terms of the stock price.
10.2 RISK-NEUTRAL VALUATION
 
Although we do not need to make any assumptions about the probabilities of up and downmovements in order to derive equation (10.2), it is natural to interpret the variable p in equa-tion (10.2) as the probability of an up movement in the stock price. The variable 1 — p is then theprobability of a down movement, and the expression
Phi + — p)fa
is the expected payoff from the option. With this interpretation of p, equation (10.2) then statesthat the value of the option today is its expected future value discounted at the risk-free rate.204 CHAPTER 10
We now investigate the expected return from the stock when the probability of an up movementis assumed to be p. The expected stock price, E(Sr), at time T is given by
E(Sp) = pSou + (1 — p)SodorE(Sy) = pSo(u — d) + Sod
Substituting from equation (10.3) for p, we obtainE(Sp) = Se (10.4)
showing that the stock price grows on average at the risk-free rate. Setting the probability of the upmovement equal to p is therefore equivalent to assuming that the return on the stock equals therisk-free rate.
In a risk-neutral world all individuals are indifferent to risk. In such a world investors require nocompensation for risk, and the expected return on all securities is the risk-free interest rate.Equation (10.4) shows that we are assuming a risk-neutral world when we set the probability ofan up movement to p. Equation (10.2) shows that the value of the option is its expected payoff in arisk-neutral world discounted at the risk-free rate.
This result is an example of an important general principle in option pricing known as risk-neutral valuation. The principle states that we can assume the world is risk neutral when pricing anoption, The price we obtain is correct not just in a risk-neutral world but in the real world as well.
The One-Step Binomial Example Revisited
We now return to the example in Figure 10.1 and illustrate that risk-neutral valuation gives thesame answer as no-arbitrage arguments. In Figure 10.1, the stock price is currently $20 and willmove either up to $22 or down to $18 at the end of three months. The option considered is aEuropean call option with a strike price of $21 and an expiration date in three months. The risk-free interest rate is 12% per annum.
We define p as the probability of an upward movement in the stock price in a risk-neutral world.We can calculate p from equation (10.3). Alternatively, we can argue that the expected return onthe stock in a risk-neutral world must be the risk-free rate of 12%. This means that p must satisfy
22p + 18(1 — p) = 20 P*7/?orAp = 20e%!23/2 _ 1gThat is, p must be 0.6523.At the end of the three months, the call option has a 0.6523 probability of being worth 1 anda 0.3477 probability of being worth zero. Its expected value is therefore
0.6523 x 1 + 0.3477 x 0 = 0.6523
In a risk-neutral world this should be discounted at the risk-free rate. The value of the optiontoday is therefore0.6523e7 012 *3/12
or $0.633. This is the same as the value obtained earlier, demonstrating that no-arbitragearguments and risk-neutral valuation give the same answer.Introduction to Binomial Trees 205
Real World vs. Risk-Neutral World
It should be emphasized that p is the probability of a up movement in a risk-neutral world. Ingeneral this is not the same as the probability of an up movement in the real world. In our example,p = 0.6523. When the probability of an up movement is 0.6523, the expected return on the stock isthe risk-free rate of 12%. Suppose that in the real world the expected return on the stock is 16%,and q is the probability of an up movement in the real world. It follows that
22g + 18(1 — q) = 202°?
~¢
a= NALq = 0.7041.The expected payoff from the option in the real world is then
qxl+(l-qx9This is 0.7041. Unfortunately it is not easy to know the correct discount rate to apply to the expectedpayoff in the real world. A position in a call option is riskier than a position in the stock. As a resultthe discount rate to be applied to the payoff from a call option is greater than 16%. Without knowingthe option’s value, we do not know how much greater than 16% it should be.! The risk-neutralvaluation solves this problem. We know that in a risk-neutral world the expected return on all assets(and therefore the discount rate to use for all expected payoffs) is the risk-free rate.
10.3 TWO-STEP BINOMIAL TREES
 
We can extend the analysis to a two-step binomial tree such as that shown in Figure 10.3. Here thestock price starts at $20 and in each of two time steps may go up by 10% or down by 10%. Wesuppose that each time step is three months long and the risk-free interest rate is 12% per annum.As before, we consider an option with a strike price of $21.
aeeo
20
/
— 19.8oN
Figure 10.3 Stock prices in a two-step tree
/
16.2
' Because the correct value of the option is 0.633, we can deduce that the correct discount rate is 42.6%. This isdecause 0.633 = 0.704 16704758 3/12206 CHAPTER 10
D24.23.2
2.0257
 
0.0
Figure 10.4 Stock and option prices in a two-step tree. The upper numberat each node is the stock price; the lower number is the option price
The objective of the analysis is to calculate the option price at the initial node of the tree. Thiscan be done by repeatedly applying the principles established earlier in the chapter. Figure 10.4shows the same tree as Figure 10.3, but with both the stock price and the option price at each node.(The stock price is the upper number and the option price is the lower number.) The option pricesat the final nodes of the tree are easily calculated. They are the payoffs from the option. At node Dthe stock price is 24.2 and the option price is 24.2 — 21 = 3.2; at nodes E and F the option is out ofthe money and its value is zero.
At node C, the option price is zero, because node C leads to either node E or node F and atboth nodes the option price is zero. We calculate the option price at node B by focusing ourattention on the part of the tree shown in Figure 10.5. Using the notation introduced earlier inthe chapter, u = 1.1, d= 0.9, r=0.12, and T = 0.25, so that p = 0.6523, and equation (10.2)gives the value of the option at node B as
e°Px3/12 (96523 x 3.2 + 0.3477 x 0) = 2.0257
D
eo oe22B2.0257
E19.80.0
Figure 10.5 Evaluation of option price at node BIntroduction to Binomial Trees 207
It remains for us to calculate to option price at the initial node A. We do so by focusing on the firststep of the tree. We know that the value of the option at node B is 2.0257 and that at node C it iszero. Equation (10.2) therefore gives the value at node A as
023/129 6523 x 2.0257 + 0.3477 x 0) = 1.2823
The value of the option is $1.2823.
Note that this example was constructed so that u and d (the proportional up and downmovements) were the same at each node of the tree and so that the time steps were of the samelength. As a result, the risk-neutral probability, p, as calculated by equation (10.3) is the same ateach node.
A Generalization
We can generalize the case of two time steps by considering the situation in Figure 10.6, The stockpice is initially Sp. During each time step, it either moves up to u times its initial value or movesdown to d times its initial value. The notation for the value of the option is shown on the tree. (Forexample, after two up movements the value of the option is f,,,.) We suppose that the risk-freeinterest rate is r and the length of the time step is 4r years.
Repeated application of equation (10.2) gives
fa = OO LD fau + = P) Sua (10.5)fa= ep fea + — p) faa (10.6)f=e pf, + (1 — p) ful (10.7)
Substituting from equations (10.5) and (10.6) into (10.7), we get
Fae Ly faa + 2D — PV fea + — PY Fuad (10.8)
S gu?
uuu
 
Figure 10.6 Stock and option prices in a general two-step tree208 CHAPTER 10
This is consistent with the principle of risk-neutral valuation mentioned earlier. The variables D,2p(1 — p), and (1 — p)* are the probabilities that the upper, middle, and lower final nodes will bereached. The option price is equal to its expected payoff in a risk-neutral world discounted at therisk-free interest rate.
As we add more steps to the binomial tree, the risk-neutral valuation principle continues to hold.The option price is always equal to its expected payoff in a risk-neutral world, discounted at the
risk-free inierest rate.
10.4 A PUT EXAMPLE
 
The procedures described in this chapter can be used to price any derivative dependent on a stockwhose price changes are binomial. Consider a two-year European put with a strike price of $52 ona stock whose current price is $50. We suppose that there are two time steps of one year, and ineach time step the stock price either moves up by a proportional amount of 20% or moves down bya proportional amount of 20%. We also suppose that the risk-free interest rate is 5%.
The tree is shown in Figure 10.7. The value of the risk-neutral probability, p, is given by
eo x! —08
P= 7-087 0.6282
The possible final stock prices are: $72, $48, and $32. In this case fy, = 0, fuga = 4, and fig = 20.From equation (10.8),
f = 67 2*00*1(9 6282? x 0-42 x 0.6282 x 0.3718 x 4 +0.37187 x 20) = 4.1923The value of the put is $4.1923. This result can also be obtained using equation (10.2) and working
72
ae
601.41475044.1923 _” :409.4636
3220
Figure 10.7 Use of two-step tree to value European put option. At each node,the upper number is the stock price and the lower number is the option priceIntroduction to Binomial Trees 209
back through the tree one step at a time. Figure 10.7 shows the intermediate option prices that arecalculated.
10.5 AMERICAN OPTIONS
Up to now all the options we have considered have been European. We now move on to considerhow American options can be valued using a binomial tree such as that in Figure 10.4 or 10.7. Theprocedure is to work back through the tree from the end to the beginning, testing at each node tosee whether early exercise is optimal. The value of the option at the final nodes is the same as forthe European option. At earlier nodes the value of the option is the greater of
1. The value given by equation (10.2)2. The payoff from early exercise
Figure 10.8 shows how Figure 10.7 is affected if the option under consideration is American ratherthan European. The stock prices and their probabilities are unchanged. The values for the option atthe final nodes are also unchanged. At node B, equation (10.2) gives the value of the option as1.4147, whereas the payoff from early exercise is negative (= —8). Clearly early exercise is not optimalat node B, and the value of the option at this node is 1.4147. At node C, equation (10.2) gives thevalue of the option as 9.4636, whereas the payoff from early exercise is 12. In this case, early exerciseis optimal and the value of the option at the node is 12. At the initial node A, the value given byequation (10.2) is°° (9 6282 x 1.4147 + 0.3718 x 12.0) = 5.0894
and the payoff from early exercise is 2. In this case early exercise is not optimal, and the value of
072
 
oT, 2
20
Figure 10.8 Use of two-step tree to value American put option. At each node,the upper number is the stock price and the lower number is the option price210 CHAPTER 10
the option is therefore $5.0894. More details on the use of binomial trees to value Americanoptions are given in Chapter 18.
10.6 DELTA
 
At this stage it is appropriate to discuss de/ta, an important parameter in the pricing and hedgingof options.
The delta of a stock option is the ratio of the change in the price of the stock option to thechange in the price of the underlying stock. It is the number of units of the stock we should holdfor each option shorted in order to create a riskless hedge. It is the same as the A introduced earlierin this chapter. The construction of a riskless hedge is sometimes referred to as de/ta hedging. Thedelia of a call option is positive, whereas the delta of a put option is negative.
From Figure 10.1, we can calculate the value of the delta of the call option being considered as
1-0
m1 9
This is because when the stock price changes from $18 to $22, the option price changes from $0 to $1.In Figure 10.4, the delta corresponding to stock price movements over the first time step is
2.0257 — 0
> a = 0.5064
The delta for stock price movements over the second time step is
3.2-0
a2 —198 7°?
if there is an upward movement over the first time step and0-0 |19.8 — 16.2 —
if there is a downward movement over the first time step.
From Figure 10.7, delta is1.4147 — 9.4636
60 — 40 = —0,4024at the end of the first time step and either0-4a7 4g = —0.1667or4-20Ro —1.0000
at the end of the second time step.
The two-step examples show that delta changes over time. (In Figure 10.4, delta changes from0.5064 to either 0.7273 or 0; in Figure 10.7, it changes from —0.4024 to either —0.1667 or —1.0000.)Thus, in order to maintain a riskless hedge using an option and the underlying stock, we need toIntroduction to Binomial Trees 211
adjust our holdings in the stock periodically. This is a feature of options that we will return to inChapter 14.
10.7_ MATCHING VOLATILITY WITH uw AND d
 
In practice, when constructing a binomial tree to represent the movements in a stock price, wechoose the parameters uw and d to match the volatility of the stock price. To see how this is done, wesuppose that the expected return on a stock (in the real world) is yz and its volatility is o. Figure 10.9ashows stock price movements over the first step of a binomial tree. The step is of length 5r. The stockprice either moves up by a proportional amount uw or moves down by a proportional amount d. Theprobability of an up movement (in the real world) is assumed to be q.
The expected stock price at the end of the first time step is Sye“*. On the tree the expected stockprice at this time is
gSyu + (1 — q)Sod
In order to match the expected return on the stock with the tree’s parameters, we must thereforehaveSou + (1 — Sod = Sye*™orett g
1a (10.9)
As we will explain in Chapter 11, the volatility o of a stock price is defined so that oV/6r is thestandard deviation of the return on the stock price in a short period of time of length 6.Equivalently, the variance of the return is o°ét. On the tree in Figure 10.9a, the variance of thestock price return is?
qu’ + (1 ~ q)d° — [gu + (1 — gdPIn order to match the stock price volatility with the tree’s parameters, we must therefore havequ’ + (1 — q)d ~ [qu + (1 — q)dP = 07 bt (10.10)Substituting from equation (10.9) into equation (10.10), we getey 4+ d)— ud — * = 6 bt
When terms in 8t? and higher powers of dr are ignored, one solution to this equation is
uae (10.11)d= (10.12)
These arc the values of w and d proposed by Cox, Ross, and Rubinstein (1979) for matching uand d.
2 This uses the result that the variance of a variable Q equals E(Q2) — [E(Q)°, where E denotes the expected value.212 CHAPTER 10
Sout Sou
So So
1-q l-p
Sod Sod(a) (b)Figure 10.9 Change in stock price in time St in (a) the real world and (b) the risk-neutral worldThe analysis in this chapter shows that we can replace the tree in Figure 10.9a by the tree in
Figure 10.9b, where the probability of a up movement is p, and then behave as though the world isrisk neutral. The variable p is given by equation (10.3) as
 
It is the risk-neutral probability of an up movement. In Figure 10.9b, the expected stock price atthe end of the time step is Sye“, as shown in equation (10.4). The variance of the stock pricereturn is
dr 85
pu? + (1 = p)d? — [pu + (1 pid? = [eu +) ~ ud —
Substituting for u and d from equations (10.11) and (10.12), we find this equals o* ¢ when termsin 5r° and higher powers of Sr are ignored.
This analysis shows that when we move from the real world to the risk-neutral world theexpected return on the stock changes, but its volatility remains the same (at least in the limit as éttends to zero). This is an illustration of an important general result known as Girsanov’s theorem.When we move from a world with one set of risk preferences to a world with another set of riskpreferences, the expected growth rates in variables change, but their volatilities remain the same.We will examine the impact of risk preferences on the behavior of market variables in more detailin Chapter 21. Moving from one set of risk preferences to another is sometimes referred to aschanging the measure.
10.8 | BINOMIAL TREES IN PRACTICE
 
 
The binomial models presented so far have been unrealistically simple. Clearly an analyst canexpect to obtain only a very rough approximation to an option price by assuming that stock pricemovements during the life of the option consist of one or two binomial steps.
When binomial trees are used in practice, the life of the option is typically divided into 30 ormore time steps of length 5r. In each time step there is a binomial stock price movement. WithIntroduction to Binomial Trees 213
30 time steps, this means that 31 terminal stock prices and 20) or about | billion, possible stockprice paths are considered.The parameters # and d are chosen to match the stock price volatility. A popular way of doingthis is by settinguae and d= en ovh
as indicated in the previous section. The complete set of equations defining the tree is then
r btoVlt da oot _e@v ad: ~e , P= 1d
Chapter 18 provides a further discussion of these formulas and the practical issues involved in theconstruction and use of binomial trees. DerivaGem provides a way of valuing options withbetween 2 and 500 time steps (set the option type as Binomial European or Binomial American).Trees with up to 10 time steps can be displayed by the software.
SUMMARY
 
 
This chapter has provided a first look at the valuation of stock options. If stock price move-ments during the life of an option are governed by a one-step binomial tree, it is possible to setup a portfolio consisting of a stock option and the stock that is riskless. In a world with noarbitrage opportunities, riskless portfolios must earn the risk-free interest. This enables the stockoption to be priced in terms of the stock. It is interesting to note that no assumptions arerequired about the probabilities of up and down movements in the stock price at each node ofthe tree.
When stock price movements are governed by a multistep binomial tree, we can treat eachbinomial step separately and work back from the end of the life of the option to the beginning toobtain the current value of the option. Again only no-arbitrage arguments are used, and noassumptions are required about the probabilities of up and down movements in the stock price ateach node.
Another approach to valuing stock options involves risk-neutral valuation. This very importantprinciple states that it is permissible to assume the world is risk neutral when valuing an option interms of the underlying stock. This chapter has shown, through both numerical examples andalgebra, that no-arbitrage arguments and risk-neutral valuation are equivalent and lead to thesame option prices.
The delta of a stock option, A, considers the effect of a small change in the underlying stockprice on the change in the option price. It is the ratio of the change in the option price to thechange in the stock price. For a riskless position, an investor should buy A shares for each optionsold. An inspection of a typical binomial tree shows that delta changes during the life of an option.This means that to hedge a particular option position, we must change our holding in theunderlying stock periodically.
In Chapter 12 we examine the Black-Scholes analytic approach to pricing stock options. InChapter 13 we cover the use of binomial trees for other types of options. In Chapter 18 we give amore complete discussion of how binomial trees are implemented.214
CHAPTER 10
SUGGESTIONS FOR FURTHER READING
 
Cox, J., S. Ross, and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of FinancialEconomics, 7 (October 1979), 229-64.
Rendleman, R., and B. Bartter, “Two State Option Pricing,” Journal of Finance, 34 (1979), 1092-1110.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
10.2.
10.3.10.4.
—oOa
10.6.
10.8.
10.9.
10.10.
10.11.
. A stock nrice is currently $100. Over each of the next two six-month neriods it is ex
. A stock price is currently $40. It is known that at the end of one month it will be either $42
or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is thevalue of a one-month European call option with a strike price of $39?
Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European optionusing a one-step binomial tree.
What is meant by the delta of a stock option?
A stock price is currently $50. It is known that at the end of six months it will be either $45or $55. The risk-free interest rate is 10% per annum with continuous compounding. What is thethe value of a six-month European put option with a strike price of $50?
tad to
. stock price 1s Current) necexpected to
go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuouscompounding. What is the value of a one-year European call option with a strike price of $100?
each of fhe next six-month period:
For the situation considered in Problem 10.5, what is the value of a one-year European putoption with a strike price of $100? Verify that the European call and European put prices satisfyput-caii parity.
. Consider the situation in which stock price movements during the life of a European option are
governed by a two-step binomial tree. Explain why it is not possible to set up a position in thestock and the option that remains riskless for the whole of the life of the option.
A stock price is currently $50. Tt is known that at the end of two months it will be either $53or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is thevalue of a two-month European call option with a strike price of $492 Use no-arbitragearguments.
A stock price is currently $80. It is known that at the end of four months it will be either $75or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is thevalue of a four-month European put option with a strike price of $80? Use no-arbitragearguments.
A stock price is currently $40. It is known that at the end of three months it will be either $45or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate thevalue of a three-month European put option on the stock with an exercise price of $40. Verifythat no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
A stock price is currently $50. Over each of the next two three-month periods it is expectedto go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuouscompounding. What is the value of a six-month European call option with a strike priceof $51?Introduction to Binomial Trees 215
10.12.
10.13.
10.14.
10.15.
10.16
10.17.
10.18.
10.19.
For the situation considered in Problem 10.11, what is the value of a six-month European putoption with a strike price of $51? Verify that the European call and European put prices satisfyput-call parity. If the put option were American, would it ever be optimal to exercise it early atany of the nodes on the tree?
A stock price is currently $25. It is known that at the end of two months it will be either $23or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose S7is the stock price at the end of two months. What is the value of a derivative that pays off S} atthis time?
ASSIGNMENT QUESTIONS
A stock price is currently $50. It is known that at the end of six months it will be either $60or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculatethe value of a six-month European call option on the stock with an exercise price of $48. Verifythat no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
A stock price is currently $40, Over each of the next two three-month periods it is expected to goup by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuouscompounding.
a. What is the value of a six-month European put option with a strike price of $42?
b. What is the value of a six-month American put option with a strike price of $42?
Tising a “trial-and-error” approach, estimate how high the strike price has to be in Problem 10.15for it to be optimal to exercise the option immediately.
Footnote | shows that the correct discount rate to use for the real-world expected payoff in thecase of the call option considered in Section {0.2 is 42.6%. Show that if the option is a put ratherthan a call then the discount rate is —52.5%. Explain why the two real-world discount rates are
eo diferentS80 Ginerent.
A stock price is currently $30. Each month for the next two months it is expected to increase by8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the valueof a derivative that pays off max[(30 ~ S;)*, 0], where Sz is the stock price in two months? If thederivative is American-style, should it be exercised early?
Consider a European call option on a non-dividend-paying stock where the stock price is $40,the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, andthe time to maturity is six months.
a. Calculate u, d, and p for a two-step tree.
b. Value the option using a two-step tree.
c. Verify that DerivaGem gives the same answer.
d. Use DerivaGem to value the option with 5, 50, 100, and 500 time steps. 
CHAPTER 11
MODEL OF THE BEHAVIOR OFSTOCK PRICES
Any variable whose value changes over time in an uncertain way is said to follow a stochasticprocess. Stochastic processes can be classified as discrete time or continuous time. A discrete-timestochastic process is one where the value of the variable can change only at ceriain fixed points intime, whereas a continuous-time stochastic process is one where changes can take place at anytime. Stochastic processes can also be classified as continuous variable or discrete variable. Inacontinuous-variable process, the underlying variable can take any value within a certain range,whereas in a discrete-variable process, only certain discrete values are possible.
This chapter develops a continuous-variable, continuous-time stochastic process for stock prices,An understanding of this process is the first step to understanding the pricing of options and othermote complicated derivatives. Note that, in practice, we do not observe stock prices followingcontinuous-variable, continuous-time processes. Stock prices are restricted to discrete values(e.g., multiples of a cent) and changes can be observed only when the exchange is open. Nevertheless,the continuous-variable, continuous-time process proves to be a useful model for many purposes.
Many people feel that continuous-time stochastic processes are so complicated that they shouldbe left entirely to “rocket scientists’. This is not so. The biggest hurdle to understanding theseprocesses is the notation. Here we present a step-by-step approach aimed at getting the reader overthis hurdle. We also explain an important result, known as /té’s lemma, that is