A DiscreteTime Approach to Evaluate PathDependent Derivatives in a RegimeSwitching Risk Model
Abstract
:1. Introduction
2. The RegimeSwitching Framework
3. The LatticeBased Model
3.1. The Discretization in Each Regime
3.2. The PathDependent Function Values
 to work with effective values for the pathdependent function that we call representative values;
 not to resort to simulated values as in Yuen and Yang (2010);
 to reduce the computational complexity of the evaluation problem;
 to guarantee that the discrete time model converges to the continuous time one.
 Step 1:
 among nodes $(s,{j}_{s})$ belonging to $\tau (i,j)$, we consider only the ones where the asset has registered the maximum value, ${S}_{{0}_{max}}(s,{j}_{s})$;
 Step 2:
 among them, we select the node in correspondence of the minimum value assumed by s, ${s}_{min}$ (i.e., node $({s}_{min},{j}_{{s}_{min}})$), in a way that the new trajectory obtained by substituting node $({s}_{min},{j}_{{s}_{min}})$ with node $({s}_{min},{j}_{{s}_{min}}1)$ in $\tau (i,j)$ still reaches node $(i,j)$;
 Step 3:
 ${f}_{0}(i,j;k+1)$ is computed on this new trajectory.
3.3. The Evaluation Scheme in a TwoRegime Economy
4. Numerical Results
4.1. Asian Options
4.2. EquityIndexed Annuities
4.3. Currency Lookback Options
5. Conclusions
 while the pricing of standard options under regimeswitching models has been widely studied in finance, few are the contributions developed for pricing pathdependent derivatives;
 the increasing importance of such derivatives, which are widely used not only in the financial field but also in the actuarial one;
 the difficulties encountered when managing the complex pathdependent structures of these derivatives that deeply influence the pricing problem tractability.
 by providing a simple and flexible binomial lattice algorithm, useful for practitioners, because different specifications of the pathdependent function may be easily managed and both European and Americanstyle contingent claims or insurance policies may be evaluated in the developed model;
 by reducing the problem computational complexity and overcoming the drawbacks evidenced for the Yuen and Yang (2010) model.
Funding
Acknowledgments
Conflicts of Interest
References
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1.  We recall that the spanning function used in the Hull and White (1993) method is of the form $S{e}^{mh}$, where m assumes all the integer values in a certain interval and h is a fixed parameter governing the fineness of the grid. 
2.  By considering an economy based on L regimes, ${c}_{l}(n,j;k),l=0,\dots ,L1$, is the European pathdependent option payoff in correspondence of ${f}_{l}(n,j;k)$ on the ending node $(n,j)$ for the lth volatility regime. 
3.  In an L regime economy, we have to take into account every possible pair $(l,w),l,w=0,1,\dots ,L1,l\ne w$, of regimes to calculate the option values. The option price in correspondence with the kth average, ${f}_{l}(i,j;k)$ for each node $(i,j)$, of the lattice in the lth regime, is computed as follows
$$\begin{array}{c}\hfill {c}_{l}(i,j;k)={e}^{{r}_{l}\Delta t}\left(\right)open="\{"\; close>\left(\right)open="["\; close="]">1\sum _{w=0,w\ne l}^{L1}{a}_{l,w}\Delta t[{p}_{l}{c}_{l}(i+1,j+1;{k}^{u})+{q}_{l}{c}_{l}(i+1,j;{k}^{d})]+\end{array}\hfill \left(\right)open\; close="\}">\sum _{w=0,w\ne l}^{L1}{a}_{l,w}\Delta t[{p}_{l}{\overline{c}}_{w}(i+1,{j}^{u};{k}^{u})+{q}_{l}{\overline{c}}_{w}(i+1,{j}^{d};{k}^{d})].$$

4.  In an Lregime economy, the quantity ${R}_{l}(n,j;k)={\displaystyle \frac{{f}_{l}(n,j;k)}{S}}1$, with $j=0,\dots ,n$, and $k=1,\dots ,1+j(nj)$, represents the average return of the equityindex values registered from time 0 to time $T=n\Delta t$, under the lth regime with $l=0,\dots ,L1$. 
5.  Generally, ${r}_{l}^{D}$ and ${r}_{l}^{F}$ are the domestic and foreign interest rates in the lth regime with $l=1,\dots ,L1$. Consequently, in the lth regime, the probability of an upward movement is computed as ${p}_{l}=\frac{{e}^{\left(\right)open="("\; close=")">{r}_{l}^{D}{r}_{l}^{F}}}{}{u}^{{\varsigma}_{l}}{d}^{{\varsigma}_{l}}$, while ${q}_{l}=1{p}_{l}$. 
6.  A numerical example showing the lack of convergence of the Hull and White (1993) model has been already reported in Costabile et al. (2006). 
7.  The convergence of the proposed algorithm to the continuous time model is guaranteed if both the truncation error and the interpolation error tend to zero as the time step of the lattice $\Delta t\to 0$. To evaluate the truncation error affecting the solution of the valuation problem when the backward scheme in (3) is used, Jiang and Dai (2004) provide useful findings for what concerns the convergence of CRR models for pricing pathdependent options that may be applied to our algorithm. To evaluate the interpolation error, the Forsyth et al. (2002) methodology may be easily adapted. 
S  Method  Regime 0 : ${\mathit{r}}_{0}=0.05$, ${\mathit{\sigma}}_{0}=0.25$  Regime 1 : ${\mathit{r}}_{1}=0.05$, ${\mathit{\sigma}}_{1}=0.15$  

$\mathit{K}=90$  $\mathit{K}=100$  $\mathit{K}=110$  $\mathit{K}=90$  $\mathit{K}=100$  $\mathit{K}=110$  
90  B  4.6288  1.1238  0.1979  5.8834  2.1773  0.6613 
BD  4.6204  1.1172  0.1966  5.8747  2.1808  0.6694  
YY  4.5964  1.0970  0.1899  5.8655  2.1688  0.6600  
95  B  8.1034  2.6274  0.5882  9.1608  3.9879  1.4210 
BD  8.1132  2.6288  0.5809  9.1475  3.9850  1.4281  
YY  8.0937  2.6014  0.5668  9.1380  3.9731  1.4165  
100  B  12.3322  5.1387  1.4592  13.0469  6.5339  2.7008 
BD  12.3374  5.1338  1.4574  13.0381  6.5274  2.7010  
YY  12.3253  5.1071  1.4331  13.0294  6.5172  2.6876  
105  B  16.9559  8.5702  3.1007  17.3633  9.7703  4.6079 
BD  16.9523  8.5831  3.0956  17.3580  9.7659  4.6041  
YY  16.9453  8.5608  3.0651  17.3506  9.7554  4.5911  
110  B  21.7454  12.7266  5.6185  21.9580  13.5841  7.1908 
BD  21.7353  12.7242  5.5472  21.9435  13.5774  7.1802  
YY  21.7306  12.7091  5.6179  21.9372  13.5675  7.1689 
$\mathit{\lambda}$  Method  Regime 0 : ${\mathit{r}}_{0}=0.05$, ${\mathit{\sigma}}_{0}=0.25$  Regime 1 : ${\mathit{r}}_{1}=0.05$, ${\mathit{\sigma}}_{1}=0.15$  

$\mathit{K}=90$  $\mathit{K}=100$  $\mathit{K}=110$  $\mathit{K}=90$  $\mathit{K}=100$  $\mathit{K}=110$  
0.5  B  12.2671  4.9673  1.3068  13.1136  6.6570  2.8194 
BD  12.2651  4.9609  1.3053  13.1165  6.6669  2.8325  
YY  12.2538  4.9340  1.2817  13.1073  6.6559  2.8183  
MZ  12.2545  4.9351  1.2844  13.1049  6.6493  2.8160  
1  B  12.3322  5.1387  1.4592  13.0469  6.5339  2.7008 
BD  12.3374  5.1338  1.4574  13.0381  6.5274  2.7010  
YY  12.3253  5.1071  1.4331  13.0294  6.5172  2.6876  
MZ  12.3267  5.1092  1.4369  13.0265  6.5093  2.6841 
S  Method  Regime 0 : ${\mathit{r}}_{0}=0.05$, ${\mathit{\sigma}}_{0}=0.25$  Regime 1 : ${\mathit{r}}_{1}=0.05$, ${\mathit{\sigma}}_{1}=0.15$  

$\mathit{K}=90$  $\mathit{K}=100$  $\mathit{K}=110$  $\mathit{K}=90$  $\mathit{K}=100$  $\mathit{K}=110$  
90  B  5.0704  1.1689  0.1999  6.5118  2.2839  0.6760 
YY  5.0370  1.1333  0.1921  6.5067  2.2815  0.6757  
95  B  9.2362  2.7858  0.5997  10.4983  4.2901  1.4711 
YY  9.2197  2.7548  0.5787  10.4824  4.2813  1.4694  
100  B  14.2030  5.6172  1.5106  15.2987  7.2308  2.8437 
YY  14.1980  5.5967  1.4861  15.2867  7.2297  2.8385  
105  B  19.2812  9.7365  3.2841  20.4216  11.1607  4.9672 
YY  19.2790  9.7214  3.2542  20.4078  11.1561  4.9590  
110  B  24.3732  14.6667  6.1838  25.5736  15.8847  7.9644 
YY  24.3641  14.6491  6.1563  25.5604  15.8711  7.9526 
$\mathit{K}=90$  Regime 0  Regime 1  

$\mathit{\lambda}$  n  Price  Difference  Price  Difference 
0.5  50  12.2498  0.0114  12.9910  0.0814 
100  12.2612  0.0059  13.0724  0.0412  
200  12.2671  13.1136  
1  50  12.3059  0.0173  12.8326  0.1428 
100  12.3232  0.0090  12.9754  0.0715  
200  12.3322  13.0469  
$\mathbf{K}=\mathbf{100}$  Regime 0  Regime 1  
$\mathbf{\lambda}$  $\mathbf{n}$  Price  Difference  Price  Difference 
0.5  50  4.9429  0.0164  6.5645  0.0613 
100  4.9593  0.0080  6.6258  0.0312  
200  4.9673  6.6570  
1  50  5.1172  0.0141  6.3693  0.1105 
100  5.1313  0.0074  6.4798  0.0541  
200  5.1387  6.5339  
$\mathbf{K}=\mathbf{110}$  Regime 0  Regime 1  
$\mathbf{\lambda}$  $\mathbf{n}$  Price  Difference  Price  Difference 
0.5  50  1.2767  0.0204  2.7542  0.0433 
100  1.2971  0.0097  2.7975  0.0219  
200  1.3068  2.8194  
1  50  1.4322  0.0171  2.5902  0.0741 
100  1.4493  0.0099  2.6643  0.0365  
200  1.4592  2.7008 
g  Method  Regime 0 : ${\mathit{r}}_{0}=0.05$, ${\mathit{\sigma}}_{0}=0.25$  Regime 1 : ${\mathit{r}}_{1}=0.07$, ${\mathit{\sigma}}_{1}=0.15$  

$\mathit{\zeta}=5\%$  $\mathit{\zeta}=10\%$  $\mathit{\zeta}=15\%$  $\mathit{\zeta}=5\%$  $\mathit{\zeta}=10\%$  $\mathit{\zeta}=15\%$  
0  B  0.97522  0.99995  1.02035  0.97243  0.98652  0.99963 
YY  0.96885  0.98307  0.99085  0.96079  0.97747  0.98880  
1%  B  0.97837  1.00311  1.02351  0.97449  0.98942  1.00099 
YY  0.97294  0.98716  0.99494  0.96500  0.98168  0.99301  
2%  B  0.98173  1.00646  1.02686  0.97675  0.99548  1.00103 
YY  0.97743  0.99165  0.99943  0.96949  0.98617  0.99750  
3%  B  0.98528  1.01001  1.03041  0.97921  1.00072  1.00349 
YY  0.98230  0.99652  1.00430  0.97425  0.99093  1.00227 
n  $\mathit{\sigma}=0.1$  $\mathit{\sigma}=0.2$  $\mathit{\sigma}=0.3$  

B  CV  B  CV  B  CV  
50  4.2449  4.24  8.9693  8.97  13.5217  13.52 
100  4.3673  4.37  9.2007  9.20  13.8501  13.85 
500  4.5371  4.54  9.5216  9.52  14.3051  14.31 
1000  4.5784  4.58  9.5997  9.60  14.4157  14.42 
n  Regime 0 : ${\mathit{\sigma}}_{0}=0.3$  Regime 1 : ${\mathit{\sigma}}_{1}=0.1$  

European  American  European  American  
50  9.4592  11.0479  4.2636  4.6337 
100  9.8685  11.4756  4.4764  4.8401 
500  10.3854  12.0330  4.7531  5.1122 
1000  10.4948  12.1482  4.8171  5.1750 
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Russo, E. A DiscreteTime Approach to Evaluate PathDependent Derivatives in a RegimeSwitching Risk Model. Risks 2020, 8, 9. https://doi.org/10.3390/risks8010009
Russo E. A DiscreteTime Approach to Evaluate PathDependent Derivatives in a RegimeSwitching Risk Model. Risks. 2020; 8(1):9. https://doi.org/10.3390/risks8010009
Chicago/Turabian StyleRusso, Emilio. 2020. "A DiscreteTime Approach to Evaluate PathDependent Derivatives in a RegimeSwitching Risk Model" Risks 8, no. 1: 9. https://doi.org/10.3390/risks8010009