7.2 A Simulation and a Comparison of the SPDs

The example used here to show the procedure of generating the IBT, is taken from Derman and Kani (1994). Assume that the current value of the stock is $ S=100$, the annually compounded riskless interest rate is $ r=3\%$ per year for all time expirations, the stock has zero dividend. The annual BS implied volatility of an at-the-money call is assumed to be $ \sigma=10\%$, and the BS implied volatility increases (decreases) linearly by 0.5 percentage points with every 10 point drop (rise) in the strike. From the assumptions, we see that $ \sigma_{imp}(K,\tau)=0.15-0.0005\,K$.

In order to investigate the precision of the SPD estimation obtained from the IBT, we give a simulation example assuming that the stock price process is generated by the stochastic differential equation model (7.3), with an instantaneous local volatility function $ \sigma(S_{t},t)=0.15-0.0005\,S_{t}$, $ \mu_{t}=r=0.03$. We may then easily compare the SPD estimations obtained from the two different methods.


7.2.1 Simulation using Derman and Kani algorithm

With the XploRe quantlet XFGIBT01.xpl, using the assumption on the BS implied volatility surface, we obtain the following one year stock price implied binomial tree, transition probability tree, and Arrow-Debreu price tree.

14860 XFGIBT01.xpl

Derman and Kani one year (four step) implied binomial tree

stock price
                                      119.91
                             115.06
                    110.04            110.06
           105.13            105.13
  100.00            100.00            100.00
            95.12             95.12
                     89.93             89.92
                              85.22
                                       80.01
transition probability
                       0.60
                0.58
         0.59          0.59
  0.56          0.56
         0.59          0.59
                0.54
                       0.59
Arrow-Debreu price
                                  0.111
                          0.187
                  0.327           0.312
          0.559           0.405
  1.000           0.480           0.343
          0.434           0.305
                  0.178           0.172
                          0.080
                                  0.033

This IBT is corresponding to $ \tau =1$ year, and $ \triangle t=0.25$ year, which shows the stock prices, and the elements at the $ j$th column are corresponding to the stock prices of the nodes at the $ (j-1)$th level in the tree. The second one, its $ (n,j)$ element is corresponding to the transition probability from the node $ (n,j)$ to the nodes $ (n+1,j+1)$. The third tree contains the Arrow-Debreu prices of the nodes. Using the stock prices together with Arrow-Debreu prices of the nodes at the final level, a discrete approximation of the implied distribution can be obtained. Notice that by the definition of the Arrow-Debreu price, the risk neutral probability corresponding to each node should be calculated as the product of the Arrow-Debreu price and the factor $ e^{r\tau}$.

If we choose small enough time steps, we obtain the estimation of the implied price distribution and the implied local volatility surface $ \sigma_{loc}
(s,\tau)$. We still use the same assumption on the BS implied volatility surface as above here, which means $ \sigma_{imp}(K,\tau)=0.15-0.0005\,K$, and assume $ S_{0}=100, r=0.03, T=5$ year.

14866 XFGIBT02.xpl

Two figures are generated by running the quantlet XFGIBT02.xpl, Figure 7.2 shows the plot of the SPD estimation resulting from fitting an implied five-year tree with 20 levels. The implied local volatilities $ \sigma_{loc}
(s,\tau)$ in the implied tree at different time to maturity and stock price levels is shown in Figure 7.3, which obviously decreases with the stock price and increases with time to maturity as expected.

Figure 7.2: SPD estimation by the Derman and Kani IBT.
\includegraphics[width=1\defpicwidth]{dkspd.ps}

Figure 7.3: Implied local volatility surface estimation by the Derman and Kani IBT.
\includegraphics[width=1.7\defpicwidth]{dkvola.ps}



7.2.2 Simulation using Barle and Cakici algorithm

The Barle and Cakici algorithm can be applied in analogy to Derman and Kani's. The XploRe quantlets used here are similar to those presented in Section 7.2.1, one has to replace the quantlet 14970 IBTdk by 14973 IBTdc . The following figure displays the one-year (four step) stock price tree, transition probability tree, and Arrow-Debreu tree. Figure 7.4 presents the plot of the estimated SPD by fitting a five year implied binomial tree with 20 levels to the volatility smile using Barle and Cakici algorithm, and Figure 7.5, shows the characteristics of the implied local volatility surface of the generated IBT, decreases with the stock price, and increases with time.

Barle and Cakici one year implied binomial tree

stock price
                                      123.85
                             117.02
                    112.23            112.93
           104.84            107.03
  100.00            101.51            103.05
            96.83             97.73
                     90.53             93.08
                              87.60
                                       82.00


transition probability
                       0.46
                0.61
         0.38          0.48
  0.49          0.49
         0.64          0.54
                0.36
                       0.57


Arrow-Debreu price
                                  0.050
                          0.111
                  0.185           0.240
          0.486           0.373
  1.000           0.619           0.394
          0.506           0.378
                  0.181           0.237
                          0.116
                                  0.050

Figure 7.4: SPD estimation by the Barle and Cakici IBT.
\includegraphics[width=1\defpicwidth]{bcspd.ps}

Figure 7.5: Implied local volatility surface by the Barle and Cakici IBT.
\includegraphics[width=1.7\defpicwidth]{bcvola.ps}


7.2.3 Comparison with Monte-Carlo Simulation

We now compare the SPD estimation at the fifth year obtained by the two IBT methods with the estimated density function of the Monte-Carlo simulation of $ S_{t}, t=5$ generated from the model (7.3), where $ \sigma(S_{t},t)=0.15-0.0005\,S_{t}$, $ \mu_{t}=r=0.03$. We use the Milstein scheme, Kloeden, Platen and Schurz (1994) to perform the discrete time approximation in (7.3). It has strong convergence rate $ \delta^{1}$. We have set the time step with $ \delta=1/1000$ here.

In order to construct the IBT, we calculate the option prices corresponding to each node at the implied tree according to their definition by Monte-Carlo simulation.

15084 XFGIBT03.xpl 15087 XFGIBTcdk.xpl 15090 XFGIBTcbc.xpl

Figure 7.6: SPD estimation by Monte-Carlo simulation, and its $ 95\%$ confidence band, the B & C IBT, from the D & K IBT (thin), level =20, $ T=5$ year, $ \triangle t=0.25$ year.
\includegraphics[width=1\defpicwidth]{mcspdcom.ps}

Here we use the quantlets 15094 XFGIBTcdk.xpl and 15097 XFGIBTcbc.xpl . These two are used to construct the IBT directly from the option price function, not starting from the BS implied volatility surface as in quantlets 15100 IBTdk and 15103 IBTbc . In the data file "IBTmcsimulation20.dat", there are 1000 Monte-Carlo simulation samples for each $ S_{t}$ in the diffusion model (7.3), for $ t=i/4 $ year, $ i=1,...20$, from which we calculate the simulated values of the option prices according to its theoretical definition and estimate the density of $ S_{t},\,
T=5$ year as the SPD estimation at the fifth year.

From the estimated distribution shown in the Figures 7.2.3, we observe their deviation from the log-normal characteristics according to their skewness and kurtosis. The SPD estimation obtained from the two IBT methods coincides with the estimation obtained from the Monte-Carlo simulation well, the difference between the estimations obtained from the two IBTs is not very large.

On the other hand, we can also estimate the implied local volatility surface from the implied binomial tree, and compare it with the one obtained by the simulation. Compare Figure 7.7 and Figure 7.8 with Figure 7.9, and notice that in the first two figures, some edge values cannot be obtained directly from the five-year IBT. However, the three implied local volatility surface plots all actually coincide with the volatility smile characteristic, the implied local volatility of the out-the-money options decreases with the increasing stock price, and increase with time.

Figure 7.7: Implied local volatility surface of the simulated model, calculated from D& K IBT.
\includegraphics[width=1.7\defpicwidth]{mcvoladk.ps}

Figure 7.8: Implied local volatility surface of the simulated model, calculated from B& C IBT.
\includegraphics[width=1.7\defpicwidth]{mcvolabc.ps}

Figure 7.9: Implied local volatility surface of the simulated model, calculated from Monte-Carlo simulation.
\includegraphics[width=1.7\defpicwidth]{mcvola.ps}

We use the data file "IBTmcsimulation50.dat" to obtain an estimated BS implied volatility surface. There are 1000 Monte-Carlo simulation samples for each $ S_{t}$ in the diffusion model (7.3), for $ t=i/10 $ year in it, $ i=1,...50$, because we can calculate the BS implied volatility corresponding to different strike prices and time to maturities after we have the estimated option prices corresponding to these strike price and time to maturity levels. Figure 7.10 shows that the BS implied volatility surface of our example reflects the characteristics that the BS implied volatility decrease with the strike price. But this BS implied volatility surface does not change with time a lot, which is probably due to our assumption about the local instantaneous volatility function, which only changes with the stock price.

15110 XFGIBT04.xpl

Figure 7.10: BS implied volatility surface estimation by Monte-Carlo simulation.
\includegraphics[width=1.3\defpicwidth]{mcbsvola.ps}