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Stochastic Analysis and Applications Volume 37, 2019 - Issue 2

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Representations for conditional expectations and applications to pricing and hedging of financial products in Lévy and jump-diffusion setting

Catherine DavelooseDepartment of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium; View further author information

Asma KhedherKorteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The NetherlandsCorrespondence[email protected]
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Michèle VanmaeleDepartment of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium; View further author information

Pages 281-319 | Received 02 Nov 2016, Accepted 06 Dec 2018, Published online: 19 Mar 2019

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https://doi.org/10.1080/07362994.2018.1561306
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Figure 2. Estimates for the Bermudan put option price obtained through the CDM and MM representations with control variate and with localization technique, against the number of simulated paths. In the right graph the vertical axis is restricted to $[9.5, 12]$ .

Table 1. Estimates of European put option prices $P^{Eu} (s, α)$ (5.5) via the CDM and MM approach, with relative errors to the Merton prices in percentages.

Display Table

Table 2. Estimates of European put option deltas $Δ^{Eu} (s, α)$ (5.6) via the CDM and MM approach, with relative errors to the Merton deltas in percentages.

Display Table

Figure 1. Estimates for the Bermudan put option price obtained through the CDM and MM representations without control variate and without localization technique, against the number of simulated paths. In the right graph the vertical axis is restricted to $[0, 100]$ .

Table 3. Estimates of Bermudan put option prices and confidence intervals for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization and by the Longstaff–Schwartz method [Citation8] with a control variate (LSM). n = 10 and $N = 100000$ .

Display Table

Table 4. CPU time in seconds for the different methods to compute the estimates of Bermudan put option prices for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization and by the Longstaff–Schwartz method [Citation8] with control variate (LSM). n = 10 and $N = 100000$ .

Display Table

Table 5. Estimates of Bermudan put option deltas for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization. n = 10 and $N = 500000$ .

Display Table

Table 6. Estimates of American put option prices for parameter set (5.8), obtained through the sorted CDM approach with exponential localization. n = 10 and $N = 10000$ .

Display Table

Longstaff, F. A., Schwartz, E. S. (2001). Valuing American options by simulations: a simple least squares approach. Rev. Finan. Stud. 14(1):113–148. DOI: 10.1093/rfs/14.1.113.

Web of Science ®Google Scholar

Hilliard, J., Schwartz, A. (2005). Pricing European and American derivatives under a jump-diffusion process: a bivariate tree approach. J. Finan. Quant. Anal. 40(03):671–691.

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Representations for conditional expectations and applications to pricing and hedging of financial products in Lévy and jump-diffusion setting

Table 1. Estimates of European put option prices $P^{Eu} (s, α)$ (5.5) via the CDM and MM approach, with relative errors to the Merton prices in percentages.

Table 2. Estimates of European put option deltas $Δ^{Eu} (s, α)$ (5.6) via the CDM and MM approach, with relative errors to the Merton deltas in percentages.

Table 5. Estimates of Bermudan put option deltas for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization. n = 10 and $N = 500000$ .

Table 6. Estimates of American put option prices for parameter set (5.8), obtained through the sorted CDM approach with exponential localization. n = 10 and $N = 10000$ .

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Representations for conditional expectations and applications to pricing and hedging of financial products in Lévy and jump-diffusion setting

Figures & data

Table 1. Estimates of European put option prices P Eu(s,α) (5.5) via the CDM and MM approach, with relative errors to the Merton prices in percentages.

Table 2. Estimates of European put option deltas ΔEu(s,α) (5.6) via the CDM and MM approach, with relative errors to the Merton deltas in percentages.

Table 3. Estimates of Bermudan put option prices and confidence intervals for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization and by the Longstaff–Schwartz method [Citation8] with a control variate (LSM). n = 10 and N=100000.

Table 5. Estimates of Bermudan put option deltas for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization. n = 10 and N=500000.

Table 6. Estimates of American put option prices for parameter set (5.8), obtained through the sorted CDM approach with exponential localization. n = 10 and N=10000.

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Table 1. Estimates of European put option prices $P^{Eu} (s, α)$ (5.5) via the CDM and MM approach, with relative errors to the Merton prices in percentages.

Table 2. Estimates of European put option deltas $Δ^{Eu} (s, α)$ (5.6) via the CDM and MM approach, with relative errors to the Merton deltas in percentages.

Table 5. Estimates of Bermudan put option deltas for parameter set (5.4), obtained through the sorted CDM and MM approach with control variate and exponential localization. n = 10 and $N = 500000$ .

Table 6. Estimates of American put option prices for parameter set (5.8), obtained through the sorted CDM approach with exponential localization. n = 10 and $N = 10000$ .