6.3 Implied Trinomial Trees


6.3.1 Basic Insight

A trinomial tree with $ N$ levels is a set of nodes $ s_{n,i}$ (representing the underlying price), where $ n=1,\ldots,N$ is the level number and $ i=1,\ldots,2n-1$ indexes nodes within a level. Being at a node $ s_{n,i}$, one can move to one of three nodes (see Figure 6.4a): (i) to the upper node with value $ s_{n+1,i}$ with probability $ p_i$; (ii) to the lower node with value $ s_{n+1,i+2}$ with probability $ q_i$; and (iii) to the middle node with value $ s_{n+1,i+1}$ with probability $ 1-p_i-q_i$. For the sake of brevity, we omit the level index $ n$ from transition probabilities unless they refer to a specific level; that is, we write $ p_i$ and $ q_i$ instead of $ p_{n,i}$ and $ q_{n,i}$ unless the level has to be specified. Similarly, let us denote the nodes in the new level with capital letters: $ S_i$(=$ s_{n+1,i}$), $ S_{i+1}$(= $ s_{n+1,i+1}$) and $ S_{i+2}$(= $ s_{n+1,i+2}$), respectively (see Figure 6.4b).

Figure 6.4: Nodes in a trinomial tree. Left panel: a single node with its branches. Right panel: the nodes of two consecutive levels $ n-1$ and $ n$.

\includegraphics[width=1.3\defpicwidth]{ITT_fig07-generalTT.ps}

Starting from a node $ s_{n,i}$ at time $ t_n$, there are five unknown parameters: two transition probabilities $ p_{i}$ and $ q_{i}$ and three prices $ S_i$, $ S_{i+1}$, and $ S_{i+2}$ at new nodes. To determine them, we need to introduce the notation and main requirements a tree should satisfy. First, let $ F_i$ denote the known forward price of the spot price $ s_{n,i}$ and $ \lambda_{n,i}$ the known Arrow-Debreu price at node $ (n,i)$. The Arrow-Debreu prices for a trinomial tree can be obtained by the following iterative formulas:

$\displaystyle \lambda_{1,1}$ $\displaystyle =\!$ $\displaystyle 1,$ (6.6)
$\displaystyle \lambda_{n+1,1}$ $\displaystyle =\!$ $\displaystyle e^{-r\Delta t} \lambda_{n,1} p_{1},$ (6.7)
$\displaystyle \lambda_{n+1,2}$ $\displaystyle =\!$ $\displaystyle e^{-r\Delta t} \!\left\{ \lambda_{n,1}(1-p_{1}-q_{1}) + \lambda_{n,2} p_{2} \right\}\!,$ (6.8)
$\displaystyle \lambda_{n+1,i+1}$ $\displaystyle =\!$ $\displaystyle e^{-r\Delta t} \!\left\{ \lambda_{n,i-1}q_{i-1} \!+ \lambda_{n,i}(1-p_{i}-q_{i}) + \lambda_{n,i+1} p_{i+1} \right\}\!,$ (6.9)
$\displaystyle \lambda_{n+1,2n}$ $\displaystyle =\!$ $\displaystyle e^{-r\Delta t} \!\left\{ \lambda_{n,2n-1}(1-p_{2n-1}\!-q_{2n-1}) \!+\! \lambda_{n,2n-2} q_{2n-2} \right\}\!,$ (6.10)
$\displaystyle \lambda_{n+1,2n+1}$ $\displaystyle =\!$ $\displaystyle e^{-r\Delta t} \lambda_{n,2n-1}q_{2n-1}.$ (6.11)

An implied tree provides a discrete representation of the evolution process of underlying prices. To capture and model the underlying price correctly, we desire that an implied tree:

  1. reproduces correctly the volatility smile,
  2. is risk-neutral,
  3. uses transition probabilities from interval $ (0,1)$.
To fulfill the risk-neutrality condition, the expected value of the underlying price $ s_{n+1,i}$ in the following time period $ t_{n+1}$ has to equal its known forward price:

$\displaystyle \mathrm{E}s_{n+1,i} = p_i S_i + (1-p_i-q_i) S_{i+1} + q_i S_{i+2} = F_i = e^{r\Delta t}s_{n,i},$ (6.12)

where $ r$ denotes the continuous interest rate and $ \Delta t$ is the time step from $ t_n$ to $ t_{n+1}$. Additionally, one can specify such a condition also for the second moments of $ s_{n,i}$ and $ F_i$. Hence, one obtains a second constraint on the node prices and transition probabilities:

$\displaystyle p_i(S_i - F_i)^2 + (1-p_i-q_i)(S_{i+1} - F_i)^2 + q_i(S_{i+2} - F_i)^2 = F_i^2 \sigma_i^2 \Delta t + {\scriptstyle\cal O}(\Delta t),$ (6.13)

where $ \sigma_i$ is the stock or index price volatility during the time period.

Consequently, we have two constraints (6.12) and (6.13) for five unknown parameters, and therefore, there is no unique implied trinomial tree. On the other hand, all trees satisfying these constraints are equivalent in the sense that as the time spacing $ \Delta t$ tends to zero, all these trees converge to the same continous process. A common method for constructing an ITT is to choose first freely the underlying prices and then to solve equations (6.12) and (6.13) for the transition probabilities $ p_i$ and $ q_i$. Afterwards one only has to ensure that these probabilities do not violate the above mentioned Condition 3. Apparently, using an ITT instead of an IBT gives us additional degrees of freedom. This allows us to better fit the volatility smile, especially when inconsistent or arbitrage-violating market option prices make a consistent tree impossible. Note, however, that even though the constructed tree is consistent, other difficulties can arise when its local volatility and probability distributions are jagged and ``implausible.''


6.3.2 State Space

There are several methods we can use to construct an initial state space. Let us first discuss a construction of a constant-volatility trinomial tree, which forms a base for an implied trinomial tree. As already mentioned, binomial and trinomial discretization of the constant-volatility Black-Scholes model have the same continous limit, and therefore, are equivalent. Hence, we can start from a constant-volatility CRR binomial tree and then combine two steps of this tree into a single step of a new trinomial tree. This is illustrated in Figure 6.5, where thin lines correspond to the original binomial tree and the thick lines to the constructed trinomial tree.

Figure 6.5: Constructing a constant-volatility trinomial tree (thick lines) by combining two steps of a CRR binomial tree (thin lines).

\includegraphics[width=1\defpicwidth]{ITT_fig06-bi2tri.ps}

Consequently, using formulas (6.2) and (6.3), we can derive the following expressions for the nodes of the constructed trinomial tree:

$\displaystyle S_i$ $\displaystyle = s_{n+1,i}$ $\displaystyle = s_{n,i}\ e^{\sigma\sqrt{2\Delta t}},$ (6.14)
$\displaystyle S_{i+1}$ $\displaystyle = s_{n+1,i+1}$ $\displaystyle = s_{n,i},$ (6.15)
$\displaystyle S_{i+2}$ $\displaystyle = s_{n+1,i+2}$ $\displaystyle = s_{n,i}\ e^{-\sigma\sqrt{2\Delta t}},$ (6.16)

where $ \sigma$ is a constant volatility (e.g., an estimate of the at-the-money volatility at maturity $ T$). Next, summing the transition probabilities in the binomial tree given in (6.4), we can also derive the up and down transition probabilities in the trinomial tree (the ``middle'' transition probability is equal to $ 1-p_i-q_i$):
$\displaystyle p_i$ $\displaystyle =$ $\displaystyle \left( \frac{e^{r\Delta t / 2} - e^{-\sigma\sqrt{\Delta t / 2}}}
{e^{\sigma\sqrt{\Delta t / 2}} - e^{-\sigma\sqrt{\Delta t / 2}}} \right)^2,$  
$\displaystyle q_i$ $\displaystyle =$ $\displaystyle \left( \frac{e^{\sigma\sqrt{\Delta t / 2}} - e^{r\Delta t / 2}}
{e^{\sigma\sqrt{\Delta t / 2}} - e^{-\sigma\sqrt{\Delta t / 2}}} \right)^2.$  

Note that there are more methods for building a constant-volatility trinomial tree such as combining two steps of a Jarrow and Rudd's (1983) binomial tree; see Derman, Kani, and Chriss (1996) for more details.

When the implied volatility varies only slowly with strike and expiration, the regular state space with a uniform mesh size, as described above, is adequate for constructing ITT models. On the other hand, if the volatility varies significantly with strike or time to maturity, we should choose a state space reflecting these properties. Assuming that the volatility is separable in time and stock price, $ \sigma(S,t)=\sigma(S)\sigma(t)$, an ITT state space with a proper skew and term structure can be constructed in four steps.

First, we build a regular trinomial lattice with a constant time spacing $ \Delta t$ and a constant price spacing $ \Delta S$ as described above. Additionally, we assume that all interest rates and dividends are equal to zero.

Second, we modify $ \Delta t$ at different time points. Let us denote the original equally spaced time points $ t_0=0,t_1,\ldots,t_n=T$. We can then find the unknown scaled times $ \tilde{t}_0=0,\tilde{t}_1,\ldots,\tilde{t}_n=T$ by solving the following set of non-linear equations:

$\displaystyle \tilde{t}_k \sum_{i=1}^{n-1} \frac{1}{\sigma^2(\tilde{t}_i)} + \t...
...T)}= T \sum_{i=1}^{k} \frac{1}{\sigma^2(\tilde{t}_i)}, \qquad k = 1,\ldots,n-1.$ (6.17)

Next, we change $ \Delta S$ at different levels. Denoting by $ S_1,\ldots,S_{2n+1}$ the original (known) underlying prices, we solve for rescaled underlying prices $ \tilde{S}_1,\ldots,\tilde{S}_{2n+1}$ using

$\displaystyle \frac{\tilde{S}_k}{\tilde{S}_{k-1}}$ $\displaystyle =$ $\displaystyle \exp \left\{ \frac{c}{\sigma(S_k)} \ln \frac{S_k}{S_{k-1}} \right\}, \qquad k=2,\ldots,2n+1,$ (6.18)

where $ c$ is a constant. It is recommended to set $ c$ to an estimate of the local volatility. Since there are $ 2n$ equations for $ 2n+1$ unknown parameters, an additional equation is needed. Here we always suppose that the new central node equals the original central node: $ \tilde{S}_{n+1}=S_{n+1}$. See Derman, Kani, and Chriss (1996) for a more elaborate explanation of the theory behind equations (6.17) and (6.18).

Finally, one can increase all node prices by a sufficiently large growth factor, which removes forward prices violations, see Section 6.3.4. Multiplying all zero-rate node prices at time $ \tilde{t}_i$ by $ e^{r\tilde{t}_i}$ should be always sufficient.


6.3.3 Transition Probabilities

Once the state space of an ITT is fixed, we can compute the transition probabilities for all nodes $ (n,i)$ at each tree level $ n$. Let $ C(K,t_{n+1})$ and $ P(K,t_{n+1})$ denote today's price of a standard European call and put option, respectively, struck at $ K$ and expiring at $ t_{n+1}$. These values can be obtained by interpolating the smile surface at various strike and time points. The values of these options given by the trinomial tree are the discounted expectations of the pay-off functions: $ \max(S_j-K,0) = (S_j-K)^+$ for the call option and $ \max(K-S_j,0)$ for the put option at the node $ (n+1,j)$. The expectation is taken with respect to the probabilities of reaching each node, that is, with respect to transition probabilities:

$\displaystyle C\left(K, t_{n+1}\right)$ $\displaystyle =$ $\displaystyle e^{-r\Delta t} \sum_j
\left\{ p_{j} \lambda_{n,j} + (1-p_{j-1}-q_{j-1}) \lambda_{n,j-1} \right.$ (6.19)
    $\displaystyle \qquad \qquad \qquad ~ \left. + q_{j-2} \lambda_{n,j-2} \right\} (S_j - K)^+,$  


$\displaystyle P\left(K, t_{n+1}\right)$ $\displaystyle =$ $\displaystyle e^{-r\Delta t} \sum_j
\left\{ p_{j} \lambda_{n,j} + (1-p_{j-1}-q_{j-1}) \lambda_{n,j-1} \right.$ (6.20)
    $\displaystyle \qquad \qquad \qquad ~ \left. + q_{j-2} \lambda_{n,j-2} \right\} (K - S_j)^+.$  

If we set the strike price $ K$ to $ S_{i+1}$ (the stock price at node $ (n+1,i+1)$), rearrange the terms in the sum, and use equation (6.12), we can express the transition probabilities $ p_i$ and $ q_i$ for all nodes above the central node from formula (6.19):
$\displaystyle p_i$ $\displaystyle =$ $\displaystyle \frac{e^{r\Delta t} C(S_{i+1},t_{n+1}) - \sum_{j=1}^{i-1}
\lambda_{n+1,j}(F_j - S_{i+1})}{\lambda_{n+1,i}(S_{i}-S_{i+1})},$ (6.21)
$\displaystyle q_i$ $\displaystyle =$ $\displaystyle \frac{F_i - p_i(S_{i}-S_{i+1}) - S_{i+1}}{S_{i+2} - S_{i+1}}.$ (6.22)

Similarly, we compute from formula (6.20) the transition probabilities for all nodes below (and including) the center node $ (n+1,n)$ at time $ t_n$:
$\displaystyle q_i$ $\displaystyle =$ $\displaystyle \frac{e^{r\Delta t} P(S_{i+1},t_{n+1}) - \sum_{j=i+1}^{2n-1}
\lambda_{n+1,j}(S_{i+1} - F_j)}{\lambda_{n+1,i}(S_{i+1}-S_{i+2})},$ (6.23)
$\displaystyle p_i$ $\displaystyle =$ $\displaystyle \frac{F_i - q_i(S_{i+2}-S_{i+1}) - S_{i+1}}{S_{i} - S_{i+1}}.$ (6.24)

A detailed derivation of these formulas can be found in Komorád (2002). Finally, the implied local volatilities are approximated from equation (6.13):

$\displaystyle \sigma_i^2 \approx \frac{p_i(S_i - F_i)^2 + (1-p_i-q_i)(S_{i+1} - F_i)^2 + q_i(S_{i+2} - F_i)^2 } {F_i^2 \Delta t}.$ (6.25)

Figure 6.6: Two kinds of the forward price violation. Left panel: forward price outside the range of its daughter nodes. Right panel: sharp increase in option prices leading to an extreme local volatility.

\includegraphics[width=1.7\defpicwidth]{ITT_fig02-csgraph.ps}


6.3.4 Possible Pitfalls

Formulas (6.21)-(6.24) can unfortunately result in transition probabilities which are negative or greater than one. This is inconsistent with rational option prices and allows arbitrage. We actually have to face two forms of this problem, see Figure 6.6 for examples of such trees. First, we have to check that no forward price $ F_{n,i}$ at node $ (n,i)$ falls outside the range of its daughter nodes at the level $ n+1$: $ F_{n,i} \in (s_{n+1,i+2}, s_{n+1,i})$. This inconsistency is not difficult to overcome since we are free to choose the state space. Thus, we can overwrite the nodes causing this problem.

Second, extremely small or large values of option prices, which would imply an extreme value of local volatility, can also result in probabilities that are negative or larger than one. In such a case, we have to overwrite the option prices which led to the unacceptable probabilities. Fortunately, the transition probabilities can be always corrected providing that the corresponding state space does not violate the forward price condition $ F_{n,i} \in (s_{n+1,i+2}, s_{n+1,i})$. Derman, Kani, and Chriss (1996) proposed to reduce the troublesome nodes to binomial ones or to set

$\displaystyle p_i=\frac{1}{2}\left( \frac{F_i-S_{i+1}}{S_i-S_{i+1}} + \frac{F_i...
...i+2}} \right), \quad q_i=\frac{1}{2}\left( \frac{S_i-F_i}{S_i-S_{i+2}} \right),$ (6.26)

for $ F_i \in (S_{i+1}, S_i)$ and

$\displaystyle p_i=\frac{1}{2}\left( \frac{F_i-S_{i+2}}{S_i-S_{i+2}} \right), \q...
...eft( \frac{S_{i+1}-F_i}{S_{i+1}-S_{i+2}} + \frac{S_i-F_i}{S_i-S_{i+2}} \right),$ (6.27)

for $ F_i \in (S_{i+2}, S_{i+1})$. In both cases, the ``middle'' transition probability is equal to $ 1-p_i-q_i$.