An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_9 pot

Suppose{V i+1 n }i+1 n=0 are known; that is, we have the option values corresponding to time t = ti+1 and all possible asset prices.. The big idea in the binomial method is to multiply t

16.3 Deriving the parameters 153

Our task is to find V00, the option value at time zero We may do this by workingbackwards through the tree Suppose{V i+1

n }i+1

n=0 are known; that is, we have the

option values corresponding to time t = ti+1 and all possible asset prices Then

consider the option value V n i corresponding to asset price S i n at time t = ti Because

of our up/down assumption about the asset price movement, working from right to

left, the asset price S n i comes either from S i+1

n+1, with probability p, or from S n i+1,with probability 1− p Now, recall the definition (3.1) for the expected value of

a discrete random variable The big idea in the binomial method is to multiply

the two possible values V i+1

n+1 and V n i+1 by their associated probabilities to get an

expected value In this way the option value V n i corresponding to asset price S n i is

taken to be pV i+1

n+1 + (1 − p)V i+1

n , scaled by the appropriate factor that allows for

the interest rate, r This gives the fundamental relation

Once the parameters u, d, p and M have been chosen, the formulas (16.1)–

(16.3) completely specify the binomial method The recurrence (16.1) shows how

to insert the asset prices in the binomial tree Having obtained the asset prices at

time t = tM = T , (16.2) gives the corresponding option values at that time The relation (16.3) may then be used to step backwards through the tree until V00, the

option value at time t = t0= 0, is computed

16.3 Deriving the parameters

Since the discrete asset price model in the binomial method fits into the framework

of (6.2), by appealing to Exercise 6.2 we could tune the parameters by asking for

the corresponding Y i to have zero mean and unit variance This would lead totwo constraints However, to give more insight into the workings of the method,

we will derive those constraints from first principles Exercise 16.5 asks you toconfirm that the two approaches lead to the same conclusion

As a means to write down an expression for the up/down asset price model used

in the binomial method, we define a random variable Ri such that Ri = 1 if theasset price goes up from time(i − 1)δt to iδt and R i = 0 if the asset price goes

down Hence, Ri = 1 with probability p and Ri = 0 with probability 1 − p This means that Ri is a Bernoulli random variable with parameter p, so from (3.2) and

(3.14) we see thatE(R i ) = p andvar(R i ) = p(1 − p) After n time increments

the asset has undergonen

i=1R i upward movements and n−n

Taking logs gives

a normal random variable Hence, for large n, log (S(nδt)/S0) will be close to

normal To match the continuous asset price model (6.8) used in the Black–Scholesanalysis, we thus require the mean of log(S(nδt)/S0) to be (µ −1

2σ2)nδt and the

variance to beσ2n δt Further, as the binomial method works with expected values,

we impose the risk neutrality assumptionµ = r This leads to the conditions

see Exercise 16.2 Regardingδt = T/M as pre-specified, we now have two

equa-tions in the three unknowns, p, u and d In general, we can fix one of the three

and solve for the other two To pick out a particular solution this way, we may set

p= 1

2 and solve to find that

u = e σ√δt+(r−1σ2)δt , d = e −σ√δt+(r−1σ2)δt , (16.7)see Exercise 16.3

16.4 Binomial method in practice

The arguments in the previous section suggest that the binomial method assetmodel matches that used in the Black–Scholes analysis for smallδt, that is, large

M We may thus hope that the option values computed from the binomial method

agree well with those from the Black–Scholes formulas, and that the agreement

improves if M is increased.

Computational example We use the binomial method to value a European

put with S0= 9, E = 10, T = 3, r = 0.06 and σ = 0.3 Table 16.1 shows the results for M = 100, M = 200 and M = 400, along with the Black–Scholes

value 1.4728 Our first observation is that with all three choices of M the

bi-nomial method approximation is correct to at least two decimal places The

16.4 Binomial method in practice 155

Table 16.1 European put value

approximations from binomial method

Fig 16.2 Convergence of the binomial method for a European put as the

num-ber of time points, M, increases Upper picture: M goes from 20 to 250 in steps

of 5 Dashed line is ‘exact’ solution Lower picture: M goes from 200 to 400 in

steps of 1.

most accurate approximation of the three comes from the largest value of M, which is intuitively reasonable However, it is perhaps surprising that M = 200

gives less accuracy that M = 100 To check whether this is simply a quirk,

the upper picture in Figure 16.2 shows the computed option value for M =

20, 25, 30, , 250, with the Black–Scholes value superimposed as a dashed

line We see that although the binomial method approximations do appear to

converge as M increases, the convergence is by no means monotonic – ing a slightly bigger M may worsen the error – and there is a general ‘saw- tooth’ pattern to the sequence of approximations as M increases The lower plot

tak-156 Binomial method

emphasizes the waviness Here we have plotted the computed solution for all M

between 200 and 400 The result appears to oscillate between two smooth curves,

Two features stand out in Figure 16.2

(i) The binomial method approximation converges to the Black–Scholes value as

M → ∞.

(ii) The convergence is not monotonic.

These may be shown to be generic Moreover, it is possible to describe the rate at

which convergence takes place Letting e M = |V0

0 − P(S0, 0)| denote the error in

the binomial method approximation withδt = T/M, it can be shown that there is

a constant K such that

e M ≤ K

In the upper picture of Figure 16.3 we display the errors in the example above for

M between 100 and 400 The points have been joined by straight lines for clarity.

The curve 1/M is added as a solid line, and we see that (16.8) appears to hold with

K = 1 Taking logs in (16.8) gives log eM ≤ log K − log M, showing that the log

of the error as a function of log M should lie below a straight line of slope −1.The lower picture of Figure 16.3 re-scales the axes logarithmically to confirm thisbehaviour

16.5 Notes and references

Cox, Ross and Rubinstein (Cox et al., 1979) wrote the original binomial method

paper Since then numerous authors have analysed and extended the ideas

It is possible to derive the parameters u, d and p from a number of different viewpoints For example, with p= 1

is common; see (Kwok, 1998; Wilmott et al., 1995) Exercise 16.4 shows that this

is very close to the choice (16.7) for smallδt.

Although much literature has been devoted to establishing that the error in

vari-ous classes of binomial methods tends to zero as M → ∞, surprisingly little

atten-tion was initially paid to the rate of convergence Leisen and Reimer (Leisen and

Reimer, 1996) developed a general convergence rate theory, and the bound (16.8)follows from their results A more detailed analysis, with explicit error constants,appears in (Walsh, 2003)

16.5 Notes and references 157

Fig 16.3 Upper picture: Error in the binomial method for a European put as the

number of time points, M, increases from 100 to 400 Solid line is 1 /M Lower

picture: same data on a log–log scale.

The odd–even ripples in the error, as depicted in Figures 16.2 and 16.3, havebeen widely reported The references (Leisen and Reimer, 1996; Rogers and Sta-pleton, 1998) give explanations for the effect and propose fixes

Applying the binomial method may be shown to be equivalent to using a finitedifference method to approximate the Black–Scholes PDE, a point that we pursue

in Section 24.4 This is one means of proving that the binomial method solution

converges to the Black–Scholes value as M→ ∞, see (Kwok, 1998), for example,and numerical analysis insights can also be used to explain the odd-even ripples.The book (Clewlow and Strickland, 1998) covers a number of practical issues

in the implementation of the binomial method, and provides pseudo-code listings

A case study with the aim of making the binomial method run as quickly aspossible in MATLAB is given in (Higham, 2002), along with downloadable codes

It is possible to compute Greeks via the binomial method For partial derivatives

with respect to S or t, approximations can be obtained using information from the

tree Exercise 16.8 illustrates the idea Other partial derivatives can be treated byre-running the method with perturbed data, in the manner outlined in Section 15.4.Further details can be found in (Hull, 2000), for example, and (Walsh, 2003) showsthat delta can be approximated to the same order of accuracy as the option value

158 Binomial method

E X E R C I S E S

16.1.  Consider the discrete asset price model used in the binomial method.

Show that it may be written in the form (6.2) if we let Yi be defined as

Note that these values agree with those in Exercise 16.4 up to O (δt3/2 ).

16.6.    Returning to the recurrence (16.3) we see that for M = 1

16.6 Program of Chapter 16 and walkthrough 159

Similarly for M = 3 we find that

V00 = e −3rδt
p3V33+ 3p2(1 − p)V3

2 + 3p(1 − p)2V13+ (1 − p)3V03



.

The coefficients{1, 1}, {1, 2, 1}, {1, 3, 3, 1} are familiar from Pascal’s

tri-angle Having spotted this connection, prove by induction that

denotes the binomial coefficient,



M k

write down the form of the i by (i + 1) matrix B i such that W i = B i W i+1.

16.8.  Explain why the ratio (V1

1 − V1

0)/(S1

1− S1

0) can be regarded as an

ap-proximation to the time-zero delta

16.6 Program of Chapter 16 and walkthrough

The program ch16 implements the binomial method for a European call It is listed in Figure 16.4 First, parameters are initialized, using (16.7) for u and d The quantity

S*d.^([M:-1:0]’).*u.^([0:M]’)

is an M+1 by 1 array whose components cover the values S0M , S M

1 , , S M

M in the expiry-time level

of the asset price tree in Figure 16.1 Hence, the line

W = max(S*d.^([M:-1:0]’).*u.^([0:M]’)-E,0);

contains the expiry time option values, as in (16.2) We then work through the iteration (16.3) by exploiting MATLAB’s colon notation to extract subarrays The syntax

exp(-r*dt)*(p*W(2:i+1) + (1-p)*W(1:i));

160 Binomial method

%CH16 Program for Chapter 16

%

% Implements binomial method for European call

%%%%%%%% Problem and method parameters %%%%%%%%%%%

disp(’Option value is’), disp(W)

Fig 16.4 Program of Chapter 16: ch16.m.

around the loop On exit, W is a scalar, whose value is V00.

Running ch16.m produces the value 1.1175 To check, we may call ch08.

>> [C, Cdelta, P, Pdelta] = ch08(3,2,0.05,0.3,1)

C = 1.1175 Cdelta = 0.9524

P = 0.0200 Pdelta = -0.0476

P R O G R A M M I N G E X E R C I S E S

P16.1 Alterch16 so that the choice (16.9) for u and d is used.

P16.2 Implement the binomial method via the formula (16.11).

16.6 Program of Chapter 16 and walkthrough 161

Quotes

‘Would you tell me, please, which way I ought to go from here?’

‘That depends a good deal on where you want to get to,’ said the Cat.

‘I don’t much care where ’ said Alice.

‘Then it doesn’t matter which way you go,’ said the Cat.

L E W I S C A R R O L L, Alice in Wonderland Sir, In your otherwise beautiful poem (The Vision of Sin)

there is a verse which reads

‘Every moment dies a man,

every moment one is born.’

Obviously, this cannot be true and I suggest that in the next edition you have it read

‘Every moment dies a man,

every moment 1161 is born.’

Even this value is slightly in error

but should be sufficiently accurate for poetry.

C H A R L E S B A B B A G E (in a letter to Lord Tennyson), source (Fr¨oberg, 1985)

In the literature,

there are numerous contributions with limit proofs to European type options.

Astonishingly, however, the convergence speed of binomially computed option prices has, so far, rarely been examined technically.

Here, we present a theorem

D I E T M A R L E I S E N A N D M A T T H I A S R E I M E R (Leisen and Reimer, 1996)

cash-or-• They are widely traded, and hence of practical importance.

• The corresponding Black–Scholes values can be found analytically.

• They give us another opportunity to investigate the risk neutrality idea.

where A > 0 is fixed Holding this option amounts to making a straight bet that

the terminal asset price will exceed the exercise price, E, that is, the European call will finish in-the-money Winning the bet gets you A, losing the bet gets you

nothing Unlike the European case, there is no added value to be had from the asset

exceeding the strike by a wide margin; the upside is limited to A.

We have not yet specified the payoff for the case S (T ) = E This is an

excep-tional event, technically it occurs with zero probability, so the resulting payoff is

163

164 Cash-or-nothing options

E A/2

A

S(T )

Cash-or-nothing put

Fig 17.1 Payoff diagrams for cash-or-nothing call and put.

not important But to be consistent with the formula that we derive, we will assume

that A /2 is paid off in this at-the-money scenario, S(T ) = E.

Analogously, a cash-or-nothing put option differs from a European put option

in that the payoff at expiry is

0, if S (T ) > E, and

A , if S (T ) < E.

Holding this option amounts to making a straight bet that the European put will

finish in-the-money As for the call, we assume that A /2 is paid off if S(T ) = E.

Cash-or-nothing call and put payoff diagrams are shown in Figure 17.1

Cash-or-nothing options are sometimes called binary, or digital options,

al-though these phrases are also used more generally when there is a discontinuouspayoff diagram

17.3 Black–Scholes for cash-or-nothing options

We will let Ccash(S, t) and Pcash(S, t) denote the values of the cash-or-nothing

call and put options, respectively, for asset price S and time t.

The hedging argument used in Chapter 8 is very general – it requires only that

the option value is a smooth function of S and t Hence, we may ask for Ccash(S, t)

17.3 Black–Scholes for cash-or-nothing options 165

and Pcash(S, t) to satisfy the Black–Scholes PDE (8.15) Specifying appropriate

final time and boundary conditions is then sufficient to characterize the valuationformulas

There is a simple put–call parity relation connecting Ccash(S, t) and Pcash(S, t),

see Exercise 17.1, and hence we will focus on finding a formula for Ccash(S, t).

The cash-or-nothing call payoff function gives final time conditions

Ccash(0, t) = 0, for all 0≤ t ≤ T. (17.2)

When S is very large, the option is almost certain to pay off the amount A So,

after discounting for interest, we find that

Ccash(S, t) ≈ Ae −r(T −t) , for large S (17.3)Just as for the European case, imposing the final time and boundary conditions

is enough to specify a unique solution The solution turns out to have the simpleform

Ccash(S, t) = Ae −r(T −t) N (d2), (17.4)

where d2 is the quantity (8.21) that appears in the European formulas Our proach for confirming that (17.4) is an appropriate solution will be to check that theformula satisfies the Black–Scholes PDE and the extra conditions (17.1)–(17.3).Exercise 17.2 asks you to do the latter

ap-It is a straightforward exercise in differentiation to show that the partial tives appearing in the Black–Scholes PDE have the following form: