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# Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

• The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

Mathematics Subject Classification: 91-08, 91B25, 91B70, 91G20, 91G60, 91G80.

 Citation: • • Figure 1.  Relative early exercise contribution, diffusion contribution to the early exercise premium, and absolute early exercise premium of the geometric down-and-out step call as functions of the underlying price $S_{0} \in [85,115]$ and the knock-out rate $\rho_{L} \in [-1000,0]$, when the remaining parameters are chosen as: $\mathcal{T} = 1.0$, $\sigma_{X} = 0.2$, $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\lambda = 5$, $p = 0.5$, $\xi = 25$, $\eta = 50$

Figure 2.  Difference in the prices, deltas, and gammas for the geometric down-and-out step calls with and without jumps as functions of the underlying price $S_{0} \in [85.115]$ and the intensity parameter $\lambda \in [0,20]$, when the remaining parameters are chosen as: $\mathcal{T} = 1.0$, $\sigma_{X} = 0.2$, $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$, $\eta = 50$

Figure 3.  Difference in the prices, deltas, and gammas for the geometric down-and-out step calls with and without jumps as functions of the underlying price $S_{0} \in [85.115]$ and the positive jump parameter $\xi \in [5,100]$, when the remaining parameters are chosen as: $\mathcal{T} = 1.0$, $\sigma_{X} = 0.2$, $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $\lambda = 5$, $p = 0.5$, $\eta = 50$

Table 1.  Theoretical (down-and-out) call values and diffusion contributions to the early exercise premium for $r = 0.05$, $\delta = 0.07$, $S_{0} = 100$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.7$, $\xi = 25$ and $\eta = 50$

 (Down-and-Out) Call Option Prices Parameters Standard Call Price Step Call Price Barrier Call Price $\lambda$ Euro Amer DC (%) Euro Amer DC (%) Euro Amer DC (%) $1$ $6.833$ $7.040$ $91.52 \%$ $4.596$ $4.789$ $91.71 \%$ $3.374$ $3.551$ $91.88 \%$ $S_{0} = 100$ $0.1$ $6.622$ $6.822$ $99.07 \%$ $4.519$ $4.706$ $99.09 \%$ $3.338$ $3.514$ $99.12 \%$ $\sigma_{X}=0.2$ $0.01$ $6.600$ $6.800$ $99.91 \%$ $4.511$ $4.698$ $99.91 \%$ $3.334$ $3.510$ $99.91 \%$ $\mathcal{T} = 1.0$ $0.001$ $6.598$ $6.797$ $99.99 \%$ $4.510$ $4.697$ $99.99 \%$ $3.333$ $3.509$ $99.99 \%$ $0.0001$ $6.598$ $6.797$ $100.00 \%$ $4.510$ $4.697$ $100.00\%$ $3.333$ $3.509$ $100.00 \%$ B & S Values - 6.598 6.885 - 4.511 4.745 - 3.332 3.529 - Rel. Error (%) - 0.001% -1.277% - 0.015% -1.025% - 0.025% -0.568% -

Table 2.  Theoretical American-type (down-and-out) call values and structure of the early exercise premium for $\delta = 0.05$, $K = 100$, $\sigma_{X} = 0.2$, and $\mathcal{T} = 0.25$

 (Down-and-Out) Barrier Call Option Prices Parameters Barrier Option Values $\lambda$ $p$ $\xi$ $\eta$ FST Amer DC (%) $3.029$ $0.381$ $26$ $24$ $1.016$ $1.017$ $80.08\%$ $3.051$ $0.386$ $51$ $24$ $0.920$ $0.921$ $96.04\%$ (1) $2.991$ $0.386$ $26$ $49$ $0.934$ $0.934$ $78.03\%$ $S_{0}=90$ $3.013$ $0.390$ $51$ $49$ $0.837$ $0.838$ $95.22\%$ $r = 0.07$ $7.067$ $0.381$ $26$ $24$ $1.337$ $1.338$ $69.81\%$ $L = 78.261$ $7.120$ $0.386$ $51$ $24$ $1.126$ $1.127$ $93.44\%$ $6.978$ $0.386$ $26$ $49$ $1.141$ $1.141$ $67.16\%$ $7.031$ $0.390$ $51$ $49$ $0.924$ $0.924$ $92.04\%$ $3.029$ $0.381$ $26$ $24$ $3.850$ $3.850$ $85.02\%$ $3.051$ $0.386$ $51$ $24$ $3.741$ $3.728$ $96.73\%$ (1) $2.991$ $0.386$ $26$ $49$ $3.693$ $3.677$ $84.91\%$ $S_{0}=100$ $3.013$ $0.390$ $51$ $49$ $3.580$ $3.555$ $96.66\%$ $r = 0.00$ $7.067$ $0.381$ $26$ $24$ $4.316$ $4.366$ $72.44\%$ $L = 86.957$ $7.120$ $0.386$ $51$ $24$ $4.089$ $4.103$ $93.13\%$ $6.978$ $0.386$ $26$ $49$ $3.970$ $3.977$ $71.33\%$ $7.031$ $0.390$ $51$ $49$ $3.719$ $3.699$ $92.77\%$

Table 3.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 50$ and $\eta = 25$

 (Down-and-Out) Call Option Prices Parameters Standard Call Price Step Call Price Barrier Call Price $S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) $90$ $3.500$ $0.062$ $1.74 \%$ $94.20 \%$ $0.268$ $0.009$ $3.07 \%$ $94.32 \%$ $0$ $0$ - - (1) $95$ $5.241$ $0.112$ $2.09 \%$ $94.27 \%$ $1.757$ $0.059$ $3.23 \%$ $94.33 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $7.416$ $0.190$ $2.50 \%$ $94.34 \%$ $4.992$ $0.178$ $3.45 \%$ $94.36 \%$ $3.686$ $0.165$ $4.28 \%$ $94.37 \%$ $\lambda = 5.0$ $105$ $10.011$ $0.305$ $2.96 \%$ $94.40 \%$ $8.309$ $0.330$ $3.82 \%$ $94.39 \%$ $7.305$ $0.353$ $4.61 \%$ $94.40 \%$ $\mathcal{T} = 1.0$ $110$ $12.992$ $0.469$ $3.48 \%$ $94.46 \%$ $11.804$ $0.535$ $4.34 \%$ $94.44 \%$ $11.037$ $0.597$ $5.13 \%$ $94.44 \%$ $115$ $16.314$ $0.691$ $4.07 \%$ $94.52 \%$ $15.492$ $0.811$ $4.98 \%$ $94.50 \%$ $14.914$ $0.920$ $5.81 \%$ $94.54 \%$ $90$ $4.098$ $0.065$ $1.57 \%$ $89.68 \%$ $0.344$ $0.010$ $2.79 \%$ $89.87 \%$ $0$ $0$ - - (2) $95$ $5.933$ $0.113$ $1.87 \%$ $89.80 \%$ $2.012$ $0.061$ $2.93 \%$ $89.89 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $8.169$ $0.186$ $2.22 \%$ $89.90 \%$ $5.413$ $0.175$ $3.12 \%$ $89.93 \%$ $3.990$ $0.161$ $3.88 \%$ $89.95 \%$ $\lambda = 10.0$ $105$ $10.791$ $0.290$ $2.62 \%$ $90.00 \%$ $8.791$ $0.314$ $3.44 \%$ $89.99 \%$ $7.683$ $0.334$ $4.17 \%$ $89.99 \%$ $\mathcal{T} = 1.0$ $110$ $13.767$ $0.435$ $3.06 \%$ $90.08 \%$ $12.313$ $0.497$ $3.88 \%$ $90.05 \%$ $11.442$ $0.552$ $4.60 \%$ $90.05 \%$ $115$ $17.056$ $0.628$ $3.55 \%$ $90.17 \%$ $16.004$ $0.738$ $4.41 \%$ $90.12 \%$ $15.325$ $0.835$ $5.16 \%$ $90.13 \%$

Table 4.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 50$ and $\eta = 50$

 (Down-and-Out) Call Option Prices Parameters Standard Call Price Step Call Price Barrier Call Price $S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) $90$ $3.163$ $0.064$ $1.98 \%$ $93.97 \%$ $0.232$ $0.008$ $3.46 \%$ $94.10 \%$ $0$ $0$ - - (1) $95$ $4.835$ $0.117$ $2.37 \%$ $94.05 \%$ $1.588$ $0.060$ $3.63 \%$ $94.12 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $6.958$ $0.202$ $2.82 \%$ $94.12 \%$ $4.679$ $0.188$ $3.87 \%$ $94.14 \%$ $3.432$ $0.174$ $4.82\%$ $94.15 \%$ $\lambda = 5.0$ $105$ $9.523$ $0.328$ $3.33 \%$ $94.19 \%$ $7.949$ $0.355$ $4.28 \%$ $94.18 \%$ $6.983$ $0.382$ $5.18 \%$ $94.18 \%$ $\mathcal{T} = 1.0$ $110$ $12.498$ $0.509$ $3.91 \%$ $94.25 \%$ $11.430$ $0.583$ $4.85 \%$ $94.23 \%$ $10.702$ $0.654$ $5.76 \%$ $94.24 \%$ $115$ $15.835$ $0.758$ $4.57 \%$ $94.33 \%$ $15.122$ $0.891$ $5.57 \%$ $94.31 \%$ $14.586$ $1.017$ $6.52 \%$ $94.43 \%$ $90$ $3.441$ $0.068$ $1.94 \%$ $88.95 \%$ $0.268$ $0.009$ $3.40 \%$ $89.16 \%$ $0$ $0$ - - (2) $95$ $5.155$ $0.121$ $2.30 \%$ $89.08 \%$ $1.685$ $0.062$ $3.55 \%$ $89.19 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $7.303$ $0.204$ $2.72 \%$ $89.20 \%$ $4.836$ $0.190$ $3.77 \%$ $89.23 \%$ $3.522$ $0.174$ $4.71 \%$ $89.25 \%$ $\lambda = 10.0$ $105$ $9.875$ $0.325$ $3.19 \%$ $89.30 \%$ $8.138$ $0.352$ $4.14 \%$ $89.29 \%$ $7.107$ $0.377$ $5.04 \%$ $89.29 \%$ $\mathcal{T} = 1.0$ $110$ $12.839$ $0.497$ $3.72 \%$ $89.40 \%$ $11.636$ $0.569$ $4.66 \%$ $89.36 \%$ $10.845$ $0.638$ $5.56 \%$ $89.37 \%$ $115$ $16.152$ $0.729$ $4.32 \%$ $89.53 \%$ $15.330$ $0.859$ $5.31 \%$ $89.46 \%$ $14.737$ $0.981$ $6.24 \%$ $89.57 \%$

Table 5.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$ and $\eta = 50$

 (Down-and-Out) Call Option Prices Parameters Standard Call Price Step Call Price Barrier Call Price $S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) $90$ $3.645$ $0.080$ $2.15 \%$ $75.53 \%$ $0.294$ $0.012$ $3.75 \%$ $76.36 \%$ $0$ $0$ - - (1) $95$ $5.362$ $0.137$ $2.49 \%$ $75.97 \%$ $1.685$ $0.067$ $3.82 \%$ $76.45 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $7.501$ $0.222$ $2.88 \%$ $76.37 \%$ $4.854$ $0.202$ $3.99 \%$ $76.61 \%$ $3.506$ $0.182$ $4.94\%$ $76.81 \%$ $\lambda = 5.0$ $105$ $10.054$ $0.345$ $3.31 \%$ $76.71 \%$ $8.177$ $0.368$ $4.31 \%$ $76.94 \%$ $7.110$ $0.391$ $5.21 \%$ $77.25 \%$ $\mathcal{T} = 1.0$ $110$ $12.994$ $0.514$ $3.80 \%$ $76.98 \%$ $11.685$ $0.585$ $4.77 \%$ $77.55 \%$ $10.861$ $0.652$ $5.67 \%$ $78.24 \%$ $115$ $16.279$ $0.740$ $4.35 \%$ $77.14 \%$ $15.381$ $0.870$ $5.35 \%$ $78.79 \%$ $14.759$ $0.989$ $6.28 \%$ $80.55 \%$ $90$ $4.347$ $0.096$ $2.16 \%$ $62.58 \%$ $0.391$ $0.015$ $3.78 \%$ $63.44 \%$ $0$ $0$ - - (2) $95$ $6.141$ $0.155$ $2.45 \%$ $63.00 \%$ $1.865$ $0.074$ $3.82 \%$ $63.47 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $8.321$ $0.238$ $2.78 \%$ $63.38 \%$ $5.152$ $0.212$ $3.94 \%$ $63.56 \%$ $3.649$ $0.188$ $4.89\%$ $63.68 \%$ $\lambda = 5.0$ $105$ $10.878$ $0.354$ $3.15 \%$ $63.72 \%$ $8.549$ $0.374$ $4.19 \%$ $63.76 \%$ $7.328$ $0.393$ $5.09 \%$ $63.90 \%$ $\mathcal{T} = 1.0$ $110$ $13.788$ $0.508$ $3.55 \%$ $64.01 \%$ $12.099$ $0.577$ $4.55 \%$ $64.10 \%$ $11.126$ $0.640$ $5.44 \%$ $64.46 \%$ $115$ $17.019$ $0.709$ $4.00 \%$ $64.18 \%$ $15.809$ $0.834$ $5.01 \%$ $64.74 \%$ $15.047$ $0.946$ $5.91 \%$ $65.93 \%$

Table 6.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$ and $\eta = 25$

 (Down-and-Out) Call Option Prices Parameters Standard Call Price Step Call Price Barrier Call Price $S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) $90$ $3.966$ $0.077$ $1.91 \%$ $76.61 \%$ $0.330$ $0.012$ $3.37 \%$ $77.40 \%$ $0$ $0$ - - (1) $95$ $5.745$ $0.131$ $2.23 \%$ $77.04 \%$ $1.845$ $0.066$ $3.44 \%$ $77.48 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $7.931$ $0.210$ $2.58 \%$ $77.42 \%$ $5.151$ $0.192$ $3.60 \%$ $77.62 \%$ $3.748$ $0.174$ $4.44\%$ $77.79 \%$ $\lambda = 5.0$ $105$ $10.514$ $0.323$ $2.98 \%$ $77.75 \%$ $8.516$ $0.346$ $3.90 \%$ $77.89 \%$ $7.415$ $0.366$ $4.70 \%$ $78.11 \%$ $\mathcal{T} = 1.0$ $110$ $13.463$ $0.479$ $3.43 \%$ $78.01 \%$ $12.037$ $0.544$ $4.32 \%$ $78.35 \%$ $11.178$ $0.603$ $5.12 \%$ $78.81 \%$ $115$ $16.740$ $0.685$ $3.93 \%$ $78.17 \%$ $15.730$ $0.803$ $4.85 \%$ $79.21 \%$ $15.069$ $0.907$ $5.68 \%$ $80.34 \%$ $90$ $4.950$ $0.091$ $1.81 \%$ $64.97 \%$ $0.468$ $0.016$ $3.20 \%$ $65.77 \%$ $0$ $0$ - - (2) $95$ $6.842$ $0.144$ $2.07 \%$ $65.37 \%$ $2.166$ $0.073$ $3.25 \%$ $65.81 \%$ $0$ $0$ - - $\sigma_{X}=0.2$ $100$ $9.098$ $0.220$ $2.36 \%$ $65.74 \%$ $5.678$ $0.198$ $3.37 \%$ $65.91 \%$ $4.077$ $0.177$ $4.16\%$ $66.01 \%$ $\lambda = 5.0$ $105$ $11.704$ $0.322$ $2.68 \%$ $66.08 \%$ $9.138$ $0.341$ $3.60 \%$ $66.09 \%$ $7.852$ $0.358$ $4.35 \%$ $66.17 \%$ $\mathcal{T} = 1.0$ $110$ $14.634$ $0.458$ $3.03 \%$ $66.37 \%$ $12.709$ $0.518$ $3.92 \%$ $66.34 \%$ $11.668$ $0.571$ $4.67 \%$ $66.50 \%$ $115$ $17.857$ $0.632$ $3.42 \%$ $66.60 \%$ $16.419$ $0.741$ $4.32 \%$ $66.73 \%$ $15.583$ $0.834$ $5.08 \%$ $67.20 \%$
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