British Strangle Options

Transcription

1 British Strangle Options Shi Qiu First version: 1 May 216 Research Report No. 2, 216, Probability and Statistics Group School of Mathematics, The University of Manchester

2 1. Introduction British Strangle Options Shi Qiu Following the economic rationale of [1] (British put options) and [11] (British call options) we present a new class of strangle options where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the maximum between the best prediction of European call payoff and the best prediction of European put payoff under the hypothesis that the true drift of the stock price equals to the contract drift. Inherent in this is a protection features which is the key to British options. The practical implications of this protection feature are most remarkable as not only can the option holder exercise between the double strike price to a substantial reimbursement of the original option price but also when the stock price movements are favourable he will generally receive higher returns at a lesser price. The British strangle options are designed for stocks with high volatility. We derive a closed form expression for the arbitrage-free price in terms of two exercise boundaries and show that the boundaries do not intersect. Using the result we perform a financial analysis of the British strangle option that leads to the conclusions above and shows that with the properly chosen contract drift, the British strangle option becomes a very attractive alternative to the classic American strangle options [13]. The purpose of the present paper is to introduce and examine the British options with strangle payoff. The payoff of strangle options in American type is the maximum between the American call payoff and American put payoff (see [2], [4] and [13]). Using the payoff of British call and put options in [1] and [11], we can define the payoff of British strangle options enjoying the benefit of British options with non-intersecting free boundaries. Following the economic rationale for the family of British options ([1, British put options], [11, British call options], the British options was expanded into [9, British Russian Options], [8, British Asian Options], [1, British Barrier Options] and [5, British Lookback Options]. We refer to such contracts as British strangle for the reason that the payoff is the maximum between British call payoff with one contract drift and the British put payoff with another contract drift. Similarly to [1] and [11], the most remarkable about British strangle options are not only that they provide a protection against the unfavourable stock price movements but also when the stock price movements are favourable they enable its holder to obtain higher returns than American strangle options. In this paper, we study the British strangle options which are suitable for the underlying asset with high volatility and inherit the higher returns from the recognised characteristics of British options. The paper mainly consists two parts: analytical solution of the value function and the financial analysis. As we know, the contract drift of British options ([1], [11], [9], [8] and [5]) is characterized as the tolerance drift. However, the discussion of the theoretical Key words and phrases: British option,strangle options, continuous dividend yield, change-of-variable formula on curves, optimal stopping, double free-boundaries, Option returns 1

3 solution of British strangle options begins with the classification of the contract drift into tolerance and preference. For t [,T) given and fixed, we denote the intersection between x G µ 1 (t,x) (British put payoff) and x G µ 2 (t,x) (British call payoff) by g(t) and prove that its graph lies inside the continuation region. The nonempty continuation region guarantees that the two free boundaries can not intersect before T. Because the payoff function of the British strangle option is undifferentiated at t g(t), not only the tolerance drift but also the preference drift can be chosen in the British strangle option. By the local time-space formula on curve, we derive the closed form formula of early exercised premium (EEP) representation for British strangle options and show that the two free boundaries in the EEP representation form a unique solution pair to the system of two nonlinear integral equations. The second part of this paper presents the analysis of the returns of British strangle options with preference drift and tolerance drift. We also compare the returns of exercising British strangle options with returns of exercising or selling American strangle options. After observing the returns, we may conclude the remarkable feature of British strangle options: (i) the tolerance drift can provide higher returns than the preference drift, particularly when the time is approaching to maturity; (ii) if the option is exercised rationally (immediately exercising the option when the stock price is inside the optimal stopping region), the returns of British strangle options with tolerance drift perform better than the American strangle option; (iii) the British strangle option enjoys significant returns when the option is out of money. However, if the American strangle option is in the liquid market (i.e. the option holder can sell the option at arbitrage-free price without any tax or transaction cost), the returns of American strangle options are higher than the British type. The paper is organized as follows. In Section 2 we present the motivation to design the British strangle option. In Section 3 we provide the mathematical definition of British strangle option and discuss how to separate the optimal stopping region into two parts by the proof that t g(t) is inside the continuation region. In Section 4 we derive the EEP representation for British strangle options by applying the local time-space formula and prove that the free boundaries in the EEP representation can be characterized as a unique solution pair to the system of two integral equations. From the financial analysis appeared in Section 5 we compare the returns of British strangle options with the returns of American strangle options to find the financial feature of British strangle options. 2. Motivation for British Strangle Options The economic motivation for British strangle options is parallel to that of British put and call options (see [1] and [11]). We will briefly review the key part of this motivation following another motivation for avoiding the two free boundaries of British strangle option from intersecting. 1. For the traditional strangle options introduced in [4], we can long a British call option and a British put option with same maturity to construct a traditional British strangle option. However, reviewing the free boundaries of British call and put options, when the the contract drift is close to risk-free rate, the free boundary for British call options and the free boundary for British put options may intersect (see Figure 1). 2

4 45 4 Stock Price D BC D BP t b BP (t) 15 1 t b BC (t) Time Figure 1. Shows the free-boundary of British call b BC (t) and put options b BP (t). D BC D BP are the intersection of the optimal stopping region of British call options and the optimal stopping region of British put options. We set the parameter for British call andput options is: L = 15, K = 2, σ =.6, δ = r =.1, µ put =.12, µ call =.8. Since the free-boundaries of British call and put options are non-monotonic and may intersect, both the call and put in the traditional British strangle options are optimal to exercise in D BC D BP and change into European type options. So the British strangle is unsuitable to be constructed in the traditional way. The purpose of this paper is to design the strangle options in British type enjoying the benefit of British options and the free boundaries will never intersect. 2. Consider the financial market consisting of a risky stock X and a riskless bound B whose prices respectively evolve as (2.1) (2.2) dx t = (µ δ)x t dt+σx t dw t db t = rb t dt (B = 1), (X = x), where µ is the appreciation rate (drift), δ is the continuous dividend, σ > is the volatility coefficient, W = (W t ) t is a standard Wiener process, and r > is the interest rate. Recall that the payoff of European(American) strangle options in[4] and[13] is the maximum between the call option payoff and put options payoff. By the definition of British call and put options, the payoff of British strangle options can be defined as (2.3) G µ 1,µ 2 (t,x) = G µ 1 (t,x) G µ 2 (t,x) where G µ 1 (t,x) = E µ 1 t,x(l X T ) + isthepayoffforbritishputoptionsand G µ 2 (t,x) = E µ 2 t,x(x T K) + is the payoff for the British call options. The contract drift µ 1 and µ 2 can take any real number. E µ 1 and E µ 2 is the expectation under the measure P µ 1 and P µ 2, respectively. 3

5 These two measure will be discussed in the following section. Recall that the European strangle holder has the right to exercise the options at maturity and receive the payoff (2.4) (L X T (µ)) + (X T (µ) K) +, where X T (µ) represents the stock price at time T under the actual probability measure. If µ = r, the payoff is fair for the option buyer and (2.5) V ES = Ee rt( (L X T (r)) + (X T (r) K) +), V ES is the arbitrage-free price of European strangle options under the risk neutral measure. On the other hands, from the standpoint of true buyer who has no ability or desire to sell the option nor to hedge his position (see the analysis of true buyer in [1, section 2] and [6]), if µ < r, the payoff is favourable for the put part in strangle options (2.5) and unfavourable for the call part in strangle options (2.5), in the sense that (2.6) (2.7) E(L X T (µ)) + > E(L X T (r)) +, E(X T (µ) K) + < E(X T (r) K) +. If µ > r, the payoff is unfavourable for the put part and favourable for the call part, in the sense that (2.8) (2.9) E(L X T (µ)) + < E t,x (L X T (r)) +, E(X T (µ) K) + > E t,x (X T (r) K) +. Notethattheactualdrift µ isunknownattime t = anddifficulttoestimateatthelatertime t (,T]. And the real drift cannot make both parts in strangle options become favourable. 3. The brief analysis above shows that the actual drift µ of the underlying asset is irrelevant in determining the arbitrage-free price of the option. To the (true) buyer, he will buy the strangle options expected to exercise at the call part if he believes that µ > r. The true µ is unknown constant, if µ is smaller than r, the strangle buyer will lose the profit from the call part until keep waiting to the maturity. The British strangle option gives the right for buyer to use constant contract drift µ 2 > r to replace unpredictable µ in the payoff immediately and grabs the higher profit in the call part than keep holding the option to the maturity. In another case, the true buyer will buy the strangle options expected to exercise at the put part if he believes that true drift µ < r. The British strangle option gives the right for buyer to use constant contract drift µ 1 < r to replace unpredictable µ in the payoff immediately and grabs the higher profit in the put part than keep holding the options to the maturity. In above case, we discuss that when the true drift is favourable for call or put part in the strangle options, the contract drift in the British strangle options cloud lock the favourable drift and give profit controlled by favourable drift to the buyer. Since the contract drift is favourable to call part or put part in the strangle options and it represents the preferred level for the drift that the true buyer expect, so we name these contract drift as preference drift. What if the options holder observes the stock movement which causes him to believe µ > r andthepayofffromtheput partishigher thanthecallpart(i.e. X t < L)? Thestrangleoptions holder is effectively able to substitute the unfavourable drift with the contract drift µ 1 > r in 4

6 the put part L X T (µ 1 ) to minimise the loss. So for the put part in strangle options, the contract drift µ 1 > r is selected to represent the buyer s expected level of tolerance for the deviation of the actual drift from the original belief. What if the stock price movement causes the options buyer to believe µ < r and the payoff from the call part is higher than the put part (i.e. X t > K )? The options holder is able to use contract drift µ 2 < r to replace the unfavourable drift in the call part X T (µ 2 ) K to minimise the loss. So for the call part in strangle options, the contract drift µ 2 < r is selected to be tolerance drift. It will be shown that the practical implication of these tolerance contract drifts are most remarkable at not only the British strangle options holder exercise between the strike price ( L < K ) but also when the stock price with large fluctuation he will generally receive higher returns. 4. The true buyer of the strangle options who believes that the options is out-of-money chooses to sell the strangle contact. However, the price to sell is determined by real market and involves transaction cost or tax. The barrier from the market improves the difficulty for selling contract out-of-money in the market, particularly the over the counter (OTC) market. From the payoff of British strangle options, we write the value of European options inside the payoff to guarantee the payoff is always strictly positive. From the financial analysis part in this paper, we find that exercising the out-of-money British strangle options at the starting period, the payoff could offset the majority of the options payment. 3. The British Strangle Options: Definition and Basic Properties We begin this section by the definition of British strangle options in (3.21). Then we will prove that the two free boundaries do not intersect. This is followed by the free-boundary problem characterising the arbitrage-free price and the rational exercise strategy. 1. Consider the financial market consisting of a risky stock X and a riskless bond B given in formula (2.1) and (2.2), where µ is the appreciation rate, σ is the volatility coefficient, W = (W t ) t is the standard Wiener Process, and r > is the interest rate. The upper strike price K and lower strick price L, satisfying L K, and T is the expiry for the option. Definition 1. The British strangle options is a financial contract permit to give the payoff as the maximum between the British call payoff and British put payoff at any stopping time τ before the maturity T. The British call payoff E µ 2 [(X T K) + F τ ] is the best prediction of the European call payoff given all the information up to time τ under the hypothesis that the true drift of stock price equals µ 2. The British put payoff E µ 1 [(L X T ) + F τ ] is the best prediction of the European put payoff given all the information up to time τ under the hypothesis that the true drift of stock price equals µ 1. The quantity µ 1 and µ 2 are defined in the option contract (see (2.3)) and we refer to it as the contract drift. The discussion in section 2 shows that (3.1) (3.2) µ 1 > r µ 2 < r is the tolerance level for the deviation of the true drift µ from the original belief. And (3.3) µ 1 r 5

7 (3.4) µ 2 r is the preference level for the deviation of the true drift µ from the original belief. 2. Denoting by (F t ) t T the natural filtration generated by X, the call part G µ 2 in the strangle payoff is (3.5) E µ 2 [(X T K) + F τ ], where the conditional expectation is taken in the probability measure P µ 2 under which the stock price is (3.6) dx t = (µ 2 δ)x t dt+σx t dw t. The put part G µ 1 in the strangle payoff is (3.7) E µ 1 [(L X T ) + F τ ], where the conditional expectation is taken in the probability measure P µ 1 under which the stock price is (3.8) dx t = (µ 1 δ)x t dt+σx t dw t. Comparing (2.1) with (3.6) and (3.8), when the stock price with true drift µ hits the upper boundary, we exercise the option and insert the contract drift µ 2 into the stock price for the remaining term of the contract. When the stock price hits lower boundary, we will exercise the options and use the contract drift µ 1 in the stock price for the remaining life of the contract. 3. The property (stationary increments, independent increments and Markov property) of Brownian Motion W, the payoff of British strangle options G µ 1,µ 2 (t,x) = G µ 1 (t,x) G µ 2 (t,x) defined in (2.3) is (3.9) G µ 1 (t,x) = E(L xz µ 1 T t )+, and (3.1) G µ 2 (t,x) = E(xZ µ 2 T t K)+, where Z µ 1 T t and Z µ 2 T t is given by (3.11) (3.12) ( ) Z µ 1 T t = exp σw T t +(µ 1 δ σ2 2 )(T t), ( ) Z µ 2 T t = exp σw T t +(µ 2 δ σ2 2 )(T t). By the scaling property of Brownian motion, the formula (3.9) and (3.1) can be written as (3.13) G µ 1 (t,x) = LN( d L 2(t,x)) xe (µ 1 δ)(t t) N( d L 1(t,x)), (3.14) G µ 2 (t,x) = xe (µ 2 δ)(t t) N(d K 1 (t,x)) KN(dK 2 (t,x)), (3.15) G µ 1 x (t,x) = e(µ 1 δ)(t t) N( d L 1 (t,x)), 6

8 (3.16) G µ 2 x (t,x) = e (µ 2 δ)(t t) N(d K 1 (t,x)), where (3.17) (3.18) (3.19) (3.2) d K 1 (t,x) = ln x +(µ K 2 δ + σ2 (T t)) 2 σ, T t d K 2 (t,x) = d K 1 (t,x) σ T t, d L 1(t,x) = ln x +(µ L 1 δ + σ2 (T t)) 2 σ, T t d L 2(t,x) = d L 1(t,x) σ T t, for t [,T) and x (, ) where N( ) is the standard normal distribution function given by N(x) = (1/ 2π) x /2 dy for x R. e y2 4. Standard hedging arguments based on self-financing portfolio (with consumption) imply that the arbitrage-free price of British strangle option is given by ] (3.21) V BS (t,x) = sup Ẽ t,x [e rτ E µ 1 [(L X T ) + F τ ] E µ 2 [(X T K) + F τ ], τ T t where the supremum is taken over all the stopping times τ of X = X(µ) in [,T] and Ẽ t,x is the expectation with respect to the unique martingale measure P t,x for X t = x. By Law(X(µ) P) is the same as Law(X(r) P), the formula (3.21) can be written as (3.22) V BS (t,x) = sup E t,x [e rτ G µ 1,µ 2 (t+τ,x t+τ )], τ T t the process X = X(r) under the P solves that (3.23) dx t = (r δ)x t dt+σx t dw t. The formula (3.22) shows that the value of British options is the optimal stopping problem. 5. Using the established probabilistic techniques in book [12], the optimal stoping time for V BS (t,x) is (3.24) τ D BS = inf{s [,T t] (t+s,x x s ) DBS }, where D BS is the optimal stopping region (3.25) D BS = {(t,x) [,T) (, ) V BS (t,x) = G µ 1,µ 2 (t,x)}, and C BS is the continuation region (3.26) C BS = {(t,x) [,T) (, ) V BS (t,x) > G µ 1,µ 2 (t,x)}. Since x G µ 1 (t,x) is continuously (strictly) decreasing function with G µ 1 (t,) = L and G µ 1 (t, ) =,and x G µ 2 (t,x) iscontinuously(strictly)increasingfunctionwith G µ 2 (t,) = 7

9 and G µ 2 (t, ) =, so there exist a unique intersection for G µ 1 (t,x) and G µ 2 (t,x) for t given and fixed. Then we can define a function t g(t) where (3.27) G µ 2 (t,g(t)) = G µ 1 (t,g(t)). Figure 2 may give an intuitive understanding of the function g(t). By the implicit function theorem, the function t g(t) is continuous and the first derivative exists. The following part will prove that graph of function g(t) is inside the continuation region C BS. 6. To gain a deep insight into the solution to the optimal stopping problem (3.21), we apply the local time-space formula on e rs G µ 1,µ 2 (t+s,x t+s ) and take E t,x from both sides (3.28) E t,x e rs G µ 1,µ 2 (t+s,x t+s ) s = G µ 1,µ 2 (t,x)+e t,x e ru H µ 1 (t+u,x t+u )I(X t+u < g(t+u)))du s + E t,x e ru H µ 2 (t+u,x t+u )I(X t+u > g(t+u)))du + E t,x s ) e (G ru µ 2 x (t+u,g(t+u)+) G µ 1 x (t+u,g(t+u) ) dl g u(x), where l g u (X) is the local time for process X at function g with the mathematical definition l g 1 u u(x) = P lim ε I(g(t+r) ε<x 2ε t+r<g(t+r)+ε)d X,X r (see the abstract in [7]). Moreover (3.29) (3.3) H µ 1 (t,x) = G µ 1 t (t,x)+(r δ)xg µ σ2 1 x (t,x)+ 2 x2 G µ 1 xx (t,x) rgµ 1 (t,x), H µ 2 (t,x) = G µ 2 t (t,x)+(r δ)xg µ σ2 2 x (t,x)+ 2 x2 G µ 2 xx (t,x) rgµ 2 (t,x). It is clear that the function G µ 1 (t,x) and G µ 2 (t,x) satisfy the Kolomogorov backward equation i.e. (3.31) (3.32) G µ 1 t +(µ 1 δ)xg µ 1 x + σ2 2 x2 G µ 1 xx =, G µ 2 t +(µ 2 δ)xg µ 2 x + σ2 2 x2 G µ 2 xx =. So the equation H µ 1 (t,x) and H µ 2 (t,x) can be transformed into (3.33) (3.34) H µ 1 (t,x) = (r µ 1 )xg µ 1 x (t,x) rgµ 1 (t,x), H µ 2 (t,x) = (r µ 2 )xg µ 2 x (t,x) rgµ 2 (t,x). Since G µ 1 x (t,x) and G µ 2 x (t,x), the function H µ 1 (t,x) for µ 1 r and H µ 2 (t,x) for µ 2 r. For µ 1 > r, section 3 in [1] shows that there exists a function t h 1 (t) such that (3.35) H µ 1 (t,h 1 (t)) =, and H µ 1 (t,x) > for x > h 1 (t) and H µ 1 (t,x) < for x < h 1 (t). On the other hands, when µ 1 r, function H µ 1 (t,x) < for all the x R + so that we set h 1 (t) =. For µ 2 < r, section 3 in [11] shows that there exists a function t h such that (3.36) H µ 2 (t,h ) =, 8

10 and H µ 2 (t,x) < for x > h and H µ 2 (t,x) > for x < h. On the other hands when µ 2 r, function H µ 2 (t,x) < for all the x R + so that we set h =. Since the local time term is always positive in (3.28), the region {(t,x) x h 1 (t) and x h } is inside C EA. 7. In the following theorem, we will try to prove that the function g(t) is inside the continuation region. Theorem 1. The function t g(t) is inside the continuation region C BS. Proof. To prove that (t, g(t)) is inside the continuation region, we need to find a nonzero stopping time τ such that E t,g(t) e rτ G µ 1,µ 2 (t+τ,x t+τ ) > G µ 1,µ 2 (t,g(t)). For time t given and fixed, we define a stopping time (3.37) τ ε = inf{s [,T t] X t+s g(t+s) ε or X t+s g(t+s)+ε}. Using τ ε to replace s in formula (3.28), we get that (3.38) E t,g(t) e rτε G µ 1,µ 2 (t+τ ε,x t+τε ) τε = G µ 1,µ 2 (t,g(t))+e t,g(t) e ru H µ 1 (t+u,x t+u )I(X t+u < g(t+u)))du τε + E t,g(t) e ru H µ 2 (t+u,x t+u )I(X t+u > g(t+u)))du + E t,g(t) τε ) e (G ru µ 2 x (t+u,g(t+u)+) G µ 1 x (t+u,g(t+u) ) dl g u(x). Since u is from [,τ ε ], the value of X t+u [g(t + u) ε,g(t + u) + ε] for u before τ ε. Meanwhile H µ 1 (t,x) and H µ 2 (t,x) are binary continuous function in the domain, so there must exist constant C 1, C 2, C 3 and C 4 such that C 1 e ru H µ 1 (t+u,x t+u ) C 2 and C 3 e ru H µ 2 (t+u,x t+u ) C 4 for u [,τ ε ]. If C 1 and C 3 are positive, it is obvious that (t,g(t)) C BS by (3.38). We will assume that C 1 < and C 3 < for the following proof. From (3.38), we have the inequality that (3.39) E t,g(t) e rτε G µ 1,µ 2 (t+τ ε,x t+τε ) G µ 1,µ 2 (t,g(t))+c 1 E t,g(t) τ ε +C 3 E t,g(t) τ ε + E t,g(t) τε Since (t,x) G µ 1 x (t,x), (t,x) G µ 2 e ru (G µ 2 x (t+u,g(t+u)+) Gµ 1 x (t+u,g(t+u) ) )dl g u (X). x (t, x) and t g(t) are continuous function, and x (t,x) > for (t,x) [,T] (, ), so we can find constant C 5 and C 6 such x (t+u,g(t+u)) C 6 for u [,τ ε ] G µ 2 x (t,x) Gµ 1 that < C 5 G µ 2 x (t+u,g(t+u)) Gµ 1 (3.4) E t,g(t) e rτε G µ 1,µ 2 (t+τ ε,x t+τε ) G µ 1,µ 2 (t,g(t))+c 1 E t,g(t) τ ε +C 3 E t,g(t) τ ε +C 5 E t,g(t) l g τ ε (X). In the following content, we will discuss the convergent speed of E t,g(t) l g τ ε (X) when τ ε. We apply the local time space formula on the X t+s g(t+s) X t+s g(t+s) = X t g(t) + s 9 (r δ)x t+u I(X t+u > g(t+u))du

11 + + (3.41) + s s s (r δ)x t+u I(X t+u < g(t+u))du σx t+u I(X t+u > g(t+u))dw u σx t+u I(X t+u < g(t+u))dw u +l g s (X). Replacing s in (3.41) by τ ε and taking E t,g(t) from both sides, we get E t,g(t) l g τ ε (X) = E t,g(t) X t+τε g(t+τ ε ) τε E t,g(t) (r δ)x t+u I(X t+u > g(t+u))du τε (3.42) + E t,g(t) (r δ)x t+u I(X t+u < g(t+u))du. By the basic mathematical analysis, we know the convergent speed of τε (3.43) E t,g(t) (r δ)x t+u I(X t+u > g(t+u))du O(τ ε ) τε (3.44) E t,g(t) (r δ)x t+u I(X t+u < g(t+u))du O(τ ε ) when τ ε. By the mean value theorem, we can find ξ [,τ ε ] such that (3.45) g(t+τ ε ) = g(t)+g (ξ)τ ε Since g (ξ) goes to a constant as ξ by the implicit function theorem, the convergent rate of (3.46) g(t+τ ε ) g(t) O(τ ε ) as τ ε. BythedefinitionofgeometricBrownianmotion,weknowthat X t+τε = g(t)exp(σw τε + (r δ σ 2 /2)τ ε ). Since the the convergent speed of W τε is the same as τ ε, the L Hospital s rule shows the convergent rate of (3.47) g(t)exp(σw τε +(r δ σ 2 /2)τ ε ) g(t) O( τ ε ), as τ ε. The formula (3.46) and (3.47) show that (3.48) E t,g(t) X t+τε g(t+τ ε ) O( τ ε ). The formula (3.43), (3.44) and (3.48) contribute to the conclusion (3.49) E t,g(t) l g τ ε (X) O( τ ε ). Going back to the equation (3.4), since the convergent rate of E t,g(t) l g τ ε (X) is same as τ ε as τ ε, which is slower than the convergent rate of τ ε. So we can find a ε small enough to make the term in (3.4) (3.5) G µ 1,µ 2 (t,g(t))+c 1 E t,g(t) τ ε +C 3 E t,g(t) τ ε +C 5 E t,g(t) l g τ ε (X) >. 1

12 Value x V BS (t,x) x G µ 1,µ 2 (t,x) b BS Stock Price 1 (t) g(t) b BS Figure 2. Shows the the value function V BS (bold line) and payoff function G µ 1,µ 2 (slender line) of British strangle options when t is given and fixed. The function b BS 1 (t) is the lower free-boundary and the function b BS is the upper free-boundary. Function g(t) defined in (3.27) which is the intersection of function G µ 1 (t,x) and G µ 2 (t,x). Then we find the stopping time τ ε such that (3.51) E t,g(t) e rτε G µ 1,µ 2 (t+τ ε,x t+τε ) > G µ 1,µ 2 (t,g(t)) So the function t g(t) is inside the continuation region C BS for t [,T]. 8. By the result from theorem 1, we can separate the optimal stopping region D BS into two regions (3.52) (3.53) D BS 1 = {(t,x) [,T] (, ) V BS (t,x) = G µ 1 (t,x)}, D2 BS = {(t,x) [,T] (, ) V BS (t,x) = G µ 2 (t,x)}, and D BS 1 D BS 2 =. Then we can find the optimal stopping boundary b BS 1 and b BS 2 to separate the continuation region and stopping region (3.54) (3.55) (3.56) D1 BS = {(t,x) [,T] (, ) x b EA 1 (t)}, D2 BS = {(t,x) [,T] (, ) x b EA }, C BS = {(t,x) [,T] (, ) b EA 1 (t) < x < b EA }. For t is given and fixed, the position of b EA 1 (t) and b EA are displayed in Figure 2. Theorem 2. Recall the definition of the optimal stopping region and continuation region in [11, British call options] and [1, British put options], we find that D BS 1 D BP when µ 1 > r 11

13 and D BS 2 D BC when < µ 2 < r, where the region D BP is the optimal stopping region for the British put options with contract drift µ 1 and strike price L; the region D BC is the optimal stopping region for the British call options with contract drift µ 2 and strike price K. Proof. (t,x) D2 BS, we have V BS (t,x) = G µ 2 (t,x). By the definition of the value of British strangle options in (3.21), it is clear that V BS (t,x) V BC (t,x), where V BC (t,x) is the value of British call options with contract drift µ 1 and strike price K. By the above analysis we have V BC (t,x) V BS (t,x) = G µ 2 (t,x), i.e. V BC (t,x) = G µ 2 (t,x). So (t,x) D BC and it is proved that D2 BS D BC. Similarly, we can prove that D1 BS D BP. For t [,T], the result from Theorem 2 shows that b BS b BC (t) h for < µ 2 < r and b BS 1 (t) b BP (t) h 1 (t) for µ 1 > r. The function t b BC (t) and t b BP (t) is the free boundaries for British call and put options, respectively. From the above analysis on H µ 1 (t,x) and H µ 2 (t,x), we know that h 1 (T) = r µ 1 L and h 2 (T) = r µ 2 K. So when time t approaching to T, we can find t such that t [t,t] satisfying h 1 (t) < h and b BS 1 (t) < h 1 (t) < h < b BS. In the view of formula (3.28) this implies that: i)for h 2 (T) > g(t), as t is sufficiently closed to T, if x > h, we have H 2 (t,x) < and it is optimal to stop immediately (since the obtained from being below h 2 cannot offset the cost of getting there due to lack of time). This shows that b BS 2 (T) = h 2 (T). ii) For h 2 (T) g(t), it is optimal to exercise the options as x > g(t) immediately (since the obtained from the local time term in (3.28) cannot offset the cost of get there due to lack of time). This shows that b BS 2 (T) = g(t). Combining the result i) and ii), we have that b BS 2 (T) = max{h 2 (T),g(T)}. Similar analysis, wecanconcludetheresult b BS 1 (T) = min{h 1 (T),g(T)}. Ontheotherhands, when µ 2 r, the H µ 2 term in (3.28) is always negative, so when x > g(t) and t is sufficiently close to T then it is optimal to stop immediately (since the gain from local time term and the H µ 1 term in (3.28) cannot offset the cost of getting there due to lack of time). So b BS 2 (T) = g(t). Similar argument, we can show that b BS 1 (T) = g(t) for µ 1 r. 9. We see from the definition of payoff function in (2.3) and the formula (3.13) and (3.14) that (3.57) x G µ 1,µ 2 (t,x) is convex, with G µ 1,µ 2 (t,) = L and G µ 1,µ 2 (t, ) = for any t [,T] given and fixed. As t T, the formula (3.13) and (3.14) shows that G µ 1,µ 2 (T,x) = (x K) + (L x) + for x (, ) showing that the British strangle payoff is the same as the European/American strangle payoff at the time of maturity. Moreover, if the contract drift µ 2 for the call part in British BS strangle options, when (t,x) D 2, from (3.1) and (3.12) we have that (3.58) V BS (t,x) = G µ 2 (t,x) < e r(t t) E(xZ r T t K)+ V ES (t,x), where V ES is the value of European strangle options. As we know, when we chooser τ = T t in (3.22), the British strangle option will be changed into European strangle options so the maximum payoff of exercising the call part can be obtained as waiting to the expiry. Therefore, it is not optimal to exercise the British strangle option to grasp the call part payoff 12

14 before the maturity. So when µ 2, we can set b BS = for t [,T). Since V BS (t,x) = V ES (t,x) as selecting τ = T t in (3.22), the analysis confirms that the British strangle options is more expensive than European strangle options no matter what the contract drift is selected. Finally, from (3.22) and (3.57), we can find that (3.59) x V BS (t,x) is convex. The definition of payoff function G µ 1,µ 2 shows that it is strictly increasing on the call part G µ 2 for x and strictly decreasing on the put part G µ 1 for x. Additionally, the value function V BS is the smallest envelope curve over the payoff function. We can guess the shape of the value function x V BS (t,x) is similar to the value of European strangle options (a snapshot of British strangle options is shown in Figure 2). The most important technical difference is that the American strangle boundaries b ST 1 and b ST this is not the case for British strangle boundaries b BS 1 and b BS 2 is monotonic shown in ([13]), however 1. By the Standard argument based on the strong Markov property, we can define the free-boundary problem of British strangle options, (3.6) (3.61) (3.62) (3.63) (3.64) (3.65) (3.66) (3.67) V BS t +L X V BS = rv BS in C BS, V BS (t,x) > G BS (t,x) in C BS, V BS (t,x) = G µ 1 (t,x) V BS (t,x) = G µ 2 (t,x) V BS (t,x) = G µ 1 (t,x) V BS (t,x) = G µ 2 (t,x) Vx BS (t,x) = G µ 1 x (t,x) Vx BS (t,x) = G µ 2 x (t,x) in D BS 1, in D BS 2, 2. for x = b BS 1 (t), and t [,T), for x = b BS, and t [,T), for x = bbs 1 (t), and t [,T), for x = bbs, and t [,T). where b BS 2 (T) = max{h 2 (T),g(T)} and b BS 1 (T) = min{h 1 (T),g(T)}. Thedetailsfortheproof that the free boundaries are continuous, satisfies smooth-fit and etc will be omitted because when µ 1 is negative or µ 2 is large the two free boundaries are not the monotonic functions for time (see Figure 3 and Figure 4), which makes analysis more complicated than the American strangle options [13]. Comparing with the free boundary of British put options [1], the lower free boundary b BS 1 will not go to infinity as µ 2 approaching to r or µ 2 r. From the Figure 3, the lower free boundary seems increasing for positive µ 2 and the monotonicity is uncertain for negative µ 2. For the upper free boundary b BS 2 in Figure 4, when the contract drift µ 1 =.8, the upper free boundary is not monotonic. Additionally, from the analysis of British call [11], the free boundary will be as the contract drift µ c r. However, for any contract drift µ 2 R +, the upper free boundary is nonzero. The reason for the difference in the free boundaries between the British strangle options and British call (put) options is based on Theorem 1, which guarantees the two free boundaries cannot intersect such that the free boundaries cannot get extreme value. We assume that the underlying asset satisfying the geometric Brownian motion. The constant volatility appeared in the dynamic should be agreed in the contract, which can be estimated by statistic method or implied volatility. However the estimation of true drift µ is not easily obtained, therefore the contract drift µ 1 and µ 2 seem the tolerance or preference level for the put part and call put to protect the risk of the true drift. 13

15 22 2 µ 1 = 4 18 µ 1 Stock Price µ 1 = 2 1 µ 1 µ 1 = 8 µ 1 = Time Figure 3. Shows the lower free boundary of British strangle options with contract drift µ 1 =, µ 1 =.1 and µ 1 =.12. The region below the lower free boundary is the optimal stopping region D BS 1. We set the remaining parameter as: L = 15, K = 2, σ =.6, δ = r =.1, µ 2 = µ 2 =.8 µ 2 Stock Price 3 25 µ 2 = µ 2 =.8 µ Time Figure 4. Shows the upper free boundary of British strangle options with contract drift µ 2 =.8, µ 1 =.2 and µ 1 =.8. The region above the upper free boundary is the optimal stopping region D BS 2. We set the remaining parameter as: L = 15, K = 2, σ =.6, δ = r =.1, µ 2 =

16 4. The Arbitrage-free Price and the Rational Exercise Boundaries In this section, we will use the local time-space formula [7] to get the arbitrage-free price V BS for the British strangle option in terms of two free boundaries b BS 1 and b BS 2. Moreover, the free boundaries can be characterised as the unique solution to a system of two nonlinear integral equations. Let V BS : [,T] (, ) R +, b BS 1 : [,T] R + and b BS 2 : [,T] R + are the solution to the free-boundary problem (3.6)-(3.67). The function V BS, b BS 1 and b BS 2 are continuous functions satisfying: (i) V BS is C 1,2 on C BS D BS ; (ii) b BS 1 and b BS 2 are bounded variation; (iii) Vt BS +L X V BS rv BS is locally bounded on C BS D BS ; (iv) x V BS (t,x) is convex on (, ); and t Vx BS (t,b BS 1 (t)±) = G µ 1 x (t,bbs 1 (t)) and t Vx BS (t,b BS ±) = G µ 2 x (t,bbs ) are continuous. So the local time-space formula [7] can be applied on e rs V BS (t+s,x t+s ) for t [,T] given and fixed (4.1) E t,x e rs V BS (t+s,x t+s ) s = V BS (t,x)+e t,x e rv H µ 1 (t+v,x t+v )I(X t+v b BS 1 (t+v))dv + E t,x s Let s replace by T t, we have e rv H µ 2 (t+v,x t+v )I(X t+v b BS 2 (t+v))dv. (4.2) V BS (t,x) = E t,x e r(t t) V BS (T,X T ) + E t,x T t + E t,x T t e rv H µ 1 (t+v,x t+v )I(X t+v b BS 1 (t+v))dv e rv H µ 2 (t+v,x t+v )I(X t+v b BS 2 (t+v))dv. By the analysis in [11, Section 4] and [1, Section 4], the formula (4.2) can be simplified into computable form (4.3) (4.4) where (4.5) V BS (t,x) = E t,x e r(t t) V BS (T,X T ) + + T t T t e rv b BS 1 (t+v) e rv + b BS 2 (t+v) = E t,x e r(t t) V BS (T,X T )+ + T t H µ 1 (t+v,y)f(v,x,y)dydv H µ 2 (t+v,x t+v )f(v,x,y)dydv, T t J 2 (t,x,v,b BS 2 (t+v))dv, z J 1 (t,x,v,z) = e rv H µ 1 (t+v,y)f(v,x,y)dy, J 1 (t,x,v,b BS 1 (t+v))dv 15

17 (4.6) (4.7) f(v,x,y) = 1 ( 1 [ ( y ]) σy v ϕ σ log δ v x) (r σ2 2 )v, ( J 1 (t,x,(t t),z) =rle r(t t) 1 [ ( z L ]) Φ σ log ) (r δ σ2 T t x 2 )(T t) ( 1 [ ( z L ]) µ 1 xφ σ log ) (r δ + σ2 T t x 2 )(T t), and (4.8) (4.9) J 2 (t,x,v,z) = e rv H µ 2 (t+v,y)f(v,x,y)dy, ( J 2 (t,x,(t t),z) =µ 2 xφ z 1 [ ( x ]) σ log )+(r +δ + σ2 T t z K 2 )(T t) ( rke r(t t) 1 [ ( x Φ σ log T t z K )+(r+δ σ2 2 )(T t) ]). By the definition of free boundaries (see (3.64) and (3.65)), we have that V BS (t,b BS 1 (t)) = G µ 1 (t,b BS 1 (t)) and V BS (t,b BS ) = G µ 2 (t,b BS ). The free boundaries of British strangle options are a solution pair of the following two equations: (4.1) (4.11) G µ 1 (t,b BS 1 (t)) =E t,b BS 1 (t)e r(t t) V BS (T,X T )+ + T t T t J 2 (t,b BS 1 (t),v,b BS 2 (t+v))dv, G µ 2 (t,b BS ) =E t,b BS e r(t t) V BS (T,X T )+ + T t T t J 2 (t,b BS,v,b BS 2 (t+v))dv, J 1 (t,b BS 1 (t),v,b BS 1 (t+v))dv J 1 (t,b BS,v,b BS 1 (t+v))dv The above analysis shows the existence of the solution pair to the system (4.1)-(4.11) and we now turn to its uniqueness. Theorem 3. The optimal stopping boundaries of British strangle options (3.21) can be characterized as the unique solution pair of the system (4.1)-(4.11). One of the solution is in the class of continuous functions c 1 : [,T] R + satisfying c 1 (t) min(h 1 (t),g(t)) for t [,T] and c 1 (T ) = min(h 1 (T),g(T)); the other solution is in the class of continuous function c 2 : [,T] R + satisfying c max(h,g(t)) for t [,T] and c 2 (T ) = max(h 2 (T),g(T)). Proof. Assume that there exists two functions c 1 and c 2 satisfying the condition mentioned in the statement of Theorem 3. They are another solution pair of the system (4.1)-(4.11). Inserting them back to the value function (4.2), we get a new value function U c (t,x) = e r(t t) E t,x [G µ 1,µ 2 (T,X T )] T t e rv E t,x H µ 1 (t+v,x t+v )I(X t+v < c 1 (t+v))dv 16

18 (4.12) T t e rv E t,x H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv. i) This part will prove that U c (t,x) = G µ 1 (t,x) for x c 1 (t) and U c (t,x) = G µ 2 (t,x) for x c. The Markov property of X implies that e rs U c (t+s,x t+s ) (4.13) s s e rv H µ 1 (t+v,x t+v )I(X t+v < c 1 (t+v))dv e rv H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv is a martingale under P t,x for s [,T t]. For x c, considering the stopping time (4.14) σ c2 = inf{s [,T t] X t+s c 2 (t+s)}. Since U c (t,c ) = G µ 2 (t,c ) for all t [,T] and U c (T,x) = G µ 2 (T,x) for x > g(t), we see that U c (t+σ c2,x t+σc2 ) = G µ 2 (t+σ c2,x t+σc2 ). Replacing s by σ c2 in (4.13), taking E t,x from both sides and using the optional sampling theorem, we have that σc2 U c (t,x) = E t,x [e rσc 2 U c (t+σ c2,x t+σc2 )] E t,x e rv H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv (4.15) σc2 = E t,x [e rσc 2 G µ 2 (t+σ c2,x t+σc2 )] E t,x e rv H µ 2 (t+v,x t+v )dv = G µ 2 (t,x) The last equality is from (3.28) using σ c2 to replace s and c > g(t) for t [,T]. Similarly, we can prove that U c (t,x) = G µ 1 (t,x) for x c 1 (t). ii) We want to show that U c (t,x) V BS (t,x) for (t,x) [,T] (, ). Take any such (t, x) and consider the stopping time (4.16) τ c = inf{s [,T t] X t+s c 2 (t+s) or X t+s c 1 (t+s)}. Since c 1 and c 2 is the solution of the system (4.1)-(4.11), we know that U c (t+τ c,x t+τc ) = G µ 1,µ 2 (t+τ c,x t+τc ). If x c 1 (t) or x c, the stopping time τ c is zero. So U c (t,x) = G µ 1,µ 2 (t,x) V BS (t,x). If c 1 (t) < x < c, then the claim follows since U c (t+τ c,x t+τc ) = G µ 1,µ 2 (t+τ c,x t+τc ). Replacing s by τ c in (4.13), taking E t,x from both side and using the optional sampling theorem, we have τc U c (t,x) = E t,x [e rτc U c (t+τ c,x t+τc )] E t,x e rv H µ 1 (t+v,x t+v )I(X t+v < c 1 (t+v))dv (4.17) τc E t,x e rv H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv = E t,x [e rτc G µ 1,µ 2 (t+τ c,x t+τc )] V BS (t,x). iii) We will prove that c b BS and c 1 (t) b BS. Firstly, we will prove c b BS for t [,T]. Suppose that there exists t [,T] such that c > b BS. Select a point x > c and set the stopping time (4.18) σ b2 = inf{s [,T t] X t+s b BS 2 (t+s)}. 17

19 Replacing s by σ b2 in(4.1)and(4.13), take E t,x forthese two formulaeandapplying optional sampling theorem, we have σb2 E t,x [e rσ b 2V BS (t+σ b2,x t+σb2 )] = V BS (t,x)+e t,x e rv H µ 2 (t+v,x t+v )dv, (4.19) σb2 E t,x [e rσ b 2U c (t+σ b2,x t+σb2 )] = U c (t,x)+e t,x e rv H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv. (4.2) Since x > c > b BS, the result from i) tells us U c (t,x) = G µ 2 (t,x) = V EA (t,x). By ii), we know that V BS (t + σ b2,x t+σb2 ) U c (t + σ b2,x t+σb2 ). So the equality (4.19) and (4.2) imply that (4.21) σb2 E t,x e rv H µ 2 (t+v,x t+v )I(X t+v c 2 (t+v))dv. The fact that c > b BS and the continuity of these functions imply that there exist ε such that c 2 (t+v) > b BS 2 (t+v) for v [,ε]. Consequently, the process X = (X t+v ) v ε spend strictly positive time before hitting b BS 2. With the fact that b BS 2 lies above h 2, then the value of inequality on the left hand side is strictly negative, which is contradictory with (4.21). So the assumption is incorrect. Then we prove that c b BS for t [,T]. For the proof of c 1 (t) b BS 1 (t), the analysis can be established in the same way. iv) We will show that b BS 1 (t) = c 1 (t) and b BS = c for t [,T]. Firstly, we will prove b BS = c. Supposed that there exists t [,T] such that b BS > c. Take a point x (c,b BS ) and set stopping time (4.22) τ b = inf{s [,T t] X t+s b BS 2 (t+s) or X t+s b BS 1 (t+s)}. Replacing s with τ b in (4.1) and (4.13), taking E t,x for these two formulae and applying the optional sampling theorem, we find that (4.23) E t,x [e rτ b V BS (t+τ b,x t+τb )] = V BS (t,x), τb E t,x [e rτ b U c (t+τ b,x t+τb )] = U c (t,x)+e t,x e rv H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv τb (4.24) + e rv H µ 1 (t+v,x t+v )I(X t+v < c 1 (t+v))dv. The result ii) shows that U c (t,x) V BS (t,x). From iii), we know that b BS 1 (t) c 1 (t) and b BS c for t [,T], so V BS (t+τ b,x t+τb ) = U c (t+τ b,x t+τb ) = G µ 1,µ 2 (t+τ b,x t+τb ). So (4.23) and (4.24) imply that (4.25) τb E t,x e rv H µ 2 (t+v,x t+v )I(X t+v > c 2 (t+v))dv τb +E t,x e rv H µ 1 (t+v,x t+v )I(X t+v < c 1 (t+v))dv. 18

20 Bythecontinuity of c 2 and b BS 2, weknowthat τ b >. Withthefactthat c 2 (t+v) > h 2 (t+v) and c 1 (t + v) < h 1 (t + v), we know that e rv H µ 2 (t + v,x t+v )I(X t+v > c 2 (t + v)) < and e rv H µ 1 (t+v,x t+v )I(X t+v < c 1 (t+v)) < for v [,τ b ]. So the formula on the left hand sides of inequality (4.25) is strictly negative and provides the contradiction. Thus c 2 = b BS 2 has been proved. The proof of c 1 = b BS 1 can be shown in the same way. 5. The Financial Analysis of British Strangle Options In this section we will firstly present the free boundaries of British strangle options and traditional British strangle options. After that, we present the analysis on the financial returns of British strangle options and show the practical feature of this option. Finally, we make the comparison between the American strangle option and the British strangle option, since the former option has been discussed in the previous research ([4], [3] and [13]). We will apply the skeleton analysis used in [11] and [1], which mainly address the question as to what the returns would be if the stock price enters the given region at a given time. Such skeleton analysis gives the option holder an intuitive assessment of option risk and performance without the mathematical proof. 1. At the beginning of this paper, Figure 1 shows that the free boundaries of traditional British strangle options (long a British call and a British put option) may intersect as the contract drift closed to the risk-free rate r. After the analysis of the British strangle options, we will plot the free boundaries of this option. Figure 5 uses the same parameter between traditional British strangle options and British strangle options. We found that two free boundaries of British strangle options do not intersect so that the continuation region C BS is non-empty for anytime before maturity. And the graph of function g(t) represented by the red line in Figure 5 is inside the continuation region which has been proved by Theorem 1. So the British strangle can be used to replace traditional British strangle options and make profit for the underlying asset with high volatility. 2. Fromthe economic analysis of the British strangle options in section 3, when µ 1 > r and µ 2 < r, we know that the contract drift µ 1 and µ 2 is tolerant level for the option holder. From the Figure 3 and Figure 4, the free boundaries look like the monotonic function. At initial point, b BS 1 () and b BS 2 () will not go to extreme value as contract drift converges to r. So that British strangle option will not appear overprotected case illustrating in [11] and [1]. This fact is proved by Theorem 1, which shows that g() is inside the continuation region such that b BS 1 () < g() and b BS 2 () > g(). On the other hands, since b BS 1 (T) = min{ r µ 1 L,g(T)} and b BS 2 (T) = max{ r µ 2 K,g(T)}, as µ 1 and µ 2, the lower free boundary b BS 1 (t) and the upper free boundary b BS. It is never optimal to exercise the the options until the maturity, i.e. the British strangle option changes into European strangle options. 3. From the formula (3.28), since there exists the local time term making that µ 1 can take the value less than r and µ 2 can take the value bigger than r. For this case, we name the drift µ 1 and µ 2 as the preference drift. When the British strangle option exercises rationally (exercising the options when the stock price hits either free boundaries), the options will give the benefit controlled by contract drift more than benefit under the risk neutral measure. From 19

21 t b BS Stock Price t g(t) t b BP (t) 15 1 t b BC (t) t b BS 1 (t) Time Figure 5. The free boundary of British call options b BC (t), British put options b BP (t) and the double free boundaries of British strangle options b BS 1 (t) and b BS. The red line indicates function g(t) defined in (3.27). The region between b BS 1 (t) and b BS is the continuation region C BS, the region below b BS 1 (t) is the optimal stopping region D 1 and the region over b BS is the optimal stopping region D 2. The parameter for British strangle is: L = 15, K = 15, σ =.6, δ = r =.1, µ 1 =.12, µ 2 =.8. The contract driftfor British call options is µ call =.8 andbritish putoptions is µ put =.12. 2

22 Time (months) Exercise at 3 with µ 1 = % 211% 24% 196% 189% 182% 18% Exercise at 3 with µ 1 =.7 215% 28% 21% 194% 186% 179% 177% Exercise at 25 with µ 1 = % 139% 131% 122% 111% 99% 93% Exercise at 25 with µ 1 =.7 145% 137% 129% 12% 11% 98% 92% Exercise at 2 with µ 1 =.13 85% 78% 7% 6% 49% 35% 25% Exercise at 2 with µ 1 =.7 84% 77% 69% 6% 49% 35% 25% Exercise at 15 with µ 1 =.13 61% 56% 5% 44% 36% 26% 18% Exercise at 15 with µ 1 =.7 66% 6% 54% 46% 38% 27% 19% Exercise at 1 with µ 1 =.13 16% 13% 99% 96% 93% 9% 9% Exercise at 1 with µ 1 =.7 111% 17% 13% 99% 95% 91% 89% Exercise at 5 with µ 1 = % 178% 178% 178% 178% 178% 178% Exercise at 5 with µ 1 =.7 181% 18% 179% 179% 178% 178% 177% Table 1. The returns of British strangle options observed by tolerance contract drift µ 1 =.13 and preference contract drift µ 1 =.7 respectively, when the other contract drift µ 2 = r =.1 is given and fixed. The remained parameters used in the above table are: K = 2, L = 15, r = δ =.1, σ =.6 and T = 1 (year). Figure 3, we can see the reverse U-skew for the contract drift µ 2 =.8, which is much bigger than the interest rate r =.1. As µ 2, the value of b BS 2 () < K. It means that the preference drift is so high that the real stock price cannot achieve and it is optimal to exercise immediately when stock price lies in the upper strike price. We call this kind of British strangle options is over-benefit for the call part with contract drift µ 2. From Figure 2, the U-skew is appeared for µ 1 = 2 and µ 1 = 4, where the lower free boundary at t = is over the lower strike price L = 15 (i.e. b BS 1 () > L). If the stock price starts at L, it will be optimal to exercise the option to get the payoff controlled by µ 1. We said that the preference drift µ 1 for the put part is so low that the strangle option is over-benefit. 4. From the definition of contract drift µ 1 and µ 2, we can distinguish the contract drift as preference drift and tolerance drift. This section will compare the returns under preference drift and tolerant drift to verify how to select the contract drift to improve the options returns. The options returns at time t and stock price x is defined as (5.1) R(t,x) = G(t,x) V(, L+K 2 ) where G is the options payoff and V is the initial price for buying options at the middle of the double strike price. From the second column to the seventh column in Table 1, the returns of preference drift µ 1 =.7 is higher than the returns of tolerance drift µ 1 =.13 when the option is exercised below the lower strike price L = 15. The reason is that the payoff G µ 1 (t,x) is higher for the preference drift, i.e. µ 1 < r. However, the preference drift will improve the initial price of British strangle options, which may reduce the returns of preference drift when the option exercises above the higher strike price K = 2. The same result can be observed from Table 2 21

23 Time (months) Exercise at 3 with µ 2 =.7 26% 21% 196% 191% 185% 181% 18% Exercise at 3 with µ 2 = % 217% 28% 199% 189% 18% 177% Exercise at 25 with µ 2 =.7 138% 132% 125% 118% 19% 98% 93% Exercise at 25 with µ 2 = % 144% 135% 124% 112% 99% 92% Exercise at 2 with µ 2 =.7 79% 73% 66% 58% 48% 34% 25% Exercise at 2 with µ 2 =.13 89% 81% 72% 62% 5% 35% 25% Exercise at 15 with µ 2 =.7 64% 59% 53% 47% 37% 26% 19% Exercise at 15 with µ 2 =.13 62% 57% 51% 44% 36% 26% 18% Exercise at 1 with µ 2 =.7 11% 16% 13% 99% 95% 92% 91% Exercise at 1 with µ 2 =.13 16% 13% 1% 96% 92% 89% 88% Exercise at 5 with µ 2 =.7 182% 182% 181% 181% 181% 181% 181% Exercise at 5 with µ 2 = % 176% 176% 176% 175% 175% 175% Table 2. The returns of British strangle options observed by tolerance contract drift µ 2 =.7 and preference contract drift µ 2 =.13 respectively, when the other contract drift µ 1 = r =.1 is given and fixed. The remained parameters used in the above table are: K = 2, L = 15, r = δ =.1, σ =.6 and T = 1 (year). (from the second column to the seventh column). The returns of preference drift µ 2 =.13 is higher thanthe returns of tolerance drift µ 2 =.7 when the optionis exercised over the upper strike price K = 2. The returns of tolerance drift outperform the returns of preference drift when theoptionisexercised belowthelower strike price L = 15. Moreover, astimeapproaches to the maturity, the returns of tolerance drift outperforms the returns of preference drift (see the last column of Table 1 and Table 2). As time close to maturity, the difference in G µ 1,µ 2 between tolerance drift and preference drift become smaller. However the difference in initial value between tolerance drift and preference drift do not change. So the returns of tolerance drift is significantly higher as time approaching to the maturity. Considering that nearly 7% options will be exercised close to maturity, we may conclude the returns of tolerance drift is better than the returns used preference drift. This is an important result used to determine the contract drift for British strangle options. In the following part, we will keep on analyzing the returns of British strangle options with preference drift or tolerance drift, which will restate that the tolerance drift will give option holder more cash back rate. 5. For the definition of tolerance drift and preference drift, we can classify the combination of µ 1 and µ 2 into four cases: µ 1 > r and µ 2 r (case one), µ 1 r and µ 2 r (case two), µ 1 r and µ 2 < r (case three), µ 1 > r and µ 2 < r (case four). Following here, we will compare the returns between American strangle options and British strangle options in above four cases. By the skeleton analysis used in [1], we can find how to determine the contract drift µ 1 and µ 2 to make the returns of British strangle options outperform American strangle options. 22