4.2 Constructing the State Space
4.2.1 Term Structure Adjustments
pn,2nλn,2n for i = 2n + 2
pn,2n−1λn,2n−1+ (1 − pn,2n− qn,2n) λn,2n for i = 2n + 1 pn,i−2λn,i−2+ (1 − pn,i−1− qn,i−1) λn,i−1+ qn,iλn,i for 2 ≤ i ≤ 2n (1 − pn,0− qn,0) λn,0+ qn,1λn,1 for i = 1
qn,0λn,0 for i = 0
(4.9)
4.2 Constructing the State Space
The choice of a trinomial scheme provides an additional degree of freedom which allows us significant freedom in choosing the state space. Depending on the relationship between implied volatility, strike and time to expiration, the choice of state space may vary from being regular to being skewed. Uniform mesh sizes are generally adequate when the implied volatility varies quite slowly. If it varies significantly with strike or time to maturity, it may be necessary to choose a node spacing that changes accordingly and is skewed. Negative transition probabilities can be avoided by selecting node spacing that incorporates the skew evident in the market prices at each maturity.
Our strategy will be to first generate a regular trinomial lattice, assuming interest rates and dividend yields are zero. This translates into a constant time spacing ∆t and logarithmic mesh spacing ∆S. Then we modify ∆t or ∆S at different time and stock levels to capture the basic term- and skew-structures of local volatility in the market.
In certain cases, it may not be possible to avoid negative probabilities, even if the forward value at a particular node lies between Sn+1,i and Sn+1,i+2. In such cases, the option price that produces the negative probability can be overwritten. The implied tree will not fit all option price data but will necessarily fit the forward prices and hence provide transition probabilities that are in the correct range.
4.2.1 Term Structure Adjustments
First consider the case when there is a significant term structure of implied volatility but no skew structure.
The local volatility is a function of time, σ(t). For some constant c, rubber time ˜t is implicitly defined by
t = c Z ˜t
0
σ2(u)du, (4.10)
where σ(u) is the instantaneous (local) volatility at time u. There is no skew structure.
If, for example, we have that σ2(u) = a + bu, where a and b are positive constants, then
t = c Z ˜t
0
(a + bu) du
= c£
au + 12bu2¤t˜ 0
= ca˜t+ 12cb˜t2
⇒ ˜t = −ca +√
c2a2+ 2cbt cb
Alternatively, if
σ(u) =
( σ1 if u ≤ u1
σ2 if u > u1
as in raw interpolation of yield curves. Then t = c˜tσ12if ˜t ≤ u1. So,
˜t = t cσ12. If ˜t = u1, then t = cu1σ21. Lastly, if ˜t > u1, then
t = c ÃZ u1
0
σ2(u)du + Z ˜t
u1
σ2(u)du
!
= cu1σ21+¡
˜t− u1¢ σ22
⇒ ˜t = t − cu1σ12 σ22 + u1.
Using rubber time as opposed to standard time transforms the evolution process into a constant volatility process. This can be shown by defining a new stock price variable ˜S by ˜S(t) := S(˜t) and a new Brownian motion ˜Z by
Z(t) = ˜˜ Z Ã
c Z ˜t
0
σ2(u)du
! :=√
c Z ˜t
0
σ(u)dZ(u), (4.11)
for some constant c; we will make a convenient choice later. So,
d ˜Z (t) =√
cσ(˜t)dZ(˜t) (4.12)
We need to verify that ˜Z(t) is indeed a Brownian motion.
Suppose we have a probability triple (Ω, F, P). Recall the definition of Brownian motion.
Definition 1 (Rogers & Williams 2000) A real-valued stochastic process {Wt: t ∈ R+} is a Brownian motion if it has the properties
(i) W0= 0, ∀ ω;
(ii) t 7−→ Wt(ω) is a continuous function of t ∈ R+, ∀ ω;
(iii) For every t, h ≥ 0, Wt+h− Wtis independent of {Wu: 0 ≤ u ≤ t}, and has a Gaussian distribution with mean 0 and variance h.
In (4.12), it is clear ˜t exists and is unique.
Clearly (i) and (ii) are satisfied. For the distributional properties:
E
This is a result of the martingale property of the Itˆo integral.
V
The second line follows from the Itˆo isometry (Oksendal 2004, §3.1.5).
For the independent increments: let 0 ≤ τ1≤ τ2≤ τ3≤ τ4and consider
This is as a result of the linearity of E [·], Fubini’s Theorem (Rogers & Williams 2000, §II.12) and the fact that Z(t) is a standard Brownian motion so
E [Z(u)Z(v)] = 0
Using the definition of scaled time and (4.11), d ˜S(t)
S(t)˜ = d¡ S(˜t)¢ S(˜t)
= . . . + σ(˜t)d¡ Z(˜t)¢
= 1
√cd ˜Z(t),
by (4.12). Hence, the new stock price variable has a constant volatility of √1c.
Now c is chosen to ensure that the rescaled and standard times coincide at a fixed future time (usually the last maturity of the input data), that is we want T = ˜t(T ). Thus using (4.10)
c = T ,Z T
0
σ2(u)du (4.14)
In the trinomial tree with N known equally-spaced time points 0 = t0, t1, . . . , tN = T , the requirement is to find the unknown scaled time points 0 = ˜t0, ˜t1, . . . , ˜tN = T such that σ(˜tk)2∆˜tk is a constant for all times tk. This ensures the tree will recombine. (Derman, Kani & Chriss 1996) show that this can be done by solving for 1 ≤ k ≤ N :
˜tk= TPk
i=1 1 σ2(˜ti)
PN
i=1 1 σ2(˜ti)
(4.15)
The formula for the term structure (4.15) is implicit and hence quite difficult to implement. We now derive an alternative iterative scheme which will enable all scaled times to be determined explicitly.
The notation for the remainder of this section for variance is as follows:
• σI2(t): Implied Black-Scholes variance for an option with maturity t.
• σf2(t): Forward variance which will be defined below.
• σl2(t): Local variance as a function of time.
It will be required that for 1 ≤ k ≤ N
ˆc ≡ Z ˜tk
t˜k−1
σl2(˜t(s))ds
is independent of t, for some new constant ˆc; again, we will choose this in due course.
Since there is no strike structure, the local variance reduces to the forward Black-Scholes implied variance (Gatheral 2004, §2.4).
So,
Z t˜k
˜tk−1
σ2f(˜t(s))ds = ˆc (4.16)
where ˆc is a constant.
The implied forward variance at time 0 between t and t + ∆t is given by σ2f(0; t, t + ∆t) = σI2(t + ∆t)(t + ∆t) − σI2(t)t
∆t Taking the limit as ∆t → 0
σ2f(t) = d dsσ2I(s)s
¯¯
¯¯
s=t
= d
dtσ2I(t)t (4.17)
Then
N ˆc = XN k=1
Z t˜k
˜tk−1
σ2f¡
˜t(s)¢ ds =
Z T
0
σ2f¡
˜t¢ d˜t
= Z T
0
σ2f(t)dt
= Z T
0
µd dtσI2(t)t
¶ dt
= σ2I(T )T.
But,
Z T
0
σf2(t)dt = Z T
0
σf2(˜t)d˜t
= Z T
0
1 cdt
= T c. So we have that
ˆc = σI2(T ) · T N, c = 1
σI2(T ).
Given σI(t1) and σI(t2) at maturities t1 and t2 respectively, it is the case that σI2(t1)t1 < σI2(t2)t2. This must be true to ensure the forward ATM implied volatility between t1 and t2 is always positive.
Performing linear interpolation on σI(t) or σI2(t) does not always give rise to positive forward volatilities.
The problem is analogous to that of yield curve interpolation where the interpolation method must be carefully chosen to ensure that forward rates cannot be negative (Hagan & West 2005). So, σ2f(t) ↔ f (t) and σ2I(t) ↔ r(t), where r(t) is the risk free yield-to-maturity of a discount instrument maturing at time t and f (t) is the instantaneous forward rate. The relationship between r(t) and f (t) is
f (t) = d dtr(t)t which generalizes to (4.17). So,
r(t) =1 t
Z t
0
f (s)ds
and
σI2(t) = 1 t
Z t
0
σf2(s)ds (4.18)
Given two points, r(tj−1) and r(tj), the discrete forward rate that is applicable between tj−1and tj, fd,j
is given by
fd,j= r(tj)tj− r(tj−1)tj−1
tj− tj−1
and the discrete forward implied volatility applicable between tj−1and tj, σd,f ;j, is given by
σd,f ;j= s
σI2(tj)tj− σI2(tj−1)tj−1
tj− tj−1 (4.19)
Two satisfactory methods prescribed for yield curve interpolation are raw and monotone convex interpo-lation (Hagan & West 2005). The raw interpointerpo-lation method is selected here, as it is by definition that method which has piecewise constant forward curves, which enables us to find closed-form solutions for the ATM volatility. The interpolation function for σI2(t) turns out to be σ2I(t) = B +Ct where B and C are to be derived below. Given σI(t1) and σI(t2) (two endpoints), the discrete forward implied volatility σd,f ;2 is calculated using (4.19). The implied volatility at any time t ∈ [t1, t2) can be found using (4.18):
σI2(t)t = Z t
0
σ2f(s)ds
= Z t1
0
σ2f(s)ds + Z t
t1
σf2(s)ds
= σI2(t1)t1+ (t − t1)σd,f ;22 since the interpolation method is raw. Therefore,
σI(t)
= r
σ2I(t1)t1
t +(t − t1) t σd,f ;22
= s
σ2I(t1)t1
t +(t − t1) t
µσ2I(t2)t2− σI2(t1)t1
t2− t1
¶
= s
σ2I(t1)t1t2− t21σ2I(t1) + t (σI2(t2)t2− σI2(t1)t1) − σ2I(t2)t1t2− σ2I(t1)t21 t (t2− t1)
= s
σ2I(t2)t2− σI2(t1)t1
t2− t1 +t1t2(σI2(t1) − σI2(t2)) t (t2− t1) :=
r B + C
t (4.20)
Thus, the implied ATM volatilities for all times between any two points can be found using this method of interpolation. This method guarantees that the ATM forward volatilities are positive.