Essay on Financial Options

Introduction

The study of options as derivative products from financial assets have gained no less academic and commercial importance than other older alternative forms of financial derivatives, such as currency forwards, commodity futures contracts, interest rate swaps, and the underlying assets themselves especially since the institutionalization of the Chicago Board Options Exchange (CBOE) and the formulation of option valuation models by Black, Scholes, and Merton all in 1973, which are later simplified by Cox, Ross, and Rubinstein in 1979.

Standard financial option contracts on securities include calls and puts. Arditti (1996) provides a concise definition for options that a European call (C_E) contract grants its owner the right, but not the obligation, to purchase (or to be long in) a specified quantity of security at a specified price (the strike price, K) on or before a specified date (the expiration date, T). A European put (P_E) contract grants its owner the right, but not the obligation, to sell (or to be short of) a specified number of stocks at the strike price on or before the expiration date. If the option can be exercised prior to its expiration date, then it is called an American option (C_A or P_A). There are several variations of option contracts which are embedded in the underlying securities and which the option writer engineers to fit the specific requirements of the option owner. The former are often referred to as the option products and the latter the exotic options. Option contracts that are derived from the interest rate are called the interest rate options including caps, floors, collars, and corridors.

The next section introduces discrete and continuous option pricing models and their variations. Section three provides empirical evidence of option's mispricing on various underlying assets. Applications of financial options are provided in section four including option portfolio, portfolio insurance, PRIME & SCORE, LEAP, FLEX, PERCS, SuperShares, Quantos, exchange and interest rate options. Exotic options are discussed in section five. Section six concludes this essay.

Theoretical Basis for Financial Options Valuation Back to Top

Option Value and the Underlying Parameters

The value of an option can be decomposed into two parts: the intrinsic value and the time value. The intrinsic value of a European call is given by the maximum of either the difference between security price and strike price or zero; the intrinsic value of a European put is the maximum of either the difference between strike price and security price or zero:

C_E = Max[(S-K),0]
P_E = Max[(K-S),0]

The option is said to be in the money when its intrinsic value is strictly positive, at the money when the intrinsic value is zero, and out of the money when the intrinsic value is strictly negative. The call value has a direct relationship with security price, interest rate, time to expiration, and security price volatility, and has an inverse relationship with its strike price. On the other hand, the put value is directly related to its strike price, time to expiration, and stock price volatility, but inversely related to security price and interest rate. Noticeably, both call and put are directly related to their time to expiration and the price volatility. These two parameters add the time value to the option's intrinsic value to arrive at the overall option value.

The Put-Call Parity Relationship

A put-call parity relates the value of the European put to that of the European call through their identical parameters namely, the security price (S), the strike price (K), the risk-free interest rate (i), and the time to expiration date (T), and is given by the following equation:

C_E- P_E = S - e^-iTK

For a security which pays the present value of cash dividends (D) at T, the put-call parity relationship becomes:

C_E - P_E = S - e^-iTK - D

Certain characteristics of the put-call parity should be noted. First, if dividends are increased, the value of security decreases. This would lower the call value, other things remain unchanged. Second, if D = 0, buying a security, buying a put to obtain insurance, and borrowing the strike price to obtain leverage is equivalent to buying a call. Therefore, the owner of the call is willing to pay, in addition to the option's intrinsic value, the time premium, because the call gives her something she cannot obtain through the straight security purchase: an insurance and a leverage. As time to expiration decreases, insurance and leverage benefit also decrease thereby reducing the overall value of the call.

The same logic can be applied to the put with respect to the parameters determining the put's time premium. Hence, the put's value can simply be derived from the call's value through their parity relationship. The put owner can be seen as being short a security with an insurance call plus leverage from lending the strike price. The higher the dividends, the higher the put's value. Arbitrage opportunity exits if the put-call parity does not hold under identical parametric values.

The Discrete-Time Binomial Option Pricing Model (BOPM)

Even though the Black-Scholes Option Pricing Model (BSOPM) has been introduced since 1973, it is worthwhile to explore a simpler derivation of option price developed by Cox, Ross, and Rubinstein (1979) which is based upon a stochastic binomial process. For a one-period option:

Let: r_u = security rate of return at the up node; r_d = security rate of return at the down node; r = risk-neutral return = (1+i); i = risk-free interest rate = (r-1)

At t = 1:

If: C_u = NSu + Br; C_d = NSd + Br
where: u = (1+r_u); d = (1+r_d)

then: N = (C_u- C_d)/S(u-d) = number of ex-dividend shares to be bought.; B = r-1(uC_d- dC_u)/(u-d) = amount to be borrowed.

At t = 0:

If: C_E = r^-1[{(r-d)/(u-d)}C_u + {(u-r)/(u-d)}C_d] = NS + B; C_E = r^-1[pC_u + (1-p)C_d]
where: p = (r-d)/(u-d), (1-p) = (u-r)/(u-d)
: C_E= Max[S-K, r^-1{pC_u + (1-p)C_d}]; P_E = Max[K-S, r^-1[pP_u + (1-p)P_d}] based on put-call parity

For the n-period in-the-money call, the multiplicative binomial distribution function is applied:
: C_E = r^-n{Sⁿ(n!/(n-j)!j!)p^j(1-p)^n-j[u^jd^n-j(S-K)]}; = S[Sⁿ(n!/(n-j)!j!)pⁿ(1-p)^n-jr^-t(u^jd^n-j)] - r^-nK[Sⁿ(n!/(n-j)!j!)pⁿ(1-p)^n-j]; = S[B(n,p)r^-n(u^jd^n-j)] - r^-nK[B(n,p)]

The Continuous-Time Black-Scholes Option Pricing Model (BSOPM)

Following Cox, Ross, and Rubinstein (1979) above, the Black-Scholes Option Pricing Model is the limiting case of the Binomial Option Pricing Model being derived as follow:

Let: q = r^-nu(p); (1-q) = r^-nd(1-p)

Then,: C_E = S[B(n,q)] - r^-nK[B(n,p)]

In the limit as n approaches infinity, the discrete binomial distribution converges to the continuous lognormal distribution of security returns with mean mT and variance s²T. The B(n,q) approaches N(d1) and B(n,p) approaches N(d1 - sÖT). The discount factor, r^-n, also approaches e^-iT as n approaches infinity. Therefore, the value of a European call becomes:
: C_E = SN(d1) - e^-iTKN(d2); P_E = e^-iTK[1-N(d2)] - S[1-N(d1)] based on put-call parity

where: d1 = [ln(S/K) + (i+½s²)T]/sÖT, and d2 = d1 - sÖT

Black and Scholes (1973) accomplish their BSOPM by the assuming the following conditions:

Stock returns are lognormally distributed with constant mean and variance,
Value of stock returns is known and is directly proportional to the passage of time.
Risk-free interest rate is known and time-invariant.
There are no cash dividends paid during the life of the option.
The option is a European option, i.e., no early exercise is allowed.
There are no transaction costs to allow for the riskless hedging between the option and its underlying security at no sunk cost.

Let x be a normally distributed random variable with mean µx and variance s2x. The density function of x, f(x), is given by:

f(x) = (2ps²x)^-½exp{½(x-mx)²/s²x}

Define a lognormal random variable y = e^x. The distribution functions of x and y are given by:
: F(x) = G(y); g(y) = dF(x)/dy; = [dF(x)/dx][dx/dy]; = f(x)y-1

The mean and standard deviation of y, µy and sy, can then be derived from µx and sx as follows:
: µy = exp{µx + ½s²x}; sy = [exp{2µx + s²x}][exp{s²x} - 1]

Relating the above notations to the European call, the in-the-money call's value is given by:

C_E= e^-iTE[S_t-K]

where: S_t = stock price at time t
C_E = e^-iTò_x^¥ (S_t-K)h(S_t)dS_t

where h(S) = lognormal density function of S_t

C_E= e^-iT[ò_x^¥ S_th(S_t)dS_t - Kò_x^¥ h(S_t)dS_t]
= e^-iTE_x[S_t] - e^-iTK[1-H(K)]

where: E_x[S_t] = partial expectation of S_t; H[K] = ò_-¥^x h(K)dS_t

Expressing S_t in terms of price ratio S_t/S = e^kT, where k = rate of return on S_t, yields:
: C_E = e^-iT[Sò_x/S^¥(S_t/S)g(S_t/S)d(S_t/S) - Kò_x/S^¥ g(S_t/S)d(S_t/S)]; C_E= e^-iTSE_x/S[S_t/S] - e^-iTK[1-G(K/S)]

where: g(S_t/S) = lognormal density function of S_t/S, and G(K/S) = ò_x/S^¥g(K/S)d(S_t/S)

Then,: C_E = e^-iTS[òln_(x/S)^¥ f(kT)e^kTdk] - e^-iTK[òln_(x/S)^¥ f(kT)Tdk]; C_E= S[òd1(2p)^-½exp{½d1}d(d1)] - e^-iTK[òd2(2p)^-½exp{½d2}d(d2)]; C_E= SN(d1) - e^-iTKN(d2)

and: P_E = e^-iTK[1-N(d2)] - S[1-N(d1)] based on put-call parity

where: d1 = [ln(S/K) + (i+½s²)T]/sÖT; d2 = d1 - sÖT

The Black-Scholes-Merton Option Pricing Model for Dividend-Paying Stock

Merton (1973) relaxes the no-dividend assumption by modifying the BSOPM for a European call on a security paying continuous dividends in proportion to the security price at that time. The dividend paid at time t is dS_t where d is a proportionality constant that represents the annual dividend yield. The modification to the BSOPM is expressed as:
: C_E = e^-dTSN(d1) - e^-iTKN(d2)
and: P_E = e^-iTK[1-N(d2)] - e^-dTS[1-N(d1)]

where: d1 = [ln(S/K) + (i^-d+½s²)T]/sÖT; d2 = d1 - sÖT

The Black Pseudo-American Option Pricing Model for Dividend-Paying Stock

Black (1975) adjusts the Black-Scholes-Merton Option Pricing Model by allowing the option on a dividend-paying security to be exercised before its expiration date. The model compares the value of an option that is exercised before the ex-dividend date (called a pseudo-American option) with the usual European option. The correct call's value will be the maximum of the two calls: C = Max[C_A,C_E], where C_A = f(S-De^-iT,T₁,i,s,K-D) and C_E = f(S-DeⁱT,T₂,i,s,K).

Other Variations to the Standard Option Pricing Models

There are various alterations to the BSOPM around the option's parameters beside dividends and early exercise as mentioned earlier. Roll (1978), Geske (1979), and Whaley (1981, 1982) attempt to value an American option when its time value is less than the dividend yield from security. Hull and White (1988) apply the control-variance approach to the BOPM and find the difference between the value of American and European puts. Geske and Johnson (1984) estimate such difference based on BSOPM but find their model to be rather complicated and slow. Barone, Adesi, and Whaley (1987) improve upon Geske-Johnson Model and offer a simpler equation to derive such difference.

Some actually traded option prices such as of the CBOE European options on the S&P 500 index seem to be substantially different from those predicted by the BSOPM. This violation has been discovered through study of the estimates of the S&P 500's volatility through the concept of implied volatility. There are two ways from which the stock's volatility can be estimated: first, by computing the standard deviation of a recent series of stock returns, and second, by using the BSOPM to solve for the standard deviation given the stock price. The volatility derived from the BSOPM is called the stock's implied volatility. However, this implied volatility is assumed to be constant over time. Cox and Ross (1976) are skeptic about this constant variance assumption and develop the constant elasticity of variance model (CEVM) to resolve this issue. Merton (1976) uses the jump-diffusion model, i.e., the BSOPM with a Poisson process, to account for the time-varying variance and the leptokurtic characteristic of security return.

The risk-neutral probability density function can also be inferred from the standard option pricing models. Rubinstein (1994) has devised a numerical procedure for obtaining the risk-neutral density function f(S_j,T). He finds that the resulting f(S_j,T) produces a density function that is very different from the normal density function. The inferred continuous rate of return distribution, ln(S_T/S), has a higher mode and exhibits some skewness. Therefore, the BSOPM which assumes the normality of continuous returns is not appropriate for pricing options on the S&P 500 index. With the inferred risk-neutral probabilities, the theoretical European call price can be determined by calculating the option's terminal value as expected by a risk-neutral individual and discounting that number by the risk-free interest rate:
: C_E = e^-iTE[Max(S_T-K,0)]; C_E = e^-iTS_j=0 f(S_j,T)[Max(S_j,T-K,0)]
and: P_E = e^-iTS_j=0 f(S_j,T)[Max(K-_Sj,T,0)]

Empirical Evidence on Option Mispricing Back to Top

The empirical tests of the BSOPM were conducted by Black and Scholes (1973) themselves on the over-the-counter (OTC) stock options and by Galai (1977) on the CBOE options. Black and Scholes find that different degrees of variance of the stock returns can cause the options to be mispriced. Galai confirms the pricing accuracy of the BSOPM without the inclusion of transaction costs. MacBeth and Merville (1979) test both BSOPM and CEVM and find that the former tends to underprice in-the-money options and overprice out-of-the-money options, while the latter offers a higher accuracy. Their results are consistent with that of Black and Scholes's (1973) study that BSOPM is not robust when variances are not constant. Beckers (1980), Emanuel and MacBeth (1982) and Lauterbach and Schultz (1990) all confirm that BSOPM does not perform very well under time-varying variances and that CEVM is a better alternative pricing model. However, Rubinstein (1985) argues that CEVM does not consistently outperform the BSOPM. For the tests on American options, Sterk (1982) compares Black's (1975) Pseudo-American option pricing model with Roll-Geske-Whaley American put pricing model and concludes that the latter is better. All empirical tests point to the assumption of constant variance as the main source of pricing error of BSOPM with lesser emphasis on the no-early exercise assumption.

Applications of Financial Options: Selected Excerpts from Arditti (1996) Back to Top

Option Portfolio and Risk Management

Using the abovementioned pricing models, the firm might identify options that are mispriced and have its traders sell (buy) the overpriced (underpriced) options. The trader is asked to hedge each trade by immediately effecting an opposite position in a fairly priced option or in the underlying security, so that the position composed of the initial and hedge trades is insensitive to a small change in price of the underlying security. This type of hedging is known as the delta (D) neutral hedge. The firm might also be interested in hedging other pricing model's parameters. These other sensitivities have also been given Greek letter names: gamma (G) for the sensitivity of D to a small change in the price of the underlying asset; kappa (K) for the sensitivity of the option price to a small change in the volatility of the underlying asset; theta (T) for the change in the option price with respect to time to expiration; and rho (r) for the option price sensitivity with respect to a small change in the short-term risk-free interest rate.

Delta Hedge: The change in the option price for a small change in the current stock price.

D_call = e^-dTN(d1) > 0
D_put = D_call - e^-dT < 0

Gamma Hedge: The change in the option delta for a small change in the current stock price.

G_call = G_put = e^-dTn(d1)/SsÖT > 0

where: n(d1) is the height of the standard normal density function at the horizonal coordinate d1.

Kappa Hedge: The change in the option price for a small change in the return volatility (s).
K_call = K_put = e^-dTST^½n(d1) > 0

Theta Hedge: The change in the option price for a small change in the time to expiration (T).
T_call = e^-dTSsn(d1)/2ÖT - de^-dTSN(d1) + ie^-dTKN(d2) > 0
T_put = T_call + dSe^-dT - ie^-dTK ¹ 0

Rho Hedge: The change in the option price for a small change in the short-term interest rate.
r_call= e^-dTKTN(d1) > 0
r_put = r_call - Te^-dTK < 0

Portfolio Insurance

Portfolio insurance is a strategy that promises to protect the capital gains of the previous few years without conceding the opportunity for future capital gains. If the portfolio's value rises, the fund gains, but should it fall, the fund's principal is preserved. The portfolio is insured against loss of principal. The position is constructed as follow: long stock and long put.

S + P = S - S(1 - N(d1)) + e^-iTK(1 - N(d2))
= SN(d1) + e^-iTK(1 - N(d2)

If the stock price rises, then N(d1) increases and [1 - N(d2)] decreases, so the funds must be reallocated from risk-free asset to stocks. If the stock price falls, the reverse occurs; stocks must be sold and T-bills are purchased. Theoretically, this method of portfolio insurance means that one must continuously adjust the proportion of stock to T-bills.

PRIME & SCORE

The PRIME (Prescribed Right to Income and Maximum Equity) and the SCORE (Special Claim On the Residual Equity) are two parts of a five-year unit investment trust issued by an investment banking firm in exchange of the blue-chip shares deposited by the larger financial institutions. These option-like instruments are used to assist portfolio insuree in replicating a long-term put through continuous rebalancing. The termination claim is set at a level such that the financial institution considers the odds long of the stock price exceeding that level. Since the SCORE is a five-year European call on the underlying stock, and such calls are difficult to replicate, those who wish to include them in their portfolio are willing to pay premium for them. Therefore, the sum of the PRIME and SCORE prices exceed the price of the underlying stock. High transaction costs prevent arbitrage between PRIME & SCORE and stock to occur.

LEAP

The LEAP (Long-term Equity pArticipation Product) is American puts and calls listed with an original maturity of 39 months, during which a new series of LEAPs is listed every 6 months. LEAPs are used to solve the difficulty in borrowing to short the SCOREs, since positions in LEAPs are like other CBOE options which allows traders to enter into a short position.

FLEX

The FLEX (FLexible EXchange traded options) are offered by CBOE to the institutional investors and traders whose transaction size is at least $10 million. FLEXs provide institutional traders with the flexibility of the OTC market, as opposed to the more rigid standardized contract terms set by the exchange, while permitting the traders to receive the credit guarantee of the Options Clearing Corporation (OCC) and the greater liquidity associated with a central marketplace.

PERCS

The PERCS (Performance Equity Redemption Cumulative Stock) is a composite security consisting of a long stock and a short call, written with a strike that is frequently 30% to 60% higher than the stock price on issue date. The life of PERCS is finite, usually maturing in three to four years from the issue date. The PERCS issuers pays for the embedded call through the dividend stream of the PERCS. Consequently, the current value of that dividend stream exceeds the current value of the stock's dividend stream by the cost of the call. Since the call is paid for through the dividend stream, the PERCS is initially offered at a price equal to the then current stock price. PERCS provides a vehicle for raising new equity when some firms could not do so through conventional common stock offerings. The investor who has a limited view of the stock's upside potential and desired an exceptional dividend finds the PERCS attractive. Some issuers find PERCS attractive because they believe that the market would underprice the PERCS's embedded call and therefore overprice the PERCS.

SuperShares

SuperShares are option-like instruments created by splitting the payoffs from two underlying securities, the Index Trust SuperUnit, a share in a portfolio that mimics the S&P 500, and the Money Market Trust SuperUnit, a share in a short-term portfolio composed of U.S. Treasury securities and loans collateralized by U.S. Treasury securities.

Both types of SuperUnits are issued at their net asset values (NAV), are traded on the American Stock Exchange, and expire on expiration date. Each may be redeemed on or before that date at its NAV. Each type of SuperUnit can be divided into two types of SuperShares, each of which expires on the date that the underlying SuperUnit expires and trades on the CBOE.

The Index SuperUnit can be divided into 1) Appreciation SuperShare, which receives all of the Index SuperUnit's value above $125 one expiration day, and 2) Priority SuperShare, which receives all dividends paid to the Index SuperUnit and that part of the Index SuperUnit's value not paid to the Appreciation SuperShare.

The Money Market SuperUnit can be divided into 1) Protection SuperShare, which receives the difference between $100 and the price of the Index SuperUnit if that difference is positive, and zero otherwise on expiration date, and 2) Income and Residual SuperShare, which receives the interest earned by the Money Market SuperUnit plus any remaining value of the Money Market SuperUnit not paid out to the Protection SuperShare on expiration date.

Binational Options and Quantos

A binational option is the option on foreign underlying assets such as stock indices which are traded domestically using the home currency. The binational options that use fixed currency exchange rate are called quantos. With differences in home and foreign interest rates and exchange rates, standard option pricing models cannot accommodate the pricing of binational options and quantos. Wei (1992) develops a European binational option (BCE or BPE) pricing model which allows for fixed and floating exchange rates to be factored into the standard option pricing models as follow:

BC_E = Y[e^(i-l)TSN(b₁) - e^-iTKN(b₂)]
and: BP_E = Y[e^-iTK{1-N(d₂)}] - e^(i-l)TS{1-N(b₁)}]

where: b₁ = [ln(S/K) + (½s²-l)T]/sÖT; b₂ = [ln(S/K) - ½s²T]/sÖT; Y = fixed currency exchange rate; l = d_F + s_S,K - i_F; d_F = dividend yield from foreign stock index; s_S,K = covariance of foreign stock index and domestic strike price; i_F = foreign risk-free interest rate

Chen, Laiss, and Sears (1993) and Curcio and Byron (1995) independently develop an American binational option pricing model by combining Wei's model with Barone-Adesi-Whaley's American put pricing model.
Exchange Options

An exchange option is the option to exchange one underlying asset for another. It is used on the T-bond futures contract which allows the seller of the futures to choose any one of a number of T-bonds to deliver and currency exchange futures contract. If two bonds are deliverable with value B1 and B2, the one with the lower value will be selected for delivery. Consequently, if B1 is currently cheaper than B2, the long futures position may be viewed as the ownership of B1 with payment deferred until delivery and an option that was sold to the short. This option that the short futures holder possess grants the right to call B1 away from the long in exchange for B2. The valuation of exchange options is based upon the Black-Scholes-Merton's (1973) model for pricing the dividend-paying stock. The currency exchange European option (ECE or EPE) pricing is modeled by German and Kohlhagen (1983) which substitutes the continuous dividend yield parameter (d) with the foreign risk-free rate (Rf) as follow:

EC_E= e^-RfTSN(d₁) - e^-RdTKN(d₂)
and: EP_E = e^-RdTK[1-N(d₂)] - e^-RfTS[1-N(d₁)]

where: Rf = foreign risk-free interest rate; Rd = domestic risk-free interest rate; d₁ = [ln(S/K) + (Rd-Rf+½s²)T]/sÖT; d₂ = d₁ - sÖT

Stock Index Options

Stock index options are options whose underlying assets are stock indices such as the S&P 500. Its valuation problem is that dividends are not continuous and early exercises are usual. Thus, Merton's and Black's models are not appropriate; the Binomial model should be used instead.

Interest Rate Options

Interest rate options are generally subcategorized into 6 kinds including caps, floors, collars, corridors, callable bond, and sinking fund. Their characteristics are discussed below:

A cap is used to limit the interest cost of a floating rate loan. The cap's life, the length of time from initiation date (usually two days after the trade date and called the effective date) to expiration date is divided into a number of payment periods, each of which ends with a payment day. Two business days prior to the beginning of each payment period, or the reset date, the cap's reference rate level is determined. If the cap's reference rate exceeds a previously agreed upon figure, termed the cap rate, the seller of the cap makes a cash payment equal to the difference between the reference rate and the cap rate, multiplied by the principal amount on which the cap is written. The principal amount is used only for calculation purposes. It is often referred to as the cap's notional principal. The cap ensures that the floating rate borrower pays the lesser of the reference rate and the cap rate. For the benefits conferred by cap ownership, the buyer pays the seller an amount termed the cap premium.

A floor is used to set the minimum rate that would be earned on a floating rate loan. This is ensured for the life of the floor. If the reference rate is below the floor rate on a reset date, the seller makes a payment to the buyer at period's end equal to the notional principal. Just as a cap can be viewed as a strip of European calls on the reference rate, a floor can be viewed as a strip of European puts on the reference rate. The price paid for it is called the floor premium.

A collar is a long position in cap and a short position in floor. Whereas the cap rate is normally set at the reference rate prevailing on the cap's trade date, the floor rate is set below the current reference rate level. In a collar, the floor is out-of-the-money when initiated. At reference rate levels below the floor rate, the short floor loses money and the long cap is out-of-the-money. Any gain realized from paying a lower borrowing rate is offset by the payment made on the short floor. The difference in premiums is lost. If the reference rate falls between the floor and cap rates, both options finish out-of-the-money. There is a loss on the collar equal to the difference in premiums. At reference rate above the cap rate, the cap is in-the-money and the floor is out-of-the-money. Payoffs from the long cap serve to offset the higher costs of borrowing. In short, a collar is a trade-off of the lower cost of insuring against higher rates in exchange for parting with the reduced borrowing costs to be realized if rates are lower.

A corridor is a position that trades off some possible gains from a higher interest for a lower insurance premium. A corridor is formed by purchasing a cap with a low cap rate and selling another cap with a high cap rate.

A callable bond is the corporate bond which has a provision written into the bond indenture that allows the firm to repurchase the bonds at a price, referred to as the call price, that is set above the bond's face value. If rates fall, the issuer has the option of repurchasing the bonds at the call price and financing the purchase with a new issue of lower coupon bonds. Alternatively, the firm could finance the call by issuing bonds with the old, higher-then-current coupon rate and collect an amount above the call price. The amount that is in excess of the call price is the value upon exercise of the bond's call provision. The price of a callable bond is equal to the price of a noncallable bond minus the price of a call option on the noncallable bond.

A sinking fund is a requirement that the bond issuer repay a number, M, of the outstanding bonds, U, at face value on each of a sequence of scheduled dates (redemption dates). The sinking fund may grant the issuer a set of options with respect to the number of bonds that can be retired on a redemption date. Clearly, such options will serve to lower a bond's market value. These embedded options assumed various forms: 1) Purchase Substitution, 2) Voluntary Redemption, and 3) Leftover Substitution. In purchase substitution, the issuer purchases a number of bonds, P, of the outstanding bonds, U, and substitutes the purchased bonds for all or part of the required purchase of the M bonds. In voluntary redemption, on redemption date the bond issuer chooses to retire the mandatory number, M, plus a voluntary number, V, for a total of M+V bonds redeemed. Finally for leftover substitution, the issuer chooses to substitute L number of bonds that were voluntarily retired at an earlier redemption date for some or all of the M bonds required to be retired at the current redemption date.

Exotic Options Back to Top

Asian Options

An Asian option is the option that allows the holder to choose the larger of the average value of the underlying asset price over a given time period, or the face value of the asset. The final payoffs of the Asian option is dependent on an average of prices (path dependent) rather than the price on termination. Asian options are used by corporate treasurers to cover the exposure of a series of transactions in a foreign currency or in an interest rate instrument. They can be employed by a dealer who makes a continuous market in a particular stock. Whereas standard options are appropriate for reducing the risk of a particular position whose size is known or of a cash flow to be made or received on a specific date, the Asian options are suitable for reducing the risk of one's average asset position or that of a stream of continuous cash flows whose size and timing cannot be easily predicted.

Barrier Options

A barrier option is applied to any option that is extinguished or comes to life if the underlying security price crosses a prespecified level, or barrier, during the option's life. There are two general categories of barrier options: 1) in-barriers (or knock-in options), and 2) out-barriers (or knock-out options). Since the barrier may be set above or below the current underlying security price, knock-out options can be further sorted into up-and-out and down-and-in options. Likewise, knock-in options can be either up-and-in and down-and-out options.

Lookback Option

A lookback call (put) has its strike price set to the minimum (maximum) of the underlying asset price attained during the life of the option. The value of a lookback option is path dependent. What the option holder receives depends upon where the stock price has been. We apply the risk-neutral valuation method to obtain the lookback option's current price.

Compound Option

A compound option is the option that permits the holder to purchase another option on a stock (option on option). The classic use of a compound option is in tendering. A U.S. firm bids for a U.K. firm in sterling. The venture has two uncertainties: 1) whether or not the bid will be accepted, and, if so, 2) what will be the dollar cost of sterling at the future date. To mitigate the associated financial risk, the U.S. firm buy a European call on another European call, a compound call. The second call in the sequence is written on a stipulated amount of sterling at an agreed upon strike price. Should the tender not be accepted, the firm lets the call die. If the bid is accepted, the firm pays the first call's strike in dollars and accepts in return another call on the pound with a strike price that was set in $/£. This second call, whose expiration date is most likely set to the date that the pounds are to be delivered to complete the purchase, will be exercised at termination if the pound's dollar price exceeds its strike price.

Conclusion Back to Top

Financial options, both standardized contracts and exotic products, have lent tremendous impacts on capital markets in terms of increasing allocative and informational efficiencies through risk reduction and control. They are no longer the derivatives in the eyes of financial engineers, since financial options themselves are their basic tools for managing financial risk exposed by their corporate and individual clients. The theoretical significance of financial options also overflows to the theory of capital budgeting under uncertainty in what the academicians in such field call real options. The analysis of real options is drawn from the generalized options valuation and application frameworks of contingent claims analysis (CCA). Without the theoretical foundation contributed by Black and Scholes, advancement in the area of CCA and its related issues would not be as much as what we have experienced today.

As Debreu (1959) and Arrow (1964) state, the necessary condition for markets to be complete is the existence of the number of state contingent claims to be equal to the number of long-lived securities traded in such markets. Both real and financial options function just as what they would like them to do in economic reality. Despite its mathematical rigors and derivational sophistications, CCA and options pricing models should be studied and researched along side with the conventional asset pricing models in order to bridge the dynamic gaps between the two. With the emergence of nonlinear assets and options pricing paradigms and high-speed computation, we no longer have a sharp delineation between the analyses of the security markets and that of the derivatives markets.

Risk reduction while enhancing returns through financial engineering, active assets allocation, and portfolios management is possible and makes more sense should individual investors be aware of and familiar with the basic option pricing tools and their exotic applications. Sophisticated investors, speculators, and hedgers who are well-informed and have been equipped with these analytical capabilities already squeezed extra returns from their investments in real and financial assets, let alone the institutional traders who seek arbitrage opportunities around the globe. Until all heterogeneous market participants are fluent with the language of derivative products and fine-tuned their expectations accordingly, the normative assumptions underlying modern finance theory will prove to be relevant in the real world.

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* Worapot Ongkrutaraksa is a lecturer in Finance and Strategic Management at Maejo University's Faculty of Agricultural Business, Chiang Mai, Thailand. He used to conduct his post-graduate research in financial economics at Kent State University and international political economy at Harvard University through the Fulbright sponsorship between 1995 and 1998.

E-mail: [email protected]

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