6.4 Examples

To illustrate the construction of an implied trinomial tree and its use, we consider here ITTs for two artificial implied-volatility functions and an implied-volatility function constructed from real data.


6.4.1 Pre-specified Implied Volatility

Let us consider a case where the volatility varies only slowly with respect to the strike price and time to expiration (maturity). Assume that the current index level is 100 points, the annual riskless interest rate is $ r=12\%$, and the dividend yield equals $ \delta = 4\%$. The annualized Black-Scholes implied volatility is assumed to be $ \sigma =11\%$, and additionally, it increases (decreases) linearly by 10 basis points (i.e., 0.1%) with every 10 unit drop (rise) in the strike price $ K$; that is, $ \sigma_{I} = 0.11 - \Delta K * 0.001$. To keep the example simple, we consider three one-year steps.

Figure 6.7: The state space of a trinomial tree with constant volatility $ \sigma =11\%$. Nodes A and B are reference points for which we demonstrate constructing of an ITT and estimating of the implied local volatility.

\includegraphics[width=1.3\defpicwidth]{ITT_ex02stsp1.ps}

First, we construct the state space: a constant-volatility trinomial tree as described in Section 6.3.2. The first node at time $ t_0 = 0$, labeled $ A$ in Figure 6.7, has the value of $ s_A=100$, today's spot price. The next three nodes at time $ t_1$, are computed from equations (6.14)-(6.16) and take values $ S_{1}=116.83$, $ S_{2}=100.00$, and $ S_{3}=85.59$, respectively. In order to determine the transition probabilities, we need to know the price $ P(S_{2},t_1)$ of a put option struck at $ S_{2}=100$ and expiring one year from now. Since the implied volatility of this option is $ 11\%$, we calculate its price using a constant-volatility trinomial tree with the same state space and find it to be $ 0.987$ index points. Further, the forward price corresponding to node $ A$ is $ F_A = Se^{(r^*-\delta^*)\Delta
t}=107.69$, where $ r^* = \log(1+r)$ denotes the continuous interest rate and $ \delta^* = \log(1+\delta)$ the continuous dividend rate. Hence, the transition probability of a down movement computed from equation (6.23) is

$\displaystyle q_A = \frac{e^{\log(1+0.12) \cdot 1}0.987 - \Sigma}{1 \cdot (100.00-85.59)} = 0.077,$

where the summation term $ \Sigma$ in the numerator is zero because there are no nodes with price lower than $ S_{3}$ at time $ t_1$. Similarly, the transition probability of an upward movement $ p_A$ computed from equation (6.24) is

$\displaystyle p_A=\frac{107.69+0.077\cdot(100.00-85.59)-100}{116.83-100.00}=0.523.$

Finally, the middle transition probability equals $ 1-p_A-q_A=0.4$. As one can see from equations (6.6)-(6.11), the Arrow-Debreu prices turn out to be just discounted transition probabilities: $ \lambda_{1,1}=e^{-\log(1+0.12)\cdot 1}\cdot0.523=0.467$, $ \lambda_{1,2}=0.358$, and $ \lambda_{1,3}=0.069$. Finally, we can estimate the value of the implied local volatility at node A from equation (6.25), obtaining $ \sigma_A=9.5\%$.

Figure 6.8: Transition probabilities for $ \sigma _{I} = 0.11 - \Delta K \cdot 0.001$.

\includegraphics[width=1.5\defpicwidth]{ITT_ex03probs.ps}

Let us demonstrate the computation of one further node. Starting from node $ B$ in year $ t_2=2$ of Figure 6.7, the index level at this node is $ s_B=116.83$ and its forward price one year later is $ F_B=e^{(r^*-\delta^*)\cdot 1}\cdot 116.83=125.82$. From this node, the underlying can move to one of three future nodes at time $ t_3=3$, with prices $ s_{3,2}=136.50$, $ s_{3,3}=116.83$, and $ s_{3,4}=100.00$. The value of a call option struck at $ 116.83$ and expiring at time $ t_3=3$ is $ C(s_{3,3},t_3)=8.87$, corresponding to the implied volatility of $ 10.83\%$ interpolated from the smile. The Arrow-Debreu price computed from equation (6.8) is

$\displaystyle \lambda_{2,2} = e^{-\log(1+0.12) \cdot 1}
\{0.467 \cdot (1-0.517-0.070) + 0.358 \cdot 0.523\}=0.339.$

The numerical values used here are already known from the previous level at time $ t_1$.

Figure 6.9: Arrow-Debreu prices for $ \sigma _{I} = 0.11 - \Delta K \cdot 0.001$.

\includegraphics[width=1\defpicwidth]{ITT_ex04ad.ps}

Now, using equations (6.21) and (6.22) we can find the transition probabilities:
$\displaystyle p_{2,2}$ $\displaystyle =$ $\displaystyle \frac{e^{\log(1+0.12) \cdot 1} \cdot 8.87 -
\Sigma}{0.339 \cdot (136.50-116.83)} = 0.515,$  
$\displaystyle \vspace{\baselineskip} q_{2,2}$ $\displaystyle =$ $\displaystyle \frac{125.82 - 0.515 \cdot
(136.50-116.83) - 116.83}{100 - 116.83} = 0.068,$  

where $ \Sigma$ contributes only one term $ 0.215 \cdot
(147-116.83)$, that is, there is one single node above $ S_B$ whose forward price is equal to $ 147$. Finally, employing (6.25) again, we find that the implied local volatility at this node is $ \sigma_B = 9.3\%$.

Figure 6.10: Implied local volatilities for $ \sigma _{I} = 0.11 - \Delta K \cdot 0.001$.

\includegraphics[width=1\defpicwidth]{ITT_ex05locvol.ps}

Figure 6.11: Transition probabilities for $ \sigma _{I} = 0.11 - \Delta K \cdot 0.005$. Nodes $ C$ and $ D$ had inadmissible transition probabilities (6.21)-(6.24).

\includegraphics[width=1.5\defpicwidth]{ITT_ex07probs.ps}

The complete trees of transition probabilities, Arrow-Debreu prices, and local volatilities for this example are shown in Figures 6.8-6.10.

As already mentioned in Section 6.3.4, the transition probabilities may fall out of the interval $ (0,1)$. For example, let us slightly modify our previous example and assume that the Black-Scholes volatility increases (decreases) linearly 0.5 percentage points with every 10 unit drop (rise) in the strike price $ K$; that is, $ \sigma _{I} = 0.11 - \Delta K \cdot 0.005$. In other words, the volatility smile is now five times steeper than before. Using the same state space as in the previous example, we find inadmissable transition probabilities at nodes $ C$ and $ D$, see Figures 6.11-6.13. To overwrite them with plausible values, we used the strategy described by (6.26) and (6.27) and obtained reasonable results in the sense of the three conditions stated on page [*].

Figure 6.12: Arrow-Debreu prices for $ \sigma _{I} = 0.11 - \Delta K \cdot 0.005$. Nodes $ C$ and $ D$ had inadmissible transition probabilities (6.21)-(6.24).

\includegraphics[width=1\defpicwidth]{ITT_ex08ad.ps}

Figure 6.13: Implied local volatilities for $ \sigma _{I} = 0.11 - \Delta K \cdot 0.005$. Nodes $ C$ and $ D$ had inadmissible transition probabilities (6.21)-(6.24).

\includegraphics[width=1\defpicwidth]{ITT_ex09locvol.ps}


6.4.2 German Stock Index

Following the artificial examples, let us now demonstrate the ITT modeling for a real data set, which consists of strike prices for DAX options with maturities from two weeks to two months on January 4, 1999. Given such data, we can firstly compute from the Black-Scholes equation (6.1) the implied volatilities at various combinations of prices and maturities, that is, we can construct the volatility smile. Next, we build and calibrate an ITT so that it fits this smile. The procedure is analogous to the examples described above - the only difference lies in replacing an artificial function $ \sigma_{I}(K,t)$ by an estimate of implied volatility $ \sigma_I$ at each point $ (K,t)$.

Figure 6.14: The state space of the ITT constructed for DAX on January 4, 1999. Dashed lines mark the transitions with originally inadmissible transition probabilities.

\includegraphics[width=1.1\defpicwidth]{ITT_realStateSpace3.ps}

For the purpose of demonstration, we build a three-level ITT with time step $ \Delta t$ of two weeks. First, we construct the state space (Section 6.3.2) starting at time $ t_0 = 0$ with the spot price $ S=5290$ and riskless interest rate $ r=4\%$, see Figure 6.14. Further, we have to compute the transition probabilities. Because option contracts are not available for each combination of price and maturity, we use a nonparametric smoothing procedure to model the whole volatility surface $ \sigma_I(K,t)$ as employed by Aït-Sahalia, Wang, and Yared (2001) and Fengler, Härdle, and Villa (2003), for instance. Given the data, some transition probabilities fall outside interval $ (0,1)$; they are depicted by dashed lines in Figure 6.14. Such probabilities have to be corrected as described in Section 6.3.4 (there are no forward price violations). The resulting local volatilities, which reflect the volatility skew, are on Figure 6.15.

Probably the main result of this ITT model can be summarized by the state price density (the left panel of Figure 6.16). This density describes the price distribution given by the constructed ITT and smoothed by the Nadaraya-Watson estimator. Apparently, the estimated density is rather rough because we used just three steps in our tree. To get a smoother state-price density estimate, we doubled the number of steps; that is, we used six one-week steps instead of three two-week steps (see the right panel of Figure 6.16).

Figure 6.15: Implied local volatilities computed from an ITT for DAX on January 4, 1999.

\includegraphics[width=0.8\defpicwidth]{ITT_realVol3.ps}

Figure 6.16: State price density estimated from an ITT for DAX on January 4, 1999. The dashed line depicts the corresponding Black-Scholes density. Left panel: State price density for a three-level tree. Right panel: State price density for a six-level tree.

\includegraphics[width=0.80\defpicwidth]{ITT_realSPD3b.ps}\includegraphics[width=0.80\defpicwidth]{ITT_realSPD6b.ps}

Finally, it possible to use the constructed ITT to evaluate various DAX options. For example, a European knock-out option gives the owner the same rights as a standard European option as long as the index price $ S$ does not exceed or fall below some barrier $ B$ for the entire life of the knock-out option; see Härdle et al. (2002) for details. So, let us compute the price of the knock-out-call DAX option at maturity $ T = 6$ weeks, strike price $ K= 5200$, and barrier $ B=4800$. The option price at time $ t_j$ ( $ t_0=0,t_1=2,t_2=4,$ and $ t_3=6$ weeks) and stock price $ s_{j,i}$ will be denoted $ V_{j,i}$.

Figure 6.17: Transition probabilities of the ITT constructed for DAX on January 4, 1999.

\includegraphics[width=1.5\defpicwidth]{ITT_realP3b.ps}

At the maturity $ t = T= 6$, the price is known: $ V_{3,i} = \max\{0, s_{j,i} - K\},
i=1,\ldots,7$. Thus, $ V_{3,1} = \max\{0, 4001.01-5200\} = 0$ and $ V_{3,5} =
\max\{0, 5806.07-5200\} = 606.07$, for instance. To compute the option price at $ t_j < T$, one just has to discount the conditional expectation of the option price at time $ t_{j+1}$

$\displaystyle V_{j,i} = e^{-r^* \Delta t} \{p_{j,i} V_{j+1,i+2} + (1-p_{j,i}-q_{j,i}) V_{j+1,i+1} + q_{j,i} V_{j+1,i}\}$ (6.28)

provided that $ s_{j,i} \ge B$, otherwise $ V_{j,i} = 0$. Hence at time $ t_2=4$, one obtains $ V_{2,1} = 0$ because $ s_{2,1} = 4391.40 < 4800 = B$ and

$\displaystyle V_{2,3} = e^{-\log(1+0.04) \cdot 2/52} (0.22\cdot 606.07 + 0.55 \cdot 90 + 0.23 \cdot 0) = 184.33$

(see Figure 6.17). We can continue further and compute the option price at times $ t_1 = 2$ and $ t_0 = 0$ just using the standard formula (6.28) since prices no longer lie below the barrier $ B$ (see Figure 6.14). Thus, one computes $ V_{1,1} = 79.7$, $ V_{1,2}=251.7$, $ V_{1,3}=639.8$, and finally, the option price at time $ t_0 = 0$ and stock price $ S=5290$ equals

$\displaystyle V_{0,1} = e^{-\log(1+0.04) \cdot 2/52} (0.25 \cdot 639.8 + 0.50 \cdot 251.7 + 0.25 \cdot 79.7) = 303.28.$