Understanding N d ) and N d ): Black-Scholes Model

1 UnderstandingN(d1) andN(d2):Risk-Adjusted Probabilities in theBlack- scholes Model1 Lars Tyge NielsenINSEADB oulevard de Constance77305 Fontainebleau CedexFranceE-mail: nielsen@freiba51 October 19921 Thanks to Pierre Hillion and Jes us Sa a-Requejo for comments on a previousversionAbstractThis paper uses risk-adjusted lognormal probabilities to derive the Black-Scholes formula and explain the factorsN(d1)andN(d2). It also showshow the one-period and multi-period binomial option pricing formulas canbe restated so that they involve analogues ofN(d1)andN(d2) which havethe same interpretation as in the Black-Scholes article utilise les probabilit es lognormaux corrig ees du risque pour d eriverla formule de Black-Scholes et expliquer les facteursN(d1)etN(d2). Ilmontre aussi comment les mod`eles binomiaux des prix d options d une et deplusieurs p eriodes peuvent etre exprim es d une fa con telle qu ils impliquentdes analogues deN(d1)etN(d2)quiontlam eme interpr etation que dans lemod` IntroductionThe Black-Scholes formula is an expression for the current value of a Euro-pean call option on a stock which pays no dividends before expiration of theoption.

2 The formula expresses the call value as the current stock price timesa probability factorN(d1), minus the discounted exercise payment times asecond probability factorN(d2).ExplainingN(d1)andN(d2), and in particular explaining why they are dif-ferent from each other, usually presents some difficulties. Among the majorresearch papers, black and scholes (1973) did not explain or interpretN(d1)andN(d2). Neither did Merton (1973, 1990 Chapter 8), Cox and Ross (1976),or Rubinstein (1976). As for the textbooks, Jarrow and Rudd (1983) heuris-tically derive the Black-Scholes formula using risk-adjusted probabilities, andin the process they do interpretN(d1)andN(d2). Cox and Rubinstein (1985)state that the stock price timesN(d1) is the present value of receiving thestock if and only if the option finishes in the money, and the discounted exer-cise payment timesN(d2) is the present value of paying the exercise price inthat event.

3 They do not explain why this is so or relate it to the probabilitythat the option finishes in the money. Hull (1989) and Hull (1991) do notexplainN(d1)andN(d2), although the necessary mathematics is availablein the earlier purpose of the present paper is to explain whereN(d1)andN(d2)comefrom and why they are different from each other. This is done by relatingthem to risk-adjusted probabilities in both the Black-Scholes and in the bi-nomial Model of Cox, Ross and Rubinstein (1979). The argument relating toBlack- scholes expands on that of Jarrow and Rudd (1983). The commentson the binomial Model involve simple manipulations and reinterpretations ofwell-known stated,N(d2) is the risk-adjusted probability that the option willbe exercised. The interpretation ofN(d1) is a bit more complicated.

4 Theexpected value, computed using risk-adjusted probabilities, of receiving thestock at expiration of the option, contingent upon the option finishing inthe money, isN(d1) multiplied by the current stock price and the risklesscompounding factor. Thus,N(d1) is the factor by which the present value1of contingent receipt of the stock exceeds the current stock present value of contingent receipt of the stock is not equal to but largerthan the current stock price multiplied byN(d2), the risk-adjusted proba-bility of exercise. The reason for this is that the event of exercise is notindependent of the future stock price. If exercise were completely randomand unrelated to the stock price, then indeed the present value of contingentreceipt of the stock would be the current stock price multiplied byN(d2).

5 Actually the present value is larger than this, since exercise is dependent onthe future stock price and indeed happens when the stock price is organization of the paper is as follows. Section 2 states the black -Scholesformula. Section 3 contains the substance of the argument. It splits the payoffto the call option into two components, shows how their future expected val-ues (computed using the risk-adjusted probabilies) and present values involvethe probability factorsN(d1)andN(d2), and explains whyN(d1) is largerthanN(d2). Section 4 shows how the one-period binomial option pricing for-mula can be restated in a form which resembles the Black-Scholes involves analogues ofN(d1)andN(d2) with similar interpretations as inthe Black-Scholes Model . Section 5 does the same analysis of the multiperiodbinomial Model .

6 The rest of the paper contains the documentation to backup the Black-Scholes Model : Section 6 explains the probabilistic assump-tions behind the Model , Section 7 describes how the risk-adjustment of theprobabilities is carried out, and Section 8 uses the risk-adjusted probabilitiesto derive the Black-Scholes formula by computing the present values of thecomponents of the call option payoff. Section 9 contains the The Black-Scholes FormulaThe Black-Scholes formula is an expression for the current value of a Euro-pean call option on a stock which pays no dividends before expiration of theoption. The formula isC=SN(d1) e r XN(d2),whereCis the current value of the call,Sis the current value of the stock,ris the interest rate (assumed constant), is the remaining time to expirationof the option,Xis the exercise price, andN(d1)andN(d2) are probabilityfactors:Nis the cumulative standard normal distribution function,d2= log(X/S) (r 12 2) ,d1=d2+ ,and is a parameter measuring the volatility of the stock (interpreted moreprecisely below in Section 6).

7 3 The Payoff and Value of the CallTo value the call option, I shall use the concept of risk-adjusted turns out that one can adjust the probability distribution of the stock pricein such a way that the current value of any stock-price contingent claimequals the expected future payoff to the claim, computed using the adjustedprobabilities, discounted at the riskless demonstration of this involves mathematical analysis of dynamic tradingand arbitrage between the stock and the riskless asset, something I shall notfocus on here. However, the practical aspect of exactly how the probabilityadjustment is performed, is described below in Section payoff to the call option at maturityTwill beCT=max{0,ST X}={ST XifST X0otherwise3It is useful to split this payoff into two components.}

8 The first component isthe payment of the exercise price, contingent on the option finishing in themoney. It will be referred to as contingent exercise payment, for short. Itis a claim with payoffC1T={ XifST X0otherwiseThe second component is the receipt of the stock, again contingent on theoption finishing in the money. It will be referred to as contingent receipt ofthe stock. The payoff isC2T={STifST X0otherwiseThe various payoffs are shown in Figure can value the option by valuing each of the two components current value of the contingent payment of the exercise price will bethe expected future payment, computed on the basis of the risk-adjustedprobability distribution, discounted at the riskless rate. The expected futurepayoff isEC1T= XP{ST>X}.wherePis the risk-adjusted probability, and so the value is e r XP{ST>X}.}}

9 It turns out that the risk-adjusted probability of the event that the optionwill finish in the money isP{ST>X}=N(d2). Therefore, the expected payoff is XN(d2), and the present discounted value of this payoff is e r XN(d2).So, this is the current value of the first component of the option, the contin-gent exercise current value of the second component of the option, the contingentreceipt of the stock, will also equal the expected future value, computed usingthe adjusted probabilities, and discounted at the riskless rate. The expected4 Payoff at expiration ST 0 X Call option X Contingent receipt of the stockExpected value =Ser N(d1)Present value =SN(d1) XContingent exercise paymentExpected value = XN(d2)Present value = e r XN(d2)Figure 1: Payoff to the call and its components5future value of this component of the payoff is not simply the conditionalexpectation of the stock price given exercise.

10 Rather, it is the conditionalexpectation of the stock price given exercise times the probability of exercise,EC2T=E[ST|ST>X]P{ST>X}.It turns out that this equalsE[ST|ST>X]P{ST>X}=er SN(d1),and so the current value isSN(d1).So,N(d1) is the factor by which the discounted expected value of contingentreceipt of the stock exceeds the current value of the putting together the values of the two components of the option payoff,we get the Black-Scholes formula:C=SN(d1) e r XN(d2).Why is the present value of the contingent receipt of the stock notSN(d2),corresponding to an expected future value (computed using risk-adjustedprobabilities) ofer SN(d2)?The argument would be this. The present value of unconditionally receivingthe stock at timeTis obviously equalS, the current stock value.