EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1830-1847 ISSN 1307-5543 – ejpam.com Published by New York Business Global British Put Option On Stocks Under Regime-Switching Model Felipe R. Sumalpong, Jr.1,∗, Michael B. Frondoza1, Noel Lito B. Sayson3 1 Department of Mathematics and Statistics, Faculty/Mindanao State University - Iligan Institute of Technology, Iligan City, Philippines 2 Department of Physics, Faculty/Mindanao State University - Iligan Institute of Technology, Iligan City, Philippines Abstract. In a plain vanilla option, its holder is given the right, but not the obligation, to buy or sell the underlying stock at a specified price (strike price) at a predetermined date. If the exercise date is at maturity, the option is called a European; if the option is exercised anytime prior to maturity, it is called an American. In a British option, the holder can enjoy the early exercise feature of American option whereupon his payoff is the ‘best prediction’ of the European payoff given all the information up to exercise date under the hypothesis that the true drift of the stock equals a specified contract drift. In this paper, in contrast to the constant interest rate and constant volatility assumptions, we consider the British option by assuming that the economic state of the world is described by a finite state continuous-time Markov chain. Also, we provide a solution to a free boundary problem by using PDE arguments. However, closed form expression for the arbitrage-free price are not available in our setting. 2020 Mathematics Subject Classifications: 62P05, 97M30 Key Words and Phrases: British put option, american put option, european put option, arbitrage-free price, rational exercise boundary, geometric Brownian motion, optimal stopping time, free boundary problem, regime-switching 1. Introduction Plain vanilla options such as European options and American options are widely used in the market and their pricing mechanisms are well studied. An option gives the holder the right, but not the obligation, to buy or sell an underlying asset for a specified price, called strike price, on or before a specified future date, called maturity date or expiration date. The option is European if the holder can exercise it only at expiration date; it is American if the option can be exercised anytime even prior to the expiration date. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4830 Email addresses: felipejr.sumalpong@g.msuiit.edu.ph (F.Jr. Sumalpong), michael.frondoza@g.msuiit.edu.ph (M. Frondoza), noellito.sayson@g.msuiit.edu.ph (N.L. B. Sayson) https://www.ejpam.com 1830 © 2023 EJPAM All rights reserved. F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1831 One of the pricing mechanisms for European option is provided by the well-known Black-Scholes-Merton formula. This mathematical model assumes, among other things, the absence of arbitrage opportunities and that lending and borrowing are possible at the same risk-free rate. Such method falls within the so-called risk-neutral pricing framework. In [5], G. Peskir and F. Samee introduced a new type of option, called British option, which is American in nature because it can be exercised prior to maturity but with Eu- ropean payoff. The motivation for this new financial product stems from the disparity between the expected value of the option buyer’s investment, in the form of premium paid, and the expected value of his payoff when the actual drift rate of the underlying stock price deviates from the risk-free rate. An added feature is built into this instrument which aim at both providing protection against unfavourable price movements as well as securing higher returns when these movements are favourable [5]. The derivation of the British option price in [5] assumes the usual model as in the Black-Scholes-Merton formula: a geometric Brownian motion for the dynamics of the underlying stock, a constant risk-free interest rate and a constant volatility. In [2], Yao, Zhang and Zhou priced the European options in continuous-time regime-switching via a recursive algorithm. This paper aims to extend the result in [5] by assuming that the economic state of the world is described by a finite state continuous-time Markov chain. The paper is organized as follows. In Section 2 we present the definition of the British put option as given in [5] and the financial setting. In Section 3 we define the stopping set and boundary function and provide results involving these two. In particular, we show that the boundary function satisfies the Volterra type equation, then conclude. 2. Setting of the Problem In this paper, we assume that the economic state of the world is described by a finite state continuous-time Markov chain α = (αt)t∈R+ on M = {1, 2, . . . ,m}. Suppose that the volatility σ : M → (0,∞) depends on the state α of the economy. Under the real world probability measure P, we assume that the dynamics of the stock price process follows a geometric Brownian motion: dXt = µXtdt+ σ(αt)XtdWt, X0 = x > 0, (1) where µ ∈ R is the true drift, W = (Wt)t≥0 denotes the standard Brownian motion defined on a probability space (Ω,F ,P). Here, we assume that W is independent of the Markov-chain α and the filtration F = (Ft)t∈R+ is generated by W and α. We will consider the British put option on stocks in the aforementioned financial market. The British put option with strike price K and time to maturity T (in years) is defined in [5] as follows: Definition 1. [5] The British put option is a financial contract between a seller/hedger and a buyer/holder entitling the latter to exercise at any (stopping) time τ prior to maturity T whereupon his payoff (deliverable immediately) is the ’best prediction’ of the European F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1832 payoff (K − XT ) + given all the information up to time τ under the hypothesis that the true drift µ of the stock price equals the contract drift µc. In [5], the price of the British put option is derived under the hypothesis that the volatility is constant for all t ∈ [0, T ]. Hence, this paper presents an extension of the results in [5]. For 0 ≤ t ≤ T , let β(t) := µc − µ σ(αt) , (2) where µc ̸= µ. Define an equivalent measure Pµc via the following: dPµc dP = ZT , (3) where Zt := exp [∫ t 0 β(u)dWu − 1 2 ∫ t 0 β2(u)du ] (4) and E[Zt] = 1 for 0 ≤ t ≤ T . Then by Itô′s formula, dZt Zt = β(t)dWt. (5) This shows that Zt is a local martingale. From Lemma 1 in [2], we have Wµc t = Wt − ∫ t 0 β(u)du (6) is a Pµc-Brownian motion. Under the probability measure Pµc , (1) becomes dXt = µcXtdt+ σ(αt)XtdW µc t (7) where 0 ≤ t ≤ T with X0 = x ∈ (0,∞). Thus, making use of (3), we have Eµc(X) = E(ZTX) = E(ZT )E(X) = E(X) for any random variable X. The payoff of the British put option at a given stopping time t = τ is given by Eµc [ (K −XT ) + | Fτ ] (8) where the conditional expectation is taken with respect to a new (equivalent) probability measure Pµc under which the stock price X evolves as in (7) with X0 = x ∈ (0,∞). Thus, the effect of exercising the British put option is to substitute the contract drift µc to the true (unknown) drift µ of the stock price for the remaining time of the contract. Note that the value of the contract drift µc must be equivalent to the buyer’s tolerance level for the deviation of the true drift µ from his original belief. Moreover, to avoid arbitrage F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1833 opportunity, the contract drift naturally satisfies (See [5]) µc > r. (9) Note that by Itô’s formula, the solution to equation (7) is Xt = XsZ µc s,t (10) for 0 ≤ s ≤ t ≤ T where Zµc s,t = exp [∫ t s ( µc − σ2(αu) 2 ) du+ ∫ t s σ(αu)dW µc u ] (11) so that the payoff (8) can be written as Eµc [ (K −XτZ µc τ,T ) + | Fτ ]. (12) Let α0 be given. Applying the usual hedging scheme, then the arbitrage-free price of the British put option at deal date (time 0) is given by V = V (0, X0, α0) = sup 0≤τ≤T Ẽ [ e−rτEµc ( (K −XT ) + ∣∣Fτ ) ∣∣∣F0 ] (13) = sup 0≤τ≤T Ẽ [ e−rτEµc ( (K −XT ) + ∣∣Fτ )] where the supremum is taken over all stopping time τ ∈ [0, T ] of X and the Ẽ is taken with respect to the (unique) equivalent martingale measure P̃. Now, fix t ∈ [0, T ]. We want a general expression for the price, denoted by V (t,Xt, αt), of the British put option at any time t at which the stock price Xt = x > 0. Denote the payoff in (8) at τ = s by Gµc(s, y, j) = Eµc [ (K − yZµc s,T ) + | αs = j,Xs = y ] (14) for s ∈ [0, T ] where Zµc s,T is given in (11) . If the exercise date of the British put option is at time t+ τ , where τ ∈ [0, T − t], then extending the argument in (13), we have V (t, x, i) = sup 0≤τ≤T−t Ẽt,x [ e−rτGµc(t+ τ,Xt+τ , j) | αt = i,Xt = x ] (15) where the supremum is taken overall stopping time τ ∈ [0, T−t] ofX and Ẽt,x is taken with respect to the (unique) equivalent martingale measure P̃t,x under which Xt = x ∈ R+. Using the same argument as above with µc is replaced with r in relations (2) through (7) and that Zr t,t+τ (as defined in (11) with µc is replaced with r) has stationary and independent increments (i.e., Zr t,t+τ is a version of Zr 0,τ ), the option price in (15) can be F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1834 rewritten as V (t,Xt, αt) = sup 0≤τ≤T−t E [ e−rτGµc(t+ τ,XtXτ , j) | Ft ] (16) where the process X = X(r) under P solves dXt = rXtdt+ σ(αt)XtdW r t with X0 = 1. Note that they are equivalent because Ft knows the values of Xt and αt. Proposition 1. For any t ∈ [0, T ] and j ∈ M given and fixed, the mapping x 7→ Gµc(t, x, j) (17) is convex on (0,∞). Proof. Let 0 ≤ λ ≤ 1 and x2 = λx1 + (1− λ)x3 for some x1, x3 ∈ (0,∞) with x1 < x3. We have Gµc(t, x2, j) = Gµc(t, λx1 + (1− λ)x3, j) = Eµc [( K − λx1Z µc t,T − (1− λ)x3Z µc t,T )+ ∣∣ Xt = x, αt = j ] = Eµc [( λK + (1− λ)K − λx1Z µc t,T − (1− λ)x3Z µc t,T )+ ∣∣ Xt = x, αt = j ] ≤ λEµc [( K − x1Z µc t,T )+ ∣∣ Xt = x, αt = j ] + (1− λ)Eµc [( K − x3Z µc t,T )+ ∣∣ Xt = x, αt = j ] = λGµc(t, x1, j) + (1− λ)Gµc(t, x3, j), which completes our proof. It can also be verified that the mapping in (17) is strictly decreasing on (0,∞) with Gµc(T, x, j) = (K −XT ) +, Gµc(t, 0, j) = K and lim x→+∞ Gµc(t, x, j) = 0. By Proposition 1 and equation (15) above, it follows that the mapping x 7→ V (t, x, i) (18) is convex for any t ∈ [0, T ] and i ∈ M given and fixed and strictly decreasing on (0,∞) with V (T, x, i) = (K − XT ) +, V (t, 0, i) = K and lim x→+∞ V (t, x, i) = 0. Hence, both mappings (17) and (18) are continuous on (0,∞) for any t ∈ [0, T ] and αt ∈ M given and fixed. Define the set D := {(t, x, j) ∈ [0, T ]× (0,∞)×M : V (t, x, j) = Gµc(t, x, j)}. (19) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1835 Let (T, x, j) ∈ {T} × (0,∞)×M. We note that V (T,XT , j) = (K −XT ) + = Gµc(T,XT , j). (20) Hence, {T} × (0,∞) × M ⊂ D, which is consistent with the fact that the supremum in (15) is taken over (Ft)t∈[0,T ]-stopping times τ ∈ [t, T ]. Furthermore, by Corollary 2.9 page 46 in Peskir and Shiryaev [6], the (Ft)t∈[0,T ]-stopping time τD(t,Xt, αt) := inf {s ∈ [0, T − t] : (t, x, j) ∈ D} (21) with Xt = x ∈ (0,∞) and αt = j ∈ M, is an optimal stopping time for option price in (15) since x 7→ V (t, x, j) and x 7→ Gµc(t, x, j) are both continuous on (0,∞) and Gµc(t, x, j) ≤ K for all t ∈ [0, T ] and j ∈ M. Moreover, by using the equivalent expression for the stopping set D in Relation (46) in Proposition (2), τD(t,Xt, αt) can be rewritten in terms of the optimal stopping boundary function as τD(t, x, j) := inf {s ∈ [0, T − t] : x ≤ bD(t, j)} , (22) where bD(t, j) is defined in (44) below at which Xt = x and αt = j. We next derive the following continuity results to show that the set D in (24) is closed. Lemma 1. The mapping (t, x) 7→ Gµc(t, x, j) is jointly continuous on [0, T ]× (0,∞). Proof. The continuity of the mapping x 7→ Gµc(t, x, j) follows from the fact that Gµc(t, x, j) is convex with respect to x ∈ (0,∞) for any time t ∈ [0, T ] given and fixed. It remains to show the uniform continuity of the mapping t 7→ Gµc(t, x, j) at time t = t1. Let x ∈ (0,∞) be given and fixed and 0 ≤ t1 < t2 ≤ T . Then we have, 0 ≤ ∣∣∣Gµc(t2, x, j)−Gµc(t1, x, j) ∣∣∣ ≤ Eµc [∣∣∣(K − xZµc t2,T )+ − (K − xZµc t1,T )+ ∣∣∣ ∣∣∣∣∣ Ft2 ] ≤ xEµc [∣∣∣(Zµc t1,T − Zµc t2,T )+ ∣∣∣ ∣∣∣∣∣ Ft2 ] = xEµc [∣∣∣∣∣Zµc t1,T ( 1− Zµc t2,T Zµc t1,T )+ ∣∣∣∣∣ ∣∣∣∣∣ Ft2 ] = xEµc ∣∣∣∣∣Zµc t1,T ( 1− e − ∫ t2 t1 ( µc−σ2(αu) 2 ) du− ∫ t2 t1 σ(αu)dW µc u )+ ∣∣∣∣∣ ∣∣∣∣∣ Ft2  . Therefore, as t2 − t1 → 0, we have Gµc(t2, x, j) − Gµc(t1, x, j) → 0 uniformly, which completes our proof. F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1836 Lemma 2. For any j ∈ M, the mapping (t, x) 7→ V (t, x, j) is jointly continuous on [0, T ]× (0,∞). Proof. The continuity of the mapping x 7→ V (t, x, j) at a point x0 follows from the fact that V (t, x, j) is convex with respect to x ∈ (0,∞) for any time t ∈ [0, T ] given and fixed. It remains to show that the mapping t 7→ V (t, x, j) is continuous at t1 uniformly over x ∈ R. Let x ∈ (0,∞) be given and fixed and suppose 0 ≤ t1 < t2 ≤ T . Let τ1 = τD(t, x, i) be the optimal stopping time for (15) and τ2 = τ1 ∧ (T − t2). Then 0 ≤ ∣∣∣V (t1, x, i)− V (t2, x, i) ∣∣∣ ≤ ∣∣∣E [e−rτ1Gµc(t1 + τ1, Xt1+τ1 , j) ∣∣ Ft1 ] −E [ e−rτ2Gµc(t2 + τ2, Xt2+τ2 , j) ∣∣ Ft2 ] ∣∣∣ ≤ ∣∣∣E [e−rτ2Gµc(t1 + τ1, Xt1+τ1 , j) ∣∣ Ft1 ] −E [ e−rτ2Gµc(t2 + τ2, Xt2+τ2 , j) ∣∣ Ft2 ] ∣∣∣ ≤ ∣∣∣E [e−rτ2 {Gµc(t1 + τ1, Xt1+τ1 , j)−Gµc(t2 + τ2, Xt2+τ2 , j)} ∣∣ Ft2 ] ∣∣∣ ≤ E [ e−rτ2 ∣∣Gµc(t1 + τ1, Xt1+τ1 , j)−Gµc(t2 + τ2, Xt2+τ2 , j) ∣∣ ∣∣∣ Ft2 ] . By the continuity of the mapping t 7→ Gµc(t, x, j) from Lemma 1, the mapping t 7→ V (t, x, i) is continuous on [0, T ], uniformly in x ∈ (0,∞). 3. Stopping set and boundary function Define F (t, x, j) = V (t, x, j)−G(t, x, j) ≥ 0, (23) which is nonnegative for t ∈ [0, T ], x ∈ (0,∞) and j ∈ M, so that we have D = {(t, x, j) ∈ [0, T ]× (0,∞)×M : F (t, x, j) = 0}. (24) By the continuity of both mappings (t, x) 7→ V (t, x, i) and (t, x) 7→ Gµc(t, x, j) on [0, T ]× (0,∞), the set D is closed. Thus, the continuation set C = Dc = {(t, x, j) ∈ [0, T ]× (0,∞)×M : F (t, x, j) > 0} (25) is open. Lemma 3. For any (t, x, j) ∈ D, we have lim sup ϵ↘0 F (t, x+ ϵ, j)− F (t, x, j) ϵ ≤ 0. (26) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1837 Proof. For all x ∈ (0,∞) and ϵ > 0, consider the (Fs)s∈[t,T ]-stopping time τ+ϵ = τD(t, x+ ϵ, j) ∈ [0, T − t] (27) defined in (21), which solves the optimal stopping problem V (t, x+ ϵ, αt) = sup 0≤τ≤T−t E [ e−rτGµc(t+ τ,Xt+τ , j) ∣∣∣Ft ] = E [ e−rτ+ϵ Gµc(t+ τ+ϵ , Xt+τ+ϵ , j) ∣∣∣Ft ] . (28) We first claim that τ+ϵ → 0 as ϵ → 0. (29) From the definition of τD(t, x+ ϵ, j), we have, on the event {αt = j}, τD(t, x+ ϵ, j) = inf {s ∈ [0, T − t] : (t, x+ ϵ, j) ∈ D} = inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − (x+ ϵ)XsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] = Eµc [ (K − (x+ ϵ)XsZ µc t+s,T ) + ∣∣∣Ft+s ]} ≤ inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − xXsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] ≥ Eµc [ (K − (x+ ϵ)XsZ µc t+s,T ) + ∣∣∣Ft+s ]} ≤ inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − xXsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] ≥ Eµc [ 1 2 ( K − xZµc t+s,T − ϵZµc t+s,T + ∣∣K − xZµc t+s,T − ϵZµc t+s,T ∣∣) ∣∣∣Ft+s ]} This implies that lim ϵ→0 τD(t, x+ ϵ, j) ≤ lim ϵ→0 inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − xXsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] ≥ Eµc [ 1 2 ( K − xZµc t+s,T − ϵZµc t+s,T + ∣∣K − xZµc t+s,T − ϵZµc t+s,T ∣∣) ∣∣∣Ft+s ]} = inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − xXsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] ≥ Eµc [ 1 2 ( K − xZµc t+s,T + ∣∣K − xZµc t+s,T ∣∣) ∣∣∣Ft+s ]} F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1838 = inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − xXsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] ≥ Eµc [ (K − xXsZ µc t+s,T ) + ]} = inf { s ∈ [0, T − t] : sup 0≤s≤T−t E [ e−rsEµc [ (K − xXsZ µc t+s,T ) + ∣∣∣Ft+s ] ∣∣∣Ft ] = Eµc [ (K − xXsZ µc t+s,T ) + ]} = inf {s ∈ [0, T − t] : (t, x, j) ∈ D} = 0. Now to prove (26), we use (28). Thus, we have lim sup ϵ↘0 V (t, x+ ϵ, j)− V (t, x, j) ϵ = lim sup ϵ↘0 1 ϵ { E [ e−rτ+ϵ Gµc ( t+ τ+ϵ , x+ ϵ, j ) ∣∣∣Ft ] − sup 0≤τ≤T−t E [ e−rτ+ϵ Gµc ( t+ τ+ϵ , x, j) ) ∣∣∣Ft ]} ≤ lim sup ϵ↘0 1 ϵ { E [ e−rτ+ϵ Gµc ( t+ τ+ϵ , x+ ϵ, j ) ∣∣∣Ft ] − E [ e−rτ+ϵ Gµc ( t+ τ+ϵ , x, j) ) ∣∣∣Ft ]} ≤ lim sup ϵ↘0 1 ϵ { Gµc ( t+ τ+ϵ , x+ ϵ, j ) −Gµc ( t+ τ+ϵ , x, j) )} = ∂Gµc ∂x (t, x, j), (30) hence we conclude (26). It is well-known that every convex functions on the open interval I are differentiable almost everywhere, e.g. [3]. In the following Lemmas, we use the fact that both V (t, x, j) and Gµc(t, x, j) are differentiable P-almost surely for all x on (0,∞). Lemma 4. The functions ∂V ∂x (t, x, j) and ∂Gµc ∂x (t, x, j) are continuous on (0,∞) P-almost surely for fixed t ∈ [0, T ] and j ∈ M. Proof. Let ϵ and c ∈ (0,∞) be arbitrary. Since V (t, x, j) is differentiable for all x ∈ (0,∞) for fixed t ∈ [0, T ] and j ∈ M, we know that there exists δ > 0 such that∣∣∣∣∣V (t, x, j)− V (t, c, j) c− x − ∂V ∂x (t, c, j) ∣∣∣∣∣ < ϵ 2 (31) whenever 0 < |c − x| < δ/2. Moreover, by Mean-Value Theorem, there is an element F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1839 y ∈ (x, c) such that V (t, x, j)− V (t, c, j) c− x = ∂V ∂x (t, y, j) and inequality (31) becomes∣∣∣∣∣∂V∂x (t, y, j)− ∂V ∂x (t, c, j) ∣∣∣∣∣ < ϵ 2 . (32) Note that we have 0 < |y − c| < |x− c| < δ/2. For t ∈ [0, T ] and j ∈ M given and fixed, we know from Proposition 1 that V (t, x, j) is convex for all x ∈ (0,∞), then ∂V ∂x (t, x, j) is monotonically increasing. Thus, if 0 < |x− y| < δ/2 we have∣∣∣∣∣∂V∂x (t, x, j)− ∂V ∂x (t, y, j) ∣∣∣∣∣ < ϵ 2 . (33) Therefore, combining inequalities (32) and (33) we have∣∣∣∣∣∂V∂x (t, x, j)− ∂V ∂x (t, c, j) ∣∣∣∣∣ < ϵ, whenever 0 < |x− c| ≤ |x− y|+ |y − c| < δ. Furthermore, by Lemma 6 and the fact that ∂Gµc ∂x is monotonically increasing and that V = Gµc in the stopping set D which is defined in (24) above, we have V (t, x2, j)− V (t, x1, j) x2 − x1 = ∂V ∂x (t, x, j) ≥ ∂Gµc ∂x (t, x, j) = Gµc(t, x2, j)−Gµc(t, x1, j) x2 − x1 ≥ 0 for x1, x2 ∈ (0,∞). Therefore, continuity of ∂Gµc ∂x (t, x, j) follows from the continuity of ∂V ∂x (t, x, j) on (0,∞). This completes our proof. Define the infinitesimal generator Lf(s, x, αs) = ( ∂ ∂t + rx ∂ ∂x + 1 2 σ2(αs)x 2 ∂2 ∂x2 − r ) f(s, x, j) + m∑ i=1 qjif(s, x, i) (34) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1840 of the Markov process (Xs)s∈[0,T ], where Q = (qij)i,j=1,2,...,m is the infinitesimal matrix generator of the Markov process (αs)s∈[0,T ], for any sufficiently differentiable function f of (s, x, j) ∈ [0, T ]× (0,∞)×M. Lemma 5. For all (t, x, j) ∈ [0, T ]× (0,∞)×M, we have LGµc(t, x, j) < 0 (35) when the contract drift µc satisfies µc < r. Proof. The payoff function in (14) can be rewritten as Gµc(t, x, j) = Eµc [(K −XT ) + ∣∣Xt = x, αt = j] for all j ∈ M and hence a martingale by tower property. By (7) and the Itô’s formula we have dGµc(t, x, j) = ∂Gµc ∂t (t, x, j) + µcx ∂Gµc ∂x (t, x, j) + 1 2 σ2(j)x2 ∂2Gµc ∂x2 (t, x, j) + m∑ i=1 qjiG µc(t, x, i) + σ(j)x ∂Gµc ∂x dWµc t . Since Gµc(t, x, j) is a martingale, we find ∂Gµc ∂t (t, x, j) + µcx ∂Gµc ∂x (t, x, j) + 1 2 σ2(j)x2 ∂2Gµc ∂x2 (t, x, j) + m∑ i=1 qjiG µc(t, x, i) = 0. (36) Substituting (36) to (34) we have LGµc(t, x, j) = (r − µc)x ∂Gµc ∂x (t, x, j)− rGµc(t, x, j). (37) Since Gµc(t, x, j) is convex and decreasing with respect to x ∈ (0,∞), then we have ∂Gµc ∂x (t, x, j) < 0. This completes our proof. Lemma 6. For all (t, y, j) ∈ C, we have ∂V ∂y (t, y, j) > ∂Gµc ∂y (t, y, j). (38) Proof. Let (t, xb, j) be a fixed point on the boundary function bD(t, j) so that xb = bD(t, j). Let xb < y ≤ K so that (t, y, j) ∈ C. Since x 7→ V (t, x, j) is continuous on (0,∞) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1841 by Lemma 2 and differentiable P-almost surely, by Mean Value Theorem, there exists at least one c ∈ (xb, y) such that V (t, y, j)− V (t, xb, j) y − xb = ∂V ∂x (t, c, j). Similarly, we have Gµc(t, y, j)−Gµc(t, xb, j) y − xb = ∂Gµc ∂x (t, c, j). Since V (t, y, j) > Gµc(t, y, j) for all (t, y, j) ∈ C, we have ∂V ∂x (t, c, j) = V (t, y, j)− V (t, xb, j) y − xb > Gµc(t, y, j)−Gµc(t, xb, j) y − xb = ∂Gµc ∂x (t, c, j). Since V and Gµc are continuous and convex, the above inequality holds for all (t, c, j) ∈ C. Lemma 7. For any (t, x, j) in the optimal stopping boundary ∂C ⊂ D, we have ∂V ∂x (t, x+, j) = ∂V ∂x (t, x−, j). (39) Proof. For nay ϵ > 0, consider the stopping time τ+ϵ = τD(t, x+ ϵ, j) as in (27). Noting that τ+ϵ → 0 as ϵ → 0 as claimed in (29), by (30) we have ∂Gµc ∂x (t, x, j) ≥ lim sup ϵ↘0 V (t, x+ ϵ, j)− V (t, x, j) ϵ . On the other hand, since (t, x, j) ∈ ∂C ⊂ D, we have lim inf ϵ↘0 V (t, x+ ϵ, j)− V (t, x, j) ϵ ≥ lim inf ϵ↘0 Gµc(t, x+ ϵ, j)−Gµc(t, x, j) ϵ = ∂Gµc ∂x (t, x, j). Since V = Gµc on a closed set D, we have ∂V ∂x (t, x−, j) = ∂Gµc ∂x (t, x, j) = ∂V ∂x (t, x−, j). (40) Lemma 8. We have {(t, x, i) ∈ [0, T ]× (0,∞)×M : LGµc(t, x, i) > 0} ⊂ C where C = Dc is the continuation set. F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1842 Proof. Let (t, x, i) ∈ [0, T ]× (0,∞)×M be such that LGµc(t, x, i) > 0. By Lemma 1 in [2] we have e−rsGµc(t+ s,Xt+s, αt+s) = Gµc(t, x, i) + ∫ t+s t e−ruLGµc(u,Xu, αu)du+Ms, (41) where Ms = ∫ t+s t e−ruσ(αu)Xu ∂Gµc ∂x (u,Xu, αu)dW µc u defines a continuous martingale for s ∈ [0, T−t] with t ∈ [0, T ). By Lemma (1), Lemma (4) and equation (37), the infinitesimal generator LGµc(t, x, j) is continuous with respect to (t, x) ∈ [0, T ] × (0,∞). Thus there exists an open neighborhood U × V ⊂ [0, T )× (0,∞) of (t, x) such that LGµc(s, y, j) > 0 for all (s, y) ∈ U × V . Let τU = inf{τ : (t+ τ,Xt+τ ) ∈ U × V, (Xt, αt) = (x, i) ∈ V ×M}. By Optional Sampling Theorem, the Relation (41) with s = τU shows that E [ e−rτUGµc(t+ τU , Xt+τU , αt+τU ) ∣∣∣Ft ] = Gµc(t, x, i) + E [∫ t+τU t e−ruLGµc(u,Xu, αu)du ∣∣∣Ft ] . (42) Since LGµc(u,Xu, αu) > 0 for u ∈ (t, t+τU ), the right hand side of equation (42) is strictly greater than Gµc(t, x, i), while from equation (15) we have V (t, x, i) ≥ E [ e−rτUGµc(t+ τU , Xt+τU , αt+τU ) | Ft ] showing that V (t, x, i) > Gµc(t, x, i), which implies that (t, x, i) ∈ C. This completes our proof. Next, we define the boundary function bD(t, j) via the following: For any stopping time τ ∈ [0, T − t], it can be verified from equations (37) and (42) that there is a continuous function h : [0, T ]×M → R such that the infinitesimal generator (37) satisfies LGµc(t, h(t, j), j) = 0. (43) Since µc > r, we see that LGµc(t, h(t, j), j) > 0 for x > h(t, j) and LGµc(t, h(t, j), j) < 0 for x < h(t, j) when t ∈ [0, T ] and j ∈ M are given and fixed. In view of equation (42), this implies that for any stopping time τ ∈ [0, T−t], there is no point (t, x) ∈ [0, T ]×(0,∞) with x > h(t, j) is a stopping point. From here, we define the optimal stopping boundary as follows: bD(t, j) := sup {x ∈ (0,∞) : (t, x, j) ∈ D} . (44) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1843 Now, we characterize the stopping set defined in (24) in terms of the boundary function bD(t, j). Proposition 2. For any (t, x, j) ∈ [0, T ]× (0,∞)×M such that (t, x, j) ∈ D we have {t} × (0, x]× {j} ⊂ D (45) and D = {(t, x, j) ∈ [0, T ]× (0,∞)×M : x ≤ bD(t, j)}. (46) Proof. Let (t, y, j) ∈ {t} × (0, x]× {j}. Since (t, x, j) ∈ D and V (t, x, j) ≥ Gµc(t, x, j) for all x ∈ (0,∞), we have V (t, x, j)− V (t, y, j) x− y = Gµc(t, x, j)− V (t, y, j) x− y ≤ Gµc(t, x, j)−Gµc(t, y, j) x− y . Taking the limit on both sides as x− y → 0 and by Lemma (3), we have (t, y, j) ∈ D and conclude (45). From the definition of the boundary function in (44), we have the following equivalence (t, x, j) ∈ D ⇐⇒ {t} × (0, x]× {j} ⊂ D ⇐⇒ x ≤ bD(t, j). Lemma 9. For any (x, j) ∈ (0,∞)×M, the mapping t 7→ F (t, x, j) = V (t, x, j)−Gµc(t, x, j) (47) is nonincreasing in t ∈ [0, T ]. Proof. Let s1, s2 ∈ [0, T − t] with s1 < s2 and consider the stopping time τs2 = τD(s2, x, j) ∈ [0, T − s2]. From definition of the function F in (23) and replacing τU with τs2 in (42), we have F (s2, x, j) = V (s2, x, j)−Gµc(s2, x, j) = E [ e−rτs2Gµc(s2 + τs2 , Xs+τs2 , αs+τs2 ) ∣∣∣αs2 = j ] −Gµc(s2, x, j) = E [∫ s2+τs2 s2 e−ruLGµc(u,Xu, αu)du ∣∣∣αs2 = j ] = E [∫ τs2 0 e−ruLGµc(s2 + u,Xs2+u, αs2+u)du ∣∣∣α0 = j ] . (48) Combining (48) with (49) below F (s1, x, j) = V (s1, x, j)−Gµc(s1, x, j) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1844 ≥ E [ e−rτs2Gµc(s1 + τs2 , Xs1+τs2 , αs1+τs2 ) ∣∣∣αs1 = j ] −Gµc(s1, x, j) = E [∫ s1+τs2 s1 e−ruLGµc(u,Xu, αu)du ∣∣∣αs1 = j ] = E [∫ τs2 0 e−ruLGµc(s1 + u,Xs1+u, αs1+u)du ∣∣∣α0 = j ] , (49) we have F (s2, x, j)− F (s1, x, j) ≤ E [∫ τs2 0 e−ruLGµc(s2 + u,Xs2+u, αs2+u)du ∣∣∣α0 = j ] −E [∫ τs2 0 e−ruLGµc(s1 + u,Xs1+u, αs1+u)du ∣∣∣α0 = j ] ≤ E [∫ τs2 0 {LGµc(s2 + u,Xs2+u, αs2+u)du− LGµc(s1 + u,Xs1+u, αs1+u)du} ∣∣∣α0 = j ] . From Relation (37) with r < µc, since t 7→ Gµc(t, x, j) is nondecreasing on [0, T ], we say that LGµc(t, x, j) is nonincreasing in t, we find that the right hand side is nonpositive, thereby conclude that F (t, x, j) is nonincreasing in t ∈ [0, T ]. Proposition 3. The boundary function bD(t, j) is continuous in t ∈ [0, T ] for all j ∈ M. Proof. Let αt = j ∈ M be fixed. We first show that the boundary function bD(t, j) is left- continuous. Suppose to the contrary that it is not left-continuous at time t = t0. Consider the following cases: Case 1. bD(t0−, j) < bD(t0, j) Let (t′, x′, j) ∈ (0, t0)×(bD(t0−, j), bD(t0, j))×M be a point in the continuation set C with t′ close to t0 and t′ ↑ t0. We know that, by Lemma 4, x 7→ ∂V ∂x and x 7→ ∂Gµc ∂x are both continuous. Since both ∂V ∂x and ∂Gµc ∂x are bounded by −P(y ≤ K) for (t, y, j) ∈ D, by Newton-Leibniz formula and Lemma 6 we have 0 < ∫ bD(t0,j) x′ [ Vx(t ′, u, j)−Gµc x (t′, u, j) ] du = Gµc(t′, x′, j)− V (t′, x′, j) as t′ → t0. This implies that V (t0, x ′, j) < Gµc(t0, x ′, j) which contradicts the fact that (t0, x ′, j) ∈ D since x′ < bD(t0, j), i.e., V (t0, x ′, j) = Gµc(t0, x ′, j). Case 2. bD(t0−, j) > bD(t0, j) F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1845 Let (t∗, x∗, j) ∈ (0, t0) × (bD(t0, j), bD(t0−, j) ×M) be a point on the stopping set D with t∗ close to t0 and t∗ ↑ t0. By (40), we have Vx(t ∗, x∗, j) = Gµc(t∗, x∗, j) on D. Similarly, by Newton-Leibniz formula, we have 0 = ∫ x bD(t0,j) [Vx(t ∗, v, j)−Gµc x (t∗, v, j)] dv = V (t∗, x∗, j)−Gµc(t∗, x∗, j) as t∗ → t0. This shows that V (t0, x ∗, j) = Gµc(t0, x ∗, j) which contradicts the fact that (t0, x ∗, j) ∈ C since x∗ > bD(t0, j), i.e., V (t∗, x∗, j) > Gµc(t∗, x∗, j). Therefore, in either case, bD is left-continuous. To prove the right-continuity can be done similarly. Proposition 4. The boundary function bD(t, j) satisfies the Volterra type equation Gµc(t, bD(t, j), j) = F (t, bD(t, j), j)− ∫ T t J(t, bD(t, j), u, bD(u, αu), αu)du, (50) for 0 ≤ t ≤ T , where F (t, x, j) = E [ (K −XT ) + ∣∣∣αt = j,Xt = x ] (51) and J(t, x, u, bD(u, αu), αu) = E [ LXV (u,Xu, j)I(Xu < bD(u, j)) ∣∣∣αt = j,Xt = x ] , (52) for 0 ≤ t ≤ T and x ∈ (0,∞). Proof. From Relation (16), we see that V (t, x, j) ≥ Gµc(t, x, j) for all (t, x, j) ∈ [0, T ]× (0,∞)×M and recall the continuation set C = Dc = {(t, x, j) ∈ [0, T ]× (0,∞)×M ∣∣∣ V (t, x, j) > Gµc(t, x, j)}. Noting that the stopping time τD = τD(t, x, j) defined in (21) is optimal for (16), we have V (t, x, i) = E [ e−rτGµc(t+ τD, Xt+τD , j) ∣∣∣ αt = i,Xt = x ] . It is well known from the theory of Markov processes that V (t, x, i) is C1,2 in the contin- uation set and it solves the Cauchy-Dirichlet free-boundary problem{ LXV (t, x, j) = 0, (t, x, j) ∈ C V (t, x, j) = Gµc(t, x, j), (t, x, j) ∈ ∂C, (53) where ∂C is the boundary of the open set C. By the local time space formula of [4], we F. Sumalpong, M. Frondoza, N.L. Sayson / Eur. J. Pure Appl. Math, 16 (3) (2023), 1830-1847 1846 have V (T,XT , j) = E [ (K −XT ) + ∣∣∣αt = j,Xt = x ] = V (t, x, j) + E [ M b t ∣∣∣ αt = j,Xt = x ] + E [∫ T t LXV (u,Xu, αu)I(Xu ̸= bD(u, αu))du ∣∣∣αt = j,Xt = x ] + 1 2 E [∫ T t ( ∂V ∂y (u,Xu+, αu)− ∂V ∂y (u,Xu−, αu) ) I(Xu = bD(u, αu)dℓ b u(X x) ∣∣∣αt = j,Xt = x ] (54) whereM b t = ∫ T t σ(αu)Xu ∂V ∂x dWu is a continuous local martingale and ℓb = (ℓbu(X x))t≤u≤T is the local time of Xx = (Xu)t≤u≤T at the curve u 7→ bD(u, j). Using that ∂Gµc ∂x (t, x, j) = −P(x ≤ K) ≤ ∂V ∂x (t, y, j) ≤ 0 for all t ∈ [0, T ), it can easily be verified from Proposition 4.4, page 45 in [1] that E [ M b t ] = 0. By the smooth-fit property shown in Lemma 7, the last two terms in (54) above vanishes. Furthermore, by (53) above and the fact that V = Gµc in the closed set D, equation (54) becomes E [ (K −XT ) + ∣∣∣αt = j,Xt = x ] = Gµc(t, x, j) + ∫ T t E [ LXV (u,Xu, αu)I(Xu < bD(u, αu)) ∣∣∣αt = j,Xt = x ] du. (55) Substituting x with bD(t, j), we have Gµc(t, bD(t, j), j) = E [ (K −XT ) + ∣∣∣αt = j,Xt = bD(t, j) ] − ∫ T t E [ LXV (u,Xu, αu)I(Xu < bD(u, αu)) ∣∣∣αt = j,Xt = bD(t, j) ] du = F (t, bD(t, j), j)− ∫ T t J(t, bD(t, j), u, bD(u, αu), αu). 4. Conclusion and Recommendations This paper extends the results for British put option that was introduced by G. Peskir and F. Samee (2011) by considering stochastic volatility, particularly in a regime-switching. We have shown that the boundary function satisfies the Volterra equation, instead of deriving the closed form expression for the arbitrage-free price for the British put option. For further studies, a similar extension may be done for the British call option. In addition, REFERENCES 1847 one may provide a practical implication of this study. References [1] T Björk. Arbitrage theory in continuous time. Oxford University Press, New York, 2009. [2] Q Zhang D Yao and X Y Zhou. A regime-switching for european options. 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