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BLACK-SCHOLES FORMULAE

It has been about 40 years since Black-Scholes and Merton published their seminal papers on the valuation of vanilla, European-style stock options based on several simplifying assumptions. Since then, the Black-Scholes model has been modified, extended, and adapted when valuing vanilla and exotic options. Although the authors used diffusion equations to arrive at their famed results, I will in this section use heuristic arguments supported by basic calculus and probability arguments to arrive at the same results.

One of the key assumptions that Black-Scholes made was that the stock- price process of a nondividend-paying stock can be characterized by the equation[1]

(3.1)

where

S is the price of the stock.

r is the annualized continuously compounded risk-free rate.

a is the annualized volatility of the stock-price return.

dz is the random variable drawn from a standard normal probability density function.

dS is the small change in the stock price over a small time interval dt.

To obtain the formula to price a European-style, vanilla call option, the authors go about solving equation (3.1) subject to the boundary condition max[ST – K, 0] (where ST represents the price of the stock on option maturity and K represents the strike price of the option) and assuming that there is no possibility of a riskless arbitrage in continuous time using Ito's lemma. See Black and Scholes (1973) for details.

To solve equation (3.1) subject to the boundary condition, one first needs to observe that the diffusion equation is equivalent to assuming that future stock price movements follow a geometric Brownian motion distribution. More precisely, In ST is assumed to be normally distributed with a mean of and a variance of where

St is the price of the stock at time t.

rt T is the zero rate corresponding to a maturity (T) using zero curve at time t.

σt,T is the spot volatility rate corresponding to a maturity (T) using spot volatility curve at time t.

t is the time today.

T is the time when option matures.

Using the notion of risk-neutral valuation[2] (i.e., the reason for using rt,T as both the stock growth rate and discount rate), the expression to value the option is given by the present value of the expected option payoff max – K, 0]. Doing this, one gets the expression

(3.2)

where [max(ST – K, 0)] in equation (3.2) refers to the expectation of max[ST – K, 0] taken with respect to the random variable ST.6

s Another way of looking at the same problem (which will prove to be immensely valuable when trying to value an exotic option called a pay later option, discussed in Chapter 5) can be done by first writing down the profit (not the payoff!) to the option owner at time T. Doing this yields the option owner a profit of, whererepresents the value of the option premium that is paid at time t and future valued to time T. To ensure that the option is fairly priced, one needs the present value of the expected value of the profit to be 0. More succinctly, one would need thatSimplifying this yields the expression – which is essentially equation (3.2).

Letting g(ST) represent the probability density function (pdf) of ST, it readily follows from equation (3.2) that

(3.3a)

Observing thatwhereand, it readily follows that

(3.3b)

(3.3c)

Substituting equations (3.3b) and (3.3c) into equation (3.3a), one can arrive at the famed Black-Scholes equation that is used to value European- style call options on nondividend-paying stocks. More precisely, this takes the form of[3],[4]

where

and

One can go through a similar exercise to show that a European style put option (where the maturity payoff to the option buyer is of the form max[KST, 0]) can be priced using the expression

(3.4b)

Equations (3.4a) and (3.4b) have been implemented in Microsoft Excel so as to value both European-style call and put options on nondividendpaying stocks.

The value of the option given in Table 3.2 is for the case when there is only one stock underlying the option. In practice, when transacting in exchange-traded options, it is imperative for the trader to stipulate the contract size associated with the trade – where each contract size typically[5] has a multiplier of 100. Thus, the actual cost of the transaction is simply the price of the option on one share of the stock multiplied by the contract multiplier for one contract (which is 100 in this case) and the number of contracts. Table 3.3 shows the instance of valuing an option on Baidu stock (stock-trading symbol: BIDU) where each contract has a multiplier of 100. Furthermore, when transacting a similar trade on the OTC market, it suffices to simply stipulate the contract size, as the lot size for each contract is assumed to be one. As a consequence, the trade in Table 3.3 can be done in an OTC market using a contract size of 500.

Another observation that the reader needs to take note of is that most stock options that trade on the exchange tend to be American style in nature vis-a-vis their OTC counterparts. The methodology discussed in Table 3.2 can only be used to value European-style exercise types and not the Bermudan or American-style exercise types. The valuation of early-exercise features in an option is discussed later in this chapter.

TABLE 3.2 Valuing European-Style Options on Nondividend-Paying Stocks

  • [1] The intuition behind the use of equation (3.1) stems from the assumption that the continuously compounded returns of the stock are lognormally distributed. See Hull (2012) for further details.
  • [2] This is merely an artificial assumption made to ensure that no preferences of the option holder are factored into the mathematical derivation of the option pricing, hence not allowing for any arbitrage. The reader is referred to Hull (2012) for a detailed discussion.
  • [3] is sometimes presented in financial textbooks as (3.4a)
  • [4] As an astute reader will realize, N(a) is in fact the cumulative probability function associated with a standard normal probability. So, where Z represents a normal pdf with a mean of 0 and variance of 1.
  • [5] Contrary to the common belief, this multiplier is not unique to each stock-ticker symbol. For example, shares of Apple trade under the symbol AAPL on the Nasdaq stock exchange. There are two types of multipliers used when trading options on this stock. One is the usual multiplier of 100, while the other is a multiplier of 10 – where each option contract has its own unique ticker symbol that reflects the multiplier, option type, expiry date, and strike price.
 
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