American Eagle Options

Transcription

1 American Eagle Options Shi Qiu First version: 5 February 6 Research Report No., 6, Probability and Statistics Group School of Mathematics, The University of Manchester

2 . Introduction American Eagle Options Shi Qiu This paper examines the value of American eagle options, i.e., the American strangle options payoff with upper and lower caps. So it is the refinement for strangle options and designs for underlying asset with high volatility. Depending on the value of upper and lower caps, the eagle options could be classified into eagle options with balance wings and disable eagle options. The structure of optimal stopping region is examined. The properties for double free-boundaries and value function are proved. The early exercise premium (EEP) representation of eagle options is derived and the uniqueness of the free-boundary in EEP is proved. Based on the numerical method for double free-boundaries option, we plot the 3D-picture of value function, Delta and Gamma. The American eagle option also can be named as American capped strangle options, since its payoff obtains two extra caps comparing with American strangle options[] and [4]. Inheriting the double free-boundary from the strangle options, the eagle option is suitable for the underlying asset with high volatility. Written the strangle option are inherently risky because of the unlimited loss. In the contrast, the eagle options have maximum loss controlled by caps and become the attractive instruments by the options issuer. In the aspect of buyer, the eagle option has lower premium than the strangle option and the buyer could flexibly set the suitable cap on their preference. By using the lower cap and the upper cap, the eagle option is the refinement for the strangle options. The caped options is not the new birth financial instrument in the market. As two examples mentioned in [3], one is the Mexican Index-Linked Euro Security, which is the American capped call option on the Mexican stock index. The others is the European option with caps on S&P and S&P 5 indices which were introduced in 99. The research on options with caps started by[] in 989, which gives example of European capped call options ranged from forward contract, collar loans, index notes and index currency option notes. Moreover, this paper gives the pricing formula for the European capped call options and explains it is optimal to exercise American capped call options as the stock over the cap. Flesaker discusses the design and price the capped index option in [5], but it is not American style. The paper [3] gives the analytical solution for the American capped call options with constant caps and caps with constant growth rate. For the constant caps, since the free-boundary is known (i.e. the minimum between the constant cap and the free-boundary of American capped call options), it solved out the density function for the first hitting time at the free-boundary. By the density function, the author obtained the analytical valuation formula for the American capped call option. Since the American eagle options is the capped strangle options, we follow the idea from American capped call options [3] and American strangle options[4] to price and analyze the Key words and phrases: the American eagle option, continuous dividend yield, change-of-variable formula on curves, optimal stopping, double free-boundaries, Greeks analysis

3 eagle options. The paper is organised as follow. In the second section, we will introduce the motivation to design this options, why we call it eagle option and the equivalent transformation of the payoff function G (x) in Theorem. The third section, we give some notations of the free-boundary and value function for American call, put and strangle options. The forth section introduces the eagle options with balance wings. We prove that the free-boundary is monotonic and continuous and the value function is binary continuous and smooth-fit partly on the free-boundary. Moreover, we give the early exercise premium (EEP) representation and show that the pair free-boundaries in EEP is unique. From the previous in options price like American Asian options [3], Russian options [9], British options [] [] and etc., the smoothfit property is satisfied for the whole free-boundary. However, the eagle option just holds the smooth-fit for parts of the free-boundary, which leads to the local time term appearing in the EEP representation. The fifth section discusses the American disable eagle options. As the lower cap increase up to the lower strike price, parts of lower cap will be inside the continuation region. This makes the proof different from forth section. The final section shows some pictures of free-boundaries, value function, Delta value and Gamma value. These pictures give readers more intuitive understanding of the eagle option.. Motivation First of all, we will discuss the motivation to design the eagle options. By the book [6, Chapter ], it creates a bear spread by buying a European put options on the stock with a certain strike price L and selling another European put options on the same stock with a lower strike price l. Both of the European put options have the same maturity. The payoff function at expiry is (.) G (x) = (L x) + (l x) +, x represents the price of underlying asset in the option. Conversely, it also creates a bull spread by purchasing a European call options with the strike price K and selling a European call options with higher strike price k. Both of the European call options have the same maturity. The payoff function for bear spreads at expiry is (.) G (x) = (x K) + (x k) +. From the formula (.) and (.), the strike price k and l are like the cap for European call and put options, respectively. Seen the payoff at Figure, the bear or bull spread writer will reduce the maximum loss comparing the put or call options. For the buyer, the premium is cut.

4 G(x) G(x) l L x K x k (a) Figure. picture (a) shows the payoff of bear spreads at maturity with strike price K and k for k K. (b) shows the payoff of bull spreads at maturity with strike price L and l for l L. By the Figure, the bull spread or bear spread is just suitable for increasing or decreasing underlying asset. We want to combine the payoff of G (x) with G (x), and design a new option for underlying with high volatility. The playoff of new options is G (x) = G (x) + G (x), (b) (.3) G (x) = (x K) + (x k) + +(L x) + (l x) +, where l L K k. G(x) l L K k x (a) Figure. picture (a) shows the payoff (.3) struck at l, L, K and k. (b) shows that the payoff function in (.3) looks like a flying eagle. (b) 3

5 From (b) in Figure, the payoff in (a) looks like the a flying eagle, so we call the options with payoff (.3) as eagle options. From (a) in Figure, when the asset price tremendously fluctuate out of the range [L,K], the payoff in (.3) has strictly positive return. So the options with this payoff is designed for asset price with high volatility. When stock price moves below l or above k, the payoff will keep a constant value. Since l and k is like the cap to avoid the payoff from crossing the constant, we treat l as the lower cap and k as the upper cap for the eagle option. By the pay off function in (.3), the value of European eagle options at (t,x) is defined as (.4) V E (t,x) = E t,x [e r(t t) G (X T )], for the option maturity T and E t,x is the expectation under P t,x defined as P(X t = x) =. The value of American eagle options is (.5) V (t,x) = sup E t,x [e rτ G (X t+τ )], τ [,T t] where τ is the stopping time over [,T t], and stock price X satisfies geometric Brownian motion (.6) dx t = (r δ)x t dt+σx t dw t, where W = (W t ) t is the standard Brownian motion, risk free rate r >, volatility σ > and continuous dividends δ >. With the initial value x, the analytical solution of (.6) is (.7) X t = x exp(σw t +(r δ + σ t)), which is a strong Markov process with infinitesimal generator (.8) L X = (r δ)x + σ x. The European eagle option defined in(.4) is also named as reverse iron condor options/spreads, which are traded over the counter (OTC) for risk management. Section 4 will mainly discuss about American eagle options (.5) with balance wing, i.e. L l = k K. The case L l k K will leadto American disable eagleoptionsandthisproblem will be discussed insection 5. For the definition of G (x) in (.3), it is not suitable to construct the optimal stopping region. So we will use the following theorem to transform the payoff function for eagle options. Theorem. The payoff of American eagle options (.3) can be transformed into (.9) G (x) = (k x K) + (L l x) +. for l L K k. Proof. Since (x K) + (x k) ] + = for x K and ](L x) + (l x) + for x L, so G (x) = [(x K) + (x k) + [(L x) + (l x) +. For (x K) + (x k) +, we can transform it into (x K) + (x k) + 4

6 = (x K)I(x K) (x k)i(x k) = (x K)I(x k)+(x K)I(K x k) (x k)i(x k) = (k K)I(x k)+(x K)I(K x k) = (k x K)I(x k)+(x k K)I(K x k) = (k x K)I(x K) = (k x K) +, where I( ) is the indicator function. For (L x) + (l x) +, similar approach will be applied, (L x) + (l x) + = (L x)i(x L) (l x)i(x l) = (L x)i(x l)+(l x)i(l x L) (l x)i(x l) = (L l)i(x l)+(l x)i(l x L) = (L l x)i(x l)+(l x l)i(l x L) = (L l x)i(x L) = (L l x) +. From the above proof, it is concluded that G (x) = (k x K) + (L l x) +. In this paper, we will price and analyze the American eagle options with the payoff (.9). The structure of this payoff looks like the payoff of American strangle options [] and [4]. It is the capped strangle options with lower cap l and upper cap k so that the premium is less than the strangle options. And the eagle option protect the options writer from the maximum loss L l or k K. As l and k, the eagle option will turn into the strangle option. 3. Definition and Notation In the previous section, we have already defined the value function and payoff of American eagle options in(.5) and(.9). This section will give the notation of American strangle options, which follow the paper [4]. The value of American strangle options is defined as (3.) V ST (t,x) = sup E t,x [e rτ G ST (X t+τ )], τ [,T t] where G ST (x) = (x K) + (L x) +. The stopping region is (3.) (3.3) D ST = {(t,x) [,T] (, ) V ST (t,x) = (L x) + }, D ST = {(t,x) [,T] (, ) V ST (t,x) = (x K) + }, and the continuation region is (3.4) C ST = {(t,x) [,T) (, ) V ST (t,x) > G ST (x)}. Function b ST (t) is the lower free-boundary and bst (t) is the upper free-boundary. For American call options, function V C (t,x) represents the value at time t and stock price x. Function b C (t) is the free-boundary for call options. We use V C (t,x) and b P (t) to represent the value function and the free-boundary for American put options. 5

7 4. American Eagle Options with Balance Wings As k K = L l, the payoff looks like (a) in Figure. Since The height of wings are equal, it is a flying eagle with balance wings. For the case k K L l, we call it disable eagle option and discuss in the following section. 4.. The Optimal Stopping Region and Continuation Region The value function of American eagle options defined in (.5) is the optimal stopping problem. To solve this problem, we need to define the stopping region and continuation region, (4.) (4.) C = {(t,x) [,T) (, ) V (t,x) > G (x)}, D = {(t,x) [,T] (, ) V (t,x) = G (x)}. Since x G (x) is a continuous function, function (t,x) V ST (t,x) is lower semicontinuous by the statement (..8) in []. By applying the Corollary.9 in [], the optimal stopping time for problem (.5) is (4.3) τ D = inf{ s T t X t+s D } Since {(t,x) [,T) (, ) L x K} is inside the continuation region C, we can separate the exercised region D into (4.4) (4.5) D = {(t,x) [,T] (, ) V (t,x) = (L x l) + }, D = {(t,x) [,T] (, ) V (t,x) = (x k K) + }, and D D = before T. Theorem. This theorem have two parts: D is down connectedness: if (t,x) D is up connectedness: if (t,x) D, then (t,x ) D, then (t,x ) D as well for x < x ; D as well for x > x. Proof. The proof just illustrate the up connectedness, and the down connectedness can be proved in the same way. Firstly, we need to prove that D is right connectedness, i.e. if V (t.x) D, then (t,x) D for t > t. Since (t,x) D, the V (t,x) = G (x) = x k K. By the definition of value function in (.5), t V (t,x) is decreasing. Then we have G (x) = V (t,x) V (t,x). As it is known that V (t,x) G (x), so we prove that G (x) = V (t,x), i.e. (t,x) D. Secondly, we will prove that D is up connectedness. Assume that (t,x) D, and x > x. Case One: x k. Let τ is the optimal stopping time for V (t,x ), V (t,x ) = E[e rτ (X x τ k K) + (L X x τ l) + ], E[e rτ ((x x+x)n τ k K) + (L xn τ l) + ], 6

8 = E[e rτ (((x x)n τ +xn τ ) k K) + (L xn τ l) + ], Since (a+b) c a c+b and (a+b) c a c+b for a, b, c, E[e rτ (((x x)n τ +xn τ k) K) + (L xn τ l) + ], E[e rτ (xn τ k K) + (L xn τ l) + ]+E[e rτ (x x)n τ ], x k K +(x x) = x K = G (x ), where N τ = exp((r δ + σ /)τ + σw τ ) and e rt N t is a supermartingale. The Notation (X x t ) t means process X starting at x, which will be used frequently is the following text. Case Two: x > k Let τ is the optimal stopping time for V (t,x ). V (t,x ) = E[e rτ (X x τ k K) + (L X x τ l) + ] Since (t,x) D, by the right connectedness of D, we get that (s,x) D for all s [t,t]. Assuming that τ is the optimal stopping time for V (t,x ), it is the first passage time to the stopping region for X. Then we know Xτ x x K. So V (t,x ) = E[e rτ (X x τ k K) + ] k K = G (x ). Combining case one and two, Since V (t,x ) G (x ), we have V (t,x ) = G (x ) = (x k K) +, i.e. (t,x ) D. The down connectedness of D can be proved in the same way for x l and x < l From the definition of value function in (.5), when x k or x l, the option buyer obtain the most return and keep holding the option will increase the time cost. So it is optimal to exercise the option. That is to say {(t,x) x l} D and {(t,x) x k} D. With this common sense, we will prove the following theorem. Theorem 3. The continuation region and exercise region are nonempty: {(t,x) (,T) (,L) x l b ST (t)} D and {(t,x) (,T) (,L) L x > l b P (t)} C, {(t,x) (,T) (K, ) x k b ST (t)} D and {(t,x) (,T) (K, ) K x < k b C (t)} C. Using the same value of r, δ, σ and T in eagle options, function b C and b P are the free-boundary for American call struck at K and put options struck at L. b ST and b ST is the lower free-boundary and the higher free-boundary for American strangle options struck at L and K, respectively. Proof. We will prove the second item in this theorem and the first item can be proved in the same way. Firstly, we will prove that {(t,x) x k b ST (t)} D. If k b ST (t) = k and t is given and fixed, when x k, it is the most return we can get from the options payoff, and the options 7

9 holder is optimal to exercise, so {(t,x) x k b ST (t) = k}. On the other hands, for k b ST (t) and t is given and fixed, if x k bst (t), similar as above proof, it is true that (t,x) D. If k x b ST (t), V ST (t,x) = x K = x k K = G (x) V (t,x). By the definition of V (t,x) in (.5), it is obvious that V ST (t,x) V (t,x), so V (t,x) = G (x) i.e. (t,x) D. Secondly, we will prove that {(t,x) K x < k b C (t)} C. For x < k b C (t) and t is given and fixed, since k b C (t) is the free-boundary for American capped call options struck at K with cap k > K (see [3, Theorem ]), so V cap,call (t,x) > k x K = G (x), where V cap,call is the value of American capped call options. For V (t,x) V cap,call (t,x) in the domain [,T] [, ), we have V (t,x) > G (x). D Going back to the proof of Theorem in [3], Detemple shows that {(t,x) x < k b C (t)} is inside the continuation region of American capped call options and {(t,x) x k b C (t)} is inside the optimal exercise region, so the free-boundary of American capped call options is known as k b C (t) for t [,T]. However, the free-boundary for the American eagle option is unknown, because we are uncertain that {(t,x) K x < k b ST (t)} C. If we want to proof it following the method in [3, Theorem ], for a given t and x < k b ST (t) = k, we need to find an American strangle option with shorter maturity T < T such that l < b ST (t,t ) < x < b ST (t,t ) < k. But when x is very closed to k, this strangle option expiry at T may not exist. All in all, the free-boundary of American eagle options is unknown, so we cannot use the pricing method in [3, Theorem ]. In the following Section 4 and 5, we will introduce how to price the American eagle options by the local-time space formula. By Theorem and 3, we know the exercise region D and D are nonempty, and satisfies the down connectedness and up connectedness, respectively. We can define the lower and upper free-boundary as (4.6) (4.7) b (t) = sup{x (, ) V (t,x) b (t) = inf{x (, ) V (t,x) D }, D }. Theorem 3 shows the range of double free-boundaries are (4.8) (4.9) l b ST (t) b (t) l b P (t) for t [,T), k b C (t) b (t) k b ST (t) for t [,T). The value function of American eagle options (.5) is the Mayer functional problem. By the book [, Chapter III], we know V (t,x) solve the Dirichlet problem: (4.) (4.) V t +L X V = rv in C, V (t,x) = G (t,x) in D. By formulae (4.) and (4.), the Crank-Nicolson method (the extension for finite-difference method) can be used to detect the continuation region for American eagle options. 8

10 6 5 t b ST (t) k 4 3 Stock Price Time t b ST (t) l T = Figure 3. The figure shows that continuation region of American eagle options with the parameter: l = 7.5, L =, K =.5, k = 5, r = δ =.6, σ =., T=. The bar region is the continuation region and the others is the optimal exercise region. The upper dash line is b ST that the bar does not hit the dash line. (t) k and the lower dash line is bst (t) l. The red circle shows Seen from the red circle region, the lower free-boundary of eagle options b (t) is covered by b ST (t) l, and the upper free-boundary b (t) seems to coincide with b ST (t) k. The following theorem will show that parts of the free-boundary of eagle options is the same as strangle options when the condition is satisfied. Before the theorem, two notation t and t need to be defined Definition. Starting at T and going backward, t is the first time when the lower freeboundary of American strangle options b ST (t) hits the lower cap l, i.e. bst ( t ) = l and b ST (t) > l for t ( t,t). If b ST (T) l, we set t = T. Definition. Starting at T and going backward, t is the first time when the upper freeboundary of American strangle options b ST (t) hits the upper cap k, i.e. bst ( t ) = k and b ST (t) < k for t ( t,t). If b ST (T) k, we set t = T. Theorem 4. For all t (max( t, t ),T), we have b (t) = b ST (t) and b (t) = b ST (t). Proof. Firstly, We will prove that R := {(t,x) [ t t,t] (,+ ) b ST (t) < x < bst (t)} C. (t,x) R, a stopping time are defined as (4.) τ = inf{ s T t X t+s b ST (t+s) or X t+s b ST (t+s)}. Applying the product rule and Itô-Tanaka Formula on the discounted payoff of American strangle options e rs G ST (X t+s ), we use τ to replace s and take the expectation in P t,x from both sides to get the following formula E t,x e rτ G ST (X t+τ ) = G ST (x)+ E t,x 9 e ru dl L u (X)+ E t,x e ru dl K u (X)

11 (4.3) +E t,x e ru ( rl+δx t+u )I(X t+u < L)du +E t,x e ru (rk δx t+u )I(X t+u > K)du, where l L u (X) is the local time for X at L, and ll u (X) = P lim ε u I(L ε<x ε t+r< L+ε)d X,X r. From the paper [4], we know that τ is the optimal stopping for the value of American strangle options V ST (t,x), so V ST (t,x) = E t,x e rτ G ST (X t+τ ). For (t,x) C ST, the definition of the continuation region C ST implies that V ST (t,x) > G ST (x). So the integrand part in (4.3) satisfies (4.4) E t,x e ru dl L u (X)+ E t,x e ru dl K u (X) +E t,x e ru ( rl+δx t+u )I(X t+u < L)du +E t,x e ru (rk δx t+u )I(X t+u > K)du >. Applying the product rule and Itô-Tanaka Formula on the discounted payoff of American strangle options e rs G (X t+s ), we use τ to replace s and take the expectation in P t,x from both sides to get the following formula (4.5) E t,x e rτ G (X t+τ ) = G (x) E t,x E t,x e ru dl L u (X)+ E t,x e ru dl l u (X) E t,x e ru dl K u (X) +E t,x e ru ( rl+δx t+u )I(l < X t+u < L)du +E t,x e ru (rk δx t+u )I(k > X t+u > K)du. e ru dl k u (X) By the definition of τ and t + τ > max( t, t ), it is known that l < b ST (t + τ) X t+τ (t+τ) < k. The last two integrands in (4.5) can be written as b ST (4.6) (4.7) and E t,x e ru dl l u (X) = E t,x into E t,x e ru ( rl+δx t+u )I(l < X t+u < L)du = E t,x e ru ( rl+δx t+u )I(X t+u < L)du, E t,x e ru (rk δx t+u )I(k > X t+u > K)du = E t,x e ru (rk δx t+u )I(X t+u > K)du, E t,x e rτ G (X t+τ ) = G (x)+ E t,x e ru dl k u (X) =. Then the equation (4.5) can be simplified e ru dl L u (X)+ E t,x e ru dl K u (X)

12 (4.8) +E t,x e ru ( rl+δx t+u )I(X t+u < L)du +E t,x e ru (rk δx t+u )I(X t+u > K)du. comparing equation (4.3) and (4.8), we have E t,x e rτ G (X t+τ ) > G (x). Since value function for American eagle options (.5) is the spermium of the discounted payoff over all the stopping time, so V (t,x) > G (t,x), i.e. (t,x) C such that R C. For t [ t t,t], this result indicates that {(t,x) b ST (t) < x < b ST from Theorem 3 shows that {(t,x) x b ST (t)} D obvious that b (t) = b ST (t) and b (t) = b ST (t)} C, and the result and {(t,x) x b ST (t)} D. It is (t) for t [ t t,t]. 4.. The Property of Value Function and Free-Boundary In this section, we will discuss the value function is continuous and satisfies the smooth-fit property for parts of the free-boundary. For the free-boundary, the discussion is on the monotonicity, continuity and the convergence closed to maturity. Theorem 5. The value function for American eagle options (t,x) V (t,x) defined in (.5) is continuous on [,T] (, ). Proof. Firstly, we will prove that t V (t,x) is uniformly continuous on [,T] and x is given and fixed. Given that t t T and τ is the optimal stopping time for V (t,x). Set τ = τ (T t ). V (t,x) V (t,x) E[e rτ max((x x τ k K) +,(L X x τ l) + )] E[e rτ max((x x τ k K) +,(L X x τ l) + )] = E[e rτ max((x x τ K) +,(L X x τ ) + )] E[e rτ max((x x τ K) +,(L X x τ ) + )] E[e rτ max((x x τ K) + (X x τ K) +,(L X x τ ) + (L X x τ ) + )] E[e rτ max( X x τ X x τ, X x τ X x τ )] E[ X x τ X x τ ]. The first equality based on l Xτ x k for τ τ and l Xτ x k. The third inequality uses that max(a,b) max(c,d) max(a c,b d) for a, b, c, d. The fourth inequality is for (x K) + (y K) + x y and (L x) + (L y) + x y. Set Z t = σb t +(r δ σ /) and the process Z = (Z t ) t has stationary independent increment. By τ τ t t, we have [ ] E Xτ x Xτ x [ ]] = E E [ X xτ X xτ F τ [ [ = E E Xτ x Xτ x ]] Xτ x F τ [ e ]] = E Xτ x Z E[ τ Z τ Fτ

13 [ [ e ]] = E Xτ x Z E τ Z τ ] [ e ] = E [X xτ Z E τ Z τ ] E [X E[ ] xτ sup t t t e Zt =: E [ X x τ ] L(t t ) xe rt L(t t ), where we use Z τ Z τ is independent from F τ. So t V (t,x) is uniformly continuous. Secondly, we want to prove that x V (t,x) is uniformly continuous on (, ). Given that < x < y <, and setting τ is the optimal stopping time for V (t,y). V (t,y) V (t,x) E[e rτ (yn τ k K) + (L yn τ l) + ] (4.9) E[e rτ (xn τ k K) + (L xn τ l) + ], E[e rτ max( yn τ k xn τ k, yn τ l xn τ l )] = E[e rτ max( (k xn τ ) + (k yn τ ) +, (yn τ l) + (xn τ l) + )] E[e rτ max( yn τ xn τ, yn τ xn τ )] x y. The first equality use that a b = (b a) + + a and a b = a (a b) + for a, b >. Similarly, by setting τ is optimal stopping time for V (t,x), we can prove that (4.) V (t,y) V (t,x) x y. By (4.9) and (4.), x V (t,x) is uniformly continuous. Since t V (t,x) and x V (t,x) are both uniformly continuous, we have proved that (t,x) V (t,x) is continuous on [,T] (, ). Theorem 6. The lower free-boundary t b (t) is increasing function and the upper freeboundary t b (t) is decreasing function for t [,T]. Proof. By the right connectedness for D and to show this theorem. D proved in Theorem, it is not difficult Since b and b are monotonic and l and k are inside the optimal stopping region, we can define ˆt and ˆt, which will be used in the following proof. Definition 3. Starting at T and going backward, time ˆt is the first time when b (t) hits l, i.e. b (t) = l for t [,ˆt ) and b (t) > l for t (ˆt,T]. Definition 4. Starting at T and going backward, time ˆt is the first time when b (t) hits k, i.e. b (t) = k for t [,ˆt ) and b (t) > k for t (ˆt,T]. As we known, the smooth-fit property is always satisfied for the previous research on options pricing, such as American put option [8], American Asian options [3], Russian options [9] and etc. If the smooth-fit is satisfied, the local time term in the time space formula [7] will be

14 vanished such that the early exercise premium representation can be simplified. We will discuss the smooth-fit for American eagle options in the Theorem 7 and Theorem 8. From the fact in (4.8) and (4.9), we can separate the free-boundary into four parts b (t) = l, b (t) > l, b (t) < k and b (t) = k, and discuss the smooth-fit in each part. To obtain the intuition in Theorem 7 and Theorem 8, we numerically simulate the value of + V (t,x) x=b (t) and V (t,x) x=b (t), + V (t,x) x=b (t) V (t,x) x=b (t) Time (a) Time Figure 4. picture (a) plots the function t + V (t,x) x=b (t). (b) plots the function V (t,x) x=b (t). The used parameter for the figure is, l = 8, L =, K =, k = 4, r = δ =.6, σ =., T =. From the result of Figure 4, the value of t + V (t,x) x=b (t) and t V (t,x) dependsont,howeverbythedefinition G thevalueoftheleftderivative t V (t,x) is equal to either or and the right derivative t + V (t,x) or. So the smooth-fit property is destroyed for the eagle option. (b) x=b (t) x=b (t) x=b (t) is equal to either Theorem 7. As b (t) = l or b (t) = k, the value function dissatisfies the smooth-fit property, but (4.) (4.) (4.3) (4.4) V (t,x) + V (t,x) V (t,x) + V (t,x) x=b (t) x=b (t) x=b (t) x=b (t) =,,, =. Proof. The proof will emphasis on (4.3), (4.) can be illustrated in the same way. Let x = b (t) for t [,T] 3

15 . Since x = b (t) = k > K, there exists ε >, such that x ε > K. We have the below inequality (4.5) V (t,x) V (t,x ε) ε k K (k ε K) ε =, and ε approaching to makes (4.6). Here, we want to prove V (t,x). (4.7) V (t,x) with x = b ST (t) = k. is the optimal stopping time for V (t,x ε). We have V (t,x) V (t,x ε) E[e rτε [(X x k K) + (L X x l) + (X x ε E[e rτε (X x X x ε E[e rτε (X x ε k K) + (L Xτ x ε ε l) + ]] )I(K Xτ x ε ε Xτ x ε k,t+ > ˆt )] Xτ x ε )I(l Xτ x ε ε Xτ x ε L)] )I(K X x ε k, Xτ x ε > k)] E[e rτε (k X x ε E[e rτε (X x ε L)I(l X x ε E[e rτε [(X x K) (L X x ε E[e rτε [(k K) (L Xτ x ε ε E[e rτε εn τε I(K X x ε E[e rτε εn τε I(l X x ε E[e rτε εn τε I(l X x ε E[e rτε εn τε I(l X x ε L,L X x K)] )]I(l X x ε L,K Xτ x ε k)] )]I(l Xτ x ε ε L,k < Xτ x ε )] Xτ x ε k,t+ > ˆt )] Xτ x ε L)] L,L X x K)] L,K X x k)] E[e rτε εn τε I(l X x ε L,k < X x )]. The second inequality is based on l X x ε k and X x ε < Xτ x ε. Divided by ε from both sides and letting ε tend to zero, all the formulae on the right hand side of the inequality goes to zero, so we get equation (4.7). Then equation (4.3) hold. The equation (4.) can be proved in the same way. Using the definition of the payoff G, we can prove equation (4.4) and (4.) Theorem 8. As b (t) > l or b (t) < k, the value function satisfies the smooth fit property, V (t,x) (4.8) =, x=b (4.9) V (t,x) (t) x=b (t) 4 =.

16 Proof. The proof will emphasis on (4.9), (4.8) can be illustrated in the same way. Let x = b (t) < k for t [,T]. Since x = b (t) < k, then exist ε >, such that K < x ε. The same illustration from part in Theorem 7, we can prove that (4.3) V (t,x) x=b (t).. By same analysis from in Theorem 7, we can get the inequality V (t,x) V (t,x ε) E[e rτε εn τε I(K Xτ x ε ε Xτ x ε k,t+ > t )] E[e rτε εn τε I(l Xτ x ε ε Xτ x ε L)] E[e rτε εn τε I(l Xτ x ε ε L,L Xτ x ε K)] E[e rτε εn τε I(l Xτ x ε ε L,K Xτ x ε k)] E[e rτε εn τε I(l Xτ x ε ε L,k < Xτ x ε )]. Dividedby ε frombothsidesandletting ε tendtozero,thefirstexpectation E[e rτε εn τε I(K Xτ x ε ε Xτ x ε k,t + > t )] from the right hand side of the inequality goes to and the others go to zero. We get the result V (t,x). Combining with (4.3), we have x=b (t) proved that (4.3) V (t,x) x=b (t) =. By the definition of the payoff function G (x), it is obvious that (4.3) + V (t,x) x=b (t) =. Combining equation (4.3) and (4.3), we have proved the smooth-fit at b (t) for b (t) < k. Theorem 9. As approachingto the maturity T, the lowerfree-boundaryconvergesto b (T ) = max(l,min(l, r L)) andthe upper free-boundaryconvergesto b δ (T ) = min(k,max(k, rk)). δ Proof. From Theorem 3, k b C (t) b (t) b ST (t) k for t [,T). Since t b (t) is increasing, then b (t ) exists. As t T, we get k b C (T ) b (T ) b ST (T ) k. For bc (T ) = b ST (T ) = max(k, r K), it can be concluded that b δ (T ) = min(k,max(k, r K)). By Theorem 3, we can prove b δ (T ) = max(l,min(l, r L)) in the δ same way. Theorem. Forall (t,x) C = {(t,x) b (t) < x < b (t)}, wehave V x (t,x). 5

17 Proof. Since (t,x) V (t,x) is C, in the continuation region C, if we prove that (4.33) (4.34) + V (t,x) V (t,x),, the statement in Theorem is true. Firstly, we will prove the inequality (4.33), and (4.34) can be done in the same way. Take any (t,x) C and ε >, set τ is the optimal stoping time for V (t,x). V (t,x+ε) V (t,x) E[e rτ (Xτ x+ε k K) + (L l Xτ x+ε ) + ] E[e rτ (Xτ x k K)+ (L l Xτ x )+ ] E[e rn τ I(l Xτ x Xx+ε τ L)] E[e rn τ I(l Xτ x L,k Xτ x+ε )] E[e rn τ I(l Xτ x L,K Xx+ε τ k)] E[e rn τ I(l Xτ x L,L Xx+ε τ K)]. Divided by ε from both sides and let ε, just the first term on the right hand side of the inequality is survived and the others are equal to zero. We can get the inequality, + V (t,x) E[e rτ N τ I(l X x τ L)] = E[exp(σW τ στ σ τ)] E[exp(σWτ σ τ)] =. So we prove (4.33). For the inequality (4.34), we can be proved it by the similar method applying on V (t,x) V (t,x ε) Theorem. The free-boundary b (t) and b (t) are continuous function on [,T). Proof. We will prove the continuity of b (t), b (t) can be proved in the same way. i) Firstly, we will show b is right continuous. t [,T), constructing a sequence t n t as n, since b is decreasing function, the right-hand side limit b (t+) exists. Becuase (t n,b (t n )) D and D is a closed set, (t,b (t+)) D i.e. b (t+) b (t). The reverse inequality follows the fact that b is decreasing in [,T], so b (t+) = b (t). ii) Secondly, we want to prove b is left continuous. Suppose that at the point t [,T) the function b has a jump and b ( t ) > b ( t). Let us fix a point t < t. Function x Vx (t,x) is continuous in C and V (t,x). x=b (t ) (a) If >, by the property of continuity, we can find < d < (b ( t ) V (t,x) x=b (t ) b ( t)), such that x (b (t ) d,b (t )) satisfies V 6 x (t,x). As the previous proof

18 in Theorem, we know that Vx (t,x) for (t,x) C. So Vx (t,x) for x (b (t ) d,b (t )). Take x (b (t ) d,b (t )) and x b ( t ), apply the Newton-Leibniz formula: (4.35) V (t,x ) (x k K) = b ST (t ) x b ST (t ) u V xx (t,v)dvdu. Since (t,x ) in C, we have (4.36) V xx (t,x ) = σ x (rv (t,x ) Vt (t,x ) (r δ)x Vx (t,x )) σ x (rv (t,x ) (r δ)x Vx (t,x )). If δ > r, since V x (t,x ), there exists a constant c such that (4.37) V xx (t,x ) σ x rg (x ) = σ x r(x K) c >. If δ r, Since V x (t,x ), there exists a constant c such that (4.38) (b) If Vxx (t,x ) σ x (rg (x )+(r δ)x ) = σ x (δx rk) c >. V (t,x) x=b (t ) =, by the property of continuity, we can find < d < (b ( t ) b ( t)), such that x (b (t ) d,b (t )) satisfies r(k K) (r δ)xv for a constant c. As the previous proof in Theorem, we know that V x (t,x) c > x (t,x) (t )). Taking x ( t ), since (t,x ) C, we have for (t,x) C. So V x (t,x) for x (b (t ) d,b (b (t ) d,b (t )) and x b (4.39) V xx (t,x ) = σ x (rv (t,x ) Vt (t,x ) (r δ)x Vx (t,x )) σ x (rv (t,x ) (r δ)x Vx (t,x )) σ x (r(k K) (r δ)x Vx (t,x )) σ x c c >. Byformula(4.37),(4.36)and(4.39),weknowthat Vxx (t,x ) c >. Inserting Vxx (t,x ) c into the Newton-Leibniz formula in (4.35), we know thatv (t,x ) (x k K) c (x b (t )) ( t )). As t t, we get V ( t,x ) (x k K) c (x b >. This implies that V ( t,x ) > G (x ), which is contradiction to x > b (t ) d > b ( t). So the jump is not true and t b (t) is continuous in [,T). 7

19 4.3. EEP Representation of American Eagle Options with Balance Wings Standard Arguments based on the strong Markov property [] (Chapter III) derive the free boundary problem for unknown V (t,x) and unknown two boundaries b and b : (4.4) (4.4) (4.4) (4.43) (4.44) (4.45) (4.46) (4.47) (4.48) (4.49) V (t,x) + V (t,x) V t +L X V = rv in C, V (t,x) = G (x) = (L x l) for x = b (t), V (t,x) = G (x) = (x k K) for x = b (t), Vx (t,x) = for x = b (t) > l, = and + V (t,x) Vx (t,x) = = and V (t,x) V (t,x) > G (x) in C, V (t,x) = G (x) = L x l V (t,x) = G (x) = x k K for x = b (t) = l for x = b (t) < k, for x = b (t) = k in D, in D. Concluding from Theorem 5 to Theorem, the value function V and boundary function b and b have the following property (4.5) (4.5) (4.5) V (t,x) is continuous function on [,T] (, ), V is C, in C t V (t,x) is decreasing function with V (T,x) = G (x), b (t) is increasing on [,T], and continuous function with < b (t) < L, on [,T), and b (T ) = max(min(l, r δ L),l), (4.53) b (t) is decreasing on [,T], and continuous function with K < b (t) <, (4.54) on [,T), b (T ) = min(max(k, r δ K),k). From Theorem, we can find constant d such that Vxx (t,x) >, i.e. x Vxx (t,x) is convex for x [b (t),b (t)+d ]. Wealso candetermine constant d such that Vxx (t,x) >, i.e. x Vxx (t,x) isconvexfor x [b (t) d,b (t)]. Bythedefinitionof G (x), itis notdifficulttofindaconstant d suchthat x V (t,x) isconcavefor x [b (t) d,b (t)] and x [b (t),b (t)+d]. We can apply the extension of time space formula in [7, Remark 3.] on e rs V (t+s,x t+s ), and take the P t,x expectation from both sides. By the optional sampling theorem, the martingale term will be vanished. Finally, using the equation (4.4), (4.48), (4.49) and taking s = T t, we get the EEP representation of American eagle options V (t,x) = e r(t t) E t,x G (X T ) E t,x e ru (rk rk)i(x t+u > k)du E t,x e ru (rk δx t+u )I(k > X t+s > b (t+u))du 8

20 (4.55) E t,x e ru ( rl+rl)i(x t+u < l)du E t,x + E t,x E t,x e ru ( rl+δx t+u )I(l < X t+u < b (t+u))du e ru V x (t+u,k )I(b (t+u) = k)dl k u (X) e ru V x (t+u,l+)i(b (t+u) = l)dl l u (X). where e r(t t) E t,x G (X T ) isthevalueofeuropeaneagleoptionswhichisequalto V C (t,x,k)+ V P (t,x,l) V C (t,x,k) V P (t,x,l) and V C (t,x,k) means the value of American call options striking at K for time t and stock price x. The term l k u(x) is the local time for X at k, and l k u (X) = P lim ε u I(k ε<x ε t+r<k+ε)d X,X r. First of all, we would like discuss how to calculate the value of local time term. By the formula (.6) in [7], we have (4.56) = E t,x e ru V x (t+u,k )I(b (t+u) = k)dl k u (X) e ru V x (t+u,k )I(b (t+u) = k)de t,x l k u (X). We will try to estimate value of E t,x l k u (X). By the definition of local time, lk u (X) = P u lim ε I(k ε<x ε t+r<k + ε)d X,X u, so there exists a subsequence {ε n } such that l k u u(x) = lim n I(k ε ε n n<x t+r <k+ε n )d X,X u P a.s.. By the Dominated Convergent Theorem, we have (4.57) E t,x l k u (X) = E t,x = = = u u u u lim n lim n I(k ε n <X t+r <k+ε n )d X,X u ε n [ ] E I(k ε n <Xu x ε <k+ε n)σ Xu x du n lim n ε n εn+k ε n+k σ k f X x u (k)du. σ y f X x u (y)dydu In equation (4.57), the second equation given by the Fubini Theorem and Dominated Convergent Theorem, the third equation is based on the definition of expectation and the final equation used the definition of derivative. Since X follows Geometric Brownian motion defined in (.7), then the density function f X x u (k) is (4.58) f X x u (k) = n ( σ u (ln k σ (r δ x )u)) kσ u, where n( ) is the density function for standard normal distribution. So equation (4.56) can be written as E t,x e ru V x (t+u,k )I(b (t+u) = k)dl k u (X) 9

21 (4.59) = σk e ru Vx (t+u,k )n( σ u (ln k x Similarly, we can transform the last local time term in (4.55) into (4.6) = σl E t,x e ru V x (t+u,l+)i(b (t+u) = l)dl l u (X) (r δ σ )u)) u du. e ru Vx (t+u,l+)n( σ u (ln l σ (r δ x )u)) du. u The remained term on the left of equation (4.55) can be transformed following [4, Theorem 4] or [4, EEP Representation of American Strangle Options]. The EEP representation for American eagle options with balance wings is V (t,x) = e r(t t) E t,x G (X T ) (4.6) σk σl e ru (rk rk)φ(d (u,k,x))du e ru rkφ(d (u,b (t+u),x)) e rδ δxφ(d (u,b (t+u),x)+σ u)du e ru rkφ(d (u,k,x)) e rδ δxφ(d (u,k,x)+σ u)du e ru ( rl+rl)φ(d (u,b (t+u),x))du e ru rlφ(d (u,b (t+u),x))+e δu δxφ(d (u,b (t+u),x) σ u)du e ru rlφ(d (u,l,x))+e δu δxφ(d (u,l,x) σ u)du e ru V x (t+u,k )ϕ(d (u,k,x)) u du e ru V x (t+u,l+)ϕ(d (u,l,x)) u du where Φ( ) is the standard normal distribution function, and d (s,y,x) = σ s d (s,y,x) = σ s ( ln y x (r δ σ /)s ), ( ln y ) x (r δ σ /)s. Since P t,x (X t+u = k) = P t,x (X t+u = l) =, we can write the formula (4.55) into an equivalent form V (t,x) = Ee r(t t) G (X x T t) E e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u > b (t+u))du

22 (4.6) E e ru ((r δ)x x ug x (X x u+) rg (X x u))i(x x u < b (t+u))du + E e ru Vx (t+u,k )dl k u(x x ) E e ru V x (t+u,l+)dl l u(x x ), Which will simplify for writing the EEP representation for American eagle options. Let x = b (t) and x = b (t), we get the system that L b (t) l = e r(t t) EG (X b E E + (4.63) E e ru ((r δ)x b e ru ((r δ)x b (t) T t ) (t) u G x (Xb (t) u (t) u G x (Xb (t) u e ru V x (t+u,k )dl k u(x b +) rg (X b (t) u +) rg (X b (t) ) E (t) u ))I(X b (t) u > b (t+u))du ))I(X b (t) u < b (t+u))du e ru V x (t+u,l+)dl l u(x b (t) ) b (t) T t ) (t) k K = e r(t t) EG (X b E E + (4.64) E e ru ((r δ)x b e ru ((r δ)x b (t) u G (t) x (Xu b (t) u G x (Xb (t) u e ru V x (t+u,k )dl k u (Xb +) rg (X b (t) u +) rg (X b (t) ) E (t) u ))I(X b (t) u > b (t+u))du ))I(X b (t) u < b (t+u))du e ru Vx (t+u,l+)dl l u (Xb (t) ) From equation (4.63) and (4.64), we can solve out the value b (t) and b (t) for t [,T] The Uniqueness of Free Boundary The value of free boundary b and b are the solution of the system including equation (4.63) and (4.64). This section will try to show that the solution is unique. Since the equation is nonlinear, we will use the probability method to prove it. The structure of the proof follows from [, Theorem 5.3], but the detail is different. Theorem. The optimal stopping boundary (free boundary) of American eagle option (.5) can be characterized as the unique solution of the system including (4.63) and (4.64) in the class of continuous increasing function c : [,T) R satisfying l c (t) < L for t [,T) and c (T ) = max(min(l, r δ L),l) ; and continuous decreasing function c : [,T) R satisfying K < c (t) k for t [,T) and c (T ) = min(max(k, r δ K),k). Proof. The free-boundary b and b are the solution of (4.63) and (4.64) are showed in above. The aim of the proof remains to the the uniqueness. We suppose two functions c and c satisfying the condition mentioned in Theorem are the solution of (4.63) and (4.64). From the following steps, we will show c (t) and c (t) are the same as b (t) and b (t)

23 for t [,T], respectively. Since c and c are the solution of system including (4.63) and (4.64), substituting them into equation (4.6), we have a new function (4.65) U c (t,x) = e r(t t) EG (X x T t) E E e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u > c (t+u))du e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u < c (t+u))du + E e ru Ux c (t+u,k )dlk u (Xx ) E e ru Ux c (t+u,l+)dll u (Xx ), where Ux c(t,k ) and Uc x (t,l+) for t [,T]. We set that (4.66) M t+s = e rs U c (t+s,x t+s ) + e ru ((r δ)x t+u G x (X t+u+) rg (X t+u ))I(X t+u > c (t+u))du e ru ((r δ)x t+u G x (X t+u+) rg (X t+u ))I(X t+u < c (t+u))du e ru Vx (t+u,k )dl k u (X) E e ru V x (t+u,l+)dl l u (X) We will briefly introduce how to prove it is a martingale. By the definition of local time, we can find two sequences {ε n } and {ε n } such that s Mt+s n = e r(t t) E t+s,xt+s G (X T ) E t+s,xt+s F(t+s+u,X t+s+u )du + s E t+s,xt+s e r(s+u) Vx (t+s+u,k )I(k ε n < X t+s+u < k +ε n )σ X 4ε t+s+udu n s E t+s,xt+s e r(s+u) Vx (t+s+u,l+)i(k ε n < X t+s+u < k +ε n)σ Xt+s+udu (4.67) 4ε n + 4ε n 4ε n F(t+u,X t+u )du e ru V x (t+u,k )I(k ε n < X t+u < k +ε n )σ X t+udu e ru V x (t+u,l+)i(k ε n < X t+u < k +ε n)σ X t+udu, where F(t+u,X r+u ) = e ru ((r δ)x t+u G x (X t+u+) rg (X t+u ))I(X t+u > c (t+u))+ e ru ((r δ)x t+u G x (X t+u+) rg (X t+u ))I(X t+u < c (t+u)). As n, Mt+s n M t+s P a.s.. By [4, Fact 4 and Fact 5], we know that Mt+s n is a martingale i.e. (4.68) E[M n t+s F t+s ] = Mn t+s,

24 for s < s. We take lim n from both side in (4.68), the limit can go into the conditional expectation ( ) on the right hand side by the Dominated Convergent Theorem. So we prove that Mt+s is a martingale. s T t i) We want to prove that U c (t,x) = G (x) for x c (t) or x c (t). Since c (t) and c (t) are the solution of (4.63) and (4.64), so (4.69) (4.7) U c (t,c (t)) = L c (t) l = G (c (t)), U c (t,c (t)) = c (t) k K = G (c (t)), x < c (t) and set the stopping time σ c = inf{s [,T t] X x s c (t+s)}. By the optional sampling theorem and the martingale process (4.66), we get (4.7) U c (t,x) = E[e rσc U c (t+σ c,x x σ c )] E σc E σc e ru V x (t+u,l+)dl l u (Xx ) e ru ((r δ)x x u G x (Xx u +) rg (X x u ))du, and Vx (t + u,l+)dl l u(x x ) = dl l u(x x ) for u [,σ c ]. Applying the product rule and Itô Tanaka formula on e rs G (X t+s ), (4.7) e rs G (X t+s ) = G (X t ) e ru ( rl+δx t+u )I(l < X t+u < L)du e ru ( rl+rl)i(x t+u < l)du e ru (rk δx t+u )I(K < X t+u < k)du e ru (rk rk)i(k < X t+u < k)du e ru dl l u (X)+ e ru dl k u (X)+ e ru dl L u (X) e ru dl K u (X). Using σ c to replace s in formula (4.7) and taking the E t,x from both sides, we get (4.73) G (x) = E[e rσc G (Xσ x c )]+ σc E e ru dl l u (Xx ) E σc e ru ((r δ)x x u G x (Xx u +) rg (X x u ))du. Since E[e rσc G (X x σ c )] = E[e rσc U c (t + σ c,x x σ c )], equation (4.7) and (4.73) implies that U c (t,x) = G (x) for x < c (t). Similarly, we can prove that U c (t,x) = G (x) for x > c (t). ii) In this part, we want to prove that U c (t,x) V (t,x) for (t,x) [,T] [, ). If x c (t) or x c (t), we have U c (t,x) = G (x) V (t,x) showed in i). For 3

25 c (t) < x < c (t), we set the stopping time τ c = inf{s [,T t] Xs x c (t + s) or Xs x c (t+s)}. Since the process (4.66) is a martingale, through the optional sampling theorem, we get U c (t,x) = E[e rτc U c (t+τ c,xτ x c )] = E[e rτc G (Xτ x c )] V (t,x). iii) We want to prove that b (t) c (t) and b c (t) for t [,T]. Supposed that t [,T], such that c (t) < b (t). Take x c (t), set the stopping time τ b = inf{s [,T t] Xs x b (t+s)}. In the formula (4.6), using τ b to replace T t, we get (4.74) E[e rτ b V (t+τ b,x x τ b )] = V (t,x)+ E b + E b e ru V x (t+u,l+)dl l u (Xx ) e ru ((r δ)x x u G x (Xx u +) rg (X x u ))du. Applying the optional sampling theorem on the martingale (4.66), we get (4.75) E[e rτ b U c (t+τ b,xτ x b )] = U c (t,x)+ b E e ru Ux c (t+u,l+)dll u (Xx ) +E b e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u < c (t+u))du. Since E[e rτ b U c (t+τ b,xτ x b )] E[e rτ b V (t+τ b,xτ x b )], and V (t,x) = U c (t,x). For u [,τ b ), Vx (t+u,l+)dl l u(x x ) = dl l u(x x ) and Ux(t+u,l+) c. So formulae (4.74) and (4.75) imply that (4.76) E b e ru ((r δ)x x ug x (X x u+) rg (X x u))i(x x u c (t+u))du However, by the continuity of b (t) and c (t), the stopping time τ b >. Since Xu x b (t + u) max(min(l, r L),l), the integrand for the inequality (4.76) is strictly negative. δ So our assumptions is false and b (t) c (t) for t [,T]. Similarly, we can prove that b (t) c (t) for t [,T]. iv) This part will show that b (t) = c (t) for t [,T]. Supposed that t [,T), such that b (t) < c (t). Take x (b (t),c (t)) and set the stopping time τ b = inf{s [,T t] Xs x b (t+s) or Xs x b (t+s)}. Replacing T t in formula (4.6) by τ b, we get (4.77) E[e rτ b V (t+τ b,x x τ b )] = V (t,x). By the martingale defined in (4.66) and the optional sampling theorem, we get (4.78) E[e rτ b U c (t+τ b,xτ x b )] = U c (t,x)+ b E e ru Ux(t+u,l+)dl c l u(x x ) b E e ru Ux c (t+u,k )dlk u (Xx ) +E +E b b e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u < c (t+u))du e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u > c (t+u))du. 4

26 For b (t) c (t) and b (t) c (t) provediniii),so E[e rτ b U c (t+τ b,xτ x b )] = E[e rτ b V (t+ τ b,xτ x b )] and U c (t,x) V (t,x). Obviously, thelocaltimetermarenegative, sotheequation (4.77) and (4.78) show that (4.79) E +E b b e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u < c (t+u))du e ru ((r δ)x x u G x (Xx u +) rg (X x u ))I(Xx u > c (t+u))du. By the continuity of b (t) and c (t), the stopping time τ b >. The first term in (4.79) is strictly negative for Xu x < c (t+u) max(min(l, r L),l). The second term is strictly negative δ for Xu x > c (t+u) min(max(k, r K),k). so it is contradictory to formula (4.79). Then we δ can conclude that b (t) = c (t) for t [,T]. Similarly, we can prove that b (t) = c (t) for t [,T]. 5. American Disable Eagle Options For the disable eagle, we have k K L l. Without losing generality, the discussion in this section is based on the assumption k K > L l. The picture for the payoff function G (t,x) is G (x) l L K k Stock Price Figure 5. The figure shows the payoff for American disable eagle option for k K > L l. It is the eagle with bigger right wing than left wing and this eagle looks like disable. In this section, we will discuss the free-boundary and value function for the option. Since some proofs in this section is the same as the previous section (American eagle options with balance wings), we will just give the statement for these theorems. 5

27 5.. The Optimal Stopping Region and Continuation Region for Disable Eagle Options To solve the optimal stopping problem in (.5), we will define the continuation region and stopping same as previous section 4. (5.) (5.) C = {(t,x) [,T) (, ) V (t,x) > G (x)}, D = {(t,x) [,T] (, ) V (t,x) = G (x)}. The same analysis as section 4., the optimal stopping time for (.5) is (5.3) τ D = inf{ s T t X t+s D }. The stopping region D can be separated into two disjoint sets (5.4) (5.5) D = {(t,x) [,T] (, ) V (t,x) = (L x l) + }, D = {(t,x) [,T] (, ) V (t,x) = (x k K) + }. Since k K > L l, the value k K is the maximum payoff obtained from the disable eagle options so that {(t,x) x = k} D. However, some parts of {(t,x) x = l} may be inside the continuation region, which will be show in the following content. This will make the disable eagle options be different from the eagle options with balance wings. Theorem 3. This theorem have two parts: D is down connectedness: if (t,x) D is up connectedness: if (t,x) Proof. The proof is the same as Theorem. D, then (t,x ) D, then (t,x ) D as well for x < x ; D as well for x > x. Theorem 4. The continuation region and exercise region are nonempty: {(t,x) (,T) (,L) x l b ST (t) and b ST (t) l} (,L) L x > l b P (t)} C, D and {(t,x) (,T) {(t,x) (,T) (K, ) x k b ST (t)} k b C (t)} C, D and {(t,x) (,T) (K, ) K x < where b C and b P are the free-boundaryfor American call and put options. b ST and b ST is the lower free-boundary and the higher free-boundary for American strangle options, respectively. Proof. Wewill show that {(t,x) (,T) (,L) x l b ST (t) andbst (t) l} D, andthe remained part in Theorem 4 is the same as the proof of Theorem 3. Since b ST (t) l, we set x = b ST (t). V ST (t,x) = L x = G (x) V (t,x). By the definition of American strangle options and eagle options, we have V ST (t,x) V (t,x), so V ST (t,x) = V (t,x) = G (s) i.e. (t,x) D. For the down connectedness in D, we know that {(t,x) x l b ST (t) and b ST (t) l} D. 6

28 By Theorem 3 and 4, we know the exercise region D and D satisfies the down connectedness and up connectedness and C and D are nonempty. We can define the lower and upper free-boundary as (5.6) (5.7) b (t) = inf{x (, ) V (t,x) C }, b (t) = inf{x (, ) V (t,x) D }. Theorem 4 shows the range of the double free-boundaries are (5.8) (5.9) (5.) b (t) l b P (t) for t [,T), b (t) b ST (t) for b ST (t) l and t [,T) k b C (t) b (t) k b ST (t) for t [,T). From (5.8) and (5.9), we just know the range of b (t) for b ST (t) l. But for bst (t) < l, the range of lower bound b (t) is unknown. Before the mathematically proving the property for the free-boundary and value function of disable eagle options, we will numerically detect the the continuation region of this options. As we know from section 4., the eagle options satisfies the SDE and boundary conditions (5.) (5.) V t +L X V = rv in C, V (t,x) = G (t,x) in D. 3 t b ST (t) l Stock Price 9 t b ST (t) k Time T = Figure 6. The figure shows that continuation region of American disable eagle options with the parameter: l = 9, L =, K =, k = 3, r = δ =.6, σ =., T =. The bar region is the continuation region and the others is the optimal exercise region. The upper dash line is b ST (t) k and the lower dash line is bst (t) l. 7

29 As l approaches to L from below, some parts of l will be inside the continuation region shown in Figure 6. From the definition of C, the sufficient and necessary condition for (t,l) C is V (t,l) > G (l). To avoid from estimating the value V (t,l), we want to find a sufficient condition for (t,l) C. Theorem 5. If V ST (t,l,l,k) V ST (t,l,l,k) (L l) >, (t,l) is inside the continuation region C. Notation V ST (t,x,l,k) is the value of American strangle options at (t,x) with lower strike price L and upper strike price K. Proof. Applied the product rule and Itô Tanaka formula on e rs G (X t+s ), (5.3) e rs G (X t+s ) = G (X t ) e ru ( rl+δx t+u )I(l < X t+u < L)du e ru ( rl+rl)i(x t+u < l)du e ru (rk δx t+u )I(K < X t+u < k)du e ru (rk rk)i(k < X t+u )du e ru dl l u (X)+ e ru dl k u (X)+ e ru dl L u (X) e ru dl K u (X) Since I(l < X t+u < L) = I(X t+u < L) I(X t+u l) and I(K < X t+u < k) = I(X t+u < k) I(X t+u K), we can sperate the second term and fourth term on the right hand side of equation (5.3). Additionally, plus (l K) + (L l) +, (l k) + (l l) + and minus (L l) on the right hand side of equation (5.3). Take E t,l from both side, we get that (5.4) E t,l e rs G (X t+s ) = G (l) (L l) + (l K) + (L l) + E t,l e ru ( rl+δx t+u )I(X t+u < L)du + E t,l e ru (rk δx t+u )I(K < X t+u )du + E t,l e ru dl L u (X)+ E t,l e ru dl K u (X) (l k) + (l l) + E t,l e ru ( rl+δx t+u )I(X t+u < l)du E t,l e ru (rk δx t+u )I(k < X t+u )du E t,l e ru dl l u(x) E t,l e ru dl k u(x). By the product rule and Itô Tanaka formula on the discounted payoff function of American strangle option e rs G ST (x), we get the formula (5.4). Seting τ is the optimal stopping time 8