11.2 The Black-Scholes Model

In 1973, the Chicago Board of Options Exchange begun trading options in exchanges, although previously options had been regularly traded by financial institutions in over the counter markets. In the same year, Black and Scholes (1973), and Merton (1973), published their seminal papers on the theory of option pricing. Since then the growth of the field of derivative securities has been phenomenal. In recognition of their pioneering and fundamental contribution to option valuation, Scholes and Merton received in 1997 the Award of the Nobel Prize in Economics. Unfortunately, Black was unable to receive the award since he had already passed away.

In essence, the Black-Scholes model states that by continuously adjusting the proportions of stocks and options in a portfolio, the investor can create a riskless hedge portfolio, where all market risks are eliminated. The ability to construct such a portfolio relies on the assumptions of continuous trading and continuous sample paths of the asset price. In an efficient market with no riskless arbitrage opportunities, any portfolio with a zero market risk must have an expected rate of return equal to the risk-free interest rate. This approach led to the differential equation, known in physics as the "heat equation". Its solution is the Black-Scholes formula for pricing European options on non-dividend paying stocks: $\,$

$$C(S,t) = S\Phi(d_1) - K\,exp(-r\tau)\Phi(d_2),$$ (11.10)

$$P(S,t) = K\,exp(-r\tau)\Phi(-d_2) - S\Phi(-d_1),$$ (11.11)

where

$$d_1=\frac{ln(S/K) + (r + \sigma^2/2)\tau}{\sigma \sqrt{\tau}},$$

$$d_2=\frac{ln(S/K) + (r -\sigma^2/2)\tau}{\sigma \sqrt{\tau}}=d_1-\sigma \sqrt{\tau}.$$

$$C(S,t) = \textrm{price of the European the call option,}$$

$$P(S,t) = \textrm{price of the European put option,}$$

$$S = \textrm{current underlying asset (stock) price,}$$

$$K = \textrm{strike price,}$$

$$\tau = T-t \textrm{ is the current annualized time-to-expiration, where T is}$$

$$\textrm{the expiration date,}$$

$$r = \textrm{the annualized risk-free interest rate,}$$

$$\sigma = \textrm{the annualized standard deviation of underlying asset price,}$$

$$\Phi = \textrm{the cumulative distribution function for a standardized normal}$$

$$\textrm{variable.}$$

11.2.1 Software Application

The option price according to the Black-Scholes formula can be calculated with XploRe . First, the functions in library finance must be loaded by typing the command:

  library("finance")

There are mainly two ways for computing the option prices according to (11.10) and (11.11) in XploRe. One way is by giving the requested input parameters through interactive menus and the other way is by directly defining the values of the input parameters within the quantlet.

opc = BlackScholes (S,K,r,sigma,tau, task)

calculates the European option price on non-dividend paying underlying asset specifying interactively the input parameters.

{opvv,sel,ingred} = bs1 (1)

opvv = bs1 (S,K,r,sigma,tau,opt,1)

calculates European option prices on non-dividend paying underlying assets, specifying either interactively or directly the input parameters.

BlackScholes only calculates the European option price for a non-dividend paying underlying asset, whereas bs1 calculates the European option price for different kinds of dividends and is therefore more general. The input parameter of the interactive form of bs1 has a value of 1. This means that the underlying asset pays no dividends. Other values of this input parameter, in the case of dividend paying underlying asset, are explained in detail in "Software Application" in subsection 11.2.3. When the direct specification of the input parameters is selected,

bs1(S,K,r,sigma,tau,opt,1)

, the first five follow the usual notation: $\tt {S}$ for the current level of the underlying asset, $\tt {K}$ for the strike price, $\tt {r}$ for the continuously compounded risk-free interest rate, $\tt {sigma}$ for the instantaneous standard deviation of underlying asset and $\tt {tau}$ for time to expiration. $\tt {task}$ is a scalar, which specifies the type of option. For $\tt {task=1}$, a European call is selected and for $\tt {task=0}$, a European put. The computation result is assigned to the variable $\tt {opc}$ when BlackScholes is used and $\tt {opvv}$ when bs1 is used. The other two output variables $\tt {sel}$ and $\tt {ingred}$ contain input parameters (see "Software Application" in subsection 11.2.3, pp. 11.2.3).

For example, the command

BlackScholes(100,120,0.05,0.25,0.5,1)

yields the call price of $C(S,t)=1.952$, and the command

  BlackScholes(100,120,0.05,0.25,0.5,0)

yields the put price of $P(S,t)=18.898$.

11.2.2 Derivation of the Black-Scholes Formula

In the following, the basic assumptions and methodology originally employed to derive the option bs1-paprice will briefly be described. The Black-Scholes assumes the following:

• The asset price follows a geometric Brownian motion (see also subsection 11.1.2)

  $$\frac{dS_t}{S_t} = {\mu}\, dt + {\sigma}\, d W_t.$$ (11.12)

  
  

• Trading can take place continuously without any transaction costs or taxes.
• Short selling is permitted and the assets are perfectly divisible. Therefore assets can be sold that are not own and any number (not necessarily an integer) of the underlying assets can be bought and sold.
• The continuously compounded risk-free interest rate is constant.
• Investors can borrow or lend at the same risk-free rate of interest.
• There are no riskless arbitrage opportunities. It follows that all risk-free portfolios must earn the same return.

For notational convenience, the following uses $S$ for the stock price at time $t$, i.e. $S_t=S$. Suppose that the value of the option $V$ depends on the underlying asset $S$ and time $t$. The idea is to construct a portfolio, which involves short selling of one unit of the European option and holding of $\Delta$ units of underlying stock $S$. The value of this portfolio $\Pi$ is given as:

$$\Pi=-V + \Delta\,S.$$ (11.13)

The change in the value of this portfolio in one time-step is

$$d\Pi=-dV + \Delta\,dS,$$ (11.14)

where $\Delta$ is held fixed during the time-step. Since both $V$ and $\Pi$ are random variables, Ito's lemma is applied to compute their stochastic differentials. The stochastic differential for the option is written as

$$dV=\sigma S \frac{\partial V}{\partial S}\,dW + \left(\mu S \frac... ...\,S^2\frac{\partial^2 V}{\partial S^2}+ \frac{\partial V}{\partial t}\right)dt,$$ (11.15)

where it is required that $\frac{\partial V}{\partial t}$, $\frac{\partial V}{\partial S}$ and $\frac{\partial^2 V}{\partial S^2}$ exist. This expression provides the random walk followed by $V$. By placing (11.12) and (11.15) in (11.14) together, it follows

$$d\Pi= \sigma S \left(-\frac{\partial V}{\partial S} + \Delta\righ... ...tial S^2}+\left(-\frac{\partial V}{\partial S}+ \ \Delta\right)\mu S\right\}dt.$$

If $\Delta$ is chosen equal to $\frac{\partial V}{\partial S}$, then the portfolio becomes a riskless hedge, since the stochastic term $dW$ in the portfolio disappears. In an efficient market with no riskless arbitrage opportunities, any portfolio with a zero market risk, also a perfectly hedged portfolio, must earn the risk-free interest rate. The return on an amount $\Pi$ invested in riskless assets would face a growth of $r\Pi\,dt$ in the intervall $dt$. Hence, it follows

$$d\Pi= -\left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2\... ...ial S^2}\right)dt=r\Pi\,dt=r\left(-V + S \frac{\partial V}{\partial S}\right)dt$$ (11.16)

and after rearranging terms,

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2\,S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S} - rV=0$$ (11.17)

is obtained. This is the Black-Scholes partial differential equation. The solution of this equation with different auxiliary conditions, such as boundary and final conditions, provides pricing formulae for different types of derivative securities. For example the call option final condition is:

$$C(S,T)=max(S-K,0)$$

and the boundary conditions are:

$$C(0,t)=0, \hspace{1cm} \lim_{S\to\infty}C(S,t)=S$$

The Black-Scholes formulae for pricing a European call $C(S,t)$ and a European put $P(S,t)$ on non-dividend paying stocks are (11.10) and (11.11). Technical details on how to solve equation (11.17) with the auxiliary conditions can be found in Hull (2000, ch. 11) or Wilmott et al. (1997, pp. 75-80). The price of an American call on a non-dividend paying underlying asset is equivalent to its European counterpart, since such an American call will not be optimally exercised prior to expiration (Hull; 2000). Hence the Black-Scholes pricing formula (11.10) is also valid for pricing American calls.
Note, that the price of a European put option on a non-dividend paying asset (11.11) is derived by combining the call option price formula (11.10) and the put-call parity under the continuous-time assumption:

$$P(S,t)=C(S,t)-S+K\,exp(-r\tau)$$ (11.18)

Kwock (1998, pp. 52) provides a probabilistic interpretation of the option pricing formulae. For example, under the assumption of risk-neutrality the call option price formula (11.10), $\Phi(d_2)$ is seen as the probability of the call option being in-the-money at expiration. Hence, $K\Phi(d_2)$ is the risk neutral expectation of the payment made by the holder of the call option at expiration by exercising the option. On the other hand, $S\,exp(r\tau)\Phi(d_1)$ is the risk neutral expectation of the asset price at expiration, conditional on the call being in-the-money. It follows that the expectation of the call value at expiration is $S\,exp(-r\tau)\Phi(d_1) - K\Phi(d_2)$. This is (in the risk neutral world) discounted by the factor $exp(-r\tau)$ in order to obtain the present value of the call price.

11.2.3 Options on Dividend Paying Assets

The dividend received by holding an asset may be stochastic or deterministic. The modelling of stochastic dividends is complicated, since there is another random variable in addition to the underlying asset. The Black-Scholes formula, however, requires only some slight modification to remain valid under the crucial assumption that the dividend yields are deterministic. This means that over the remaining time to expiration, the option dividends are at most a known function of time and/or of the underlying asset. This assumption is not unrealistic given the short-term life of trade options (generally less than one year) and given the stable dividend policy that most companies tend to follow over a short horizon. The prices of European call and put options on continuously dividend paying underlying asset, noted as $q$, are:

$$C(S,t) = S\,exp(-q\tau)\Phi(d_1) - K\,exp(-r\tau)\Phi(d_2),$$ (11.19)

$$P(S,t) = K\,exp(-r\tau)\Phi(-d_2) - S\,exp(-q\tau)\Phi(-d_1),$$ (11.20)

where

$$d_1=\frac{ln(S/K) + (r - q + \sigma^2/2)\tau}{\sigma \sqrt{\tau}},$$

$$d_2=d_1-\sigma \sqrt{\tau}.$$

The Black-Scholes pricing formulae show that a European option, on an underlying asset $S$ paying continuous dividend yields at rate $q$, has the same value as the corresponding European option on an underlying asset with the price $S\,exp(-q\tau)$ that pays no dividend.

11.2.3.1 Software Application

The quantlets bs1 , european and optstart in XploRe can be used to price European options within the Black-Scholes framework, when dividends exist according to (11.19) and (11.20) either directly or through interactive menus.

{opvv,sel,ingred} = bs1 (typeofdiv)

{opvv} = bs1 (S,K,r,sigma,tau,opt,typeofdiv,div)

calculates European option prices, specifying either interactively or directly the input parameters.

european ()

calculates the prices of European options, or their implied volatilities, specifying interactively the input parameters.

optstart ()

calculates the prices of either European or American options, or their implied volatilities, specifying interactively the input parameters. For American options the McMillan formula or binomial trees can be used.

european calculates either the option price by using the quantlet bs1 , or the implied volatility by using the quantlet volatility . The quantlet optstart uses several interactive menus to compute either i) the price of an American option through the McMillan formula (subsection 11.2.3) or through binomial trees (subsection 11.3.1), or ii) the price of a European option using the the Black-Scholes formula, or iii) the implied volatilities (section 11.5).

The input parameter $\tt {typeofdiv}$ in bs1 is an integer that specifies the type of the dividend payment: for $\tt {typeofdiv}=1$ no dividend, for $\tt {typeofdiv}=2$ continuous dividends and for $\tt {typeofdiv}=3$ a fixed dividend at the end of T is assumed. Finally, if $\tt {typeofdiv}=4$, then an exchange rate is assumed as underlying. The call and put formulae for the foreign currency options are analog to (11.19) and (11.20), except that the dividend yield $q$ is replaced by the foreign interest rate. In this case, $S$ is replaced by the exchange rate - the domestic currency price of a unit foreign currency - that is assumed to follow the lognormal diffusion process. In addition, both domestic and foreign interest rates are assumed to be constant. In

bs1(S,K,r,sigma,tau,opt,typeofdiv,div)

, the first five parameters follow the usual notation. The parameter $\tt {opt}$ specifies the option type. It has the value 1 for a call and 0 for a put. The parameter $\tt {div}$ refers to the dividend amount. It must be given the value zero, if no dividend is assumed.

When the quantlet $\tt {bs1}$ is used interactively, the output variable $\tt {dat}$ is a scalar containing the computed option price. The option type is denoted through a (2x1) dimensional vector $\tt {sel}$: $\tt {sel}$=$1\vert$ for a European call and $\tt {sel}$=$0\vert 1$ for a European put. The third variable $\tt {ingred}$ is a (6x1) dimensional vector that contains six input parameters: the price of the underlying asset, the strike price, the time to expiration, the volatility of the underlying asset, the domestic risk-free interest rate and the dividend payment(s).

bs1(S,K,r,sigma,tau,opt,typeofdiv,div)

offers the possibility to compute simultaneously the price of more than one European call or put option, with or without dividends. The output variable $\tt {opvv}$ is a (nx9) dimensional matrix, where the first columns contain the input parameters and the last one contains the computed option prices.

In order to use (11.19) and (11.20), the quantlet bs1 converts the fixed dividend $D$ to a constant continuous dividend yield $q$ through

$$q=\frac{S(1+r)}{S(1+r)-D}-1.$$

A simple example in which the underlying asset pays continuous dividend, i.e. $\tt {typeofdiv}=2$, is given as in the following

library("finance")
 bs1(2)               ; continuous dividend payment

with default values

(230,210,5,25,0.5)

for

(S, K, r, \sigma, \tau)

and with $q=15\%$, yields a call price of $20.024$. The same result is achieved by computing a call option price with a fixed dividend payment of $D=31.5$, and default values

(230,210,5,25,0.5)

for

(S, K, r, \sigma, \tau)

, by typing the following commands:

 library("finance")
 bs1(3)                ; a fixed amount of dividend payments
 

The following XploRe code reads the input parameters and calculates directly the prices of two calls and three puts:

library("finance")
 S=aseq(230,5,10)        ; additive sequence of five underlying
 K=aseq(210,5,15)        ; additive sequence of the strike prices
 r=5                     ; the annualized risk-free interest rate
                         ; in %
 sigma=aseq(25,5,-5)     ; additive sequence of volatility
 tau=0.5                 ; annualized time to expiration
 opt=#(1,1,0,0,0)        ; options type: two calls and three puts
 typeofdiv=#(0,1,2,3,2)  ; type of dividend payments
 dividend=#(0,10,35,8,45); the value of the dividend payments
 dat= bs1(S,K,r,sigma,tau,opt,typeofdiv,dividend)
 dat.opvv                ; displays the results

XLGfindex4.xpl

 
yielding the following output matrix $\tt {dat.opvv}$:

Contents of opvv

[1,]   230   210   0.04879   0.25   0.5   1   0   0         30.986
[2,]   240   225   0.04879   0.20   0.5   1   1   0.09531   17.8
[3,]   250   240   0.04879   0.15   0.5   0   2   0.1431    10.632
[4,]   260   255   0.04879   0.10   0.5   0   3   0.076961   6.392
[5,]   270   270   0.04879   0.05   0.5   0   2   0.17284   15.991

Starting from the left, each column contains for each option the underlying price, the strike price, the continuous risk-free interest rate, the volatility, the time to expiration, the option type, the type of dividend, the dividend payment and the computed option price.

11.2.3.2 Derivation of the Formula

The concept of pricing European options on dividend paying underlying assets will now be briefly outlined.

The constant continuous dividend yield is represented by $q=q(S,t)$. In other words, it is the dividend payment per unit of time, which always represents the same fraction $q$ of the stock price. The holder then receives dividend payment(s) equal to $qSdt$ within the interval $dt$. As the price of the underlying asset falls by the amount of the dividend, the asset price dynamics based on the geometric Brownian motion model becomes

$$\frac{dS(t)}{S} = ({\mu}-q)\, dt + {\sigma}\, d W(t).$$ (11.21)

For every asset held $qSdt$ is received. The holder of the portfolio, who holds $\Delta$ assets, earns an amount equal to $\Delta\Pi$ and dividend payment(s) equal to $qS\Delta dt$ in the interval $dt$. The change in value of the portfolio $\Pi$ is given by

$$d\Pi = -dV + \Delta dS + q\Delta S dt$$

$$= -\left(\frac{\partial V}{\partial t} - \frac{1}{2}\sigma^2\,S^2\frac{\partial^2 V}{\partial S^2}+ qS\frac{\partial V}{\partial S}dt\right)dt,$$ (11.22)

where $\Delta=\frac{\partial V}{\partial S}$. The last term $qS\Delta dt$ denotes the wealth added to the portfolio due to the dividend yields. By applying the no arbitrage argument, the hedged portfolio should earn the risk-free interest rate, so that

$$-\left(\frac{\partial V}{\partial t} - \frac{1}{2}\sigma^2\,S^2\f... ...l V}{\partial S}dt\right)dt=r\left(-V+S\frac{\partial V}{\partial S}\right) dt.$$ (11.23)

This leads to the following modified form of the Black-Scholes equation:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2\,S^2\frac{\partial^2 V}{\partial S^2}+ \left(r-q\right)S\frac{\partial V}{\partial S} - rV=0$$ (11.24)

For a call option, the only change to the boundary conditions is

$$\lim_{S\to\infty}C(S,t)=S\,exp(-q\tau).$$

The final condition is still $C(S,T)=max(S-K,0)$. Without working through a similar integration procedure, the price of a European option can be obtained by a simple modification of the Black-Scholes price formula. The European option on an underlying asset with price $S$ paying continuous dividend yields at rate $q$ has the same value as the corresponding European option on an underlying asset with the price $S\,exp(-q\tau)$ that pays no dividend. This yields the option valuation formulae (11.19) and (11.20).

Note, that if the underlying asset is a commodity, like grain or livestock, there may be additional costs in holding the asset, such as storage or insurance. In simple terms, these additional costs, denoted for example with $\tt {u}$, can be considered as negative dividends paid by the underlying asset. In this case, the option price is equivalent to (11.19) and (11.20), where the continuous compound dividend $\tt {q}$ is simply replaced by $\tt {-u}$. The term $r+u$ is interpreted as the cost of carry, denoted with $b$. It consists of two parts, the costs of funds tied up in the asset that require interest for borrowing and additional costs due to storage, insurance etc. When the underlying asset pays a continuous dividend $q$, then the cost of carry $b$ equals $r-q$, and the Black-Scholes differential equation (11.17) is written as

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2\,S^2\frac{\partial^2 V}{\partial S^2}+ bS\frac{\partial V}{\partial S} - rV=0.$$ (11.25)

11.2.4 Valuation of American Options

An American option confers all the rights of its European counterpart plus the privilege of early exercise. Since this additional privilege should not be worthless, it has potentially a higher value than its European counterpart. The extra cost is usually called early exercise premium. An exception to this is an American call option on non-dividend paying underlying asset, whose price always equals its European counterpart. For other American options it may be optimal to exercise them prior to expiration. For an American call or put, this only happens when the price of the underlying asset at a given time to expiration $\tau$, rises above (falls below) a critical asset value $S^*(\tau)$, known as the optimal exercise price. The collection of these critical values for all times constitutes a curve, which is known as the optimal exercise boundary. It is not a known priori where to apply the boundary condition, so that the optimal exercise boundary has to be determined in the solution process.

The early exercise premium can be expressed in terms of the exercise boundary in a stochastic integral. A detailed explanation is given in Kwock (1998, ch. 4) and Wilmott et al. (1997, ch. 7). The direct solution of the stochastic integral equation is in many cases unmanageable, so that several analytic approximation methods for the valuation of American options and the associated optimal exercise boundaries have been developed.

In XploRe the price of the American option is computed by using the popular quadratic approximation method, which was first proposed by MacMillan (1986) for non-dividend paying stock options and later extended to commodity options by Barone-Adessi and Whaley (1987). This class of approximation methods involves the reduction of the Black-Scholes partial differential equation to an ordinary one. The idea is to transform the Black-Scholes differential equation (11.17), so that the temporal derivative term can be considered as a quadratic small term and then dropped as an approximation (Kwock; 1998, pp. 166). Then applying the boundary conditions, the prices of American call and put options follow.

The price of an American call for $b\ge r$ equals the price of its European counterpart (see subsection 11.2.4 for an example). The price of an American call for $b<r$ is:

$$C_{am}(S,t)=\left\{\begin{array}{r@{\quad when \quad}l} \ C(S,t)+... ...\right)^{\gamma_2} & S<S^* \vspace{0.5cm}\\ S-K & S \ge S^* \end{array} \right.$$ (11.26)

$$C_{am}(S,t) = \textrm{price of the American call option,}$$

$$C(S,t) = \textrm{price of the European counterpart,}$$

$$S = \textrm{current underlying stock price,}$$

$$S^* = \textrm{critical price of the underlying stock, above which the call}$$

$$\textrm{ should be exercised.}$$

The price of an American put option is given in (11.27) and holds for all values of $b$.

$$\ P_{am}(S,t)=\left\{ \begin{array}{r@{\quad when \quad}l} P(S)+A... ...)^{\gamma_2} & S>S^{**} \vspace{0.5cm}\\ K-S & S \le S^{**} \end{array} \right.$$ (11.27)

$$P_{am}(S,t) = \textrm{price of the American put option,}$$

$$P(S,t) = \textrm{price of the European counterpart,}$$

$$S = \textrm{current underlying stock price,}$$

$$S^{**} = \textrm{critical price of the underlying stock, below which the put should be}$$

$$\textrm{exercised.}$$

$S^*$ and $S^{**}$ are estimated by solving the following equations (11.28) and (11.29) iteratively

$$S^*-K = C(S^*,t)+\left[1-exp{\left\{(b-r)\tau\right\}}\;\Phi\left\{d_1(S^*)\right\}\right]\frac{S^*}{\gamma_2}$$ (11.28)

$$K-S^{**} = P(S^{**},t)-\left[1-exp{\left\{(b-r)\tau\right\}}\;\Phi\left\{-d_1(S^{**})\right\}\right]\frac{S^{**}}{\gamma_1}$$ (11.29)

The other variables used in (11.28) and (11.29) are

$$\gamma_1 = \frac{1}{2}\left\{-(\beta-1)-\sqrt{(\beta-1)^2+\frac{4\alpha}{h}}\right\}$$ (11.30)

$$\gamma_2 = \frac{1}{2}\left\{-(\beta-1)+\sqrt{(\beta-1)^2+\frac{4\alpha}{h}}\right\}$$ (11.31)

$$A_1 = -\frac{S^{**}}{\gamma_1}\left[1-exp{\left\{(b-r)\tau\right\}}\;\Phi\left\{-d_1(S^{**})\right\}\right]$$ (11.32)

$$A_2 = \frac{S^{*}}{\gamma_2}\left[1-exp{\left\{(b-r)\tau\right\}}\;\Phi\left\{d_1(S^{*})\right\}\right]$$ (11.33)

$$d_1(S) = \frac{ln(S/K)+(b+\sigma^2/2)\tau}{\sigma\sqrt{\tau}},$$ (11.34)

where $b=r-q$ is the cost of carry and the variables $\alpha$, $\beta$ are defined as in (11.36). This method is considered to be quite efficient and to have a reasonably good level of accuracy for valuing American options, particulary for valuing options with a short time to expiration. Other approximation methods, such as the interpolation between bounds, or the approximation of the optimal exercise boundary by a known analytical (e.g. exponential) curve (see Kwock ch.4), or direct numerical approaches by using finite difference algorithms, or Monte Carlo simulations (see Hull ch. 14 and Wilmott ch. 8 to 10), are not discussed.

11.2.4.1 Software Application

XploRe offers the following quantlets to calculate the price of american options using the MacMillan approximation:

mcmillan (eopv,sel,task,ingred)

calculates the American option price, specifying directly the input parameters.

american ()

calculates the American option price, specifying interactively the input parameters.

optstart ()

calculates the prices of either European or American options, or their implied volatilities, specifying interactively the input parameters. For American options the McMillan formula or binomial trees can be used.

The price of American options can be calculated directly with the quantlet mcmillan or through interactive menus by using american . The third quantlet optstart is more general. It uses several interactive menus to compute the price of American or European options, or their implied volatility. However, in spite of the different comfortability that quantlet american or optstart offers, both use the quantlet mcmillan to calculate the price of American options.

The parameter $\tt {eopv}$ specifies the price of the European option. The option type is specified through the (2x1) dimensional vector $\tt {sel}$, with $\tt {sel}$=$1\vert$ for American call and $\tt {sel}$=$0\vert 1$ for put. The third variable $\tt {ingred}$ is a (6x1) dimensional vector that contains six input parameters: the price of the underlying asset, the strike price, the time to expiration, the annualized volatility of the underlying asset, the annualized risk-free interest rate and the dividend payment(s). The following example computes the price of an American option on non-dividend paying underlying asset.

library("finance")
 eopv=12.70          ; price of the European call
 sel=1|0             ; specifying an American call
 task=1              ; no dividend payments
 ingred=230|231|0.3|0.25|0.05|0.00
 mcmillan(eopv,sel,task,ingred)

XLGfindex5.xpl

 

The XploRe output shows the expected result, that an American call on a non-dividend paying underlying asset will have the same price as its European counterpart:

Contents of aus
 [1,] " "
 [2,]"-------------------"
 [3,] " The Price of Your American Call-Option "
 [4,] " on Given Stock is "
 [5,]"12.7000"
 [6,]"-------------------"
 [7,] " "

A second example computes the price of an American option when the underlying is a commodity that involves continuous annualized costs of around 5% of the commodity. The annualized risk-free interest rate is $r=5\%$, so that the cost of carry is $b=r-q=10\%$.

library("finance")
 eopv=12.70       ; price of the European call
 sel=1|0          ; specifying an American call
 task=2           ; continuous costs, e.g. storage or insurance
 ingred=230|231|0.3|0.25|0.05|-0.05
 mcmillan(eopv,sel,task,ingred)

XLGfindex6.xpl

 
When $b>r$, the American call will never be exercised, since the present value of all dividend payments until options expiration date is less than the present value of the interest rate that can be earned on the strike price of the call during its remaining time to expiration. In this case, the value of the American call simply equals its European counterpart, as is confirmed in the XploRe output.

Contents of aus
 [1,] " "
 [2,]"-------------------"
 [3,] " The Price of Your American Call-Option "
 [4,] " on Given Commodity with cont. Costs is"
 [5,]"12.7000"
 [6,]"-------------------"
 [7,] " "

When $b<r$ the value of the American call is given in (11.26). It is always higher than the price of its European counterpart, as illustrated in a third example.

library("finance")
 eopv=12.70          ; price of the European call
 sel=1|0             ; specifying an American call
 task=2              ; continuous dividend payments
 ingred=230|231|0.3|0.25|0.05|0.05
 mcmillan(eopv,sel,task,ingred)

XLGfindex7.xpl

The Xplore output is:

Contents of aus
 [1,] " "
 [2,]"-------------------"
 [3,] " The Price of Your American Call-Option "
 [4,] " on Given Stock with cont. Dividends is "
 [5,]"12.7373"
 [6,]"-------------------"
 [7,] " "

11.2.4.2 Derivation of the Formula

The quadratic approximation method will be briefly outlined following Hull (2000, pp. 432-434) and Kwock (1998, pp. 174-177). Consider an American option on a stock, paying continuous dividends at rate $q$. The early exercise premium, defined by $v(S,t)$, is

$$v(S,t)=C_{am}(S,t)-C(S,t),$$

where $C_{am}(S,t)$ is the value of an American call option and $C(S,t)$ is its European counterpart. Within the continuation region both $C_{am}(S,t)$ and $C(S,t)$ satisfy the Black-Scholes differential equation (11.25). It follows that $v(S,t)$ also satisfies and can therefore be written as

$$\frac{\partial v}{\partial t} + \frac{1}{2}\sigma^2\,S^2\frac{\partial^2 v}{\partial S^2}+ b\;S\frac{\partial v}{\partial S} = rv,$$ (11.35)

where $b=r-q$ is the cost of carry. By writing

$$\alpha = \frac{2r}{\sigma^2},$$

$$\ \beta = \frac{2b}{\sigma^2},$$ (11.36)

$$\tau = T-t,$$

$$\ h(\tau) = 1-exp(-r\tau)$$

and defining

$$v(S,\tau)=h(\tau)f(S,h),$$

equation (11.35) can be transformed into the following form

$$S^2\frac{\partial^2f}{\partial S^2}+\beta S \frac{\partial f}{\partial S} -\frac{\alpha}{h}\left\{f-(1-h)h\frac{\partial f}{\partial h}\right\}=0.$$ (11.37)

The approximation used involves assuming that the last term $(1-h)h\frac{\partial f}{\partial h}$ equals zero, so that (11.37) is reduced to an ordinary differential equation with the error being controlled by the quadratic term $(1-h)h$. This last term, which is ignored, is fairly small. When $\tau$ is large, it moves to zero and when $\tau$ is small, $\frac{\partial f}{\partial h}$ will be nearly zero. Then the approximative equation will be reduced to

$$S^2\frac{\partial^2f}{\partial S^2}+\beta S \frac{\partial f}{\partial S} -\frac{\alpha}{h}f=0,$$ (11.38)

where h is assumed to be non-zero. When $h$ is treated as a parameter, equation (11.38) becomes a non-homogeneous second order differential equation and can be solved with the standard techniques. After applying boundary conditions, the valuation formulae for the American call and put options (11.26) and (11.27) follow respectively.