4.4 Negative Transition Probabilities
4.6.2 Constructing the required state space
Case I: Normal State Space
In this case, there is no extreme term or skew structure associated with the local volatility function. An underlying constant-volatility trinomial tree will be appropriate for the determination of the transition probabilities and Arrow-Debreu prices. A single time step of the trinomial tree is constructed from the combination of two steps of the binomial Cox-Ross-Rubinstein tree.
Consider the Cox-Ross-Rubinstein constant volatility trinomial tree2:
1We will be pricing European and some path-dependent options on the ALSI40 equity index. Most relevant information is provided by implied skew data (from the South African Futures Exchange, SAFEX, or a dealer) on the futures contracts on this index that trade. It is also convenient for interpolation purposes as the ATM implied volatility can be found using raw interpolation on the relative strikes, this is X/F , where X is the strike and F is the ATM futures level.
2Since two steps of a binomial C-R-R tree equates to one step of the C-R-R trinomial tree, then Su2= SU and Sd2= SD, where u = eσ
q∆t
2 is the upward movement in a binomial tree of time step ∆t2 and d = e−σ
q∆t
2 the downward movement.
When one time step in the trinomial tree is ∆t, then U =
“ eσ√
∆t/2”2
= eσ
√2∆t, and similarly D = e−σ
√2∆t. Given that the risk-neutral probability of an upward movement in the binomial tree over a time step ∆t2 is given by
π =er∆t2 − d u − d
= er∆t2 − e−σ
q∆t 2
eσ
q∆t 2 − e−σ
q∆t 2
,
then in the trinomial tree of time step ∆, the probabilities associated with the up, down and middle movements are given by pU= p = π2, pD= q = (1 − π)2 and pM = 1 − pU− pD.
• Su= Seσ√2∆t
• Sm= S
• Sd= Se−σ√2∆t
• p =³
er∆t/2−e−σ√
∆t/2
eσ
√∆t/2−e−σ
√∆t/2
´2
• q =
³ eσ
√∆t/2−er∆t/2 eσ√
∆t/2−e−σ√
∆t/2
´2
where S is the spot price at the current time step, p and q represent the transition probabilities of an up and down movement respectively, σ is the constant volatility of an ATM option and Su, Sm and Sd
are the spot prices at the following time step. The movement is shown explicitly in Figure 4.2. The time steps ∆t = (Expiry Date − Valuation Date)/(N · 365).
The probabilities, as stated above, will not be used. The required transition probabilities, pn,i, qn,i and 1 − pn,i− qn,i, are what is required in the implied tree approach. The state space is a platform that can be selected using the additional degree of freedom. This is a result of selecting a trinomial as opposed to a binomial tree.
Consider the upper portion of the tree (from Sn+1,n+2 to Sn+1,2n+2) at t = ˜tn+1: (4.27) and (4.28) are used to calculate pn,iand qn,ifor n + 1 ≤ i ≤ 2n. For this part of the tree, the European call option prices are required. Since there is a term structure of implied volatility and the constructed state space has time intervals which will not always coincide with the input dates, it will be necessary to perform linear interpolation in the vertical direction (on the strikes) and raw interpolation in the horizontal direction (on the implied volatilities at a date that is not an input). See Figure 4.4. This is done to obtain the implied volatility at a non-input strike for an option expiring at a non-input date. To ensure that forward rates are always positive, the raw interpolation method is required.
Refer to Figure 4.4. To calculate the implied volatility, σI, for strike X and maturity ˜tn+1, linear interpolation is first performed on the implied volatilities σ1 and σ2(which relates to strikes X1and X2) at maturity tk to obtain σ12. The next linear interpolation is performed on σ3 and σ4 at maturity tk+1
to obtain σ34. These implied volatilities relate to the strike X at maturities tk and tk+1. Consider the calculation of σ12and σ34:
σ12= X − X1
X2− X1σ2+ X2− X X2− X1σ1
σ34= X − X3
X4− X3σ4+ X4− X X4− X3σ3
Then, σI at ˜tn+1is calculated using by (4.20):
σI = s
tktk+1(σ122 (tk) − σ342 (tk+1))
˜tn+1(tk+1− tk) +σ234(tk+1)tk+1− σ122 (tk)tk
tk+1− tk .
Once the implied volatility has been evaluated, either a constant volatility trinomial tree or the Black-Scholes formula can be used to price the option.
tk ˜tn+1 tk+1 Raw Interpolation
Linear Interpolation
X, σ12
b X, σ34
σI b c
X1, σ1
r X2, σ2
r
X3, σ3
r X4, σ4
r
Figure 4.4: Interpolating implied volatility, σI, for strike X at scaled time ˜tn+1
The lower portion of the tree (from the central node at ˜tn downwards) is analogous to the upper portion.
To determine the transition probabilities pn,i and qn,i for 0 ≤ i ≤ n, equations (4.31) and (4.30) can be used respectively. For this portion of the tree, the European put option prices are required and interpolation is performed on the implied volatility and maturity.
Case II: Term Structure
For construction of the state space, incorporating a term structure of implied volatility is done by applying the results of §4.2.1. The implied ATM volatility (the strike of the option is the futures or forward price) is the only input required for construction of the state space, as it is assumed that there is no strike structure. The resulting implied trinomial tree will have unequal time steps. So, σ2(˜tk)∆˜tk is to be a constant for 1 ≤ k ≤ N , where ˜tk refers to the scaled time and σ2(˜tk) refers to the local variance over
∆˜tk = ˜tk− ˜tk−1. The procedure is described as follows:
• Calculate or read the relative strikes X/F at each of the input dates tj for 1 ≤ j ≤ ˆN .
• Linearly interpolate X to find the implied ATM volatility at that time - require the implied volatility that corresponds to the value X/F = 1, where F is the forward/ futures level.
• Perform raw interpolation on the implied ATM volatilities between each time to obtain the forward implied ATM volatilities. These will be constant between any two input dates as a result of the raw interpolation method used.
Once all the forward implied ATM volatilities have been found, use (4.16) to solve for the scaled times by induction. The necessary calculations are simplified due to the forward implied ATM volatilities being constant.
Suppose the known scaled time, ˜tk−1, falls between tj−1and tj and σd,f ;j is the constant forward implied ATM volatility between times tj−1 and tj (4.5). In order to determine ˜tk, the induction requires a ’Do While’ loop and a variable, dlocalintegral, that is reset to 0 once each scaled time has been found or until (4.16) is satisfied. The search for ˜tk begins by testing whether the area given by (tj− ˜tk−1)σd,f ;j is greater than or smaller than ˆc. If it is greater than ˆc, then ˜tk< tj. If not, the variable, dlocalintergral, starting at 0, is incremented by this area and the search continues by checking whether dlocalintergral + (tj+1− tj)σd,f ;j+1 is greater then or less than ˆc. So, dlocalintegral is incremented by the discrete amounts until such time it is equivalent to ˆc. The discrete amounts are the areas given in general by σd,f ;j2 ∆t. The loop is only terminated if an increment to dlocalintegral results in a value that is greater than ˆc. The procedure to determine ˜tk can be summerized as follows:
dlocalintegral = 0
Do While dlocalintegral < ˆc
• dlocalintegral = dlocalintegral + σ2d,f ;j(tj− ˜tk−1) This brings ˜tk up to tj, j must be incremented to j + 1.
• Once again, the condition for dlocalintegral is checked.
If dlocalintegral > ˆc, then ˜tk< tj and
˜tk = ˆc − dlocalintegral
σ2d,f ;j + ˜tk−1 (4.33)
If dlocalintegral < ˆc, then dlocalintegral is incremented by σ2d,f ;j+1(tj+1−tj). In this case, the ’while’
loop continues until ˜tk falls between tm−1 and tmfor 1 ≤ m ≤ ˆN and use (4.33) to determine ˜tk. Once the scaled times have been solved for, they can be used in the calculation of the transition probabil-ities. The state space (stock price mesh) is constructed using a constant volatility recombining trinomial tree.
Since there is no strike structure, the implied volatilities of the ATM options for ˜tk will be used to price the options, either using the Black-Scholes formula or the trinomial tree constructed with the unequal time steps. These volatilities will be interpolated between input dates using raw interpolation to ensure σ2I(t)t > 0.
Case III: Skew Structure
We now apply the results of §4.2.2. In the case that the local volatility function is of the form σ(S), the state space is constructed to accommodate a linear relationship between implied volatility and strike. We assume that there is no term structure. The input requirement is
σf2(tj)
σf2(tj+1)
σf2(tj+2)
A B C
˜tk−1 tj tj+1 ˜tk
Figure 4.5: ˜tk is calculated inductively by ensuring integrals equate to ˆc, A + B + C = ˆc
1. The ATM implied volatility Σ0. Since it is assumed that there is no term structure, the value for Σ0 can be taken as an average of all the ATM implied volatilities.
2. The slope of the function (b): this is generally the percentage point increase in ATM implied volatility per point decrease in the strike of the option. As previously mentioned, this is the Taylor series expansion.
The procedure is to first construct the nodal prices, Sj,i, at time step j for 0 ≤ i ≤ 2j. These are then to be adjusted using (4.23), (4.24) and (4.25).
Once the trinomial tree (state space) has been completed, the requirement is to determine the transition probabilities pn,i and qn,i for 0 ≤ i ≤ 2n as well as the Arrow-Debreu prices λn+1,i for 0 ≤ i ≤ 2n + 2.
Since the scaled stock price is assumed to have a constant volatility, the input to create the tree will be some constant volatility, ΣAT M which will be adjusted by using the linear relationship (4.25). We are interested in relative changes, not absolute therefore, the futures level at each input will be interpolated to find the futures at the time indicated by the node tree. Once we have this value, to obtain the local volatility, (4.25) becomes:
σ = Σ0+ 2b
µ K
FAT M − 1
¶ .
This state space is then used to find all the transition probabilities and Arrow-Debreu prices using (4.9).
Case IV Both
The trinomial scheme can easily be constructed to accommodate both a strike and term structure. We begin by constructing a skewed state space as was described in the third case above. This is then stretched in time according to the second case above. This is the simplest and most tractable method to obtain a surface which depicts the observed volatility phenomena.
4.6.3 Non-Constant Time Intervals and a Dividend Yield
If it is the case that the input data (option expiry times) is not equally spaced, the resulting trinomial tree should display such a feature. The original Derman-Kani-Chriss algorithm can be altered to allow for such a modification in the case where the option prices are calculated using Black-Scholes, not a trinomial tree.
This is done in exactly the same manner as in Chapter 3, §3.7.1.
Chapter 5
Characterization of Local Volatility and the Dynamics of the Smile
5.1 Introduction
A risk-neutral diffusion process for the evolution of the underlying is proposed in (Dupire 1994):
dS
S = r(t)dt + σ(S, t)dW, (5.1)
where r(t) is the expected instantaneous stock price return and σ(S, t) is the local volatility function. W (t) is standard Brownian motion. Here the spot follows a one dimensional diffusion process, and so the model is complete (it allows for arbitrage pricing and hedging). Option prices can be calculated by discounting an expectation with respect to a risk-neutral probability, under which the discounted spot has no drift, but retains the same diffusion coefficient. In the case of European options, the expectation is taken over terminal values of the spot, while path-dependent options are priced as discounted expected values of the terminal payoff over all paths. Knowledge of the prices of path-dependent options is equivalent to knowledge of the full risk-neutral diffusion, while knowing the European option prices only amounts to knowledge of the spot distribution at the various option expiry times. The full diffusions contain more information than the conditional laws, as distinct diffusions may generate identical conditional laws. One attempts to choose the local volatility function σ(S, t) so as to have the model replicate the prices of European options (for various strikes and maturities) seen trading in the market. The more maturities we have, the closer we are to knowledge of the full risk-neutral diffusion.