It is well known that market option prices are not consistent with theoretical prices derived from the Black–Scholes formula. This has led traders to quote option prices in terms of a volatility,imp, which makes the Black–Scholes formula value equal to the observed market price. Here we refer toimpas the implied volatility, to distinguish it from the theoretical constant volatility; essentiallyimpis another way of quoting option prices. Empirical studies have found that:
. For vanilla options of a given maturity the value ofimpdecreases with the level of the strike price, this asymmetry is termedvolatility skew.
. For vanilla options of a given strike price the value ofimpincreases with maturity, this variation is called thevolatility term structure.
Here we follow Derman and Kani (1994) and refer to both the volatility skew and the volatility term structure as thevolatility smile. The precise shape and magnitude of the volatility smile is dependent on the nature of the option being considered.
We are thus led to consider more sophisticated option pricing methods which capture the observed deviations from these Black–Scholes formula.
Instead of assuming, as in Section 8.3, that the underlying asset priceStfollows GBM with constant drift and volatility, we will now consider the more general GBM process:
dSt
St
¼ðtÞdtþðSt; tÞdZ ð10:98Þ
wheretis the current time,(t) is the time dependent risk neutral drift and(St,t) is an unknown volatility function which depends on both the stock price and time. If we make use of Ito’s lemma, and writeStþt for the asset price at timetþt, Equation 10.84 can be expressed in discretized form as:
log Stþt St
¼ ðtÞ 2ðSt; tÞ=2
tþðSt; tÞdZ ð10:99Þ or equivalently
log Stþt St
N ðtÞ 2ðSt; tÞ=2
t; 2ðSt; tÞt
ð10:100Þ In this section we will show how the volatility function(St,t) can be evaluated by ensuring that the option prices calculated using this model agree with those of the smile.
Theimpliedbinomial lattice constructed using this extended model will no longer be a regular lattice (as is the case for the simple Black–Scholes model) but will have a distorted shapesimilarto that shown in Figure 10.5 below.
It can be seen that the lattice levels are equispaced in time and aretapart. Lattice level 1, timet1, corresponds to the root node (1, 1) and is the current time, at which Numeric methods and single asset American options 159
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we want to find the value of the option. Time tn, associated with lattice level n, is given by:
tn¼t1þ ðn1Þt¼tþ ðn1Þt
sotnis (n1)tin the future relative to the current timet.
Construction of the implied lattice requires option prices for the complete range of strikes and maturities; these values can be obtained via interpolation from known option prices that are traded on the stock market.
Once the implied lattice has been created it can be used to price a range of European and American options.
Here we will describe the implied lattice technique developed by Derman and Kani (1994), and then consider the subsequent refinements proposed by Barle and Cakici (1995) and Chriss (1997). Of necessity our description of these techniques will be brief, and will mainly consist of explanatory detail and mathematical proofs that are not given in the original papers. For more information the reader should consult the original papers which are available (by kind permission of RISK Magazine) on the CD ROM which accompanies this book.
(7,6) (3,3)
(5,2) (1,1)
S
(7,1)
1 2 3 4 5 6 7
Lattice level (n)
0 1 2 3 4 5 6
Time (in units of ∆t)
Asset price
Figure 10.5 An implied binomial lattice which incorporates the volatility smile observed in traded put and call options. Theith node at thenth lattice level is denoted by (n,i). The current value (at timet) of the underlying asset isS, and this is the asset value assigned to the root lattice node (1, 1). The asset values at
the other lattice nodes depend on the technique used to construct the implied lattice
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Before discussing the details of implied binomial lattices we will first consider the local volatility associated with a particular lattice node.
Local volatility
An expression for the stock volatility at the binomial lattice node (n,i) will now be derived. At time instanttnthe stock value at this node is denoted bysi. After timet, time instant tnþ1, the stock price either has jumped up to Su, at lattice node (nþ1,iþ1), or jumped down toSd, at lattice node (nþ1,i). Applying Equation 10.100 to node (n,i) by settingt¼tnandSt¼sithen gives
vN ðtnÞ 2ðsi; tnÞ=2
t; 2ðsi; tnÞt
ð10:101Þ where the variatevcan only take the two valuesv1 ¼log(Su=si), andv2¼log(Sd=si).
We will letpidenote the probability of taking the valuev1, corresponding to an up jump. The probability of vhaving the value v2, corresponding to a down jump, is thus 1pi.
The quantity(si,tn) will be referred to as the local volatility,loc, associated with the lattice node (n,i), and using Equation 10.101 we can write
VarðvÞ ¼2loct ð10:102Þ
An expression forloccan then be obtained in terms ofSu,Sd andpi, as follows.
The variance ofvis:
VarðvÞ ¼E½v2 ðE½vÞ2 where
E½v2 ¼pi log Su
si
2
þð1piÞ log Sd
si
2
and
ðE½vÞ2¼ pilog Su
si
þ ð1piÞlog Sd
si
2
which means that ðE½vÞ2¼p2i log Su
si
2
þ ð1piÞ2 log Sd
si
2
þ2pið1piÞlog Su
si
log Sd
si
We can therefore write the variance as VarðvÞ ¼ pi log Su
si
2
þ ð1piÞ log Sd
si
2
p2i log Su
si
2
ð1piÞ2 log Sd
si
2
2pið1piÞlog Su
si
log Sd
si
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which simplifies to
VarðvÞ ¼pið1piÞ log Su
si
2
þ log Sd
si
2
2 log Su
si
log Sd
si
" #
ð10:103Þ However
log Su
si
log Sd
si
¼log Su
Sd
and
log Su
si
log Sd
si
2
¼ log Su
si
2
þ log Sd
si
2
2 log Su
si
log Sd
si
Substituting this into Equation 10.103 we obtain:
VarðvÞ ¼pið1piÞ log Su Sd
2
ð10:104Þ Therefore combining Equations 10.102 and 10.104 we have
2loct¼pið1piÞ log Su
Sd
2
and so the local volatility is given by:
Binomial lattice: local volatility loc¼log Su
Sd
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pið1piÞ
t r
ð10:105Þ
In an implied lattice the transition probabilities, pi, and the ratios Su=Sd are (in general) different for each lattice node. This generates a volatility surface in which the local volatility loc varies throughout the lattice. By contrast the CRR binomial lattice of Section 10.4.1 has the same value oflocfor all its lattice nodes. The reason for this thatpi andtare constants, and the up and down jumps are
Su¼siu and Sd¼sid; where u¼1 d This means that
Su
Sd
¼u2
and the (constant) local volatility is loc¼logðu2Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1pÞ
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CRR binomial lattice: local volatility loc¼2 logðuÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1pÞ
t r
ð10:106Þ where we have denoted the (constant) CRR up jump probability byp.
10.5.1 Derman–Kani implied lattice
We now consider the paper by Derman–Kani (1994), henceforth referred to as DK, which describes an implied binomial lattice based on the market values of European put and call options.
The implied lattice (see Figure 10.5) consists of uniformly spaced levelstapart, and is built using forward iteration. To explain this technique we will assume that the firstn lattice levels have been constructed and that they match the observed volatility smile for all strike prices and maturities out to timetn. The task is to determine thenþ1 nodes at the (nþ1)th lattice level from the previously calculatednnodes at thenth lattice level.
For convenience we will now give the notation used in the formulae for construct- ing the lattice nodes in the (nþ1)thlattice level from the known lattice node values in thenth lattice level.
r The known riskless interest rate for lattice level (nþ1).
si The known stock price at node (n,i); that is at theith node on lattice level n.
We also note thatsiis the strike price for options expiring at lattice levelnþ1.
Fi The known forward price at lattice levelnþ1 of the known pricesi at lattice leveln.
Si The unknown stock price at node (nþ1,i).
i The known Arrow-Debreu price at node (n,i).
pi The unknown risk-neutral up jump transition probability from node (n,i) to node (nþ1,iþ1).
Here theith node at levelnhas known stock price si, and is denoted by (n,i). The probability that the stock pricesiincreases toSiþ1in lattice levelnþ1 is denoted bypi, whereas the probability that the stock price decreases toSiin levelnþ1 is given by 1pi. The forward price,Fi, ofsiat lattice levelnþ1 is simply given by the risk neutral expected value ofsione time step later. That isFi¼siexp (rt), or in terms of the up and down jump probabilitiespiand (1pi) respectively we have:
Fi¼piSiþ1þ ð1piÞSi; fori¼1;. . .;n ð10:107Þ where as beforeSiþ1is the stock value at lattice levelnþ1 following an up jump and Si is the stock value at lattice levelnþ1 following a down jump.
The Arrow-Debreu price,i, at each lattice node (n,i) is defined as:the probability of reaching node (n,i)from the root lattice node (1, 1)discounted by the risk neutral interest rate between time t1and time tn.
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The Arrow-Debreu price of a lattice node is thus the value of a security that pays
$1 if the stock price reaches that node and zero otherwise. The value of i corresponding to node (n,i) is computed as the sum, over all paths from the root node (1, 1) to node (n,i), of the product of the riskless-discounted transition prob- abilities of nodes along each path from (1, 1) to (n,i). We provide more detail concerning the computation of i in the example calculation at the end of this section, and consider the following two methods:
1. Direct calculation of the Arrow-Debreu prices in lattice level nþ1 by using all paths from the root lattice node (1, 1).
2. Iterative calculation of the Arrow-Debreu prices in lattice level nþ1 from the known Arrow-Debreu prices in lattice leveln.
It is shown that direct calculation of the Arrow-Debreu prices becomes substantially more complicated as the number of lattice level increases. This is because the number of possible paths from the root node (1, 1) to any given lattice node (nþ1,i) increases dramatically with n. The iterative approach is thus the most practical method for computing Arrow-Debreu prices in lattices containing more than just a few lattice levels.
LetC(K,tnþ1) andP(K,tnþ1) be the current, timet, respective prices of European call and European put options with strike Kand maturity corresponding to lattice level nþ1; the values C(K,tnþ1) andP(K,tnþ1) can be obtained via interpolation from the known market prices. An expression forC(K,tnþ1), can also be computed by using the binomial node values at lattice level n, and this method yields the following equation:
CðK; tnþ1Þ ¼expðrtÞXn
j¼1
jfpjmaxðSjþ1K; 0Þ
þ ð1pjÞmaxðSjK; 0Þg ð10:108Þ where max (SjK, 0) is the payout for the call at thejth lattice node on lattice level nþ1 and max (Sjþ1K, 0) is the payout for the call at the (jþ1)th lattice node on lattice levelnþ1.
When the strikeKequalssithe above equation becomes expðrtÞCðsi; tnþ1Þ ¼Xn
j¼1
jfpjmaxðSjþ1si; 0Þ
þ ð1pjÞmaxðSjsi; 0Þg ð10:109Þ Since the terms that contribute to the value of the call option,C(si,tnþ1), are those with positive payouts we only need considerjindices in the rangeiton, and theith term of the summation on the right hand side of Equation 10.109 is:
ifpimaxðSiþ1si; 0Þ þ ð1piÞmaxðSisi; 0Þg ¼ipiðSiþ1siÞ ð10:110Þ where we have used (see DK Figure 4) the following:Siþ1>si(Siþ1 is the up jump stock value from lattice levelnto lattice levelnþ1) whereasSi<si (Siis the down jump stock value from lattice levelnto lattice levelnþ1).
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This means that we can rewrite Equation 10.109 as:
expðrtÞCðsi; tnþ1Þ ¼ipiðSiþ1siÞ þ Xn
j¼iþ1
jfpjðSjþ1siÞ
þ ð1pjÞðSjsiÞg ð10:111Þ If we subtract the constant termsifrom both sides of Equation 10.107 we obtain:
Fjsi¼pjðSjþ1siÞ þ ð1pjÞðSjsiÞ;j¼1; . . .;n ð10:112Þ where we used si¼pjsiþ(1pj)si. Substituting Equation 10.112 into Equation 10.111 gives:
expðrtÞCðsi; tnþ1Þ ¼ipiðSiþ1siÞ þ ð10:113Þ where ¼Pn
j¼iþ1j(Fjsi). The first term in Equation 10.113 depends on the
unknown values of the transition probability pi and stock price Siþ1. T he last term involves a summation over the known forward pricesFj and known stock prices si on lattice level n. Since both Fj and C(si,tnþ1) are known, Equations 10.107 and 10.113 can be solved to give the following expressions for Siþ1 andpi, in terms of Si:
pi¼ FiSi
Siþ1Si
ð10:114Þ and
Siþ1¼Si Cðsi; tnþ1ÞexpðrtÞ
isiðFiSiÞ Cðsi; tnþ1ÞexpðrtÞ
iðFiSiÞ ð10:115Þ We will now derive these two results.
Proof of Equation 10.114 (DK equation 7) From Equation 10.107 we have:
Fi¼piSiþ1þ ð1piÞSi
which gives:
Fi¼piðSiþ1SiÞ þSi and pi¼ FiSi
Siþ1Si
QED
Proof of Equation 10.115 (DK equation 6)
If we substitute the value ofpifrom Equation 10.115 into Equation 10.113 we obtain:
Cðsi; tnþ1ÞexpðrtÞ ¼iðFiSiÞðSiþ1siÞ Siþ1Si
þ
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Multiplying both sides bySiþ1Siyields:
expðrtÞCðsi; tnþ1ÞfSiþ1Sig ¼ iFisiiSiSiþ1þiFiSiþ1þiSiþSiþ1Si so
Siþ1 Cðsi; tnþ1ÞexpðrtÞ þiSiiFi
¼SiCðsi; tnþ1ÞexpðrtÞ isiðFiSiÞ Si or
Siþ1 Cðsi; tnþ1ÞexpðrtÞ iðFiSiÞ
¼Si Cðsi; tnþ1ÞexpðrtÞ
isiðFiSiÞ and finally gives the following expression forSiþ1:
Siþ1¼Si Cðsi; tnþ1ÞexpðrtÞ
isiðFiSiÞ Cðsi; tnþ1ÞexpðrtÞ
iðFiSiÞ QED
If we knowSiat one initial node then Equations 10.114 and 10.115 can be used to find iteratively the values ofSiþ1andpifor alln=2þ1 nodes above the centre of the lattice on the (nþ1)th lattice level.
Ifnþ1 is odd then the initial value used forSiis the stock value associated with the central lattice node, that isSi¼S. On the other hand ifnþ1 is even then we use the CRR lattice centering condition (see Section 10.4.1). Let Siþ1 denote the (nþ1)th level stock value for the node just above the centre of the lattice, and Si denote the (nþ1)th level stock value just below the centre of the lattice. For a CRR (u¼1=d) lattice these values are related to the central node stock value,S, at lattice leveln by:
Siþ1¼Su and Si¼Sd¼S u and therefore
Siþ1Si¼S2 ð10:116Þ
Substituting Equation 10.116 into Equation 10.114 gives the following formula for the stock value at the node just above the centre of the lattice whennþ1 is even:
Siþ1¼SðþiSÞ
iFi ; for i¼ nþ1 2
ð10:117Þ where¼C(S,tnþ1) exp (rt).
Whennþ1 is even, Equation 10.117 can thus be used in conjunction with Equa- tions 10.114 and 10.115 to iteratively compute the node valuesSiþ1and probabilities pifor the (nþ1)=2 nodes above the lattice centre.
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Proof of Equation 10.117 (DK equation 8)
From Equation 10.114 we have that the probabilitypiis given by:
pi¼ FiSi
Siþ1Si
¼ ðFiSiÞSiþ1
ðSiþ1SiÞSiþ1
since, in Equation 10.116,SiSiþ1¼S2 we obtain:
pi¼FiSiþ1S2
S2iþ1S2 ¼ FiSiþ1S2 ðSiþ1SÞðSiþ1þSÞ However from Equation 10.113
expðrtÞCðsi; tnþ1Þ ¼ipiðSiþ1siÞ þ so
¼ipiðSiþ1siÞ ð10:118Þ
Whensi¼Swe therefore have:
¼ipiðSiþ1SÞ ¼iðFiSiþ1S2ÞðSiþ1SÞ
ðSiþ1SÞðSiþ1þSÞ ¼iðFiSiþ1S2Þ Siþ1þS which gives:
Siþ1þS¼iFiSiþ1iS2 Siþ1ðiFiÞ ¼ SðþiSÞ and finally:
Siþ1¼SðþiSÞ
iFi QED
Similar formulae can be derived, using interpolated put prices, which enable the stock values and probabilities for nodes below the lattice centre to be computed. The formula to determine a lower node’s stock value from an upper node’s stock value is:
Si¼Siþ1fPðsi; tnþ1ÞexpðrtÞ g isiðFiSiþ1Þ Pðsi; tnþ1ÞexpðrtÞ
f g iðFiSiþ1Þ ð10:119Þ whereP(si,tnþ1) is the interpolated price of a put with strikesiand expiry timetnþ1
and
¼Xi1
j¼1
jðsiFjÞ
denotes the sum over all nodes below the node with stock pricesi at which the put was struck.
When building the lattice the computed transition probabilities,pi, for each lattice node must obey the constraint 0pi1. The upper limit pi1 is equivalent to Numeric methods and single asset American options 167
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requiring that the up-node stock price Siþ1 at the next level does not fall below the forward priceFi. This result comes from Equation 10.114
pi¼ FiSi
Siþ1Si
where it can easily be seen that if Fi>Siþ1 then pi>1. Similarly the lower limit pi0 can be shown to be equivalent to requiring that the down-node stock priceSiis above the forward priceFi. From Equation 10.114 we now have:
piþ1¼Fiþ1Siþ1 Siþ2Siþ1
and so ifSiþ1>Fiþ1 thenpiþ1<0. We thus have:
Fi<Siþ1<Fiþ1 ð10:120Þ which is illustrated in Figure 10.6.
Si+1 Si+1
Fi+1 pi+1
pi
Si+2
pi–1
Si–1
Fi–1 Si
Fi
Si
Figure 10.6 An implied lattice showing the position of the stock prices in relation to the forward prices, between thenth and (nþ1)th lattice levels. The stock prices in lattice levelnare denoted bysiþ1,si, and si1, while those in lattice levelnþ1 are represented bySiþ2,Siþ1, andSi. The transition probabilities between lattice levelnand lattice levelnþ1 arepiþ1,pi, andpi1, and the forward prices areFiþ1,Fi, and
Fi1. If the computed stock value isSiþ1then, in order to obtain valid transition probabilities, it must satisfy the constraintFi<Siþ1<Fiþ1
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Bad probabilities
Figure 10.6 shows the relative positions of the computed stock values,Si, at lattice levelnþ1, and the forward pricesFi computed from the stock values,si, at lattice leveln. If a computed stock value,Siþ1, violates the constraints imposed by Equation 10.120 then it is necessary to choose an alternative value for which the transition probabilitypi is in the permitted range 0<pi<1. DK advocates choosing Siþ1 so that the logarithmic spacing between adjacent lattice nodes is the same as that in the previous lattice level; that is:
Siþ1 Si
¼ si
si1
This means replacing the value ofSiþ1 computed using Equation 10.115 with Siþ1¼Si
si si1
ð10:121Þ If this method still fails to produce a validpithen Chriss (1997) suggests the following more drastic measure in which
Siþ1¼Fiþ ð10:122Þ
whereis a very small number (say 106). It can be seen from Equation 10.114 that the transition probabilitypiwill then be a very small positive number.
When we remove bad probabilities in this manner the impact on the implied lattice will depend on both the Arrow-Debreu price of the node and its payout. Nodes near the top and bottom of the lattice will have small Arrow-Debreu prices because few paths lead to them, and thus removing bad probabilities from these nodes will have little impact on the lattice. When building an implied lattice it is a good idea to count how many bad nodes have been encountered; this will give some idea of the expected quality of the implied lattice that has been constructed. A more quantitative method of assessing the expected performance of an implied lattice is by checking how well it prices the put and call options that were originally used to create it.
Example calculation
Here we provide more details concerning the example calculation given in the paper by Derman and Kani (1994). The implied lattice for this example is shown in Figures 10.7 and 10.8. It is assumed that the current stock value is 100.00, the dividend is zero, and the annually compounded riskless interest rate is 3 per cent a year for all option maturities. Since we have assumed a constant riskless interest of 3 per cent the forward priceFifor any node is 1.03 times the node’s stock price,si.
Computation of the Arrow-Debreu prices
We have already mentioned that the Arrow-Debreu price for node (n,i) is computed as the sum, over all paths from the root node (1, 1) to node (n,i), of the product of the riskless-discounted transition probabilities of nodes along each path from (1, 1) Numeric methods and single asset American options 169