September 2018, 8(3): 291-297. doi: 10.3934/naco.2018018

Pricing down-and-out power options with exponentially curved barrier

1. 

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

2. 

Department of Mathematics, Faculty of Science, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

* Corresponding author

Received  April 2017 Revised  July 2017 Published  June 2018

Fund Project: The first author is supported by Universiti Putra Malaysia and the Fundamental Research Grant Scheme

Power barrier options are options where the payoff depends on an underlying asset raised to a constant number. The barrier determines whether the option is knocked in or knocked out of existence when the underlying asset hits the prescribed barrier level, or not. This paper derives the analytical solution of the power options with an exponentially curved barrier by utilizing the reflection principle and the change of measure. Numerical results show that prices of power options with exponentially curved barrier are cheaper than those of power barrier options and power options.

Citation: Teck Wee Ng, Siti Nur Iqmal Ibrahim. Pricing down-and-out power options with exponentially curved barrier. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 291-297. doi: 10.3934/naco.2018018
References:
[1]

J. Andreasen, Behind the mirror, Risk Magazine, 14 (2001), 109-110.

[2]

L. AndersenJ. Andreasen and D. Eliezer, Static replication of barrier option: some general results, Journal of Computational Finance, 5 (2002), 1-25.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.

[4]

L. P. Blenman and S. P. Clark, Power exchange options, Finance Research Letter, 5 (2005), 97-106. doi: 10.1016/j.frl.2005.01.003.

[5]

P. CarrK. Ellis and V. Gupta, Static hedging of exotic options, Journal of Finance, 5 (1998), 1165-1190. doi: 10.1142/9789812812599_0005.

[6]

A. Chen and M. Suchanecki, Parisian exchange option, Quantitative Finance, 11 (2011), 1207-1220. doi: 10.1080/14697680903194577.

[7]

J. C. CoxS. A. Ross and M. Rubinstein, Option pricing: a simplified approach, Journal of Financial Economics, 7 (1979), 229-263. doi: 10.1016/0304-405X(79)90015-1.

[8]

R. J. HaberP. Schönbucher and P. Wilmott, Pricing Parisian options, Journal of Derivatives, 6 (1999), 71-79.

[9]

S. N. I. IbrahimJ. G. O'Hara and N. Constantinou, Risk-neutral valuation of power barrier options, Applied Mathematics Letter, 26 (2013), 595-600. doi: 10.1016/j.aml.2012.12.016.

[10]

N. Kunitomo and M. Ikeda, Pricing options with curved boundaries, Mathematical Finance, 2 (1992), 275-298. doi: 10.1111/j.1467-9965.1992.tb00033.x.

[11]

T. N. LeX. Lu and S. P. Zhu, An analytical solution for Parisian up-and-in calls, ANZIAM Journal, 57 (2016), 269-279. doi: 10.1017/S1446181115000267.

[12]

C. F. Lo, H. C. Lee and C. H. Hui, A simple approach for barrier options with time-dependent parameters, Quantitative Finance, 3 (2003), 98-107. doi: 10.1088/1469-7688/3/2/304.

[13]

R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143.

[14]

M. Nalholm and R. Poulsen, Static hedging and model risk for barrier options, Journal of Future Markets, 26 (2006), 449-463. doi: 10.1002/fut.20199.

[15]

M. F. M. Osborne, Brownian motion in the stock market, Operations Research, 7 (1959), 145-173. doi: 10.1287/opre.7.2.145.

[16]

S. Trippi, The mathematics of barrier options, Advances in Options and Futures, 7 (1994), 150-172.

[17]

R. Poulsen, Barrier options and their static hedges: simple derivations and extensions, Quantitative Finance, 6 (2006), 327-335. doi: 10.1080/14697680600690331.

[18]

S. P. Zhu and W. Chen, Pricing Parisian and Parisian options analytically, Journal of Economic Dynamics and Control, 37 (2013), 875-896. doi: 10.1016/j.jedc.2012.12.005.

[19]

S. P. ZhuT. N. LeW. Chen and X. Lu, Pricing Parisian down-and-in options, Applied Mathematics Letters, 43 (2014), 19-24. doi: 10.1016/j.aml.2014.10.019.

show all references

References:
[1]

J. Andreasen, Behind the mirror, Risk Magazine, 14 (2001), 109-110.

[2]

L. AndersenJ. Andreasen and D. Eliezer, Static replication of barrier option: some general results, Journal of Computational Finance, 5 (2002), 1-25.

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.

[4]

L. P. Blenman and S. P. Clark, Power exchange options, Finance Research Letter, 5 (2005), 97-106. doi: 10.1016/j.frl.2005.01.003.

[5]

P. CarrK. Ellis and V. Gupta, Static hedging of exotic options, Journal of Finance, 5 (1998), 1165-1190. doi: 10.1142/9789812812599_0005.

[6]

A. Chen and M. Suchanecki, Parisian exchange option, Quantitative Finance, 11 (2011), 1207-1220. doi: 10.1080/14697680903194577.

[7]

J. C. CoxS. A. Ross and M. Rubinstein, Option pricing: a simplified approach, Journal of Financial Economics, 7 (1979), 229-263. doi: 10.1016/0304-405X(79)90015-1.

[8]

R. J. HaberP. Schönbucher and P. Wilmott, Pricing Parisian options, Journal of Derivatives, 6 (1999), 71-79.

[9]

S. N. I. IbrahimJ. G. O'Hara and N. Constantinou, Risk-neutral valuation of power barrier options, Applied Mathematics Letter, 26 (2013), 595-600. doi: 10.1016/j.aml.2012.12.016.

[10]

N. Kunitomo and M. Ikeda, Pricing options with curved boundaries, Mathematical Finance, 2 (1992), 275-298. doi: 10.1111/j.1467-9965.1992.tb00033.x.

[11]

T. N. LeX. Lu and S. P. Zhu, An analytical solution for Parisian up-and-in calls, ANZIAM Journal, 57 (2016), 269-279. doi: 10.1017/S1446181115000267.

[12]

C. F. Lo, H. C. Lee and C. H. Hui, A simple approach for barrier options with time-dependent parameters, Quantitative Finance, 3 (2003), 98-107. doi: 10.1088/1469-7688/3/2/304.

[13]

R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143.

[14]

M. Nalholm and R. Poulsen, Static hedging and model risk for barrier options, Journal of Future Markets, 26 (2006), 449-463. doi: 10.1002/fut.20199.

[15]

M. F. M. Osborne, Brownian motion in the stock market, Operations Research, 7 (1959), 145-173. doi: 10.1287/opre.7.2.145.

[16]

S. Trippi, The mathematics of barrier options, Advances in Options and Futures, 7 (1994), 150-172.

[17]

R. Poulsen, Barrier options and their static hedges: simple derivations and extensions, Quantitative Finance, 6 (2006), 327-335. doi: 10.1080/14697680600690331.

[18]

S. P. Zhu and W. Chen, Pricing Parisian and Parisian options analytically, Journal of Economic Dynamics and Control, 37 (2013), 875-896. doi: 10.1016/j.jedc.2012.12.005.

[19]

S. P. ZhuT. N. LeW. Chen and X. Lu, Pricing Parisian down-and-in options, Applied Mathematics Letters, 43 (2014), 19-24. doi: 10.1016/j.aml.2014.10.019.

Figure 1.  Down-and-Out Power Option with ECB: Different $\delta$
Figure 2.  Down-and-Out Power Option with ECB: Different $K$
Figure 3.  Down-and-Out Power Option with ECB: Different $B$
Figure 4.  Price Comparisons: Power Call, Down-and-Out Power Barrier and Down-and-Out Power Option with ECB
Table 1.  Prices of DOPC with ECB with different curvature, $\delta$
$K$ $\delta$ $B$ $DOPC^{ECB}$
100 0.02 75 18.9037
100 0.05 75 18.7122
100 0.1 75 18.3433
100 0.2 75 17.3732
$K$ $\delta$ $B$ $DOPC^{ECB}$
100 0.02 75 18.9037
100 0.05 75 18.7122
100 0.1 75 18.3433
100 0.2 75 17.3732
Table 2.  Prices of DOPC with ECB with different strike price, $K$
$K$ $\delta$ $B$ $DOPC^{ECB}$
100 0.02 75 18.9037
125 0.02 75 10.5658
150 0.02 75 5.6286
200 0.02 75 1.5268
$K$ $\delta$ $B$ $DOPC^{ECB}$
100 0.02 75 18.9037
125 0.02 75 10.5658
150 0.02 75 5.6286
200 0.02 75 1.5268
Table 3.  Prices of DOPC with ECB with different barrier level, $B$
$K$ $\delta$ $B$ $DOPC^{ECB}$
100 0.02 55 20.5649
100 0.02 65 20.3056
100 0.02 75 18.9037
100 0.02 85 14.6799
100 0.02 90 10.9949
100 0.02 95 6.1064
$K$ $\delta$ $B$ $DOPC^{ECB}$
100 0.02 55 20.5649
100 0.02 65 20.3056
100 0.02 75 18.9037
100 0.02 85 14.6799
100 0.02 90 10.9949
100 0.02 95 6.1064
Table 4.  Price Comparisons$:$ Power Call, Down-and-Out Power Barrier, and Down-and-Out Power Option with ECB
$K$ $PC$ $DOPC$ $DOPC^{ECB}$
100 20.5851 19.0205 18.9037
125 11.0876 10.6020 10.5658
150 5.7955 5.6402 5.6286
200 1.5459 1.5279 1.5268
250 0.4213 0.4195 0.4194
$K$ $PC$ $DOPC$ $DOPC^{ECB}$
100 20.5851 19.0205 18.9037
125 11.0876 10.6020 10.5658
150 5.7955 5.6402 5.6286
200 1.5459 1.5279 1.5268
250 0.4213 0.4195 0.4194
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