J Math Anal Appl 389 (01 968 978 Contents lists available at SciVerse Scienceirect Journal of Mathematical Analysis and Applications wwwelseviercom/locate/jmaa Cross a barrier to reach barrier options oobae Jun, Hyejin Ku epartment of Mathematics and Statistics, York niversity, oronto, ON, Canada article info abstract Article history: Received 30 October 010 Available online 8 ecember 011 Submitted by JA Filar Keywords: Barrier option Hitting time Reflection principle his paper studies a new type of barrier options a regular barrier option comes into existence in the event that the underlying asset price first crosses specified barrier levels We derive closed form formulas for the prices via the reflection principle and provide numerical results to illustrate the properties of our solutions with respect to option parameters 011 Elsevier Inc All rights reserved 1 Introduction Barrier options are a widely used class of path-dependent derivative securities hese options knock in or knock out when the price of the underlying asset crosses a certain barrier level For example, an up-and-in call option gives the option holder the payoff of a call if the price of the underlying asset reaches a higher barrier level during the option s life, and it pays off zero unless the asset price reaches that level For an up-and-out call, the option becomes worthless if the underlying asset price hits a higher barrier, and its payoff at expiration is a call otherwise Options with a lower barrier level are said to be down-and-in and down-and-out options Merton [6] has derived a down-and-out call price by solving the corresponding partial differential equation with some boundary conditions Rubinstein and Reiner [10] published closed form pricing formulas for various types of single barrier options Rich [9] also provided a mathematical framework to value barrier options In these papers, the underlying asset price is monitored with respect to a single constant barrier for the entire life of the option ue to their popularity in a market, more complicated structures of barrier options have been studied by a number of authors Kunitomo and Ikeda [5] derived a pricing formula for double barrier options with curved boundaries as the sum of an infinite series Geman and Yor [1] followed a probabilistic approach to derive the Laplace transform of the double barrier option price Heynan and Kat [3] studied so-called partial barrier options the underlying price is monitored for a part of the option s lifetime For theses options, either the barrier disappears at a specified date strictly before the maturity (ie, early ending option or the barrier appears at a fixed date strictly after the start of the option (ie, forward starting option In the paper, the authors gave valuation formulas for partial barrier options in terms of bivariate normal distribution functions As a natural variation on the partial barrier structure, window barrier options have become popular wit h investors, particularly in foreign exchange markets For a window barrier option, a monitoring period for the barrier commences at the forward start date and terminates at the early ending date (We refer to Hui [4] and Guillaume [] However, all these papers are concerned with barrier options monitoring of the barrier starts at a predetermined date * Corresponding author Fax: +1 416 736 5757 E-mail addresses: dbjun@skkuedu ( Jun, hku@mathstatyorkuca (H Ku 00-47X/$ see front matter 011 Elsevier Inc All rights reserved doi:101016/jjmaa0111038
Jun, H Ku / J Math Anal Appl 389 (01 968 978 969 his paper concerns barrier options monitoring of the barrier starts at random time when the underlying asset price first crosses a certain barrier level For these options, one can consider as a secondary barrier option that is given to a primary barrier option holder in the event of the primary barrier being crossed Furthermore, this paper studies barrier options monitoring for the barrier commences at time when the underlying asset price first crosses two barrier levels in a specified order Interestingly, these options can be seen as three standard barrier options that are chained together Options with having the similar features to the former (simple case was discussed in Pfeffer [8], the price is computed by Laplace transforms through conditioning on the hitting time However, the technique adopted in [8] cannot be applied well to the latter case of this paper, in which two hitting times are involved to activate a barrier option In this paper, we derive closed form valuation formulas for various barrier options of this new type by applying the reflection principle and Girsanov s heorem in a proper way Also, the methodology we develop in this paper is easily applicable to more complicated structure, more than two hitting times are chained together to activate barrier options his paper is organized as follows Section presents a valuation formula for a down-and-in call option (IC u activated at time when the underlying asset price hits a higher barrier level he prices of the options with knock-out barrier are discussed Section 3 gives a valuation formula of an up-and-in call option (IC ud which is activated at time when the asset price crosses two barrier levels (an up-barrier followed by a down-barrier he case of knock-out options is also treated Section 4 shows the numerical results of six graphs explaining the properties of the prices IC u and IC ud with respect to option parameters he pricing formulas for an up-and-in call option reached by crossing a down-barrier (IC d and a down-and-in call option reached by crossing a down-barrier followed by an up-barrier (IC du aregiveninappendicesa and B Case of crossing a barrier Let r be the risk-free interest rate and > 0 be a constant We assume the price of the underlying asset S follows a geometric Brownian motion S t exp(μt + W t μ r and W t is a standard Brownian motion under the risk-neutral probability P Let X t 1 ln(s t/ We define the minimum and maximum for X t to be ma b inf (X t, t [a,b] Ma b sup (X t t [a,b] and denote by E m the expectation operator under the m-measure Consider a European call expiring at with strike price K Wefixanup-barrier (> and a down-barrier (< We define k 1 ln(k /, u 1 ln(/ and d 1 ln(/ Now we provide the valuation formula for a down-and-in call option commencing at time when the asset price hits the up-barrier under the assumption of K > ; ForthecaseofK, see heorem heorem 1 Suppose K > he knock-in call option value at time 0, IC u, which is activated at time τ min{t: S t, > } is IC u ( z 1 1 ln μ z 1 e r K( ( K + μ, μ z 1 μ r +, is the underlying asset spot value at time 0 beyond the down-barrier and x is the cumulative standard normal distribution function Proof he knock-in call option value at time 0 is given by the discounted expected value of its payoff under the risk-neutral measure hus IC u [ e r E P (S K + 1 {m τ d, τ Sτ }] [ ] e r E P (S K1 {m τ d,s >K, τ Sτ } 1 {} is an indicator function Let us define a new measure P such that d P dp e 1 + W
970 Jun, H Ku / J Math Anal Appl 389 (01 968 978 hen, IC u P( m τ d, S > K, τ S τ e r K P ( m τ d, S > K, τ S τ It suffices to calculate the required probability under the P -measure: a simple change of drift from μ to μ will provide the required probability under the P -measure Note that P ( m τ d, S > K, τ S τ P ( m τ d, X > k, τ, X τ u X t W t + ( μ t is a standard Brownian motion under the equivalent measure Q,definedby [ dq dp exp μ W 1 ( μ ] Let us introduce a process Xt, t [0, ], defined by the formula { Xt (t τ Xt u X t (t > τ By virtue of the reflection principle, the process Xt also follows a standard Brownian motion under Q hen P ( m τ d, X > k, τ, X τ u E P [1 {m τ d, X >k, τ, Xτ u} ] [ dp E Q dq 1 {mτ d, X >k, τ, Xτ u} [ E Q μ e X 1 μ ] 1 {m τ d, X >k, τ, Xτ u} [ E Q μ e (u X 1 μ ] 1 { M τ u d, X <u k, τ } ] (1 ( (3 (4 (5 M τ sup t [τ, ] ( Xt Since u d > u, { M τ u d, τ }{ M 0 u d} hus, ( P mτ d, X > k, τ, X τ u [ E Q μ e (u X 1 μ ] 1 { M 0 u d, X <u k} We apply the reflection principle again Let us introduce a process Xt, t [0, ], defined by the formula Xt { Xt (t τ (u d Xt (t > τ τ min{t > τ : Xt u d} By virtue of the reflection principle, the process Xt also follows a standard Brownian motion under Q and P ( m τ d, X > k, τ, X τ u [ E Q μ e ( u+d+ X 1 μ ] 1 { X >u d+k} e μ ( u+d [ E Q μ e X 1 μ 1 { X >u d+k}] Let us define an equivalent probability measure Q by setting d Q dq μ e X 1 μ so that the process Wt Xt μ t, t [0, ], follows a standard Brownian motion under Q e μ ( u+d [ E Q μ e X 1 μ ] 1 { X >u d+k} e μ ( u+d Q ( X > u d + k e μ ( u+d Q ( W > u d + k μ (6 (7 (8
Jun, H Ku / J Math Anal Appl 389 (01 968 978 971 e μ ( u+d N ( u + d k + μ (9 ( μ 1 ln ( K In the measure P, we follow the same process to obtain P( m τ d, X > k, τ, X τ ( μ u 1 ln herefore IC u ( μ 1 ln + μ (10 ( K + μ e r K ( ( K + μ μ 1 ln ( K + μ heorem Suppose K he knock-in call option value at time 0,IC u, which is activated at time τ min{t: S t, > } is IC u [( [( e r K z 1 ( S0 ln z 4 1 ln μ z + ( ( KS0 μ z 3 ( μ z + ( + μ μ, z3 1 ln μ ] z 4 μ z 3 + ( ( S0 μ, Proof We start from (1 in the proof of heorem 1 ( P mτ d, X > k, τ, X τ u ( P mτ d, X > d, τ, X τ u + ( P mτ d, k < X d, τ, X τ u By Eq (10, ( P mτ d, X > d, τ, X τ ( μ ( u 1 S0 ln + μ From (A89 in Musiela and Rutkowski [7, p 653], we obtain ( P mτ d, k < X d, τ, X τ u P(X τ u, τ, k < X d e μ {P (X u u d + μ P (X u k + μ } ( μ { 1 ln ( S0 μ ( 1 N ln ( KS0 μ z 4 + ] μ } Remark 3 o value the knock-out call (down-and-out option at time 0, OC u, which is activated at time τ min{t: S t, > }, we apply the knock-in knock-out parity relation So, we subtract IC u from the up-and-in call price IC to get OC u IC IC u he valuation formulas for an up-and-in call (IC d and an up-and-out call (OC d activated in the event that the asset price first hits the down-barrier are provided in Appendix A
97 Jun, H Ku / J Math Anal Appl 389 (01 968 978 3 Case of crossing two barrier levels We derived, in the previous section, the pricing formulas for barrier options commencing at time when the asset price crosses a specified barrier level In this section, we consider barrier options activated in the event that the asset price crosses two barrier levels in a specified order, ie, hits the up-barrier followed by reaching the down-barrier, or vice versa he following theorem presents the valuation formula for an up-and-in call option reached by crossing the down-barrier after crossing the up-barrier under the assumption of K < heorem 3 gives the formula for the case of K heorem 31 Suppose K < he knock-in call option value at time 0, IC ud, which is activated at time τ min { t > τ 1 : S t, τ 1 min{t > 0: S t, > } } is IC ud [( z 5 1 ln z 7 1 ln [( e r K μ z 5 + ( ( 3 ( K 4 μ z 6 ( μ z 5 ( + + μ, z6 1 ln μ μ ] z 7 μ z 6 + ( ( 3 μ, μ z 7 + ] is the underlying asset spot value at time 0 beyond the down-barrier, and x is the cumulative standard normal distribution function Proof nder the risk-neutral measure, the knock-in call option value at time 0 is IC ud [ e r E P (S K + ] 1 {M τ u, τ 1 <τ, Sτ 1, Sτ } [ e r E P (S K1 {M τ u, S >K, τ 1 <τ, Sτ 1, Sτ }] Let us define a new measure P such that d P dp e 1 + W hen, we have IC ud P( M τ u, S > K, τ 1 < τ, S τ1, S τ ( e r K P Mτ u, S > K, τ 1 < τ, S τ1, S τ We calculate the required probability only under the P -measure as in the proof of heorem 1 Note that P ( M τ u, S > K, τ 1 < τ, S τ1, S τ P ( M τ u, X > k, τ 1 < τ, X τ1 u, X τ d X t W t + ( μ t is a standard Brownian motion under the equivalent probability measure Q,definedby [ dq dp exp μ W 1 ( μ hen, ] P ( M τ u, X > k, τ 1 < τ, X τ1 u, X τ d E Q [ dp dq 1 {M τ u, X >k, τ 1 <τ, Xτ 1 u, Xτ d} E Q [ e μ X 1 μ 1 {M τ u, X >k, τ 1 <τ, Xτ 1 u, Xτ d}] ]
Jun, H Ku / J Math Anal Appl 389 (01 968 978 973 Let us introduce a process Xt, t [0, ], defined by the formula { Xt (t τ 1 Xt u X t (t > τ 1 By virtue of the reflection principle, the process Xt also follows a standard Brownian motion under Q, and [ E Q μ e X 1 μ ] 1 {M τ u, X >k, τ 1 <τ, Xτ 1 u, Xτ d} [ E Q μ e (u X 1 μ ] 1 { m τ u, X <u k, τ, Xτ u d} m τ inf t [τ, ]( Xt Here, we apply the reflection principle again Let us introduce a process Xt, t [0, ], defined by the formula Xt { Xt (t τ (u d Xt (t > τ hen, the process Xt also follows a standard Brownian motion under Q, and [ E Q μ e (u X 1 μ ] 1 { m τ u, X <u k, τ, Xτ u d} [ E Q μ e ( u+d+ X 1 μ ] 1 { M τ 3u d, X >u d+k, τ } M τ sup t [τ, ]( Xt Since 3u d > u d > u, { M τ 3u d, τ }{ M 0 3u d} hus, [ E Q μ e ( u+d+ X 1 μ ] 1 { M τ 3u d, X >u d+k, τ } e μ ( u+d [ E Q μ e X 1 μ ] 1 { M 0 3u d, X >u d+k} Let us define an equivalent probability measure Q by setting d Q dq μ e X 1 μ so that the process Wt Xt μ t, t [0, ], follows a standard Brownian motion under Q e μ ( u+d [ E Q μ e X 1 μ ] 1 { M 0 3u d, X >u d+k} e μ ( u+d Q ( M 0 3u d, X > u d + k e μ ( u+d[ Q ( M 0 3u d Q ( M 0 3u d, X u d + ] k From (A9 and (A90 in Musiela and Rutkowski [7, p 655], we obtain Q ( M 0 3u d Q ( X 3u d + e μ (3u d Q ( X 3u d + μ Q ( W 3u d μ + e μ (3u d Q ( W 3u d + μ and N N ( 3u + d ( 1 ln + μ ( 3 ( + e μ (3u d 3u + d N μ + μ ( ( + e μ (3u d 1 N ln μ 3 Q ( M 0 3u d, X u d + ( k e μ (3u d 4u + d + k N μ ( K e μ (3u d N ( 1 ln 4 μ
974 Jun, H Ku / J Math Anal Appl 389 (01 968 978 herefore, P ( M τ u, X > k, τ 1 < τ, X τ1 u, X τ d ( μ 1 ( ( ln 3 μ 1 ln + μ ( μ + 1 ( K 4 In the measure P,wegetthesimilarresults μ P( M τ u, X > k, τ 1 < τ, X τ1 u, X τ d ( μ 1 ( ( ln 3 μ 1 ln + μ ( μ + 1 ( K 4 μ By combining the results together, we complete the proof ( ln 3 ( ln 3 μ μ If the strike price K is greater than or equal to the up-barrier, the payoff of the call is zero unless the asset price at expiry is greater than the up-barrier Since the asset price must reach the up-barrier after crossing the down-barrier before expiry for nonzero payoff, IC ud is equal to IC u in this case heorem 3 Suppose K he knock-in call option value at time 0, IC ud, which is activated at time τ min { t > τ 1 : S t, τ 1 min{t > 0: S t, > } } is IC ud ( μ z 1 e r K( μ z 1 Remark 33 he knock-out call option value at time 0, OC ud, which is activated at time τ min{t > τ 1 : S t, τ 1 min{t > 0: S t, > }} is calculated as follows: OC ud IC u IC ud he valuation formulas for a down-and-in call (IC du and a down-and-out call (OC du activated in the event of crossing the down-barrier followed by crossing the up-barrier are provided in Appendix B 4 Numerical results In this section, we illustrate the properties of our solutions obtained in Sections and 3 Fig 1 shows how the price IC u changes when the volatility and strike price vary he volatility increases from 01 to 05 and the strike price decreases from 10 to 80 under the assumption that 100, 110, 90, r 005 and 05 We see that the option price IC u increases as the volatility increases or the strike price decreases Since the probability of the asset price hitting given barriers gets bigger as the volatility increases, the knock-in option value naturally increases Fig concerns the price IC ud, and shows the same properties as IC u in Fig 1 Fig 3 shows the comparison of IC u and IC ud with the same parameters as in Figs 1 and We observe IC ud is always lower than IC u as expected Fig 4 illustrates the changes of IC ud when the up-barrier and down-barrier vary With the parameters 100, K 100, r 005, 03, 05, changes between 100 and 10, and moves between 80 and 100 We observe from Fig 4, IC ud approximates to zero as both and drifts farther from the underlying asset price at time 0 If and are set to be 100, then IC ud is equal to the vanilla call option price, which corresponds to the highest point in Fig 4 Fig 5 shows the prices IC u and IC ud for different underlying asset prices Finally, Fig 6 represents that simulation results, up to 10,000 paths of the underlying asset price process, converge to the exact value obtained in Section 3 We simulate the monitoring frequency from 1000 to 10,000 under the assumption that 100, K 100, 110, 90, r 005, 03, 05 hen the exact value from heorem 31 is 0146 and the simulation value of IC ud is 014 with monitoring frequency of 10,000 In general, a standard Monte Carlo Method systematically underprice the knock-in call option (see Geman and Yor [1] he reason is that the underlying asset price
Jun, H Ku / J Math Anal Appl 389 (01 968 978 975 Fig 1 IC u result, varying K and (option parameters: 100, 110, 90, r 005, and 05 Fig IC ud result, varying K and (option parameters: 100, 110, 90, r 005, and 05 Fig 3 Comparison between IC ud and IC u,varyingk and (option parameters: 100, 110, 90, r 005, and 05
976 Jun, H Ku / J Math Anal Appl 389 (01 968 978 Fig 4 IC ud results when varies between 100 and 10, and varies between 80 and 100 (option parameters: 100, K 100, r 005, 03, and 05 Fig 5 Comparison between IC ud and IC u, varying (option parameters: 110, 90, K 100, r 005, 03, and 05 Fig 6 Monte Carlo simulation results of IC ud using Antithetic Variates method when monitoring frequency is increased from 1000 to 10,000 ( 100, K 100, 110, 90, r 005, 03, 05, sample paths 10,000, exact value 0146 and the value of IC ud is 014 when monitoring frequency is 10,000
Jun, H Ku / J Math Anal Appl 389 (01 968 978 977 is checked at discrete instants through simulations In fact, the barrier might have been crossed without being detected o overcome such difficulties, we applied the Antithetic Variates, a Variance Reduction Method of Monte Carlo Method, and simulated the monitoring frequency until 10,000 5 Conclusion In this paper, we derived closed-form valuation formulas for barrier options of a new type the underlying asset price should cross a specified barrier level to activate a regular barrier option hese options are popular in the over-thecounter equity and foreign exchange derivative markets We further derived explicit valuation formulas for barrier options activated at time the underlying asset price first crosses two barrier levels in a specified order ue to this contribution, one can price various knock-in and knock-out options of this type We applied the reflection principle repeatedly on the barrier crossing times in a proper way A great advantage of our methodology is that it can be easily applied to the case of more complicated structure further crossing events are chained to activate a barrier option We also presented the graphs illustrating the properties of the prices, and showed simulation results to confirm the accuracy of our solution Acknowledgments his research has been supported by MIACS and NSERC Appendix A A1 Suppose K < he knock-in call option value at time 0, IC d, which is activated at time τ min{t: S t, < } is [( μ ( IC d z 8 + [( e r K z 8 1 ( ln z 10 1 ln ( K μ μ z 9 ( z 8 + ( + μ, z9 1 μ ln μ μ ] z 10 z 9 + ( ( μ, μ z 10 + ] A Suppose K he knock-in call option value at time 0, IC d, which is activated at time τ min{t: S t, < } is IC d ( μ z 11 e r K( z 11 1 ( ln + μ K μ z 11 A3 he knock-out call option value at time 0, OC d, which is activated at time τ min{t: S t, < } is given by Appendix B OC d IC IC d B1 Suppose K > he knock-in call option value at time 0, IC du, which is activated at time τ min{t > τ 1 : S t, τ 1 min{t > 0: S t, < }} is IC du ( μ z 1 e r K( z 1 1 ( 4 ln + μ K μ z 1
978 Jun, H Ku / J Math Anal Appl 389 (01 968 978 B Suppose K he knock-in call option value at time 0, IC du, which is activated at time τ min{t > τ 1 : S t, τ 1 min{t > 0: S t, < }} is [( μ ( μ ( μ ] IC du z 13 + z 14 z 10 [( e r μ K z 13 ( μ + z 14 + ( μ z 10 + ] z 13 1 ( 3 ln + μ, z14 1 ( 3 ln μ B3 he knock-out call option value at time 0, OC du, which is activated at time τ min{t > τ 1 : S t, τ 1 min{t > 0: S t, < }} is obtained as follows: References OC du IC d IC du [1] H Geman, M Yor, Pricing and hedging double-barrier options: A probabilistic approach, Math Finance 6 (1996 365 378 [] Guillaume, Window double barrier options, Rev eriv Res 6 (003 47 75 [3] RC Heynen, HM Kat, Partial barrier options, J Financ Engrg 3 (1994 53 74 [4] CH Hui, ime-dependent barrier optionvalues, J Futures Markets 17 (6 (1997 667 688 [5] N Kunitomo, M Ikeda, Pricing options with curved boundaries, Math Finance (199 75 98 [6] RC Merton, heory of rational option pricing, Bell J Econ Manag Sci 4 (1973 141 183 [7] M Musiela, M Rutkowski, Martingale Methods in Financial Modelling, second ed, Springer, 005 [8] Pfeffer, Sequential barrier options, Algo Res Quarterly 4 (3 (001 65 73 [9] Rich, he mathematical foundations of barrier option pricing theory, Adv Futures Options Res 7 (1997 [10] E Reiner, M Rubinstein, Breaking down the barriers, Risk 4 (8 (1991 8 35