Rubinstein, M. and E. Reiner, 1991, Breaking Down the Barriers, RISK 4, 28-35.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is breached at any instant in the life of the option. Under this assumption, Merton (1973) obtained a formula for pricing a knock out call. Subsequent work on pricing continuously monitored barrier options includes Heynen and Kat (1994a, 1994b) Kunitomo and Ikeda (1992) Rich (1994a, 1994b) and Rubinstein and Reiner (1991). However, a sizable portion of real contracts with barrier features specify fixed times for monitoring of the barrier typically, daily closings. One article in the trade literature (Derivatives Week 1995a) faults existing pricing models for not addressing Manuscript received September 1995; ....

....the substitution x = y 2a) # # T ) to get (3.3) The price of a down and out call can be obtained by subtracting the down and in price from the standard Black Scholes price. For a comprehensive treatment of the other cases and of connections among the various prices, see Rich (1994) and Rubinstein and Reiner (1991). We now write V (S 0 e a ) for V (H) and calculate the derivative of the continuous option price with respect to a: PROPOSITION 3.2. For a down and in call, # #a V (S 0 e a ) e rT 2 # T # e 2a # 2 Z # [log(K S 0 ) # # T (3.4) 1 x # T # 2a # # T S ....

RUBINSTEIN, M., and E. REINER (1991): "Breaking Down the Barriers," RISK, 4 (September), 28--35.

....of [0, T ] and percentage lookbacks, 4 in which the minimum or maximum price of the underlying is multiplied by a constant in the usual lookback payoff. The continuous versions of all of these options can be priced in closed form. See, in particular, Merton [48] Rubinstein and Reiner [54], Chance Discrete path dependent options 59 [18] Boyle and Lau [10] Rich [51] Carr [16] and Heynen and Kat [31, 32] for various kinds of barrier options, and for various kinds of lookbacks see Conze and Viswanathan [23] Garman [27] Goldman, Sosin, and Gatto [29] Goldman, Sosin, and Shepp ....

Rubinstein, M., Reiner, E.: Breaking down the barriers. Risk 4, 28--35 (September, 1991)

....for background on option pricing. In the case of barrier options, this payo# is discontinuous over the space of all paths of the underlying variables. In su#ciently simple cases, there are analytical formulas for the price (e.g. 1 Merton [27] Kunitomo and Ikeda [24] Rubinstein and Reiner [30], and Sidenius [31] However, there will not be useful formulas if the specification of the barrier or stochastic processes used to model the underlying variables is too complex or high dimensional. Consequently, it is often necessary to price via simulation, which is better suited to ....

Rubinstein, M., and E. Reiner. "Breaking down the barriers," Risk, 4, pp. 28-35 (September 1991).

....put option can be written as follows: P uo #S 0 ;K#=p uo #S 0 ;K# P p uo #S 0 ;K# #11# where p uo and P p uo are the prices of the corresponding European option and the early exercise premium respectively. Speci#cally, the price of the European #up and out put option can be written as #see Rubinstein and Reiner #1991##: p uo #S 0 ;K# = p o #S 0 ;K#,p ui #S 0 ;K# = p o #S 0 ;K#,#H=S 0 # 2#,2 p o #H 2 =S 0 ;K# #12# where p o #x; K# denotes the Black Scholes price of a standard European put option with current underlying price x and strike price K, and p ui ### represents the price of a European ....

....via a reparametrization of the pricing formula #4.4# in Broadie and Detemple #1997# for American #capped exchange options with proportional cap. 16 2 Notice that both #30# and #31# apply also to the case of zero dividend. The explicit formula for the European price c uo #t# can be found in Rubinstein and Reiner #1991#. Since # H is, in general, random, the expectation in #31# might not be always carried out explicitly. The intuition behind Theorem 5 can be conveyed by considering three cases for the location of the barrier level H relative to the level of the boundary B c . # Case #a#: H # B c t 8t 2 ....

Rubinstein, M. and E. Reiner, 1991, #Breaking Down the Barriers," RISK, 4, 28-35.

....a binomial formula for which a limit result is derived. Conze, Viswanathan [1991] define several barrier options and derive exact replication and valuation formulas using the reflection principle in continuous time. In addition they derive some results for the corresponding American type options. Rubinstein, Reiner [1991] list continuous time formulas for all the eight different barrier options. Recently, Boyle and Sok Hoon Lau [1994] have pointed out the irregularities in the convergence of prices of barrier options in binomial lattices which we mentioned above. They solved this difficulty by extracting a subset ....

....call option in the case of K H. 2. For K H the first two terms of (15) are equal to the arbitrage price of a standard European put option. In this situation the correction terms corresponds to the arbitrage price of a European up and in call option. 5 see Cox, Rubinstein [1985] and Rubinstein, Reiner [1991] 6 For completeness, the remaining limit formulae are given in the appendix (Proposition 4) A DISCRETE TIME APPROACH FOR EUROPEAN AND AMERICAN BARRIER OPTIONS 7 4. Binomial approximation The binomial formulae for barrier options cover only cases where the barrier H is exactly an endpoint of ....

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Rubinstein, M.; E. Reiner [1991]: Breaking Down the Barriers, RISK, September 1991.

....Although its option premium is more expensive than the regular down andout put premium, it is still cheaper than the ordinary put, which is attractive to the investor. The length of the barrier period can be fit to a particular view on the USD JPY exchange rate in a certain period of time. Cox and Rubinstein (1985, pp. 408 411) suggested that a downand out call is used to value a bond with a safety covenant. Hudson (1992) pointed out that Nikkei linked bonds with embedded short European up and out puts were marketed with great success to Japanese investors. In some issued bonds, the embedded knock out options ....

Rubinstein, M., and Reiner, E. (1991a): "Breaking Down the Barriers," Risk Magazine, 8:28--35.

....r is the risk free interest rate and # is the volatility of the stock. For some path dependent contingent claims on a single asset, analytic solutions 28 can be formulated as well. Goldman et al. 1979) and Conze and Viswanathan (1991) study lookback options. Conze and Viswanathan (1991) Rubinstein and Reiner (1991), Kunitomo and Ikeda (1992) Geman and Yor (1996) Roberts and Shortland (1997) among many others, examine the case of barrier options. Turnbull and Wakeman (1991) price average rate options by applying an Edgeworth series expansion to approximate the distribution of the average. The numerical ....

Rubinstein, M., and E. Reiner (1991): "Breaking Down the Barriers," RISK, 4, No. 8.

....up the valuation dramatically as well as give new insightinto barrier option valuation. 1 Analytical valuation of American barrier options Closed form solutions and valuation techniques for standard European barrier options are well known from the literature, see for instance #Merton 1973, Reiner and Rubinstein 1991, Rich 1994, Haug 1997#. No closed form solution for American barrier options exists in the extant literature. The technique used to value American barrier options have therefore been numerical methods. Lattice models have been especially popular. Without doing any adjustments lattice models have ....

....options analytically. 2 Numerical comparison In this section we will compare some well known methods for barrier option valuation with our closed form solution method. Table 1 compares European barrier option values. Column one is calculated using the closed form barrier formulas derived by Reiner and Rubinstein #1991#. Column two is calculated using the formula of Black #1976# in combination with the re#ection principle. As expected, these two columns contain identical values. Column three and four contain values calculated using a trinomial tree without any adjustments. It is evident that using a tree ....

Reiner, E., and M. Rubinstein #1991#: #Breaking Down the Barriers," Risk Magazine, 4#8#.

....possibility in the options market 2 . 2 Barrier Put Call Transformation Barrier options have become extremely popular and certainly constitutes one of the most popular class of exotic options. Closed form solutions for valuation of single barrier options have been published by Merton #1973#, Reiner and Rubinstein #1991#, and Rich #1994#, and for double barrier options byIkeda and Kuintomo #1992#, and Geman and Yor #1996#. Further, the relationship between in and out options are well known as the in out barrier parity. A long out option is equal to a long plain vanilla option plus short an in option. in this ....

Reiner, E., and M. Rubinstein #1991#: #Breaking Down the Barriers," Risk Magazine, 4#8#.

....have the same the barrier value X) is equivalent to an option which has no barrier at all: Out Barrier Option In Barrier Option = Option without Barrier: This relation holds for both call option and put option, for both European type and American type and for arbitrary payoff function. See [42] [67] and [68] for previous work on barrier options. In Section 2.1, we present the analytical solutions of European barrier options for a general payoff function. In Section 2.2, we present the analytical approximate solutions for American barrier options. Our approach follows the quadratic ....

M. Rubinstein and E. Reiner. Breaking Down the Barrier. RISK, September 1991.

....value to stock prices above the barrier: DIV (S; H) V (S; S H: 3.3 Derivation from Pricing Formula In this section, we derive the static hedge in another manner. We suppose that a pricing formula for a barrier security is known, either because it exists in the literature (see e.g. [17]) or because it has been derived using dynamic replication arguments. We then show how this formula can be used to generate a static hedge using vanilla options. The advantage of approaching static hedging in this manner is that it is very simple and the approach can be used to generate static ....

....K H j p Gamma2 (p Gamma 1) i p Gamma2 K Gamma pKc H 2 j dK puts at strike K for K H 2 Kc . To show how this approach can be used to generate adjusted payoffs for other securities, consider the valuation of an American binary put paying a dollar at the first passage time to H. From [17], the valuation formula is: ABP (S; H) S H fl ffl N 0 ln i H S j Gamma ffloe 2 oe p 1 A S H fl Gammaffl N 0 ln i H S j ffloe 2 oe p 1 A ; for S H, where fl j 1 2 Gamma r Gammad oe 2 ; ffl j q fl 2 2r oe 2 : Removing the ....

Rubenstein M. and E. Reiner, 1991, "Breaking down the Barriers", Risk, 4, 8.

....and Reneby (1996) and Rich (1996) among many others) The academic literature on the pricing of barrier options dates back at least to Merton (1973) who presented a closed form solution for the price of a continuously monitored down and out European call. More recently, both Rich (1991) and Rubinstein and Reiner (1991) provide pricing formulas for a variety of standard European barrier options (i.e. calls or puts which are either up and in, up and out, down and in, or down and out) More exotic variants such as partial barrier options and rainbow barrier options have been explored by Heynen and Kat (1994a, ....

Rubinstein, M. and E. Reiner (1991). Breaking down the barriers. RISK 4 (September), 28--35.

....can also apply the second method. The portfolio of options with maturity T 2 has payoff (see Figure 4) g out (S 2 ) ae 1 if S 2 H, Gamma Gamma S2 H Delta p if S 2 H, 9 A no touch binary pays one dollar at maturity if the barrier has not been hit. 10 See Reiner and Rubinstein[19] for the formula. 80 85 90 95 100 105 110 115 120 1.5 1 0.5 0 0.5 1 1.5 Spot at Forward Start Time Adjusted Payoff at Forward Start Time 80 85 90 95 100 105 110 115 120 1.5 1 0.5 0 0.5 1 1.5 Spot at Maturity Adjusted Payoff at Maturity (r = 0:05, d = 0:03, oe = 15, H = 100, T 2 Gamma T ....

Reiner E., and M. Rubinstein, 1991, "Breaking Down the Barriers", Risk.

....at time t h is the solution ABC(S; t) to the Black Scholes p.d.e. 35) restricted to the domain S 2 (0; H) t 2 [0; T ] and subject to the terminal condition: ABC(S;T ) 0; S 2 (0; H) 37) and boundary conditions: ABC(0; t) 0 ABC(H; t) 1; t 2 [0; T ] 38) The solution is given in [13]. From (36) the number of shares held at t h is S ABC(S t ; t) Since this varies with t, the hedging strategy is dynamic. To uncover a static hedge, we would like to use the linearity principle globally, as was done in the previous section. However, we have already seen in (29) that if ....

....the payoff implicit in (56) is recovered: ABC(S;T ) S H fl ffl S H fl Gammaffl # 1(S H) 58) As shown in Carr and Chou[5] this suggests a quick way to uncover a static hedge for an American binary put, paying a dollar at the first passage time to a lower barrier L. From [13], the valuation formula in terms of the stock is: ABP (S; t) S L fl ffl N ln Gamma L S Delta Gamma ffloe 2 (T Gamma t) oe p T Gamma t S L fl Gammaffl N ln Gamma L S Delta ffloe 2 (T Gamma t) oe p T Gamma t ; 59) for S L and t ....

Rubenstein M. and E. Reiner, 1991, "Breaking down the Barriers", Risk, 4, 8.

.... approaches for other nancial instruments like stocks or options can be found in serveral papers of the Sonderforschungsbereich 303 at the university of Bonn like Reimer Sandmann [8] Sondermann [12] Further papers of Harrison Kreps [3] Karatzas [4] Kramkov [6] Rubinstein Reiner [9] R#dler [10,11] etc. Due to the risk the income of an annuity fund can be divided into two parts. The uncertain part depends on the stock prices, currencies, etc. and the certain part depends on the weighted average interest of all annuities in the fund. Let X n 1 be the income of the fund at time n 1 (future) ....

Rubinstein, M.; Reiner, E.: Breaking down the barriers, Risk, September 1991.

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Rubinstein, M. and E. Reiner, 1991, Breaking Down the Barriers, RISK 4, 28-35.

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Rubinstein, M. and E. Reiner(1991): Breaking down the barriers, Risk, September 1991, 28-35. 16

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Rubinstein, M. and Reiner, E. (1991a): "Breaking Down the Barriers", Risk, 8, 28-35.

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Rubinstein, M. and Reiner, E. (1991a) Breaking down the barriers, Risk Magazine , 4 ( 8 ), 28 35.

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