8.5 Connections to the Multi-period Model
As mentioned in Chap.4, we can price any redundant asset with priced assets by the absence of arbitrage as stated in FTAP. Thus, we can easily come up with two different methods to price any derivative asset, whose payoffs we can replicate with existing assets. One is by forming a hedge portfolio and the other is based on using risk-neutral probabilities.
As in the Black-Scholes formula we first assume that the underlying asset price process is described by a binomial model withupanddownmoves as we have done in Sect.4.2. In the absence of arbitrage, both pricing methods result in the formula for basic two period model of European option:
C0D 1 Rf
Rf d
ud C1.u/C uRf
ud C1.d/
; (8.11)
whereC1 shows the final payoff of the option. In the case of call option, the final payoff of the option is equal to.S1 K/C, so in the state ofup moveC1.u/ D .SuK/C and in the state of downmoveC1.d/ D .SdK/C. Moreover, risk- neutral probabilities are defined forupmove anddownmove by
?D Rfd
ud ; 1?D uRf
ud;
respectively. In the multi-period discrete time settings (Chap.5), by induction, the pricing formula can be expressed with risk-neutral probabilities:
C0 D 1 Rnf
2 4Xn
jD0
nŠ
jŠ.nj/Š.?/j.1?/nj.SujdnjK/C 3
5: (8.12)
To eliminate the positivity condition in the pricing equation in the formula, we can define a variablemsuch that it denotes the minimum number of up moves for the stock price to satisfy the positivity of the final payoff function as follows:
mDmin˚
j;j2 f0; 1; : : : ;ng W.SujdnjK/0
: (8.13)
Thus, we can conclude for allj moption expires in-the-money and otherwise it expires out-of-money. Thus we have the relation form:
m> lnŒK=.Sdn/ ln.u=d/ :
Then, if we use m in the pricing equation (8.12), we eliminate the positivity condition and we have
C0D 1 Rnf
2 4
Xj jDm
nŠ
jŠ.nj/Š.?/j.1?/nj.SujdnjK/
3 5;
which yields C0 DS
Xn jDm
nŠ.?/j.1?/nj jŠ.nj/Š
ujdnj
Rnf KRnf Xn jDm
nŠ.?/j.1?/nj jŠ.nj/Š :
Here, we can define a new binomial probability ??D? u
Rf
and 1??D.1?/d Rf;
since?is the risk-neutral probability. Then the pricing formula reduces to C0 DS
Xn jDm
nŠ
jŠ.nj/Š.??/j.1??/njKRnf Xn jDm
nŠ.?/j.1?/nj jŠ.nj/Š : Here, one can notice that the summations in the pricing equation correspond to the probabilities of binomially distributed random variables with parameters.n; ??/ and.n; ?/respectively, that take values larger or equal thanm, which is defined as in equation (8.13).
We find it useful to remind the binomial distribution. A random variable distributed binomially with parameters.n;p/, as n shows the number of trials or repetitions andpshows the probability of success of an event for each trial, has a probability distribution function
P.Xx/D Xx
jD0
nŠ
jŠ.nj/Š.p/j.1p/nj and
P.Xx/D1P.Xx/D Xn
jDx
nŠ
jŠ.nj/Š.p/j.1p/nj:
8.5 Connections to the Multi-period Model 303 In the pricing equation, we have probabilities?and??as in the binomial success probability. We could interpret them as the adjusted risk-neutral probabilities of the underlying asset prices up move at each node given that the option expires in-the- money. Thus we can express the probabilities asQ.mIn; ??/andQ.mIn; ?/and so the pricing equation has a very similar formula to the Black-Scholes formula in continuous time setting as follows:
C0DSQ.mIn; ??/KRnf Q.mIn; ?/:
We have the same analogy with the Black-Scholes formula expressed with current asset price and discounted strike price with the expire in-the-money probabilities Q.mIn; ?/andQ.mIn; ??/. These probabilities come from binomial distributions unlike the standard Black-Scholes model. From multiperiod to continuous time, we take the limit of the time step number to infinity by keeping the time to maturityT finite. We define time-to-maturityT D tnand we taken ! 1. When we take the limit the discount term has an exponential form with the instantaneous risk-free rater, i.e.er DRf. Then, under continuous time we can explicitly state the up move probabilities:
?D erTd
ud ; ??DuerT:
For the convergence, analogously, we would expect that asn ! 1, the expire in-the-money probabilitiesQ.mIn; ?/ andQ.mIn; ??/ would converge to the standard normal distribution with the log-normal prices. Thus, we remind a very nice result of theorem called de Moivre Laplace limit theorem, which is a special case of the central limit theorem. It states that since the binomial random variable is actually a sum of Bernoulli random variables, asn! 1, the binomial distribution converges to the normal distribution. For the proof of the theorem and the details one can refer to Grimmett et al. [GS01]. Thus, we have
Q.mIn;p/! Z 1
m
N.x/dx; asn! 1;
whereX is a binomial random variable andN./denotes the normal probability density function (see Sect.A.2). By standardizing the normal distribution, we have
yD xE.X/
pvar.X/; zD mE.X/
pvar.X/:
Then, we have
Q.mIn;p/! Z 1
z
N.y/dyDW.z/; (8.14)
where the function ./ represents the cumulative distribution of the normal distribution.
From this point, the only task left is to show thatzactually corresponds tod1(in the Black Scholes formula) under the probability??andd2under the probability ?. To show this, we first express the asset price in terms of time to maturity, where tDT=n:
STDSuxdnx: We take the logarithm to get
ln ST
S
Dxln u
d
Cnln.d/:
Thus, we can express the expected value and the variance of the random variableX as follows:
EŒXD EŒln.ST=S/nln.d/
ln.u=d/ ;
varŒXD varŒln.ST=S/
ln.u=d/2 :
Hence, we need to find the expectation and the variance of the logarithmic return variable in order to compute them for the random variableX. We define the variable magain for continuous-time setting by doing a simple trick such that we can always find"; 0" < 1:
mD lnŒK=.Sdn/
ln.u=d/ C"; (8.15) asn! 1. By using the relation given by (8.15) and the expectation and variance of the random variableX, we can express the valuezin (8.14) as follows:
zD mpCEŒX
VŒX D ln.S=pK/CEŒln.ST=S/
VŒln.ST=S/ p"ln.u=d/ varŒln.ST=S/: Moreover, we know that
pvarŒln.ST=S/Dln.u=d/p
varŒXDln.u=d/p
n?.1?/:
by binomial distribution. Thus, we have
zD ln.S=K/CEŒln.ST=S/
pvarŒln.ST=S/ ;
8.5 Connections to the Multi-period Model 305 as n ! 1. One can show that the variance does not change under equivalent probability measures (this follows from Girsanov’s Theorem, see [KS98]). We need some specifications for up move and down move and the probability:
uDe
pt; dDe
pt; pD 1 2C 1
2 p
t
;
wherepdenotes the objective probability of the up move so that we can conclude with those values suggested first by Cox, Ross and Rubinstein [CRR79], the expected log return and variance can be expressed as
EŒln.ST=S/!T; varŒln.ST=S/!2T:
Note that one can come up with different parameterizations to satisfy the same expressions. Then we can use the normality of ln.ST=S/to calculate the expectation under different probabilities that will give the corresponding values ofd1 andd2. We know that if ln.ST=S/is normally distributed thanST=S will be log-normally distributed and we have the following relation for log-normal distributions:
ln.E.ST=S//DE.ln.ST=S//C 1
2var.ln.ST=S// : (8.16) By using (8.16) and the values of the probabilities we find for the probability??
EŒln.ST=S/DrTC2T 2 and for?
EŒln.ST=S/DrT2T 2 :
Hence, we finally reach the original values of the Black-Scholes model. The value corresponding to probability??is
d1D zD ln.S=K/CEŒln.ST=S/
pvarŒln.ST=S/ D ln.S=K/CrTC2T=2 p
T and the value corresponding to the probability?is
d2D zD ln.S=K/CrTC2T=2 p
T :
Thus, the formula (8.11) of the binomial model reduces to C0DS.d1/KerT.d2/;
asn! 1, where we used that
n!1lim
1CrT n
n
D lim
n!1erT DRf 1: