead of salary increases. When is it better to ‘take the money’ over the options and vice versa. Construct a model to justify your case.

7.3.1 Warrants

Warrants are compound options, used by corporations that issue call options with their stock as an underlying asset. When the option is exercised, new stock is issued, diluting other stockholders’ holdings but adding capital to the corporation.

A warrant is valued as follows. Say that V is the firm’s value and let there be n warrants, providing the right to buy one share of stock at a price of x and assume no other source of financing. If all warrants are exercised, then the new value of the firm is V + mx and thus, each warrant must at least be worth its price x, or:

V + mx > x N+m

This means that a warrant is exercised only if:

V + mx > (N + m)x and V>Nx or x < V /N = S If the value at time t is: W (V , τ ), τ = T − t or at time t = 0,

0 V ≤ Nx Then, if we set: λ = 1/ [N + m] we have λ(V + mx) − x = λV − x(1 − mλ)

and thereby the price W (V , 0) can be written as follows: x

W (1 − mλ)

(V , 0) = Max [λV − x(1 − mλ), 0] = Max V− , 0 λ

This corresponds to an option whose price is V , the value of the firm, and whose strike (in a Black–Scholes model) is [x(1 − mλ)] /λ, thus, applying the Black–

Scholes option pricing formula, we have at any one time:

V , τ, (V , τ, λ, x) = λW (1 − mλ) = W (λV, τ, x(1 − mλ))

Therefore, it is possible to value a warrant using the Black–Scholes option for- mula. For example, say that there are m = 500 warrants with a strike price

x = 100, a time to maturity of τ = 0.25 years, the yearly risk-free interest rate is R f = 10 %, the stock price volatility is σ = 20 % a year and let there be

169 the warrant’s price is calculated by:

COMPOUND OPTIONS AND STOCK OPTIONS

[λV , τ, x(1 − mλ)] = W {1.5(10 5 )λ, 0.25, 100[1 − (500)λ]}

= W (14.285, 0.25, 95.23) where λ = 1/(10 500) = 0.095 238.

In a similar manner, other compound options such as options on a call (call on call, put on call) and options on put (put on put, put on call) etc. may be valued.

7.3.2 Other options

We consider next and briefly a number of other options in a continuous-time framework. Throughout, we assume that the underlying process is a lognormal process.

Options on dividend -paying stocks are options on stocks that pay dividends at a rate of D proportional to the stock price. Note that the underlying price process with dividends is then:

dS S = (µ − D) dt + σ dW

Thus, applying risk-neutral pricing, the partial differential equation that values the option is given by:

2 ∂ S 2 and the boundary condition for a European call option is V (S, T ) = Max

(S(T ) − K, 0). If we apply a no-arbitrage argument as we have in the previ- ous chapter, we are left with −D dt which in essence deflates the price of the stock for the option holder (since the option holder, not owning the stock, does not benefit from dividend distribution). On this basis we obtain the option price deflated by dividends.

Options on foreign currencies are derived in the same manner. Instead of dividends, however, it is the foreign risk-free rate R for that we use. In this case, the partial differential equation is:

∂ V ∂ V 1 2 2 ∂ 2 V −R f V+

2 ∂ =0 t ∂ S 2 ∂ S Again, by specifying the appropriate boundaries, we can estimate the value of the

+ (R f −R for )S

corresponding option. Unlike options on dividend paying stocks, options on commodities involve a carrying charge of, say q S dt, which is a fraction of the value of the commodity that goes toward paying the carrying charge. As a result, the corresponding differential equation is:

2 ∂ S 2 with an appropriate boundary condition, specified according to the type of option

OPTIONS AND PRACTICE

Options on futures are defined by noting that (see also Chapter 8):

F = Se R f (T F −t)

Thus, the value of an option on a stock and an option on its futures are inherently connected by the above relationship. However, futures differ from options on stock in that the underlying security is a futures contract. Upon exercise, the option holder obtains a position in the futures contract. If we apply Ito’s differential rule to determine the value of the option on the futures, we have:

dt = e which is introduced in our partial differential equation to yield:

dF = + dS +

R f (T F −t) dS ∂ t

2 (dS) or dF + R f S

V σ 2 S 2 ∂ V −R F f F +

∂ t + 2 ∂ S 2 =0 This is solved with the appropriate boundary constraint (determined by the con-

tract we seek to value). Although options on futures have existed in Europe for some time, they have only recently become available in America. In 1982, the Commodity Futures Trading Commission allowed each commodity exchange to trade options on one of its futures contracts. In that year eight exchanges intro- duced options. These contracts included gold, heating oil, sugar, T-bonds and three market indices. Options on futures now trade on every major futures ex- change. The underlying spot commodities include financial assets such as bonds, Eurodollars and stock indices, foreign currencies such as British pounds and euros, precious metals such as gold and silver, livestock commodities such as hogs and cattle and agricultural commodities such as corn and soybeans.

An option on a futures price for say, a commodity, can be related to the spot price by:

(R F = Se f −q)(T F −t)

For a financial asset, q is the dividend yield on the asset, whereas for a commodity (which can be consumed), q must be modified to reflect the convenience yield less the carrying charge. Now in a risk-neutral economy the expected growth rate

in the price of a stock which pays continuous dividends at a rate of q is R f − q. In such an economy, the expected growth rate of a futures price should be zero, because trading a futures contract requires no initial investment. This means, that

for pricing purposes, the value of q should be R f . That is, for pricing an option on futures, the futures prices can be treated in the same way as a security paying

a continuous dividend yield rate R f −R f . We substitute G(t) = F(t) e (T F −t) into Merton’s model, leading to the model above and whose solution for a European call option is (as established by Fisher–Black):

1 1 2 (0) = e

V F −R f T F ∗

[F(0)N (d 1 ) − XN(d 2 )]; d 1 = √

ln + σ T F σ T F X 2

√ and d 2 ∗ =d 1 ∗ −σ T F .

OPTIONS AND PRACTICE