The British Knock-In Put Option

Transcription

1 Research Report No. 5, 214, Probab. Statist. Group Manchester (29 pp) The British Knock-In Put Option Luluwah Al-Fagih Following the economic rationale introduced by Peskir and Samee in [21] and [22], we present a new class of barrier options within the British payoff mechanism where the holder enjoys the early exercise feature of American type options whereupon his payoff (deliverable immediately) is the best prediction of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimise his losses. Building on our earlier work on the British knock-out put option [2], in this paper we focus on the knock-in put option with an up barrier. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British knock-in put option. We spot some of the trends previously seen in [21] but observe some behaviour unique to the knockin case. 1. Introduction The purpose of the present paper is to examine the British payoff mechanism introduced by Peskir and Samee (cf. [21] and [22]) in the context of barrier options. There are various types of barrier options that may be considered: (i) calls and puts; (ii) knock-in or knock-out; (iii) up or down barriers (for more details on each of these types see e.g. [11]). The British knock-out put and call options were investigated in detail in [2] providing some interesting results. We seek to examine if the same trends apply to other British barrier option variants. In this paper we focus on up-and-in put options. Up-and-in options are activated once the price of the underlying asset hits a certain level (barrier), which lies above the spot (initial) price of the asset. An investor would buy an option with a knock-in barrier if he expects the price of the underlying asset to be rather volatile. The pricing of European knock-in options using closed-form formulae has been addressed to considerable length in a range of literature (see for example [5], [6], [11], [16], [24] and [25]). Analytic valuation formulas for up-and-in put and down-and-in call American type options are presented by Haug [12] in terms of a plain American option. Dai and Kwok [4] extend this to include analytic valuation formulas for more types of American knock-in options by using an integral representation for the option price. Mathematics Subject Classification 21. Primary 91G8, 6G4. Secondary 35R35, 45G1, 6J6. Key words and phrases: British barrier option, European/American barrier option, British knock-out option, call option, put option, maximum/minimum stock price, stopped process, arbitrage-free price, rational exercise boundary, British barrier put-call symmetry, liquid/illiquid market, geometric Brownian motion, optimal stopping, free-boundary problem, nonlinear integral equation, local time-space calculus. 1

2 AitSahlia, Imhof and Lai [1] provide the same integral representation and compare two methods for the valuation of American knock-in options, the first is by developing a modified binomial tree method and the second is to decompose the price of the American knock-in option into the sum of the price of the European knock-in option plus an early exercise premium, resulting in a more accurate numerical approximation. Knock-in options are not as popular as other types of barrier options since investors are reluctant to pay a premium for a financial asset that does not and may never exist. However, knock-in options are much cheaper than their standard counterparts; in addition, the concept of optimal prediction in the British payoff for the knock-in option means that the option holder will be able to receive some returns even if the price of the underlying asset has not hit the barrier level, endogenously providing the holder with a protection mechanism against unfavourable asset price movements. On the other hand, when the stock price movements are favourable the holder can obtain higher returns in some cases. This reaffirms the fact noted in [2], [21] and [22] that the British feature of optimal prediction aims to provide both protection against unfavourable price movements as well as securing high returns when movements are favourable. This is especially appealing as the problems of liquidity and return are addressed completely endogenously since in a real financial market, it may be increasingly difficult for the holder to sell options when they are out of the money whereas the British feature allows the holder to exercise the option at any point during its lifetime. The paper is organised as follows. In Section 2 we present a basic motivation for the British knock-in put option. In Section 3 we formally define the British knock-in put option and present some of its basic properties, which lead us to distinguish three cases to be examined. This is continued in Section 4 where we derive a closed-form expression for the arbitrage-free price in terms of the rational exercise boundary (the early-exercise premium representation) and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation (Theorem 1). Using these results, in Section 5 we present afinancialanalysisforeachofthethreecasesofthebritishknock-inputoption,making comparisons with the American and European knock-in put options as well as the British put option. This analysis provides more insight into the full scope of the conclusions briefly outlined above. 2. Basic motivation for the British knock-in option The basic economic motivation for the British knock-in option is parallel to that of the British put and call options (see [21] and [22]). In this section we briefly review key elements of this motivation. We remark that the full financial scope of the British knock-in option goes beyond these initial considerations (see Section 5 below for further details). 1. Consider the financial market consisting of a risky stock X and a riskless bond B whose prices respectively evolve as (2.1) dx t = µx t dt + σx t dw t (X = x) (2.2) db t = rb t dt (B =1) where µ R is the appreciation rate (drift), σ> is the volatility coefficient, W =(W t ) t is a standard Wiener process defined on a probability space (Ω, F, P), and r > is the 2

3 interest rate. Recall that a barrier (up-and-in put) option of European type is a financial contract between a seller/hedger and a buyer/holder entitling the latter to sell the underlying stock at a specified strike price K> at a specified maturity time T> in the European case (or any stopping time prior to T in the American case) and receive the payoff (K X T ) + only if the price of the underlying asset has hit a specified barrier level which lies above the initial price of the underlying asset, i.e. the contract is activated once this level has been hit and the holder is now entitled to sell the underlying stock. Standard hedging arguments based on self-financing portfolios (with consumption) imply that the arbitrage-free price of the option is given by (2.3) V = Ẽ e rt (K X T ) + I (M T ) where the expectation Ẽ is taken with respect to the (unique) equivalent martingale measure P and M T =max t T X t denotes the maximum of the price process X. In this section (as in [21] and [22]) we will analyse the option from the standpoint of a true buyer, i.e. a buyer who has no ability or desire to sell the option nor to hedge his own position. Thus every true buyer will exercise the option at time T in accordance with the rational performance. For more details on the motivation and interest for considering a true buyer in this context see [18] and [21]. 2. With this in mind we now return to the European knock-in put holder whose payoff is given by (K X T (µ)) + I (M T (µ) ) where X T = X T (µ) represents the stock/market price at time T under the actual probability measure P.Recallthattheuniquestrongsolutionto (2.1) is given by (2.4) X t = X t (µ) =x exp σw t +(µ σ 2 /2) t under P for t [,T] where µ R is the actual drift. Hence, the expected value of the buyer s payoff equals (2.5) P = P (µ) =E e rt (K X T (µ)) + I (M T (µ) ). Moreover, it is well known that Law(X(µ) P) andlaw(m(µ) P) arethesameaslaw(x(r) P) and Law(M(r) P),respectively,sothatthearbitrage-freepriceoftheoptionequals (2.6) V = P (r) =E e rt (K X T (r)) + I (M T (r) l). Adirectcomparisonof(2.5)and(2.6)showsthatif µ = r then the return is fair for the buyer, in the sense that V = P, where V represents the value of his investment and P represents the expected value of his payoff. It is natural that a (true) buyer would buy a knock-in put option if he believes that µ<r and that the volatility was high enough to activate the option (for a more detailed discussion regarding the drift, see the last paragraph in Section 3 of [21]). We can see immediately from (2.6) that the maximum process and hence the volatility parameter play a crucial role in determining whether the buyer would receive any payoff at all even if the drift is indeed favourable since the buyer would like to see the prices going up to hit the barrier (thereby activating the option), followed by a drop in prices in order to maximise his payoff. We recall that the volatility of the stock price is effectively known by all market participants, 3

4 since one may take any of the standard estimators for the volatility over an arbitrarily small time period prior to the initiation of the contract. In contrast, the actual drift µ is unknown at time t = andalsodifficulttoestimateatlatertimes t (,T] unless T is unrealistically large. 3. The brief analysis above shows that while the actual drift µ of the underlying stock price is irrelevant in determining the arbitrage-free price of the option, to a (true) buyer it is crucial, and he will buy the option if he believes that µ<r,providedheishappywiththe market volatility. If this turns out to be the case then on average he will make a profit. Thus, after purchasing the option, the put holder will be happy if the observed stock price movements reaffirm his belief that µ<r and that the volatility was high enough to activate the option. The British option (as seen in [21] and [22]) seeks to address the opposite scenario: what if the put holder observes stock price movements that change his belief regarding the actual drift and cause him to believe that µ>r instead? In this contingency the British knock-in holder is effectively able to substitute this unfavourable drift with a contract drift and minimise his losses. In this way he is endogenously protected from any stock price drift greater than the contract drift. The contract drift may also help to partially tackle the problem of the stock price being less volatile than predicted thereby decreasing the risk of knocking-in. The value of the contract drift is therefore selected to represent the buyer s expected level of tolerance for the deviation of the actual drift from his original belief. 3. The British knock-in option: definition and basic properties We begin this section by presenting a formal definition of the British knock-in put option. This is then followed by a brief analysis of the optimal stopping problem and the free-boundary problem characterising the arbitrage-free price and the rational exercise strategy. These considerations are continued in Section 4 below. 1. Consider the financial market consisting of a risky stock X and a riskless bond B whose prices evolve as (2.1) and (2.2) respectively, where µ R is the appreciation rate (drift), σ> is the volatility coefficient, W =(W t ) t is a standard Wiener process defined on a probability space (Ω, F, P), and r> is the interest rate. Let a strike price K>, a barrier level >K,andamaturitytime T> be given and fixed. Definition 1. The British knock-in put option is a financial contract between a seller and abuyerentitlingthelattertoexerciseatany(stopping)time τ prior to T whereupon his payoff (deliverable immediately) is the best prediction of the European payoff of the knock-in option (K X T ) + I (M T ) given all the information up to time τ under the hypothesis that the true drift of the stock price equals µ c. Similar to other British options seen in [2], [21] and [22], the quantity µ c is defined in the option contract and we refer to it as the contract drift. Recalling our discussion in Section 2, we initially put no restrictions on µ c.weshallshowbelow,however,thatthecasesofthe contract drift satisfying µ c, <µ c <r,and µ c >r produce different results and are treated separately. 2. Denoting by (F t ) t T the natural filtration generated by X, the payoff of the British 4

5 knock-in put option at a given stopping time τ with values in [,T] can be written as (3.1) E µc (K X T ) + I (M T ) F τ, where the conditional expectation is taken with respect to a probability measure P µc which the stock price X evolves as under (3.2) dx t = µ c X t dt + σx t dw t with X = x in (, ). Comparing(2.1)and(3.2), weseethattheeffectofexercisingthe British knock-in option is to substitute the true (unknown) drift of the stock price with the contract drift for the remaining term of the contract. 3. Setting M t =max s t X s and using the stationary and independent increments property of the Brownian motion W governing X we find that (3.3) E µc (K X T ) + I (M T ) F t = G µ c (t, X t,m t ) where the payoff function G µc can be expressed as (3.4) G µc (t, x, m) = E (K xz µc T t )+ I (m xm µc T t ) with Z µc T t =exp(σw T t +(µ c σ 2 /2) (T t)) and M µc T t =max u T t Z u µc for t [,T], x (, ) and m x.togainabetterunderstandingof(3.4)weusethefactthat (3.5) I (m xm µc µc T t ) =I (m <)I (xm T t )+I (m ) to help reduce the problem (at time t )fromnamelyathree-dimensionalproblemintotwodimension. Hence, note that (3.4) can be re-written as (3.6) G µc (t, x, m) =I (m <) E (K xz µc T t )+ I (xm µc T t ) + I (m ) E (K xz µc T t )+ where the first term on the right-hand side is the payoff for the holder before the price process hits the barrier level which we denote as the payoff function (3.7) G µc (t, x) :=E (K xz µc T t )+ I (xm µc T t ). Recall that by definition of a knock-in option it is a condition that M t = m<,i.e. that the option would not have been knocked in at/before the start of the option. The second term in (3.6) is the payoff for the holder after the barrier level has been hit, expressed as the indicator of m multiplied by the payoff of the British put option given by Peskir and Samee [21, Eq.(3.7)] which we denote here as G µc BP (t, x). In the remaining part of this paper we shall begin each step by looking at the three-dimensional case and use the technique above to simplify to two-dimension as above. Although the exact value of the maximum of the price process is not important, the problem remains essentially three-dimensional as we still need to know whether the maximum of the price process has hit the barrier throughout the lifetime of the option. AlengthycalculationbasedontheknownlawofadriftedBrownianmotionanditsmaximum under P µc shows that 5

6 (3.8) ( 2µc G µc (t, x) = K σ 2 1) 1 Φ x σ T t xe µc(t t) x ( 2µc σ 2 +1) Φ log xk (µ 2 c σ2 2 1 xk log σ T t 2 )(T t) (µ c + σ2 )(T t) 2 for t [,T) and x (,) where Φ is the standard normal distribution function. It may be noted here that similarly to [21] and [22], the expression for G µc t) (t, x) multiplied by eµc(t coincides with the Black-Scholes formula for the arbitrage-free price of the European up-and-in put option (written for the remaining term of the contract) where the interest rate equals the contract drift µ c. We also note that (3.9) G µc (t, x) =Gµc BP (t, x) Gµc OUT (t, x) where G µc OUT is the payoff function of the British knock-out put option previously seen in [2]. This relation tells us that although x G µc (t, x) is not necessarily convex, it can however be written as the difference of two convex functions, a property which will be useful later. We also see from (3.8) that x G µc (t, x) is not always increasing, with Gµc (t, ) = for any t [,T] given and fixed. Moreover, it is easily verified that the British knock-in payoff function goes strictly below the British put option payoff and the two functions meet at the barrier level where G µc (t, ) =Gµc BP (t, ). We also note from (3.4) that (3.1) G µc (t,,m)=i (m ) K and G µc (t,, ) = forany t [,T] given and fixed and that (3.11) G µc (T,x,m)=I (m ) (K x) + for x (, ), showing that the British knock-in payoff coincides with the European/American knock-in payoff at the time of maturity. 4. Standard hedging arguments based on self-financing portfolios (with consumption) imply that the arbitrage-free price of the British knock-in put option is given by (3.12) V = sup Ẽ e rτ E µc (K X T ) + I (M T ) F τ τ T where the supremum is taken over all stopping times τ of X with values in [,T] and Ẽ is taken with respect to the (unique) equivalent martingale measure P. From (3.3) we see that the underlying Markov process in the optimal stopping problem (3.12) is the triple (t, X t,m t ) for t [,T]. Since Law(X(µ) P) is the same as Law(X(r) P), it follows from the well-known ladder structure of M and the multiplicative structure of X that (3.12) extends as follows (3.13) V (t, x, m) = sup E t,x,m [e rτ G µc (t+τ,xx τ,m xm τ )] τ T t 6

7 Figure 1: The value/payoff functions of the standard British and the British knock-in put options plotted against the stock price. for t [,T) and x m in (, ) where the supremum is taken as in (3.12) above and the process X = X(r) under P solves (3.14) dx t = rx t dt + σx t dw t with X =1. If m, then the option would have already been knocked-in and we have a standard British put option. Thus by the definition of the knock-in option, we only consider the case where m< and the problem (3.13) becomes two-dimensional with arbitrage-free price (3.15) V (t, x) = sup E t,x e rτ (I (xm τ µc ) G µc BP (t+τ,xx τ)+i (xm τ µc <) G µc (t+τ,xx τ)) τ T t for t [,T) and x (, ). Note that that x V (t, x) and x G µc (t, x) tend to stay close together for x lower than K, however V lies above G µc as x tends to. This can be explained by the fact that at x = the option becomes a standard British put when µ c >r, we have G µc BP (t, ) <V BP(t, ) for all t. 7

8 Figure 2: A computer drawing showing how the rational exercise boundary of the British knock-in option changes as one varies the contract drift for µ c >r. 5. We know (see e.g. [17]) that the infinitesimal generator of the three-dimensional process (t, X t,m t ) t acting on a function F C 1,2,1 is given by (3.16) (LF )(t, x, m) =(L X F )(t, x, m) in <x<m (3.17) with F (t, x, m) = at x = m m (3.18) L X = t + rx x + σ2 2 x2 2 x 2 for all t [,T) and x (, ) with m x. In view of this, we gain a deeper insight into the solution to the optimal stopping problem (3.13), by noting that Itô s formula yields (3.19) e rs G µc (t+s, X t+s,m t+s )=G µc (t, x, m)+ + s s e ru H µc (t+u, X t+u,m t+u ) du e ru G µc m (t+u, X t+u,m t+u ) dm u +N s 8

9 where the function H µc = H µc (t, x, m) is given by (3.2) H µc = G µc t + rxg µc x + σ2 2 x2 G µc xx rg µc and (3.21) N s = σ s e ru X u G µc x (t + u, X t+u,m t+u ) dw u defines a continuous martingale for s [,T t] witht [,T). Wealsonotethatfrom(3.17) the integral with respect to dm u is equal to zero. Hence by the optional sampling theorem we find (3.22) E[e rτ G µc (t + τ,x t+τ,m t+τ )] = G µc (t, x, m)+e τ e ru H µc (t + u, X t+u,m t+u ) du for all stopping times τ of X solving (3.2) with values in [,T t] with t [,T) and x (, ) where m x given and fixed. Note that from (3.6) we can write (3.23) H µc (t, x, m) =I (m ) H µc BP (t, x)+i (m <) Hµc (t, x) where (3.24) H µc = Gµc t, + rxgµc x, + σ2 2 x2 G µc xx, rgµc and H µc BP is the function Hµc for the British put option [21, Eq.(3.15)]. It is clear from (3.4) that the payoff function G µc satisfies the Kolomogorov backward equation (or the undiscounted Black-Scholes equation where the interest rate equals the contract drift) so that from (3.2) we see that (3.25) H µc =(r µ c ) xg µc x rg µc. This representation shows in particular that if µ c = r then H µc < so that from (3.22) we see that it is always optimal to exercise immediately (a formal argument confirming the economic reasoning can be found in [21]). Moreover, inserting the expression for G µc from (3.8) along with its time and space derivatives into H µc we obtain a remarkably simple function H µc (t, x) = [(r µ c) 2µ ( 2µc c t) σ + r] 2 +1) log xk σ 2 xeµc(t Φ 2 (µc + σ2 )(T t) 2 (3.26) x σ T t [(r µ c ) 2µ ( 2µc c σ + µ σ c] K 2 1) log xk Φ 2 (µc σ2 )(T t) 2 2 x σ T t for t [,T) and x (,). Adirectexaminationofthefunction H µc there exists a continuous function h :[,T] R such that (3.27) H µc (t, h(t)) = 9 in (3.26) reveals that

10 for all t [,T). Define the set (3.28) {H µc } := {(t, x) [,T] (, ) :Hµc }. We distinguish two functions h 1 and h 2 such that (i) {H µc } = {(t, x) [,T] (, ) : x h 1 (t)} when µ c > r and (ii) {H µc } = {(t, x) [,T] (, ) : x h 2(t)} when µ c (,r). It can be verified that if (t, x) {H µc } and t<t is sufficiently close to T then it is optimal to stop immediately (since the gain obtained from being outside this set cannot offset the cost of getting there due to the lack of time). This shows that the optimal stopping boundary b separating the continuation set from the stopping set satisfies b(t )=h(t) and this value equals the barrier level as can be deduced from the financial aspect to this problem. Recall for comparison that the optimal stopping boundary in the British put option takes value rk/µ c at T and takes value K at T in the American put option. We conclude from the above analysis of (3.26) that the shape of the stopping region when µ c >r is different from the shape for <µ c <r. We therefore treat these two cases separately. Moreover, by rewriting (3.26) in terms of G µc it is possible to see that Hµc for any point (t, x) [,T] (,) when µ c. Pairingthiswiththefactfrom[21]that H µc BP when µ c r and inserting into (3.22) and (3.23), we deduce that the optimal strategy when m< for µ c is to continue until the price process hits the barrier level since the option holder is able to obtain gain until the option knocks into a standard British put option at which point it becomes optimal to exercise immediately. Hence, we can say that the optimal exercise boundary is equal to a constant, namely the barrier level and the following relationships hold: G µc (t, ) =Gµc BP (t, ) and V (t, ) = V BP (t, ) where on the optimal boundary b (t) = we also have V (t, b (t)) = G µc (t, b (t)). If we let τ be the first hitting time to the level,i.e. τ =inf{s [,T t] :X t+s } then we can apply the approach used in [1] and [4] to obtain a representation for the price of the knock-in British put option with µ c as follows (3.29) V(t, x) = E t,x e rs V(t+s, ) I (τ ds) = = T t T t e rs G µc (t+s, ) P (τ ds X t = x) e r(u t) G µc (u, ) f(u t, x) du for t [,T] and x (,), where v f(v, x) is the probability density function of the first hitting time of the process X v = x exp(σw v (r σ 2 /2)v) to the level given by (3.3) f(v, x) =log 1 1 ϕ x σv 3/2 σ log v x (r σ 2 )v 2 for v [,T) and u (t, T ) with x (,), where ϕ is the standard normal density function. Hence, for µ c the expression for the arbitrage-free price is given by (3.29) and can be solved for V directly. 6. Introducing the continuation set (3.31) C = {(t, x, m) [,T) (, ) (, ) V (t, x, m) <G µc (t, x, m)} 1

11 and the stopping set (3.32) D = {(t, x, m) [,T] (, ) (, ) V (t, x, m) =G µc (t, x, m)} we may infer from general theory of optimal stopping for Markov processes (cf. [23]) that the optimal stopping time in (3.13) is given by (3.33) τ D =inf{s [,T t] :(t + s, X t+s,m t+s ) D}. Standard Markovian arguments lead to the following free-boundary problem (for the value function V = V (t, x, m) and the optimal stopping boundary b = b(t) to be determined): (3.34) V t + rxv x + σ2 2 x2 V xx rv = in C (3.35) V (t, x, m) =G µc (t, x, m) forx = b(t) andt [,T](instantaneousstopping) (3.36) V x (t, x, m) =G µc x (t, x, m) for x = b(t) and t [,T] (smooth fit) (3.37) V m (t, x, m) = for x = m and t [,T] (3.38) V (T,x,m)=I (m )(K x) + = G µc (T,x,m) (3.39) V (t,, ) = for t [,T] where we also set V (t, x, m) =G µc (t, x, m) in the stopping set D. Note that the conditions (3.35) and (3.36) will be derived in the proof below while the remaining conditions are obvious. Recall that under the condition m< the problem (3.13) reduces to the two-dimensional problem (3.15) and that the shape of the stopping region D depends on the value of µ c.we consider the value function V and split this into two cases. As it will be clear from the context which V is being considered, we will not reflect these facts directly in the notation of V. (i) The case µ c >r. In view of the above, we formulate the following free-boundary problem (3.4) V t, + rxv x, + σ2 2 x2 V xx, rv = for x>b 1 (t) and t [,T) (3.41) V (t, x) =G µc (t, x) for x = b 1(t) and t [,T] (3.42) V x, (t, x) =G µc x, (t, x) for x = b 1(t) and t [,T] (3.43) V (T,x)=G µc (T,x) (3.44) V (t, ) =V BP (t, ) for t [,T] where V BP (t, ) is the value of the British put option at x =. The continuation set is given by (3.45) C = {(t, x) [,T) (,) x>b 1 (t)} and the stopping set is given by (3.46) D = {(t, x) [,T] (,) x b 1 (t)} {(T,x) x>b 1 (T )} 11

12 Figure 3: A computer drawing showing the two-part optimal exercise boundary of the British knock-in option for <µ c <r. so that the stopping time in (3.33) is given by (3.47) τ b1 =inf{s [,T t] :X t+s b 1 (t + s)}. This stopping time represents the rational exercise strategy for the British knock-in option when µ c >r and plays a key role in financial analysis of the option. Note that we encounter the same three regimes for the optimal stopping boundary b 1 as those seen in [21] for µ c >r (see Figure 2) where depending on the value of µ c, the boundary b 1 is either an increasing function of time, a skewed U-shaped function of time, or the intermediate case where b 1 can take either of the two shapes depending on the size of T. We also see that the optimal stopping boundary b 1 exhibits a singular behaviour at T in the sense that b 1(T ) =+. (ii) The case <µ c <r. Similarly, we formulate the following free-boundary problem (3.48) V t, + rxv x, + σ2 2 x2 V xx, rv = for x<b 2 (t) and t [,T) (3.49) V (t, x) =G µc (t, x) for x = b 2(t) and t [,T] (3.5) V x, (t, x) =G µc x, (t, x) for x = b 2(t) and t [,T] (3.51) V (T,x)=G µc (T,x) 12

13 (3.52) V (t, ) =V BP (t, ) =G µc BP (t, ) for t [,T] where the continuation set is given by (3.53) C = {(t, x) [,T) (,) x<b 2 (t)} and the stopping set is given by (3.54) D = {(t, x) [,T] (,) x b 2 (t)} {(T,x) x<b 2 (T )} so that the stopping time in (3.33) is (3.55) τ b2 =inf{s [,T t] :X t+s b 2 (t + s)}. We see in Figure 3 that the boundary b 2 is two-part in the sense that it represents a function of time (curve) up to a point t [t, T ] after which it becomes a constant such that b 2 (u) = for all u [t,t]. Thecriticalvalue t is the solution of (3.56) H µc (u, ) = and can be obtained numerically. Note that for t [t,t] we have the same optimal strategy as for µ c above and the arbitrage-free price for the British knock-in option for all such t with <µ c <r is equal to V (t, x) given in (3.29) with optimal boundary b 2(t) =b (t) =. In order to solve for V and b 2 in the interval t [,t ],werewritetheproblemwithnew maturity time T = t where V (T,x)=V (T,x) for x (,). We will see in Section 5 below that the above cases have different economic interpretations and their fuller understanding is important for a correct/desired choice of the contract drift µ c in relation to the interest rate r and other parameters in the British knock-in put option. We refer to the final paragraph of Section 3 of [21] for comments regarding the choice of the volatility parameter in the British payoff mechanism. We add that it is natural that an increase in volatility will increase the expected payoff of European and American knock-in options since it becomes more likely for the option to be activated. However, in the British case both the payoff and price of the option depend on volatility and thus the other parameters play an equally important role. In fact, the impact of the choice of volatility on the British knock-in option can be made more apparent if the strike K and barrier level are chosen with reference to the volatility parameter (see [5] and [6] for a more detailed discussion). In the next section we will derive simpler equations which characterise V and b uniquely and can be used for their calculation in Section The arbitrage-free price and the rational exercise boundary In this section we follow the approach used in [21] and [22] to derive a closed form expression for the arbitrage-free price V in terms of the rational exercise boundaries b 1 and b 2 (the early-exercise premium representation) and show that the rational exercise boundaries can be characterised as the unique solution to a nonlinear integral equation (Theorem 1). W λ We fix λ IR and let W λ =(Wt λ ) t T denote the Wiener process with drift λ given by t = W t + λt for t [,T]. Defining the process S λ =(St λ ) t T by St λ =max s t Ws λ, it 13

14 follows that X t+u = x exp(σw λ u ) and M t+u = x exp(σs λ u) for u [,T t] where the drift λ is given by (4.1) λ =(r σ 2 /2)/σ. Recalling (see e.g. [15, p. 368]) that the joint density function of (Wu λ,su) λ under P is given by 2 (2s w) (2s w)2 (4.2) g(u, w, s) = exp + λ(w λu π u 3/2 2u 2 ) for all u>,s and w s. Define the function (4.3) L(t, x) =E [ I (xm T t ) G µc BP (T,xX T t)+i (xm T t <) V (T,xX T t) ] for t [,T ), <x m< and where the functions G µc BP Eq.(3.7)] and (3.29) respectively. Let (4.4) F 1 (t, x) =G µc (t, x) e r(t t) G r (t, x) (4.5) F 2 (t, x) =G µc (t, x) e r(t t) L(t, x) and V are given in [21, (4.6) (4.7) J(t, x, v, z) = e r(v t) w g(v t, w, s) [H µc BP (v, xeσw )I (xe σw <z,xe σs ) + H µc (v, xeσw )I (xe σw <z,xe σs <)] dsdw K(t, x, v, z) = e r(v t) g(v t, w, s) [H µc BP (v, xeσw )I (xe σw >z,xe σs ) w + H µc (v, xeσw )I (xe σw >z,xe σs <)] dsdw for t [,T), x>, v (t, T ) with u = v t> and z>,wherethefunctions G r and G µc are given in (3.8) above (upon identifying µ c with r )andthefunctions H µc BP and H µc are given above. Finally, it can be verified using standard means that ( 2r J(t, x, T, z) =rk e r(t t) σ 2 1) log (z K)x (r σ2 )(T t) Φ 2 2 (4.8) x σ T t ( 2r σ µ c x 2 +1) log (z K)x (r + σ2 )(T t) Φ 2 2 x σ T t for t [,T), x> and z>. Theseexpressionsareusefulinacomputationaltreatment of the equations (4.1) and (4.12) below. The main result of this section may now be stated as follows. Theorem 1. Consider the problem (3.13). We distinguish two cases: 1. The case µ c >r. The arbitrage-free price of the British knock-in put option admits the early-exercise premium representation (4.9) V (t, x) =e r(t t) G r (t, x)+ 14 T t J(t, x, v, b 1 (v)) dv

15 for all (t, x) [,T] (,), where the first term is the arbitrage-free price of the European knock-in put option and the second term is the early-exercise premium. The rational exercise boundary of the British knock-in put option can be characterised as the unique continuous solution b 1 :[,T] R + to the nonlinear integral equation (4.1) F 1 (t, b 1 (t)) = T t J(t, b 1 (t),v,b 1 (v)) dv satisfying b 1 (t) h 1 (t) for all t [,T] where h 1 is defined by equation (3.27). 2. The case <µ c <r. The arbitrage-free price of the British knock-in put option admits the early-exercise premium representation (4.11) V (t, x) =e r(t t) L(t, x)+ T t K(t, x, v, b 2 (v)) dv for all (t, x) [,T ] (,), where the first term is the arbitrage-free price of the European knock-in put option and the second term is the early-exercise premium, and T is the solution of equation (3.56). The rational exercise boundary of the British knock-in option can be characterised as the unique continuous solution b 2 :[,T ] R + to the nonlinear integral equation (4.12) F 2 (t, b 2 (t)) = T t K(t, b 2 (t),v,b 2 (v)) dv satisfying b 2 (t) h 2 (t) for all t [,T ] where h 2 is defined by equation (3.27). Proof. We first derive (4.9) and (4.11) and show that the rational exercise boundaries solve (4.1) and (4.12) respectively. Then we show that (4.1) and (4.12) cannot have other (continuous) solutions. 1. Let V :[,T] (, ) (o, ) R and b :[,T] R + denote the unique solution to the free-boundary problem (3.34)-(3.39) (where V extends as G µc in D ), set C s = {(t, x) (t, x, m) C} and D s = {(t, x) (t, x, m) D}, and using (3.16) and (3.17) above, let (4.13) L X V (t, x, m) =V t (t, x, m)+rxv x (t, x, m)+ σ2 2 x2 V xx (t, x, m) for (t, x, m) C b D b with x<m and LV (t, x, m) =at x = m. Then V and b are continuous functions satisfying the following conditions: (i) V is C 1,2,1 on C b D b ;(ii) b is of bounded variation; (iii) P(X t = c) = forall t [,T] and c>) ; (iv) L X V rv is locally bounded on C b D b (recall that V solves (3.34) and coincides with G µc on D b ); (v) t V x (t, b(t)±,m)=g µc x (t, b(t),m) is continuous on [,T] (recall that V satisfies the smooth-fit condition (3.36) at b ); and finally (vi) inserting (3.16),(3.17) into (3.34), we get 2r (4.14) V xx (t, x, m) = σ 2 x V 2 2 σ 2 x V 2 t 2r σ 2 x V 2 x (t, x, m) for (t, x, m) C b fixed, on C b D b and V xx (t, x, m) = for (t, x, m) D b, so that for each m given and the function (t, x) V xx (t, x, m) can be represented as the sum of two 15

16 functions, where the first is nonnegative and the second is continuous on the closure of these sets. From these conditions we see that the local time-space formula [2, Eq. (4.22)] can be applied. This yields ( ) e rs V (t + s, X t+s,m t+s )=V (t, x, m) s s s e rv (L X V rv )(t + v, X t+v,m t+v ) I (X t+v = b(t + v)) dv + N b s e rv V m (t + v, X t+v,m t+v ) I (X t+v = b(t + v),x t+v = M t+v ) dm t+v e rv (V x (t + v, X t+v +,M t+v ) V x (t + v, X t+v,m t+v )) I (X t+v = b(t + v)) dl b v(x x ) where (4.16) N b s = σ s e rv X t+v V x (t + v, X t+v,m t+v ) I (X t+v = b(t + v))dw t+v is a continuous martingale for s [,T t] and L b (X) =L b v(x) v s X =(X t+v ) v s at the surface b given by is the local time of L b 1 u = P t,x,m lim ε 2ε u I (b(t + v) ε<x t+v <b(t + v)+ε) σ 2 X 2 t+v dv. We note that the integral with respect to dm t+v in (4.15) is equal to zero since the increment M t+v outside the diagonal x = m is equal to zero and on the diagonal we have (3.17). Moreover, since V satisfies (3.34) on C b and equals G µc on D b,andthesmooth-fitcondition (3.36) holds at b,weseethat(4.15)becomes ( ) e rs V (t + s, X t+s,m t+s )=V (t, x, m) + s e rv H µc (t + v, X t+v,m t+v ) I (X t+v D) dv + N b s. Upon inserting (3.23) we see that (4.17) simplifies to (4.18) e rs V (t + s, X t+s,m t+s )=V (t, x, m)+n b s + s e rv [I (M t+v )H µc BP (t + v, X t+v)+i (M t+v <)H µc (t + v, X t+v)] I (X t+v D) dv for s [,T t] and (t, x, m) [,T) (, ) (, ). 2. We show that N b =(Ns) b s T t is a martingale for t [,T). For this, we need to show that Vx 2 (t, x, m) <.Takeanypoint (t, x, m) : If m then the barrier has already been hit and we have the case of a British put option for which we know that VBP,x 2 (t, x) < 16

17 (see [21, eq.(4.11)]); if m<,weconsiderthefunction V in (3.15) for which we want to show that V,x 2 (t, x) < for all t [,T) and x (,). From (4.18) we see that all the terms on either side of the equation are bounded, from where it follows that Ns b is also bounded for s [,T t] and since m<,wecandeducethat V,x (t, x) < for all t [,T) and x (,). Hence we find that T t (4.19) EN b,n b T t σ 2 x 2 E e 2rv Xv 2 Vx 2 (t, x, m) dv < from where it follows that N b is a continuous martingale as claimed. 3. Replacing s by T t in (4.18), taking E on both sides and applying the optional sampling theorem, we get: ( 4. 2 ) Ee r(t t) V (T,X T,M T )=V (t, x, m) + T t e rv E[(I (M t+v )H µc BP (t + v, X t+v)+i (M t+v <)H µc (t + v, X t+v)) I (X t+v D)] dv for all (t, x) [,T) (, ). Using the fact that (4.21) V (T,x,m)=G µc (T,x,m)=I (m )(K x) + for x> when µ c >r we recognise the left-hand side of (4.2) as e r(t t) G r (t, x). Since aknock-inoptionmeans M t = m<,from(3.4)and(4.6)weseethat(4.2)yieldsthe representation (4.9). Moreover, inserting x = b 1 (t) into (4.2) and using that V (t, b 1 (t)) = G µc (t, b 1(t)) we see from (4.9) that b 1 solves (4.1). Similarly, when <µ c <r,wereplace maturity T by T and using (4.3), (3.48) and (4.7) we see that (4.2) yields the representation (4.11). Moreover, inserting x = b 2 (t) into (4.2) and using that V (t, b 2 (t)) = G µc (t, b 2(t)) we see from (4.11) that b 2 solves (4.12). This establishes the existence of the solutions to (4.1) and (4.12). We now turn to uniqueness. 4. We show that b 1 and b 2 are the unique solutions to (4.1) and (4.12) in the class of continuous functions t b 1 (t) and t b 2 (t) on [,T] satisfying b 1 (t) h 1 (t) and b 2 (t) h 2 (t) for all t [,T]. For this, take any two continuous function a 1 and a 2 on [,T] which solve (4.1) and (4.12) and satisfy a 1 (t) h 1 (t) and a 2 (t) h 2 (t) for all t [,T]. Motivatedbyequation(4.2)above,definethefunctions (4.22) U a :[,T] (, ) (, ) R and (4.23) U a :[,T] (, ) R by setting (4.24) U a (t, x, m) =e r(t t) E G µc (T,X T,M T ) T t e rv E H µc (t + v, X t+v,m t+v ) I(X t+v D a ) dv 17

18 where (4.25) D a = {(t, x, m) [,T] (, ) (, ) U a (t, x, m) =G µc (t, x, m)}. Note that when m< then D a is equal to (4.26) {(t, x) [,T] (, ) U(t, a x) =G µc (t, x)} so that D a plays the role of the stopping region for a 1 and a 2. Observe that a 1 and a 2 solving (4.1) and (4.12) respectively means exactly that U a (t, a i (t),m)=g µc (t, a i (t),m) or when m< that U a (t, a i(t)) = G µc (t, a i(t)) where i =1 when µ c >r and i =2 when µ c <r. (i) We show that U a (t, x, m) = G µc (t, x, m) for all (t, x, m) D a (i.e. U a (t, x) = G µc (t, x) for m<). For this, take any such (t, x, m) and note that the Markov structure of the pair (X, M) implies that (4.27) e rs U a (t + s, X t+s,m t+s ) s e rv H µc (t + v, X t+v,m t+v ) I(X t+v D a ) dv is a continuous martingale under P for s [,T t]. Considerthestoppingtime (4.28) σ a =inf{ s [,T t] X t+s D a } under P. Since U a (t, a i (t),m) = G µc (t, a i (t),m) for all t [,T] and U a (T,x,m) = G µc (T,x,m) for all x> we see that (4.29) U a (t + σ a,x t+σa,m t+σa )=G µc (t + σ a,x t+σa,m t+σa ). Replacing s by σ a in (4.27), taking E on both sides and applying the optional sampling theorem, we find that (4.3) U a (t, x, m) =E e rσa U a (t + σ a,x t+σa ) E = E e rσa G µc (t + σ a,x t+σa ) E = G µc (t, x, m) σa σa e rv H µc (t + v, xx t+v ) I(X t+v D a ) dv e rv H µc (t + v, X t+v,m t+v ) dv where we use (3.22) in the last equality. This shows that U a equals G µc on D a as claimed, i.e. that U a equals Gµc below a 1 when µ c >r and above a 2 when µ c <r. (ii) We show that U a (t, x, m) V (t, x, m) for all (t, x, m) [,T] (, ) (, ), i.e. that U a V µc for m<. For this, take any such (t, x, m) and consider the stopping time (4.31) τ a =inf{ s [,T t] X t+s D a } under P.Weclaimthat (4.32) U a (t + τ a,x t+τa,m t+τa )=G µc (t + τ a,x t+τa,m t+τa ). 18

19 Indeed, if (t, x, m) D a then τ a = so that U a (t, x, m) =G µc (t, x, m) by (i) above. On the other hand, if (t, x, m) D a then the claim follows since U a (t, a i (t),m)=g µc (t, a i (t),m) for all t [,T] and U a (T,x,m)=G µc (T,x,m) for all x>. Replacing s by τ a in (4.27), taking E on both sides and applying the optional sampling theorem, we find that (4.33) U a (t, x, m) =E e rτa U a (t + τ a,x t+τa,m t+τa ) E τa e rv H µc (t + v, X t+v,m t+v ) I(X t+v D a ) dv = E e rτa G µc (t + τ a,x t+τa,m t+τa ) V (t, x, m) where we used the definition of τ a in the second equality. This shows that U a V for all (t, x, m) and thus U a V µc when m< as claimed. (iii) We show that D a D b for all t [,T] (where we denote by D b the original stopping set from our problem (3.13) defined by the function b ). Suppose that this is not the case so that there exists some time t [,T) for which either b 1 (t) >a 1 (t) or b 2 (t) <a 2 (t). Choose any x<a 1 (t) or x>a 2 (t) and consider the stopping time (4.34) σ b =inf{ s [,T t] X t+s D b } under the measure P.Replacings with σ b in (4.17) and (4.27), taking E on both sides of these identities and applying the optional sampling theorem, we get (4.35) (4.36) E e rσ b V (t + σ b,x t+σb,m t+σb ) = V (t, x, m)+e E e rσ b U a (t + σ b,x t+σb,m t+σb ) = U a (t, x, m)+e σb σb e rv H µc (t + v, X t+v,m t+v ) dv e rv H µc (t + v, X t+v ) I(X t+v D a ) dv. Since x<a 1 (t) or x>a 2 (t) we see by (i) above that U a (t, x, m) =G µc (t, x, m) =V (t, x, m) where the last equality follows since x lies below b 1 (t) or above b 2 (t) for respective values of µ c. Moreover, by (ii) above we know that U a (t + σ b,x t+σb,m t+σb ) V (t + σ b,x t+σb,m t+σb ) so that (4.35) and (4.36) imply that σb (4.37) E e rv H µc (t + v, X t+v,m t+v ) I(X t+v D a ) dv. The assumption that either b 1 (t) <a 1 (t) or b 2 (t) >a 1 (t) together with the continuity of the functions a 1, a 2, b 1 and b 2 means that there exists ε> sufficiently small such that b 1 (t + v) >a 1 (t + v) or b 2 (t + v) <a 2 (t + v) for all v [,ε]. Consequently,theprobability under P of (X t+v ) v ε spending a strictly positive amount of time (with respect to Lebesgue measure) in this set before hitting b i is strictly positive. Combined with the fact that D b and D a are contained in {H µc },thisforcestheexpectationin(4.37)tobestrictlynegative and provides a contradiction. (iv) We show that D a = D b. Suppose that this is not the case so that there exists t [,T] such that b 1 (t) >a 1 (t) or b 2 (t) <a 2 (t). Choose any point x (a 1 (t),b 1 (t)) or x (b 2 (t),a 2 (t)) and consider the stopping time (4.38) τ b =inf{ s [,T t] X t+s D b } 19

20 Figure 4: A computer drawing showing the rational exercise boundaries of the British knock-in option with K = 1, T = 1, r =.1, σ =.9, µ c =.13 when the barrier equals 11, 13 and 15. under P.Replacings with τ b in (4.17) and (4.27), taking E on both sides of these identities and applying the optional sampling theorem, we find ( ) ( 4. 4 ) E e rτ b V (t + τ b,x t+τb,m t+τb ) = V (t, x, m) E e rτ b U a (t + τ b,x t+τb,m t+τb ) = U a (t, x, m) + E τb e rv H µc (t + v, X t+v,m t+v ) I(X t+v D a ) dv Since D a is contained in D b by (iii) above and U a equals G µc on D a by (i) above, we see that (4.41) U a (t + τ b,x t+τb,m t+τb )=G µc (t + τ b,x t+τb,m t+τb )=V (t + τ b,x t+τb,m t+τb ) where the last equality follows since V equals G µc on D b (recall also that U a (T,x,m)= G µc (T,x,m)=V(T,x,m) for all x>). Moreover, by(ii)weknowthat U a V so that (4.39) and (4.4) imply that τb (4.42) E e rv H µc (t + v, X t+v,m t+v ) I(X t+v D a ) dv. 2

21 Time (months) Exercise at 14 with = 15 22% 174% 142% 16% 66% 23% % Exercise at 12 with = % 125% 14% 84% 58% 26% % Exercise at 12 with = % 123% 96% 67% 36% 9% % Exercise at K with = 11 1% 9% 79% 66% 5% 29% % Exercise at K with = 13 1% 85% 68% 5% 3% 9% % Exercise at K with = 15 1% 79% 58% 36% 16% 2% % Exercise at 8 with = 11 66% 57% 48% 37% 24% 1% % Exercise at 8 with = 13 63% 5% 38% 25% 12% 2% % Exercise at 8 with = 15 59% 44% 29% 16% 5% % % Exercise at 6 with = 11 37% 3% 23% 15% 8% 1% % Exercise at 6 with = 13 32% 24% 16% 9% 3% % % Exercise at 6 with = 15 28% 19% 11% 5% 1% % % Exercise at 4 with = 11 14% 1% 7% 3% 1% % % Exercise at 4 with = 13 11% 7% 4% 1% % % % Exercise at 4 with = 15 9% 5% 2% 1% % % % Exercise at 2 with = 11 2% 1% % % % % % Exercise at 2 with = 13 1% 1% % % % % % Exercise at 2 with = 15 1% % % % % % % Table 1: Returns observed upon exercising the British knock-in option (with µ c =.2 and barrier levels = 11, 13, 15 above and below the strike price K. The returns are expressed as a percentage of the original option price paid by the buyer (rounded to the nearest integer), i.e. R(t, x)/1 = G µc (t, x)/v (,K). The parameter set is K = 1, T =1, r =.1, σ =.9 and the initial stock price equals K. But then as in (iii) above the continuity of the functions a 1, a 2, b 1 and b 2 combined with the fact that D a {H µc } forces the expectation in (4.42) to be strictly negative and provides a contradiction. Thus D a = D b as claimed and the proof is complete. 5. Financial analysis of the British knock-in put In the present section we discuss the rational exercise strategy of the British knock-in put option, and then present a numerical example to highlight practical features of the option. In particular, we analyse the behaviour of the British knock-in put and compare it with the American knock-in put. We also draw comparisons with the British put option. We spot some of the trends previously seen in [21] but observe some behaviour unique to the knock-in case. In the financial analysis of the option returns presented below we mainly address the question as to what the return would be if the underlying process enters the given region at a given time. We use the skeletal approach of [21], with a similar parameter set so that we can make reference to results seen in [21]. 1. For µ c >r, we witness the same three regimes for the optimal stopping boundary b as those in [21]. Where depending on the value of µ c, the boundary b is either an increasing function of time, a skewed U-shaped function of time, or an intermediate case where b can take either of the two shapes depending on the size of T. However, while b() can tend to 21

22 Time (months) Exercise at 12 (British knock-in with µ c =.2) 142% 125% 14% 84% 58% 26% % Exercise at 12 (British knock-in with µ c =.13) 146% 128% 18% 85% 58% 26% % Selling at 12 (American knock-in) 147% 13% 11% 88% 61% 27% % Exercise at K (British knock-in with µ c =.2) 1% 85% 68% 5% 3% 9% % Exercise at K (British knock-in with µ c =.13) 1% 85% 68% 49% 29% 9% % Selling at K (American knock-in) 1% 85% 69% 51% 3% 9% % Exercise at 8 (British knock-in with µ c =.2) 63% 5% 38% 25% 12% 2% % Exercise at 8 (British knock-in with µ c =.13) 61% 49% 37% 24% 11% 2% % Selling at 8 (American knock-in ) 6% 49% 37% 24% 11% 2% % Exercise at 6 (British knock-in with µ c =.2) 32% 24% 16% 9% 3% % % Exercise at 6 (British knock-in with µ c =.13) 31% 23% 15% 8% 3% % % Selling at 6 (American knock-in) 29% 22% 15% 8% 3% % % Exercise at 4 (British knock-in with µ c =.2) 11% 7% 4% 1% % % % Exercise at 4 (British knock-in with µ c =.13) 1% 6% 3% 1% % % % Selling at 4 (American knock-in) 9% 6% 3% 1% % % % Exercise at 2 (British knock-in with µ c =.2) 1% 1% % % % % % Exercise at 2 (British knock-in with µ c =.13) 1% % % % % % % Selling at 2 (American knock-in) 1% % % % % % % Table 2: Returns observed upon exercising the British knock-in option (with µ c =.13 and µ c =.2 ) and selling the American knock-in option with barrier level = 13. The returns are expressed as a percentage of the original option price paid by the buyer (rounded to the nearest integer), i.e. R(t, x)/1 = G µc (t, x)/v (,K) and R A (t, x)/1 = V A (t, x)/v A (,K) respectively. The parameter set is the same as in Table 1 above and the initial stock price equals K. in the British put option (see [21]) the presence of a knock-in barrier means that b() is always at or below the barrier level. In particular, as the contract drift get closer to r, we see that b() thereby expanding the stopping region, making the buyer overprotected (in the sense that the initial stock price is below b() and so it would always be optimal to exercise immediately). On the other hand, as µ c the British knock-in put effectively reduces to the European knock-in put where the buyer cannot exercise prior to maturity even if the option has been knocked-in. Hence, depending on the size of the contract drift and its proximity to the interest rate r, the buyer may exhibit an infinite tolerance of unfavourable price movements by choosing an infinite contract drift; while the opposite scenario would take place if the contract drift was indeed close to r, leading to overprotection. This brief analysis is in line with the findings of [21] in that the contract drift should not be too close to r and should also not be too large. For the remaining analysis in this section, we look into the examples where µ c =.13 and.2 with r =.1 and σ =.9 since as we can see from Figure 2, for such values of contract drift, the rational exercise strategy of the British knock-in option is similar to that of the American and British put options and so this parameter set serves well for comparison purposes. Note also that a relatively high volatility parameter has been chosen in order to reflect the fact that a lower volatility would not persuade an investor to buy the option. Recalling the discussion in Section 3, we find that the shape of the rational exercise boundary depends strongly on the volatility coefficient especially when the stock price is near the barrier. In fact, while the rational exercise boundary is also strongly dependent 22

23 Time (months) Exercise at 12 (British knock-in µ c =.2) 181% 164% 144% 121% 93% 55% % Exercise at 12 (British knock-in µ c =.13) 187% 168% 147% 123% 93% 55% % Exercise at 12 (American knock-in) % % % % % % % Sell at 12 (European knock-in) 188% 172% 153% 129% 1% 59% % Exercise at K (British knock-in µ c =.2) 223% 27% 189% 168% 141% 14% % Exercise at K (British knock-in µ c =.13) 228% 211% 191% 169% 14% 12% % Exercise at K (American knock-in) % % % % % % % Sell at K (European knock-in) 229% 215% 198% 177% 149% 11% % Exercise at 8 (British knock-in µ c =.2) 279% 266% 251% 234% 213% 186% 157% Exercise at 8 (British knock-in µ c =.13) 282% 268% 251% 232% 21% 181% 149% Exercise at 8 (American knock-in ) 155% 155% 155% 155% 155% 155% 155% Sell at 8 (European knock-in) 282% 271% 258% 242% 222% 194% 161% Exercise at 6 (British knock-in µ c =.2) 354% 346% 337% 327% 317% 39% 314% Exercise at 6 (British knock-in µ c =.13) 354% 344% 333% 321% 39% 298% 299% Exercise at 6 (American knock-in) 31% 31% 31% 31% 31% 31% 31% Sell at 6 (European knock-in) 352% 347% 341% 333% 325% 317% 321% Exercise at 4 (British knock-in µ c =.2) 458% 456% 454% 454% 456% 462% 472% Exercise at 4 (British knock-in µ c =.13) 451% 447% 443% 44% 439% 442% 448% Exercise at 4 (American knock-in) 465% 465% 465% 465% 465% 465% 465% Sell at 4 (European knock-in) 446% 448% 451% 455% 46% 469% 482% Exercise at 2 (British knock-in µ c =.2) 62% 65% 69% 613% 618% 624% 629% Exercise at 2 (British knock-in µ c =.13) 583% 584% 585% 588% 591% 594% 594% Exercise at 2 (American knock-in) 621% 621% 621% 621% 621% 621% 621% Sell at 2 (European knock-in) 571% 581% 592% 64% 616% 629% 642% Table 3: Returns observed upon exercising the British knock-in option (with µ c =.13 and µ c =.2 ), the American knock-in option and selling the European knock-in option with barrier level = 13 after knocking-in. The returns are expressed as a percentage of the original option price paid by the buyer (rounded to the nearest integer), i.e. R 2 (t, x)/1 = G µc BP (t, x)/v (,K) and R A (t, x)/1 = (K x) + /V A (,K) and R E (t, x)/1 = V E (t, x)/v E (,K) respectively. The parameter set is the same as in Table 1 above and the initial stock price equals K. on the volatility coefficient in the British put option, this is even more visible in the knock-in case. When σ is large, the holder will have a greater tolerance of unfavourable drifts since the option is more likely to be knocked-in. Although volatility is directly correlated to the price of European and American knock-in options (see e.g. [6]), this does not necessarily result in greater returns for the British knock-in option since both the payoff and the value of the option depend on the volatility parameter. An in-depth analysis of the volatility parameter and its influence on the British knock-in option is left as the subject of future development. 2. We observe in Figure 4 the effect of varying the level of the barrier on the rational exercise strategy of the British knock-in option. We see that the introduction of a barrier to the British put option raises its rational exercise boundary; and as the barrier is lifted away from the strike price K, the boundary moves up even further. This is plausible since a higher barrier means it is less likely for the option to be knocked-in and so the stopping region expands in order to prevent further losses from taking place. This is in contrast to the behaviour witnessed in the 23