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Since no closed-form expression for the value of an American put is available, in order to value American options one needs to use a numerical procedure. It appears that the use of the CRR bino-mial tree, although remarkably simple, is far from being the most efficient way of pricing American options. Various approximations of the American put price on a non-dividend-paying stock were examined in Brennan and Schwartz (1977), Johnson (1983), MacMillan (1986), and Broadie and Detemple (1996).

Let us comment briefly on the approximate valuation method proposed by Geske and Johnson (1984). Basically, the Geske-Johnson approximation relies on the discretization of the time param-eter and the application of backward induction, as in any other standard discrete-time approach.

However, in contrast to the space-time discretization used in the multinomial trees approach or in the finite difference methods, the approach of Geske and Johnson makes use of the exact distribu-tion of the vector of stock prices (S_t1, . . . , S_tn), where t₁ < . . . < t_n = T are the only admissible (deterministic) exercise times. ⁴ In other words, the decision to exercise an option can be made at any of the dates t₁, . . . , t_n only. Let us start by considering the special case when n = 2 and t₁ = T /2, t₂ = T. Note that if n = 1, the option can be exercised at t₁ = T only, so that it is

4An option which may be exercised early,but only on predetermined dates,is commonly referred to as Bermudan option.

5.5. APPROXIMATIONS OF THE AMERICAN PUT PRICE 99 equivalent to a European put. To find an approximate value for an American put, we shall argue by backward induction. Suppose that the option was not exercised at time t₁. Then the value of the option at time t₁ is equal to the value of a European put option with maturity t₂− t1= T /2, given the initial stock price St1= S_{T /2}. The price of a European put is given, of course, by the standard Black-Scholes formula, denoted by P (S_{T /2}, T /2). The critical stock price b^∗₁ at time t1= T /2 solves the equation K− ST /2 = P (S_{T /2}, T /2), hence it can be found by numerical methods. Moreover, it is clear that the value V_{T /2} of the option at time T /2 satisfies

V_{T /2} =

 P (S_{T /2}, T /2) if S_{T /2}> b^∗₁, K− ST /2 if S_{T /2}≤ b^∗1.

Note that it is optimal to exercise the option at time t1 if and only if S_{T /2} ≤ b^∗₁. To find the value of the option at time 0, we need first to evaluate the expectation V₀(S₀) = E_P∗(e^−rT/2V_{T /2}), or equivalently

V0(S0) = e^−rT/2EP^∗



e^−rT/2(ST − K)⁺I_{{ST /2> b}∗

1}+ (K− ST /2)I_{{ST /2≤ b}∗ 1}

.

Notice that the latter expectation can be expressed in terms of the probability law of the two-dimensional random variable (S_{T /2}, ST); equivalently, one exploit the joint law of W_{T /2} and WT. A specific, quasi-explicit representation of V0(S0) in terms of two-dimensional Gaussian cumulative distribution function is in fact a matter of convenience. The approximate value of an American put with two admissible exercise times, T /2 and T, equals P₂^a(S0, T ) = V0(S0). The same iterative procedure may be applied to an arbitrary finite sequence of times t₁ < . . . < t_n = T. In this case, the Geske-Johnson approximation formula involves integration with respect to a n-dimensional Gaussian probability density function. It appears that for three admissible exercise times, T /3, 2T /3 and T, the approximate quasi-analytical valuation formula provided by the Geske and Johnson method is roughly as accurate as the binomial tree with 150time steps. For any natural n, let us denote by P_n^a(S₀, T ) the Geske-Johnson option’s approximate value associated with admissible dates t_i= T i/n, i = 1, . . . , n. It is possible to show that the sequence P_n^a(S₀, T ) converges to the option’s exact price P^a(S0, T ) when the number of steps tends to infinity, so that the step length tends to zero. To estimate the limit P^a(S0, T ), one can make use of any extrapolation technique, for instance Richardson’s approximation scheme. Let us briefly describe the latter technique. Suppose that the function F satisfies

F (h) = F (0) + c1h + c2h²+ o(h²) in the neighborhood of zero, so that

F (kh) = F (0) + c1kh + c2k²h²+ o(h²) and

F (lh) = F (0) + c1lh + c2l²h²+ o(h²)

for arbitrary l > k > 1. Ignoring the term o(h²), and solving the above system of equations for F (0), we obtain (“≈ ” denotes approximate equality)

F (0)≈ F (h) +a c

F (h)− F (kh) +b

c

F (kh)− F (lh)

, (5.26)

where a = l(l− 1) − k(k − 1), b = k(k − 1) and c = l²(k− 1) − l(k²− 1) + k(k − 1). Let us write Pn^ato denote P_n^a(S₀, T ) for n = 1, 2, 3 (in particular, P₁^a(S₀, T ) is the European put price P (S₀, T )). For n = 3 upon setting k = 3/2, l = 3 and P₁^a= F (lh), P₂^a = F (kh), P₃^a = F (h), we get the following approximate formula

P^a(S₀, T )≈ P3^a+⁷₂(P₃^a− P2^a)−¹₂(P₂^a− P1^a).

Bunch and Johnson (1992) argue that the Geske-Johnson method can be further improved if the exercise times are chosen iteratively in such a way that the option’s approximate value is maximized.

5.6 Option on a Dividend-paying Stock

Since most traded options on stocks are unprotected American call options written on dividend-paying stocks, it is worthwhile to comment briefly on the valuation of these contracts. A call option is said to be unprotected if it has no contracted “protection” against the stock price decline that occurs when a dividend is paid. It is intuitively clear that an unprotected American call written on a dividend-paying stock is not equivalent to the corresponding option of European style, in general.

Suppose that a known dividend, D, will be paid to each shareholder with certainty at a prespecified date TD during the option’s lifetime. Furthermore, assume that the ex-dividend stock price decline equals δD for a given constant δ∈ [0, 1]. Let us denote by STD and P_TD = S_TD−δD respectively the cum-dividend and ex-dividend stock prices at time T_D. It is clear that the option should eventually be exercised just before the dividend is paid – that is, an instant before T_D. Consequently, as first noted by Black (1975), the lower bound for the price of such an option is the price of the European call option with expiry date T_D and strike price K. This lower bound is a good estimate of the exact value of the price of the American option whenever the probability of early exercise is large – that is, when the probability P{CTD < S_TD − K} is large, where CTD = C(P_TD, T − TD, K) is the Black-Scholes price of the European call option with maturity T − TD and exercise price K.

Hence, early exercise of the American call is more likely the larger the dividend, the higher the stock price STD relative to the strike price K, and the shorter the time-period T − TD between expiry and dividend payment dates. An analytic valuation formula for unprotected American call options on stocks with known dividends was established by Roll (1977). However, it seems to us that Roll’s original reasoning, which refers to options that expire an instant before the ex-dividend date, assumes implicitly that the holder of an option may exercise it before the ex-dividend date, but apparently is not allowed to sell it before the ex-dividend date. To avoid this discrepancy, we prefer instead to consider European options which expire on the ex-dividend date – i.e., after the ex-dividend stock price decline.

Before formulating the next result, we need to introduce some notation. Let us denote by b^∗ the cum-dividend stock price level above which the original American option will be exercised at time T_D, so that

C(b^∗− δD, T − TD, K) = b^∗− K. (5.27) It is worthwhile to observe that C(s− δD, T − TD, K) < s− K when s ∈ (b^∗,∞), and C(s − δD, T− TD, K) > s− K for every s ∈ (0, b^∗). Note that the first two terms on the right-hand side of equality (5.28) below represent the values of European options, written on a stock S, which expire at time T and on the ex-dividend date T_D, respectively. The last term, CO_t(T_D, b^∗−K), represents the price of a so-called compound option (see Sect. 6.4). To be more specific, we deal here with a European call option with strike price b^∗− K which expires on the ex-dividend date TD, and whose underlying asset is the European call option, written on S, with maturity T and strike price K.

The compound option will be exercised by its holder at the ex-dividend date T_D if and only if he is prepared to pay b^∗− K for the underlying European option. Since the value of the underlying option after the ex-dividend stock price decline equals C(PTD, T − TD, K), the compound option is exercised whenever

C(PTD, T− TD, K) = C(STD− δD, T − TD, K) > b^∗− K,

that is, when the cum-dividend stock price exceeds b^∗ (this follows from the fact that the price of a standard European call option is an increasing function of the stock price, combined with equality (5.27)).

Proposition 5.6.1 The arbitrage price ˜C_t^a(T, K) of an unprotected American call option with expiry date T > TD and strike price K, written on a stock which pays a known dividend D at time TD, equals

C˜_t^a(T, K) = ˜Ct(T, K) + Ct(TD, b^∗)− COt(TD, b^∗− K) (5.28) for t∈ [0, TD], where b^∗ is the solution to (5.27).

Chapter 6

Exotic Options

In the preceding chapters, we have focused on the two standard classes of options – that is, call and put options of European and American style. The aim of this chapter is to study examples of more sophisticated option contracts. For convenience, we give the generic name exotic option to any option contract which is not a standard European or American option. It should be made clear that we shall restrict our attention to the case of exotic spot options. We find it convenient to classify the large family of exotic options as follows:

(a) packages – options that are equivalent to a portfolio of standard European options, cash and the underlying asset (stock, say);

(b) forward-start options – options that are paid for in the present but received by holders at a prespecified future date;

(c) chooser options – option contracts that are chosen by their holders to be call or put at a prescribed future date;

(d) compound options – option contracts with other options playing the role of the underlying assets;

(e) binary options – contracts whose payoff is defined by means of some binary function;

(f) barrier options – options whose payoff depends on whether the underlying asset price reaches some barrier during the option’s lifetime;

(g) Asian options – options whose payoff depends on the average price of the underlying asset during a prespecified period;

(h) basket options – options with a payoff depending on the average of prices of several assets;

(i) lookback options – options whose payoff depends, in particular, on the minimum or maximum price of the underlying asset during options’ lifetimes;

6.1 Packages

An arbitrary financial contract whose terminal payoff is a piecewise linear function of the terminal price of the underlying asset may be seen as a package option – that is, a combination of standard options, cash and the underlying asset. Unless explicitly stated otherwise, we shall place ourselves within the classic Black-Scholes framework.

6.1.1 Collars

Let K2 > K1 > 0be fixed real numbers. The payoff at expiry date T from the long position in a collar option equals

CL_T ^def= min

max{ST, K₁}, K2

.

It is easily seen that the payoff CL_T can be represented as follows CLT = K1+ (ST − K1)⁺− (ST− K2)⁺,

101

so that a collar option can be seen as a portfolio of cash and two standard call options. This implies that the arbitrage price of a collar option at any date t before expiry equals

CLt= K1e^{−r(T −t)}+ C(St, T − t, K1)− C(St, T− t, K2),

where C(s, T − t, K) = C(s, T − t, K, r, σ) stands for the Black-Scholes call option price at time t, where the current level of the stock price is s, and the exercise price of the option equals K (see formula (3.71)).

6.1.2 Break Forwards

By a break forward we mean a modification of a typical forward contract, in which the potential loss from the long position is limited by some prespecified number. More explicitly, the payoff from the long break forward is defined by the equality

BF_T ^def= max{ST, F} − K,

where F = FS(0, T ) = S0e^rT is the forward price of a stock for settlement at time T, and K > F is some constant. The delivery price K is set in such a manner that the break forward contract is worthless when it is entered into. Since

BFT = (ST − F )⁺+ F− K, it is clear that for every t∈ [0, T ],

BFt= C(St, T− t, F ) + (F − K)e^{−r(T −t)}. In particular, the right level of K, K₀ say, is given by the expression

K0= e^rT

S0+ C(S0, T, S0e^rT) .

Using the Black-Scholes valuation formula, we end up with the following equality K₀= e^rTS₀

 1 + N

d₁(S₀, T )

− N

d₂(S₀, T ) , where d₁ and d₂ are given by (3.43)–(3.44).

6.1.3 Range Forwards

A range forward may be seen as a special case of a collar – one with zero initial cost. Its payoff at expiry is

RFT

def= max

min{ST, K2}, K1

− F = max

min{ST− F, K2− F }, K1− F ,

where K₁< F < K₂, and as before F = F_S(0, T ) = S₀e^rT. It appears convenient to decompose the payoff of a range forward in the following way

RF_T = S_T− F + (K1− ST)⁺− (ST − K2)⁺.

Indeed, the above representation of the payoff implies directly that a range forward may be seen as a portfolio composed of a long forward contract, a long put option with strike price K1, and finally a short call option with strike price K2. Furthermore, its price at t equals

RF_t= S_t− S0e^rt+ P (S_t, T − t, K1)− C(St, T − t, K2).

As mentioned earlier, the levels K1 and K2 should be chosen in such a way that the initial value of a range forward equals 0.