$K$ | $\delta$ | $B$ | $DOPC^{ECB}$ |
100 | 0.02 | 75 | 18.9037 |
100 | 0.05 | 75 | 18.7122 |
100 | 0.1 | 75 | 18.3433 |
100 | 0.2 | 75 | 17.3732 |
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: |
Table 1.
Prices of DOPC with ECB with different curvature,
$K$ | $\delta$ | $B$ | $DOPC^{ECB}$ |
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,
$K$ | $\delta$ | $B$ | $DOPC^{ECB}$ |
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,
$K$ | $\delta$ | $B$ | $DOPC^{ECB}$ |
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
$K$ | $PC$ | $DOPC$ | $DOPC^{ECB}$ |
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 |
J. Andreasen , Behind the mirror, Risk Magazine, 14 (2001) , 109-110. | |
L. Andersen , J. Andreasen and D. Eliezer , Static replication of barrier option: some general results, Journal of Computational Finance, 5 (2002) , 1-25. | |
F. Black and M. Scholes , The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973) , 637-654. doi: 10.1086/260062. | |
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. | |
P. Carr , K. Ellis and V. Gupta , Static hedging of exotic options, Journal of Finance, 5 (1998) , 1165-1190. doi: 10.1142/9789812812599_0005. | |
A. Chen and M. Suchanecki , Parisian exchange option, Quantitative Finance, 11 (2011) , 1207-1220. doi: 10.1080/14697680903194577. | |
J. C. Cox , S. 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. | |
R. J. Haber , P. Schönbucher and P. Wilmott , Pricing Parisian options, Journal of Derivatives, 6 (1999) , 71-79. | |
S. N. I. Ibrahim , J. 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. | |
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. | |
T. N. Le , X. Lu and S. P. Zhu , An analytical solution for Parisian up-and-in calls, ANZIAM Journal, 57 (2016) , 269-279. doi: 10.1017/S1446181115000267. | |
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. | |
R. C. Merton , Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973) , 141-183. doi: 10.2307/3003143. | |
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. | |
M. F. M. Osborne , Brownian motion in the stock market, Operations Research, 7 (1959) , 145-173. doi: 10.1287/opre.7.2.145. | |
S. Trippi , The mathematics of barrier options, Advances in Options and Futures, 7 (1994) , 150-172. | |
R. Poulsen , Barrier options and their static hedges: simple derivations and extensions, Quantitative Finance, 6 (2006) , 327-335. doi: 10.1080/14697680600690331. | |
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. | |
S. P. Zhu , T. N. Le , W. 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. |
Down-and-Out Power Option with ECB: Different
Down-and-Out Power Option with ECB: Different
Down-and-Out Power Option with ECB: Different
Price Comparisons: Power Call, Down-and-Out Power Barrier and Down-and-Out Power Option with ECB