The British Lookback Option with Fixed Strike

Transcription

1 The Britih Lookback Option with Fixed Strike Yerkin Kitapbayev Firt verion: 14 February 2014 Reearch Report No. 2, 2014, Probability and Statitic Group School of Mathematic, The Univerity of Mancheter

2 The Britih Lookback Option with Fixed Strike Yerkin Kitapbayev 1. Introduction We continue reearch of the new type of option called Britih that wa introduced in [11] and [12] by preenting the Britih lookback option with fixed trike. Thi paper generalie the work about the Britih Ruian option [4] and provide financial analyi of lookback option with fixed non-zero trike. The Britih holder enjoy the early exercie feature of American option whereupon hi payoff (deliverable immediately) i the bet prediction of the European lookback payoff under the hypothei that the true drift of the tock price equal a contract drift. We derive a cloed form expreion for the arbitrage-free price in term of the optimal topping boundary of twodimenional optimal topping problem with a caling trike and how that the rational exercie boundary of the option can be characteried via the unique olution to a nonlinear integral equation. We alo how the remarkable numerical example where the rational exercie boundary exhibit a dicontinuity. Uing thee reult we perform a financial analyi of the Britih lookback option with fixed trike which how that with the contract drift properly elected thi intrument provide not only an effective protection mechanim, but become a very attractive alternative to the claic European/American lookback option from peculator point of view and give high return when tock movement are favourable. The aim of thi paper i to examine the Britih payoff mechanim in the context of the lookback option with fixed trike and continue the reearch propoed in [4]. Thi mechanim provide it holder with a protection againt unfavourable cenario for tock price and i intrinically built into the option contract uing the concept of optimal prediction (ee e.g. [1]) and we refer to uch contract a Britih for the reaon outlined in [11] and [12], where the Britih put and call option were introduced. The main idea of the Britih feature i to ubtitute the true drift by a contract drift in the Black-Schole model and then it payoff i the bet prediction of the European lookback payoff. The mot intereting feature of thi protection mechanim a not only i the option buyer offered a protection againt unfavourable tock movement but alo when the price movement are favourable he will generally receive high return (ee [11] and [12] for detail). Following the rationale of the Britih put and call option, thi type of option wa extended to the path-dependant option in [3] and [4]. Particularly, the Britih Ruian option wa introduced and tudied in [4]. Herein we ue terminology the Ruian option for the lookback Mathematic Subject Claification Primary 91G20, 60G40. Secondary 60J60, 35R35, 45G10. Key word and phrae: Britih lookback option with fixed trike, American lookback option, arbitragefree price, fixed trike, optimal topping, geometric Brownian motion, parabolic free-boundary problem, local time-pace calculu, nonlinear integral equation. 1

3 option with zero trike which wa identified in [13] (ee lat paragraph of introduction in [4] for detailed explanation of thi terminology and it hitory). According to the financial theory (ee e.g. [15]) the arbitrage-free price of the option i a olution to an optimal topping problem with the gain function a the payoff of the option. The correponding optimal topping problem for the Britih Ruian option [4] wa originally a three-dimenional (time-proce-running maximum) and wa reduced fully to two dimenion uing Giranov theorem a in [14] and [7]. It wa hown that exerciing of the Britih verion provide very attractive return compared with exerciing of the American option and elling the European option (the latter can be conidered only in a liquid market). The final ection in [4] propoed to reearch different type of lookback option: (i) call and put; (ii) thoe with fixed (non-zero) or floating trike; (iii) thoe baed on the maximum or minimum; (iv) the weighting in the maximum or minimum may be equal or flexible. In thi paper we tudy the Britih lookback option with fixed trike (non-zero) of call type a we believe that thi i the mot intereting cae from a mathematical point of view. The paper include two part: analytical olution and financial analyi. The theoretical olution i baed on the method of a caling trike. It wa remarked above that the optimal topping problem for the lookback option are three-dimenional and in the cae of non-zero trike no full reduction to dimenion two appear to be poible. However we will illutrate the reduction to two-dimenional problem with a caling trike which originally wa ued in [5] for the American lookback option with fixed trike. Thi approach implifie the dicuion and expreion for the arbitrage-free price and it allow to decreae a dimenion of the integral equation for a rational exercie boundary. Uing a local time-pace calculu on curve [8] we derive a cloed form expreion for the arbitrage-free price in term of the optimal topping boundary of the two-dimenional optimal topping problem and how that the rational exercie boundary of the option can be characteried via the unique olution to a nonlinear integral equation. We alo how the remarkable numerical example where the rational exercie boundary exhibit a dicontinuity with repect to pace variable, hence it wa not poible to apply a change-ofvariable formula with local time on urface in order to olve the three-dimenional topping problem directly. Thi i another advantage of reduction by the method of a caling trike. The olution of the zero-trike cae K = 0 (the Britih Ruian option) i fully embedded into the preent problem and can be conidered a a particular cae. We perform the analyi of return of the Britih lookback option with fixed trike in comparion with it the American and the European counterpartie. After oberving the return upon exerciing or elling option we conclude the remarkable feature of the Britih lookback option: (i) the option provide an effective protection againt unfavourable tock movement unlike the American verion which give zero return in thi cae; (ii) the Britih option holder receive very high return alo when tock movement are favourable (much better than the American option and comparable with upon elling the European option); (iii) the holder enjoy two feature above in both liquid and illiquid market. The latter fact i very fruitful, ince lookback option are uually traded in iliquid market and their elling can be very problematic. We believe that thee propertie of the Britih lookback option with fixed trike make it a very attractive financial intrument. The paper i organied a follow. In Section 2 we preent a baic motivation for the Britih lookback option with fixed trike. In Section 3 we formally define the Britih lookback option with fixed trike and how ome of it baic propertie. Then in Section 4 where we derive a 2

4 cloed form expreion for the arbitrage-free price in term of the optimal topping boundary of the two-dimenional problem and how that the rational exercie boundary of the option can be characteried via unique the unique olution to a nonlinear integral equation. Uing thee reult in Section 5 we preent a financial analyi of the Britih lookback option with fixed trike (making comparion with the European/American lookback option). 2. Baic motivation for the Britih lookback option with fixed trike The baic economic motivation for the Britih lookback option with fixed trike i parallel to that of the Britih put, call, Aian and Ruian option (ee [11], [12], [3] and [4]). In thi ection we briefly review key element of thi motivation. We remark that the full financial cope of the Britih lookback option with fixed trike goe beyond thee initial conideration (ee Section 5 below for further detail). 1. Conider the financial market coniting of a riky tock S and a rikle bond B whoe price repectively evolve a (2.1) (2.2) ds t = µs t dt + σs t dw t (S 0 = ) db t = rb t dt (B 0 = 1) where µ IR i the tock drift, σ > 0 i the volatility coefficient, W = (W t ) t 0 i a tandard Wiener proce defined on a probability pace (Ω, F, P), and r > 0 i the interet rate. Recall that the lookback option with fixed trike of European type i a financial contract between a eller/hedger and a buyer/holder entitling the latter to exercie the option at a pecified maturity time T > 0 and receive the payoff (2.3) ( M T K ) + = ( max S t K 0 t T from the eller. Standard hedging argument baed on elf-financing portfolio imply that the arbitrage-free price of the option i given by (2.4) V = Ẽ [e rt (M T K) + ] where the expectation Ẽ i taken with repect to the (unique) equivalent martingale meaure P (ee e.g. [15]). In thi ection (a in [11], [12] and [4]) we will analye the option from the tandpoint of a true buyer. By true buyer we mean a buyer who ha no ability or deire to ell the option nor to hedge hi own poition. Thu every true buyer will exercie the option at time T in accordance with the rational performance. For more detail on the motivation and interet for conidering a true buyer in thi context we refer to [11]. 2. With thi in mind we now return to the holder of the lookback option whoe payoff i given by (2.3) above. Recall that the unique trong olution to (2.1) i given by ( ) (2.5) S t = S t (µ) = exp σw t + (µ σ2 )t 2 under P for t [0, T ] where µ IR i the actual drift. Inerting (2.5) into (2.3) we find that the expected value of the buyer payoff equal (2.6) P = P (µ) = E [e rt (M T (µ) K) + ]. 3 ) +

5 Moreover, it i well known that Law (S(µ) P) i the ame a Law (S(r) P) o that the arbitrage-free price of the option equal (2.7) V = P (r) = E [e rt (M T (r) K) + ]. A direct comparion of (2.6) and (2.7) how that if µ = r then the return i fair for the buyer, in the ene that V = P, where V repreent the value of hi invetment and P repreent the expected value of hi payoff. On the other hand, if µ > r then the return i favourable for the buyer, in the ene that V < P, and if µ < r then the return i unfavourable for the buyer, in the ene that V > P with the ame interpretation a above. Exactly the ame analyi can be performed for the lookback option of American type and a the concluion are the ame we omit the detail. We recall that the actual drift µ i unknown at time t = 0 and alo difficult to etimate at later time t (0, T ] unle T i unrealitically large. 3. The brief analyi above how that whilt the actual drift µ of the underlying tock price i irrelevant in determining the arbitrage-free price of the option, to a (true) buyer it i crucial, and he will buy the option if he believe that µ > r. If thi appear to be a true then on average he will make a profit. Thu, after purchaing the option, the holder will be happy if the oberved tock price movement confirm hi belief that µ > r. The Britih lookback option with fixed trike eek to addre the oppoite cenario: What if the option holder oberve tock price movement which change hi belief regarding the actual drift and caue him to believe that µ < r intead? In thi contingency the Britih lookback holder i effectively able to ubtitute thi unfavourable drift with a contract drift and minimie hi loe. In thi way he i endogenouly protected from any tock price drift maller than the contract drift. The value of the contract drift i therefore elected to repreent the buyer level of tolerance for the deviation of the actual drift from hi original belief. It will be hown below (imilarly to [11], [12] and [4]) that the practical implication of thi protection feature are mot remarkable a not only i the option holder afforded an unique protection againt unfavourable tock price movement (covering the ability to ell in a liquid option market completely endogenouly) but alo when the tock price movement are favourable he will generally receive high return. We refer to the final paragraph of Section 2 in [12] for further comment regarding the option holder ability to ell hi contract (releaing the true buyer perpective) and it connection with option market liquidity. Thi tranlate into the preent etting, ince lookback option are not o popularly traded a plain vanilla call and put option. 3. The Britih lookback option with fixed trike: Definition and baic propertie We begin thi ection by preenting a formal definition of the Britih lookback option with fixed trike. Thi i then followed by a brief analyi of the optimal topping problem and the free-boundary problem characteriing the arbitrage-free price and the rational exercie trategy. Thee conideration are continued in Section 4 and 5 below. 1. Conider the financial market coniting of a riky tock S and a rikle bond B whoe price evolve a (2.1) and (2.2) repectively, where µ IR i the appreciation rate 4

6 (drift), σ > 0 i the volatility coefficient, K > 0 i a fixed trike, W = (W t ) t 0 i a tandard Wiener proce defined on a probability pace (Ω, F, P), and r > 0 i the interet rate. Let a maturity time T > 0 be given and fixed, and let M T denote the maximum tock price given by (2.3) above. Definition 3.1. The Britih lookback option with fixed trike i a financial contract between a eller/hedger and a buyer/holder entitling the latter to exercie at any (topping) time τ prior to T whereupon hi payoff (deliverable immediately) i the bet prediction of the European payoff (M T K) + given all the information up to time τ under the hypothei that the true drift of the tock price equal µ c. The quantity µ c i defined in the option contract and we refer to it a the contract drift. We will how below that the contract drift hould atify the following inequality (3.1) E µc[ r K S 0 µ c M T ] I(MT > K S 0 ) > 0 where the expectation E µc i taken under aumption that the drift in (2.1) equal to µ c. If (3.1) doe not hold, we are not able to guarantee that it i not optimal to exercie immediately, i.e. the buyer would not be overprotected (ee Remark 3.2 below for detail). Condition (3.1) give for u relationhip between µ c, the interet rate r, and the trike K when K > 0, ince in the cae of K = 0 (Ruian option) we have imply µ c < 0 a in [4]. We denote by µ c = µ c(r, K) the unique olution (clearly it exit and 0 < µ c < r ) to equation (3.2) E µ c [ r K S 0 µ c M T ] I(MT > K S 0 ) = 0 and hence the condition (3.1) i equivalent to µ c < µ c. Recall from Section 2 above that the value of the contract drift i elected to repreent the buyer level of tolerance for the deviation of the true drift µ from hi original belief. 2. Denoting by (F t ) 0 t T the natural filtration generated by S (poibly augmented by null et or in ome other way of interet) the payoff of the Britih lookback option with fixed trike at a given topping time τ with value in [0, T ] can be formally written a (3.3) E µc [(M T K) + F τ ] where the conditional expectation i taken with repect to a new probability meaure P µc under which the tock price S evolve a (3.4) ds t = µ c S t dt + σs t dw t with S 0 = in (0, ). Comparing (2.1) and (3.4) we ee that the effect of exerciing the Britih lookback option with fixed trike i to ubtitute the true (unknown) drift of the tock price with the contract drift for the remaining term of the contract. 3. Setting that M t = max 0 t S for t [0, T ] and uing tationary and independent increment of W governing S we find that ] E µ c [(M T K) + F t = S t E c[ ( µ Mt ) ] S S t max t T S t K + Ft (3.5) S t = S t Z µ c (t, M t, S t ) 5

7 where the function Z µc can be expreed a (3.6) Z µ c (t, m, ) = E ( µ c m M ) T t K + = E µ c ( m K M T t K ) = G µ c ( ) t, m K K where G µ c (t, x) = E µ c (x M T t ) for t [0, T ], m > 0, x [1, ) and M 0 = 1. A lengthy calculation baed on the known law of M T t under P µc (ee e.g. [13, Lemma 1, p. 759]) how that (3.7) ( [ ]) G µ c 1 (t, x) = x Φ σ log x (µ T t c σ2 )(T t) 2 ( [ ]) σ2 2µ c x 2µc/σ2 Φ 1 σ log x+(µ T t c σ2 )(T t) 2 ( ( + 1+ σ2 2µ c )e µ c(t t) Φ 1 σ T t [ log x (µ c + σ2 2 )(T t) ]) for t [0, T ] and x [1, ), where Φ i the tandard normal ditribution function given by Φ(x) = (1/ 2π) x e y2 dy for x IR. The function G µ c appeared in [4] and it propertie were tudied there. We will recall and make ue of them below. Standard hedging argument baed on elf-financing portfolio (with conumption) imply that the arbitrage-free price of the Britih lookback option with fixed trike i given by (3.8) V = up 0 τ T Ẽ [ e rτ E µc( ) ] (M T K) + F τ where the upremum i taken over all topping time τ of S with value in [0, T ] and Ẽ i taken with repect to the (unique) equivalent martingale meaure P. From (3.5) we ee that the underlying Markov proce in the optimal topping problem (3.8) equal (t, S t, M t ) 0 t T for t [0, T ] thu making it three-dimenional. 4. Since Law (S(µ) P) i the ame a Law (S(r) P), it follow from the well-known ladder tructure of M and multiplicative tructure of S that (3.8) extend a follow ( ) ] (3.9) V (t, m, ) = up E e rτ S τ [G µ c t+τ, K m max 0 u τ S u S τ K S τ 0 τ T t for t [0, T ] and m in (0, ) where the upremum i taken a in (3.8) above and the proce S = S(r) under P olve (3.10) ds t = rs t dt + σs t dw t with S 0 = 1. By the Giranov theorem it follow that ( ) ] E e rτ S τ [G µ c t+τ, K m max 0 u τ S (3.11) u S τ K S τ [ ) ] = Ê G µ c (t+τ, m K M τ S τ K S τ [ ( ) ] = Ê G µ c t+τ, Xτ x K e rτ [ ( ) ] = Ê G µ c t+τ, Xτ x Kert e r(t+τ) 6

8 for every topping time τ of S and where we ued fact that ÊS 1 τ = Êe rτ and we et (3.12) X x t = x M t S t with x = (m K)/ and P i defined by d P = exp(σw T (σ 2 /2)T ) dp o that Ŵt = W t σt i a tandard Wiener proce under P for t [0, T ]. By Ito formula one find that (3.13) dx t = rx t dt + σx t dŵt + dr t with X 0 = x under P and we et (3.14) R t = t 0 I(X = 1) dm S for t [0, T ] and x [1, ). Note that the tate pace of the Markov proce X equal [1, ) where 1 i an intantaneouly reflecting boundary point. Thu (3.11) motivate u to conider the following optimal topping problem (3.15) Ṽ (t, x) = Ṽ (t, x; K) = up E [G µ c (t+τ, X xτ ) Ke ] r(t+τ) 0 τ T t for t [0, T ] and x [1, ) where the upremum i taken over all topping time τ of X with value in [0, T t] and E tand for Ê to implify the notation. It follow from (3.9) and (3.15) uing (3.11) that (3.16) V (t, m, ) = m K Ṽ (t, ; Kert ) for t [0, T ], m > 0 and uing the etablihed probabilitic technique (ee e.g. [9]) one can verify that the optimal topping et in (3.9) i given by (3.17) D = { (t, m, ) [0, T ) S : Ṽ (t, m K ; Kert ) = G µ c (t, m K ) K } where S := { (m, ) : m > 0 }. A in [5] we have reduced the three-dimenional problem (3.9) to the two-dimenional problem (3.15), but with a caling trike K, ince we will vary K to determine the olution of (3.9) uing (3.16) and (3.17). Alo we note that the olution to the Britih Ruian option problem i fully embedded into the preent olution when K = The analyi below i parallel to that of [4]. Let u now make ue of Ito formula (combined with the fact that G µc x (t, 1+) = 0 for all t [0, T ) ) and the optional ampling theorem yield E [G µc (t+τ, X xτ ) Ke ] (3.18) r(t+τ) [ τ = G µ c (t, x) + E H µ c (t+, X x ) d Ke ] r(t+τ) 0 = G µ c (t, x) Ke [ τ ( rt + E H µ c (t+, X x ) + r Ke r(t+)) d] 7 0

9 for all topping time τ of X with value in [0, T t] with t [0, T ) and x [1, ) given and fixed, where the function H µ c = H µ c (t, x) i given by (3.19) H µ c = G µ c t rx G µ c x + σ2 2 x2 G µ c xx. To implify thi expreion note that by the Giranov theorem we find ( (3.20) G µc (t, x) = E µc (x M T t ) = E [S µc x MT t ) ] ) T t S T t = e µc(t t) Ê (X µc T x t where X under P µ c olve (3.13) with µ c in place of r. Thi how that G µ c = G µ c (t, x) olve the killed verion of the Kolmogorov backward equation (ee e.g. [8, Section 7]) (3.21) G µ c t µ c x G µ c x Inerting (3.21) into (3.19) we find that + σ2 2 x2 G µ c xx + µ c G µ c = 0. (3.22) H µ c = µ c G µ c + (µ c r)x G µ c x. A direct ue of (3.7) in (3.22) lead to a complicated expreion and for thi reaon we proceed by deriving a probabilitic interpretation of the right-hand ide in (3.22). To thi end note that G µ c (t, x) = E µ c (x M T t ) = x P µ c (M T t x)+e µ c [M T t I(M T t > x)] a well a G µ c (t, x) = P µ c (x M 0 T t > z) dz = x+ x Pµ c (M T t > z) dz o that G µ c x (t, x) = 1 P µ c (M T t > x) = P µ c (M T t x). Inerting thee expreion into (3.22) we find that (3.23) H µc (t, x) = µ c E µc [M T t I(M T t > x)] rx P µc (M T t x) for t [0, T ] and x [1, ). A lengthy calculation baed on the known law of M T t P µc (recall (3.7) above) then how that ( [ ]) H µ c 1 (3.24) (t, x) = rx Φ σ log x (µ T t c σ 2 /2)(T t) ( [ ]) + (r µ c + σ2 2 )x2µc/σ2 Φ 1 σ log x+(µ T t c σ2 )(T t) 2 ( [ ]) (µ c + σ2 2 )eµ c(t t) Φ log x (µ c + σ2 )(T t) 2 1 σ T t under for t [0, T ] and x [1, ). Now we denote by Ĥµ c the integrand in (3.18) (3.25) Ĥ µc (t, x) = H µc (t, x) + r Ke rt for t [0, T ] and x [1, ). Then the expreion (3.18) read (3.26) E [G µ c (t+τ, X xτ ) Ke ] r(t+τ) = G µ c (t, x) Ke [ τ ] rt + E Ĥ µ c (t+, X x ) d. 0 From now on the analyi differ from that of [4] due to preence of non-zero trike K 0 and thu the integrand Ĥµ c i not equal to the function H µ c. The expreion (3.26) i ueful for getting ome inight into the tructure of topping and continuation et: if Ĥ µ c (t, x) > 0 then a point (t, x) belong to continuation et, however if Ĥ µ c (t, x) < 0 then it i not ufficient 8

10 that at point (t, x) it i optimal to top at once. The important property of the Britih option i an admiible et of value for contract drift, ince if thi i not properly elected, the buyer of option become overprotected at the beginning when t = 0, i.e. he hould exercie at once and the option become meaningle. Uing probabilitic repreentation (3.23) it wa hown in [4] that when K = 0 the buyer i not overprotected if and only if µ c < 0. Indeed, if µ c 0 then H i alway negative and thu it i optimal to top at once, but if µ c < 0 then at an initial point (0, 1) the function H µ c (0, 1) > 0 i poitive and thu it i optimal to continue and the buyer i not overprotected. However in the cae K 0 the analyi become more delicate. Clearly when µ c < 0 the buyer again i not overprotected at t = 0, but there are alo poitive admiible value for contract drift. For thi we need to conider Ĥµ c at financial initial point, i.e. originally to olve option pricing problem (3.8) we inert t = 0, m = = S 0 in (3.9) and aume that S 0 < K (thi i the uual aumption for lookback option and doe not implify analyi), thu uing (3.16) we have that (t, x; K) = (0, K S 0 ; K S 0 ) and the function Ĥ µc at the initial point i given by (3.27) Ĥ µ c (0, K S 0 ) = H µ c (0, K S 0 ) + r K S 0 = µ c E µ c [M T I(M T > K S 0 )] + r K S 0 P µ c (M T > K S 0 ) = E µc[ r K S 0 µ c M T ] I(MT > K S 0 ) where we ued (3.23). It follow from (3.27) that there exit a unique root µ c = µ c of equation Ĥ µc (0, K S 0 ) = 0 uch that 0 < µ c < r and Ĥµc (0, K S 0 ) > 0 if and only if µ c < µ c. A we aid above it i not certain that if Ĥ µc < 0 at initial point then it i optimal to top at once and we cannot determine it analytically, hence the bet we can do i to reaure that the buyer i not overprotected and we will require the condition µ c < µ c o that Ĥ µ c (0, K S 0 ) < 0. Remark 3.2. It i important to note that the condition µ c < µ c doe not fully decribe all admiible value for contract drift and there are value for contract drift greater than µ c uch that the holder i till not overprotected, but due to reaon outlined above, we are not to able to determine the exact threhold. Indeed from the analyi in Section 2 above we have that the contract drift i maller than r. It will be proven below that if µ c < r then there exit an optimal topping boundary b eparating the continuation et from the topping et and thu the overprotection of the buyer i equivalent to the condition b(0) K S 0. For µ c = µ c the optimal topping boundary at zero b µ c (0) > K S 0 and there are value µ c > µ c uch that b µc (0) > K S 0. However ince µ c b µc (0) i decreaing then there exit a threhold µ c < r uch that b µ c (0) = K S 0 and the holder i overprotected if and only if µ c > µ c. It mean that in order to determine the real threhold µ c for the contract drift one hould find the value of boundary b(0) by olving numerically a nonlinear integral equation (ee Theorem 4.1 below). Thu the equation Ĥµ c (0, K S 0 ) = 0 give more acceible condition for the contract drift rather than the equation b µc (0) = K S 0 and we will ue the threhold µ c further for the financial analyi. Moreover we will how in Section 5 that the Britih option with contract drift µ c < µ c provide attractive return. Fixing µ c < µ c it follow from (3.22) and (3.25) that (3.28) Ĥ µ c x (t, x) = H µ c x (t, x) = rx G µ c x (t, x) + (µ c r)x G µ c xx(t, x) < 0 9

11 x D C h b b(t) 1 0 T Figure 1. A computer drawing of the optimal topping boundary b for the problem (3.15) in the cae K = 1.2, S 0 = 1, K = K S 0 = 1.2, T = 1, µ c = 0.05 < µ c 0.075, r = 0.1, σ = 0.4 with the boundary condition b(t ) = Ke rt > 1 and the tarting point x = K S 0 < h(0). for any x 1 and fixed t [0, T ) where we ued fact that G µc x > 0, G µc xx µ c < µ c < r. Hence it give to u that > 0 and (3.29) x H µ c (t, x) i decreaing on [1, ) for any given and fixed t [0, T ) and any choice of K. It follow from (3.23), (3.25) and (3.29) that there exit a continuou (mooth) function h : [0, T ] IR uch that (3.30) Ĥ µc (t, h(t)) = 0 for t [0, T ] with Ĥµc (t, h(t)) > 0 for x [1, h(t)) and Ĥµc (t, h(t)) < 0 for x (h(t), ) when t [0, T ] given and fixed. In view of (3.26) thi implie that no point (t, x) in [0, T ) [1, ) with x < h(t) i a topping point (for thi one can make ue of the firt exit time from a ufficiently mall time-pace ball centred at the point). Likewie, it i alo clear and can be verified that if x > h(t) and t < T i ufficiently cloe to T then it i optimal to top immediately (ince the gain obtained from being below h cannot offet the cot of getting there due to the lack of time). Thi how that the optimal topping boundary b : [0, T ] [0, ] eparating the continuation et from the topping et atifie b(t ) = h(t ) and thi value equal ( Ke rt 1 ). Moreover, the fact (3.29) combined with the identity (3.26) implie that the continuation et i given by C = { (t, x) [0, T ) [1, ) : x < b(t) } and the topping et i given by D = { (t, x) [0, T ) [1, ) : x b(t) } o that the optimal topping time in the problem (3.15) i given by (ee Figure 1) (3.31) τ b = inf { 0 t T : X t b(t) }. 10

12 D x C b(t) 1 0 T Figure 2. A computer drawing howing how the optimal topping boundary b for the problem (3.15) increae a one decreae the contract drift. There are four different cae: 1) µ c = ; 2) µ c = 0.07 ; 3) µ c = 0.05 ; 4) µ c = (the latter correpond to the American lookback option problem). The et of parameter: K = 1.2, S 0 = 1, K = K S 0 = 1.2, T = 1, r = 0.1, σ = 0.4 with the boundary condition b(t ) = Ke rt > 1, the tarting point x = K S 0 and the root of (3.2) µ c It i alo clear and can be verified that if the initial point x 1 of the proce X i ufficiently large then it i optimal to top immediately (ince the gain obtained from being below h cannot offet the cot of getting there due to the hortage of time). Thi how that the optimal topping boundary b i finite valued. 6. Standard Markovian argument lead to the following free-boundary problem (for the value function Ṽ = Ṽ (t, x) and the optimal topping boundary b = b(t) to be determined): (3.32) (3.33) (3.34) (3.35) Ṽ t rxṽx + σ2 2 x2 Ṽ xx = 0 for x (1, b(t)) and for t [0, T ) Ṽ (t, x) = G µ c (t, x) Ke rt for x b(t) and for t [0, T ) Ṽ x (t, b(t)) = G µc x (t, b(t)) for t [0, T ) Ṽ x (t, 1+) = 0 for t [0, T ) where b(t ) = Ke rt 1 and Ṽ (T, x) = Gµc (T, x) Ke rt rt = x Ke for x 1. It can be hown that thi free-boundary problem ha a unique olution Ṽ and b which coincide with the value function (3.15) and the optimal topping boundary repectively (cf. [9]). Fuller detail of the analyi go beyond our aim in thi paper and for thi reaon will be omitted ince the fact that unle µ c < 0 the boundary b i not necearily a monotone function of time (ee Figure 2) make thi analyi more complicated (in comparion with the American 11

13 lookback option [5]). In the next ection we will derive equation which characterie Ṽ and b uniquely and can be ued for their calculation. Note that x Ṽ (t, x) i increaing and convex on [1, ) for every t [0, T ] (ince Gµ c i o). Note alo that if we let µ c to then the optimal topping boundary b goe to a continuou decreaing function b : [0, T ] IR atifying b (T ) = Ke rt 1 (ee Figure 2). The limiting boundary b i optimal in the problem (3.15) where G (t, x) = x for (t, x) [0, T ] [1, ). Thi problem correpond to the American lookback option with fixed trike in the cae of finite horizon (ee [5]). 7. From (3.7) we ee that the volatility parameter appear explicitly in the payoff of the Britih option and thu hould be agreed in the contract. However, ince the underlying proce i aumed to be a geometric Brownian motion, one may take any of the tandard etimator for the volatility (e.g. uing the Central Limit Theorem) over an arbitrarily mall time period prior to the initiation of the contract. It i important to note that the etimation of the tock drift µ cannot be etimated in practice, therefore it eem natural to provide a true buyer with protection an drift rather than a volatility, at leat in current model. For more detail about thi quetion we addre reader to final paragraph of Section 3 in [11], [12] and [4]. 4. The arbitrage-free price and the rational exercie boundary In thi ection we derive a cloed form expreion for the value function Ṽ for the problem (3.15) in term of the optimal topping boundary b and how that the optimal topping boundary b itelf can be characteried a the unique olution to a nonlinear integral equation (Theorem 4.1). We will make ue of the following function in Theorem 4.1 below: (4.1) (4.2) F (t, x) = G µ c (t, x) e r(t t) G r (t, x) L(t, x, v, z) = z Ĥ µc (v, y)f(v t, x, y)dy for t [0, T ), x 1, v (t, T ) and z 1, where the function G µ c and G r are given by (3.7) above (upon identifying µ c with r in the latter cae), the function Ĥµc i given by (3.25) above, and y f(v t, x, y) i the probability denity function of Xv t x under P given by (4.3) [ ( f(v t, x, y) = 1 σy φ v t 1 σy v t ( + x 1+2r/σ2 φ [ log x y 1 σy v t ( + 1+2r/σ2 Φ 1 y 2(1+r/σ2 ) σy v t σ2 (r+ )(v t)]) 2 [ log xy+(r+ σ 2 )(v t)])] 2 [ log xy (r+ σ 2 )(v t)]) 2 for y 1 (with v t and x a above) where φ i the tandard normal denity function given by φ(x) = (1/ x2 /2 2π)e for x IR (and Φ i the tandard normal ditribution function defined following (3.7) above). It hould be noted that L(t, x, v, b(v)) > 0 for all t [0, T ), x 1 and v (t, T ), ince Ĥµ c (v, y) < 0 for all y > b(v) a b lie above h (recall (3.30) above). 1. The main reult of thi ection may now be tated a follow. 12

14 for the problem (3.15) admit the following repreenta- Theorem 4.1. The value function Ṽ tion (4.4) Ṽ (t, x) = e r(t t) G r (t, x) Ke rt + T t L(t, x, v, b(v)) dv for all (t, x) [0, T ) [1, ). The optimal topping boundary (ee Figure 1 above) can be characteried a the unique continuou olution b : [0, T ] IR + to the nonlinear integral equation (4.5) F (t, b(t)) = K(e rt e rt ) + T t L(t, b(t), v, b(v)) dv atifying b(t) h(t) for all t [0, T ] where h i defined by (3.30) above. Proof. We derive (4.4) and how that the rational exercie boundary olve (4.5). We omit the proof of fact that (4.5) cannot have other (continuou) olution, ince it i parallel to imilar proof in [4] and [5]. a) Recall that the value function Ṽ : [0, T ] [1, ) IR and the rational exercie boundary b : [0, T ] IR + olve the free-boundary problem (3.32)-(3.35) (where Ṽ extend a Gµc above b ), et C b = { (t, x) [0, T ) [1, ) : x < b(t) } and D b = { (t, x) [0, T ) [1, ) : x b(t) } and let IL X Ṽ (t, x) = rx Ṽx(t, x)+ σ2 2 x2 Ṽ xx (t, x) for (t, x) C b D b. Then Ṽ and b are continuou function atifying the following condition: (i) Ṽ i C1,2 on C b D b ; (ii) b i of bounded variation; (iii) P(Xt x = c) = 0 for all c > 0 whenever t [0, T ] and x 1 ; (iv) Ṽ t +IL X Ṽ i locally bounded on C b D b (recall that Ṽ atifie (3.32) on C b and coincide with G µ c on D b ); (v) x Ṽ (t, x) i convex on [1, ) for every t [0, T ] ; and (vi) t Ṽx(t, b(t)±) = G µ c x (t, b(t)) i continuou on [0, T ] (recall that Ṽ atifie the mooth-fit condition (3.34) at b ). From thee condition we ee that the local time-pace formula [8] i applicable to (, y) Ṽ (t+, y) with t [0, T ) given and fixed. Fixing an arbitrary x 1 and making ue of (3.35) thi yield (4.6) Ṽ (t+, X x ) = Ṽ (t, x) (Ṽt+IL X Ṽ )(t+v, X x v )I(X x v b(t+v)) dv + M b 0 0 (Ṽx(t+v, X x v +) Ṽx(t+v, X x v )) I(X x v = b(t+v)) dl b v(x x ) where M b = σ 0 Xx v Ṽx(t+v, X x v ) I(X x v b(t+v)) d W v i a martingale for [0, T t] and l b (X x ) = (l b v(x x )) 0 v i the local time of X x = (X x v ) 0 v on the curve b for [0, T t]. Moreover, ince Ṽ atifie (3.32) on C b and equal G µ c Ke rt on D b, and the mooth-fit condition (3.34) hold at b, we ee that (4.6) implifie to (4.7) Ṽ (t+, X x ) = Ṽ (t, x) + Ĥ µ c (t+v, Xv x ) I(Xv x > b(t+v)) dv + M b for [0, T t] and (t, x) [0, T ) [1, ). 0 13

15 b) Replacing by T t in (4.7), uing that Ṽ (T, x) = Gµc (T, x) Ke rt = x Ke rt for x 1, taking E on both ide and applying the optional ampling theorem, we get (4.8) T t E(XT x t) Ke rt = Ṽ (t, x) + E [ Ĥ µc (t+v, Xv x ) I(Xv x > b(t+v)) ] dv 0 T = Ṽ (t, x) L(t, x, v, b(v)) dv t for all (t, x) [0, T ) [1, ) where L i defined in (4.2) above. We ee that (4.8) yield the repreentation (4.4). Moreover, ince Ṽ (t, b(t)) = Gµc (t, b(t)) Ke rt for all t [0, T ] we ee from (4.4) with (4.1) that b olve (4.5). Thi etablihe the exitence of the olution to (4.5). 2. Now we can determine the arbitrage-free price (3.9) of the Britih lookback option with fixed trike K. Indeed, from (3.16) and (4.4) we have (4.9) V (t, m, ) = e r(t t) G r( t, x ) Ke r(t T ) + T t E [( H µ c (v, X x v t)+ rk ) ( I X x v t > b(v; Kert ) )] dv for t [0, T ], m >0 where x = m K and the optimal topping boundary b i computed under aumption K = Kert in (4.5). Standard argument baed on the trong Markov property (ee [9]) how that the topping region in (3.9) ha the following form: D = { (t, m, ) [0, T ) S : V (t, m, ) = G µc( ) } (4.10) t, m K K = { ( (t, m, ) [0, T ) S : Ṽ ) t, m K ; Kert = G µ c ( ) t, m K K } = { (t, m, ) [0, T ) S : Ṽ ( ) t, m K ; Kert = G µ c ( ) t, m K Ke rt e } rt = { (t, m, ) [0, T ) S : m K b ( ) } t; Kert = { (t, m, ) [0, T ) S : m K b ( ) } t; Kert where S = {(m, ) : m > 0} and we ued (3.16) and (3.33). Thu the optimal topping time in (3.9) i given by (4.11) τ g = inf { 0 u < T t : M u g(t+u, S u ) } where the rational exercie boundary g read { ( ) ( ) b t; Ke rt g(t, ) =, if K < b t; Ke rt (4.12), if K b ( ) t; Kert for t [0, T ), > 0. Hence if there exit t [0, T ), (0, K) and ε > 0 mall enough uch that K = b ( ) ( ) t; Kert and K > b t; Ke rt for (, + ε) then the boundary g(t, ) exhibit a left-dicontinuity at =, which i a quite rare cae in the optimal 14

16 m m= D K C C 0 Figure 3. A computer drawing of the rational exercie boundary g(t, ) for 1) t = 0 (top at = 0 ); 2) t = 0.3 ; 3) t = 0.6 (bottom at = 0 ) in the cae K = 1.2, S 0 = 1, K = K S 0 = 1.2, T = 1, µ c = 0.05 < µ c 0.075, r = 0.1, σ = 0.4. The limit of g(t, ) at zero i greater than K for every t. topping theory. Below we how a numerical example where indeed the boundary ha a jump down. 3. We now provide the numerical analyi and computer drawing of the rational exercie boundarie for (3.9). It follow from (4.12) that in order to determine g(t 0, ) for given and fixed t 0 [0, T ) we need to calculate the optimal topping boundary b( ; K) a a olution to (4.5) with K = Kert 0 for every > 0 and then g(t 0, ) = b(t 0 ; K). For computer drawing of the boundarie we aume that the initial tock price equal 1, the trike price K = 1.2, the maturity time T = 1 year, the contract drift µ c = 0.05 < µ c 0.075, the interet rate r = 0.1, the volatility σ = 0.4, i.e. we conider the option out-of-the money. We dicretie the interval (0, 2) with tep h = 0.05 and for every 0 2 of thi grid we olve numerically the integral equation (4.5) with K = Kert 0 and then put g(t 0, ) = b(t 0 ; K). In Figure 3 we draw the rational exercie boundary g(t, ) for different value of t in order to gain inight how the boundary evolve over the time. The Figure 3 how that the rational exercie boundary of the Britih verion i not a monotone function of both variable unlike the American counterparty, ince it wa hown in [5] that for the American lookback option with fixed trike the rational exercie boundary ha the following pattern: t g A (t, ) i decreaing on [0, T ) for each > 0 fixed and g A (t, ) i increaing on (0, ) for t [0, T ) fixed. From (4.12) it i eaily to een that g(t, ) b 0 (t) a for fixed t [0, T ) where b 0 (t) = b(t; 0) i the optimal topping boundary for the Britih Ruian option [4]. Since µ c > 0 we have that b 0 1 (ee [4]) o that g(t, ) = for 0 large enough, i.e. the rational exercie boundary become a diagonal (ee Figure 3). The Figure 4 how how the rational exercie boundary change a one varie the contract 15

17 m D m= K C C 0 Figure 4. A computer drawing howing how the rational exercie boundary g for the problem (3.9) increae a one decreae the contract drift for fixed t = 0. There are four different cae: 1) µ c = ; 2) µ c = 0.05 ; 3) µ c = 0.05 ; 4) µ c = (the latter correpond to the American lookback option problem). All boundarie have the ame limit at = 0. The et of parameter: K = 1.2, S 0 = 1, T = 1, µ c 0.075, r = 0.1, σ = 0.4. The rational exercie boundary in the cae µ c = i dicontinuou at 1.14 < K. drift for fixed t = 0. It can be een that tronger the protection (i.e. the contract drift increae) the larger the topping region (i.e. the rational exercie boundary decreae). In the cae µ c = we oberve a remarkable feature: the rational exercie boundary i dicontinuou at point 1.14 < K. Hence it wa not poible to apply a change-of-variable formula with local time on urface [10] in order to olve the three-dimenional topping problem directly. Thi i another advantage of reduction by the method of a caling trike. Alo uing the remark from previou ection we note that if we let µ c to then the optimal topping boundary g goe increaingly to a continuou function g (ee Figure 4). The limiting boundary g i optimal in the problem (3.9) where G (t, x)=x for (t, x) [0, T ] [1, ). Thi problem correpond to the American lookback option with fixed trike in the cae of finite horizon (ee [5]). 5. The financial analyi In thi ection we preent the analyi of financial return of the Britih lookback option with fixed trike and highlight the practical feature of the option. We perform comparion with both the American lookback option with fixed trike and the European lookback option with fixed trike ince the former option ha been the ubject of much reearch activity in recent year (ee e.g. [2], [5]) whilt the latter i commonly traded and well undertood. The o-called keleton analyi wa applied to analye financial return of option in [11], [12], [3] and [4], where the main quetion wa addreed a to what the return would be if the 16

18 underlying proce enter the given region at a given time (i.e. the probability of the latter event wa not dicued nor do we account for any rik aociated with it occurrence). Such a keleton analyi i both natural and practical ince it place the quetion of probabilitie and rik under the ubjective aement of the option holder (irrepective of whether the tock price model i correct or not). In the preent etting an analyi of option performance baed on return eem epecially inightful ince lookback option are mot often ued excluively for peculation and thu for achieving high return. 1. In the end of Section 4 above we aw that the rational exercie trategy (4.12) of the Britih lookback option with fixed trike in the problem (3.9) above change a one varie the contract drift µ c. Thi i illutrated in Figure 4 above. We recall that the contract drift mut atify µ c < µ c, ince otherwie we are not able to reaure that buyer i not overprotected. On the other hand, when µ c then g tend to the American lookback boundary g and the Britih lookback option effectively reduce to the American lookback option. In the latter cae a contract drift repreent an infinite tolerance of unfavourable drift and the Britih lookback holder will exercie the option rationally in the limit at the ame time a the American lookback holder. 2. In the numerical example below (ee Figure 5 and 6) the parameter value have been choen to preent the practical feature of the Britih lookback option with fixed trike in a fair and repreentative way. We aume that the initial tock price equal 1, the trike price K = 1.2, the maturity time T = 1 year, the interet rate r = 0.1, the volatility coefficient σ = 0.4, i.e. we conider the option out-of-the money. We chooe the contract drift µ c = 0.05, which atifie the condition (5.1) E µ c [ r K S 0 µ c M T ] I(MT > K S 0 ) > 0. For thi et of parameter the arbitrage-free price of the Britih lookback option with fixed trike i 0.254, the price of the American lookback option i 0.251, and the price of the European lookback option i Oberve that the cloer the contract drift get to µ c, the tronger the protection feature provided (with generally better return), and the more expenive the Britih lookback option become. Recall alo that when µ c then the Britih lookback option effectively reduce to the American lookback option and the price of the former option converge to the price of the latter. The fact that the price of the Britih lookback option i cloe to the price of the European (and American) lookback option in ituation of interet for trading i of coniderable practical value. 3. Figure 5 and 6 below provide the analyi of comparion between the Britih lookback option with fixed trike and it American and European verion. We conider the et of parameter above, the arbitrage-free price of the Britih option in thi etting can be computed uing (4.9) o that V (0, 1, 1) = We exploit the ame method to find the price of the American option V A (0, 1, 1) = (ee [5]). The European option price V E (0, 1, 1) = can be eaily evaluated uing the following manipulation (5.2) V E (t, m, ) = e r(t t) E r( m max S u K ) + u T t = e r(t t) E r( m K max S ) u K u T t = e r(t t)( G r( ) ) t, m K K. 17

19 Exercie time (month) S = 0.6 Exercie at M 1.2 with µ c = % 4% 2% 1% 0% 0% 0% Exercie at M 1.2 (American) 0% 0% 0% 0% 0% 0% 0% Exercie at M = 1.4 with µ c = % 80% 79% 79% 79% 79% 79% Exercie at M = 1.4 (American) 80% 80% 80% 80% 80% 80% 80% S = 1.0 Exercie at M 1.2 with µ c = % 78% 62% 46% 28% 10% 0% Exercie at M 1.2 (American) 0% 0% 0% 0% 0% 0% 0% Exercie at M = 1.4 with µ c = % 121% 109% 97% 87% 80% 79% Exercie at M = 1.4 (American) 80% 80% 80% 80% 80% 80% 80% Exercie at M = 1.6 with µ c = % 180% 171% 164% 160% 157% 157% Exercie at M = 1.6 (American) 159% 159% 159% 159% 159% 159% 159% S = 1.4 Exercie at M = 1.4 with µ c = % 274% 250% 223% 193% 157% 79% Exercie at M = 1.4 (American) 80% 80% 80% 80% 80% 80% 80% Exercie at M = 1.6 with µ c = % 287% 264% 239% 212% 182% 157% Exercie at M = 1.6 (American) 159% 159% 159% 159% 159% 159% 159% Exercie at M = 1.8 with µ c = % 320% 300% 280% 260% 242% 236% Exercie at M = 1.8 (American) 239% 239% 239% 239% 239% 239% 239% S = 1.8 Exercie at M = 1.8 with µ c = % 487% 456% 422% 383% 337% 237% Exercie at M = 1.8 (American) 239% 239% 239% 239% 239% 239% 239% Exercie at M = 2.0 with µ c = % 497% 467% 434% 398% 357% 315% Exercie at M = 2.0 (American) 318% 318% 318% 318% 318% 318% 318% Figure 5. Return oberved upon exerciing the Britih lookback option with fixed trike compared with return oberved upon exerciing the American lookback option with fixed trike. The return are expreed a a percentage of the original option price paid by the buyer (rounded to the nearet integer), i.e. R(t, m, )/100 = (G µc (t, m K ) K)/V (0, 1, 1) and R A (t, m, )/100 = (m K) + /V A (0, 1, 1). The parameter et i µ c = 0.05, K = 1.2, T = 1, r = 0.1, σ = 0.4 and the initial tock price equal 1. We compare the return that the Britih lookback holder can obtain upon exerciing hi option with the return oberved upon (i) exerciing the American lookback option in the ame contingency (Figure 5) and (ii) elling the European lookback option in the ame contingency (Figure 6). The latter i motivated by the fact that in practice the European option holder may chooe to ell hi option at any time during the term of the contract, and in thi cae one may view hi payoff a the price he receive upon elling. It i important to note that the payoff of the American option depend only on the maximum proce, but both the Britih payoff and the price of the European option depend on the three-dimenional proce (maximum-tock price-time), hence in Figure 5 and 6 we fix four different value of the current tock price (0.6, 1.0, 1.4, 1.8) and then compare return for the range of the maximum proce and the time. From (3.6) and (5.2) we ee that the payoff of the Britih and the European option doe not depend on the maximum proce M when M K =1.2. From Figure 5 and 6 we ee that (i) exerciing the Britih lookback option provide generally much better return than exerciing the American lookback option: it i more appreciable for favourable tock movement rather 18

20 Exercie time (month) S = 0.6 Exercie at M 1.2 with µ c = % 4% 2% 1% 0% 0% 0% Selling at M 1.2 (European) 8% 5% 2% 1% 0% 0% 0% Exercie at M = 1.4 with µ c = % 80% 79% 79% 79% 79% 79% Selling at M = 1.4 (European) 77% 77% 77% 78% 79% 80% 82% S = 1.0 Exercie at M 1.2 with µ c = % 78% 62% 46% 28% 10% 0% Selling at M 1.2 (European) 100% 84% 68% 50% 31% 12% 0% Exercie at M = 1.4 with µ c = % 121% 109% 97% 87% 80% 79% Selling at M = 1.4 (European) 134% 122% 110% 99% 88% 82% 82% Exercie at M = 1.6 with µ c = % 180% 171% 164% 160% 157% 157% Selling at M = 1.6 (European) 184% 176% 169% 164% 160% 160% 163% S = 1.4 Exercie at M = 1.4 with µ c = % 274% 250% 223% 193% 157% 79% Selling at M = 1.4 (European) 299% 278% 255% 229% 200% 163% 82% Exercie at M = 1.6 with µ c = % 287% 264% 239% 212% 182% 157% Selling at M = 1.6 (European) 308% 288% 267% 243% 217% 188% 163% Exercie at M = 1.8 with µ c = % 320% 300% 280% 260% 242% 236% Selling at M = 1.8 (European) 335% 318% 300% 282% 263% 248% 245% S = 1.8 Exercie at M = 1.8 with µ c = % 487% 456% 422% 383% 337% 237% Selling at M = 1.8 (European) 511% 486% 458% 428% 392% 346% 245% Exercie at M = 2.0 with µ c = % 497% 467% 434% 398% 357% 315% Selling at M = 2.0 (European) 518% 494% 468% 439% 406% 366% 326% Figure 6. Return oberved upon exerciing the Britih lookback option with fixed trike compared with return oberved upon elling the European lookback option with fixed trike. The return are expreed a a percentage of the original option price paid by the buyer (rounded to the nearet integer), i.e. R(t, m, )/100 = (G µc (t, m K ) K)/V (0, 1, 1) and R E (t, m, )/100 = V E (t, m, )/V E (0, 1, 1). The parameter et i µ c = 0.05, K = 1.2, T = 1, r = 0.1, σ = 0.4 and the initial tock price equal 1. than unfavourable; (ii) exerciing the Britih lookback option provide very comparable return to elling the European lookback option: the Britih return are generally better away from expiry and the European return are better near maturity. However, a remarked in [11], [12] and [4] in a real financial market the option holder ability to ell hi contract will depend upon a number of exogenou factor. Thee include hi ability to acce the option market, the tranaction cot and/or taxe involved in elling, and in particular the liquidity of the option market itelf. For lookback option the latter factor i epecially important, ince they generally trade in over-the-counter market which have no organied exchange and a uch thee market can be illiquid and thu the elling of the European option can be problematic. Crucially, the protection feature of the Britih lookback option i intrinic to it, that i, it i completely endogenou. It i inherent in the payoff function itelf (obtained a a conequence of optimal prediction), and a uch it i independent of any exogenou factor. From thi point of view the Britih lookback option i a particularly attractive financial intrument. 19

21 Acknowledgement. The author gratefully acknowledge financial upport and hopitality from the Haudorff Reearch Intitute for Mathematic at the Univerity of Bonn under the Trimeter Programme entitled Stochatic Dynamic in Economic and Finance where the preent reearch wa developed (May 2013). The author i grateful to Profeor G. Pekir for poing the problem and fruitful dicuion concerning the writing of thi work. Reference [1] Du Toit, J. and Pekir, G. (2007). The trap of complacency in predicting the maximum. Ann. Probab. 35 ( ). [2] Gapeev, P. (2006). Dicounted optimal topping for maxima in diffuion model with finite horizon. Electron. J. Probab. 11 ( ). [3] Glover, K., Pekir, G. and Samee, F. (2010). The Britih Aian option. Sequential Anal. 29 ( ). [4] Glover, K., Pekir, G. and Samee, F. (2011). The Britih Ruian option. Stochatic 80 ( ). [5] Kitapbayev, Y. (2013). On the lookback option with fixed trike. To appear in Stochatic. [6] Pekir, G. (2005). On the American option problem. Math. Finance 15 ( ). [7] Pekir, G. (2005). The Ruian option: Finite horizon. Finance Stoch. 9 ( ). [8] Pekir, G. (2005). A change-of-variable formula with local time on curve. J. Theoret. Probab. 18 ( ). [9] Pekir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-Boundary Problem. Lecture in Mathematic, ETH Zürich, Birkhäuer. [10] Pekir, G. (2007). A change-of-variable formula with local time on urface. Sém. de Probab. XL, Lecture Note in Math. 1899, Springer (69 96). [11] Pekir, G. and Samee, F. (2011). The Britih put option. Appl. Math. Finance. 18 ( ). [12] Pekir, G. and Samee, F. (2013). The Britih call option. Quant. Finance. 13 (95 109). [13] Shepp, L. and Shiryaev, A. N. (1993). The Ruian option: Reduced regret. Ann. Appl. Probab. 3 ( ). [14] Shepp, L. and Shiryaev, A. N. (1994). A new look at the Ruian option. Theory Probab. Appl. 39 ( ). [15] Shiryaev, A. N. (1999). Eential of Stochatic Finance. World Scientific. 20