Multiple Optimal Stopping Problems and Lookback Options

Transcription

1 Multiple Optimal Stopping Problems and Lookback Options Yue Kuen KWOK Department of Mathematics Hong Kong University of Science & Technology Hong Kong, China web page: maykwok/ Joint work with Min DAI

2 Financial derivatives/instruments with multiple exercise rights: Employee reload options Fund guarantees in equity-linked insurance contracts Strike and maturity reset rights in options Exercisable at all times Finite number of exercise rights within a specified period. Refracting period (time vesting requirement) minimum length of time interval between two successive exercises.

3 Optimal stopping formulation Reward function: f(x) = (K x) + for an American put. [ P(S, t) = sup E e ] τ t r ds f(xτ t,x ) τ S(t,T) where S(t, T) is the set of stopping times taking values in [t, T]. Obviously P(S, t) f(x). The supremum is reached at the optimal stopping time τ = inf s {t s T, P(s, X s) = f(x s )}, the first time that the price of the derivative drops down to the reward function.

4 American put Reward function = (X S)+

5 Linear Complementarity Formulation The price of American put satisfies the following linear complementarity formulation: P t + σ2 2 S2 2 P S 2 + rs P S rp 0 P X S (i) (ii) [ ] P t + σ2 P 2 S2 2 S 2 + rs P S rp [P (X S)] = 0 (iii) to be solved in {[S, t) : S > 0,0 < t < T } and P(S, T) = (X S) +.

6 Sketch the proof: For any stopping time ξ, t < ξ < T, by Ito s calculus, ξ ( e rξ P(S ξ, ξ) = e rt P(S t, t) + e ru t t + σ2 2 S2 2 S 2 + rs S r P(S u, u) du τ + e ru P σs u t S (S u, u) dws. The integrand of the Riemann integral is non-positive and the expectation of the martingale term is zero, so we have E [e r(ξ t) P(S ξ, ξ)] P(S t, t). ) P For s between t and T, t + σ2 P 2 S2 2 S 2 + rs P rp = 0. By the optional S sampling theorem, if ξ = ξ, ξ is the optimal stopping time, then P(S ξ ξ ) = E [e r(ξ t) (X S ξ )] so we have equality.

7 Options with n-exercise rights V (n) (X 0 ) = sup E (τ 1,,τ n ) S (n) n i=1 e rτ if i (X τi ) where S (n) = {(τ 1,, τ n ); τ i τ i 1 δ for all i = 1,2,3,, n}. Here, δ is the refracting period, τ i is the optimal stopping time for the i th exercise right.

8 Examples of instruments with reset feature 1. S & P index bear market put warrants with a 3-month reset (traded in CBOE and NYSE) original exercise price of the warrant, X = closing index level on issue date exercise price is reset at the closing index level S t on the reset date if S t > X (automatic reset) Reset-strike warrants are available in Hong Kong and Taiwan markets

9 2. Reset feature in Japanese convertible bonds reset downward on the conversion price Sumitomo Banks 0.75% due 2001 Issue date May 96 First reset date 31 May 1997 Annual reset date thereafter 31 May Reset calculation period 20 business day period, excluding holidays in Japan, ending 15 trading days before the reset day Calculation type Simple average over calculation period

10 3. Corporate debts strong incentive for debt holders to extend the maturity of a defaulting debt if there are high liquidation costs 4. Canadian segregated fund Guarantee on the return of the fund (protective floor); guarantee level is simply the strike price of the embedded put. Two opportunities to reset per year (at any time in the year) for 10 years. Multiple resets may involve sequentially reduced guarantee levels. Resets may require certain fees.

11 Shout call options and reset strike options Single shouting/reset right c shout,1 (S, T) = max(s T X, S t X,0) where S t is the prevailing stock price at the shouting instant t. p reset,1 (S, T) = { max(s t S T,0) if reset has been exercised max(x S T,0) if no reset has occurred. Multiple shouting/reset rights Let Sl denote the asset price at the last shouting chosen by the holder, 0 l n. c shout,n (S, T) = max(s T X, Sl { X,0) max(s p reset,n (S, T) = l S T,0) if l th reset has occurred, 1 l n, max(x S T,0) if no reset has occurred. Put-call parity relation c shout,n (S, t) = p reset,n (S, t) + Se q(t t) Xe r(t t).

12 n-reset put options Let n denote the maximum number of resets allowed, X be the strike price set at initiation, t j be the optimal stopping time of the jth reset S j denote the critical stock price at t j. Monotonicity property: X = S < S j+1 < S j V n (S, τ; X) denotes the value of the n-reset put option, τ = T t V j (S, τ; S j+1 ) = sup t j S(j) t,t E [e r(t j t) V j 1 (S t j, T t j ; S t j ) S t = S ] j = n, n 1,,1, where S (j) t,t is the set of stopping times between t and T associated with the jth reset right, j = 1,, n.,

13 Upon the j th reset, the reset put becomes an-at-the-money (j 1) th -reset put reward function : V j 1 (S, τ, S) = SP n (τ). Linear complementarity formulation V n τ σ2 V n 2 S2 2 S 2 (r q)s V n S + rv n 0, V n (S, τ) SP n (τ), [ ] Vn τ σ2 V n 2 S2 2 S 2 (r q)s V n S + rv n [V n SP n (τ)] = 0. V n (S,0) = max(x S,0). In the stopping region, V n (S, τ) = SP n (τ). [ τ σ2 2 2 S2 (r q)s S2 S + r A necessary condition for optimal shouting is given by ] d dτ [eqτ P n (τ)] > 0. SP n (τ) = Se qτ d dτ [eqτ P n (τ)].

14 Properties of P n (τ) (i) If r q, then d dτ [eqτ P n (τ)] > 0 for τ (0, ). (ii) If r > q, there exists a unique critical value τ n and d dτ [eqτ P n (τ)] τ=τ n = 0, (0, ) such that d dτ [eqτ P n (τ)] > 0 for τ (0, τn ), d dτ [eqτ P n (τ)] < 0 for τ (τn, ). In addition, we have τ n < τ n+1 and lim n τ n =.

15 7 6 n= 5 4 n=4 e qτ P n (τ) 3 n=3 2 n=2 1 n= τ Plot of e qτ P n (τ) against τ, r < q.

16 n= e qτ P n (τ) n= n=3 0.1 n= n= τ Plot of e qτ q n (τ) against τ, r > q. Maximum values of P n (τ), n = 1,2,3 are attained at τ1 = 5.71, τ 2 = 9.55 and τ 3 = 13.0, respectively.

17 Perpetual n-reset put options We write W n (S, τ) = e rτ V n (S, τ), and let Wn (S) denote the limit of W n(s, τ) as τ. The governing equation for Wn (S) takes the form σ 2 W 2 S2d2 n ds 2 + (r q)sdw n ds = 0, 0 < S < Sn n,, with auxiliary conditions W n (S n, ) = β ns n, and dw n ds (S n, ) = β n, where β n = lim r e rτ P n (τ). When n = 1, it can be shown easily that In general, we have β 1 = lim r e rτ P E (1, τ;1) = 1. β n = lim τ P n (τ) = lim τ W n 1 (1, τ;1) = W n 1 (1;1).

18 ( Sn, = ) X. α β n The recurrence relation for β n is deduced to be β n = W n 1 (1;1) = 1 + α α (1 + α) 1+αβ1+α n 1. The monotonic relation β n > β n 1 leads to the monotonic property S n 1, > S n,. Taking the limit n in the recurrence relation for β n, we obtain lim n β n = α. Correspondingly, this implies lim n S n, = X.

19 Behaviors of optimal reset policies r < q 1. The optimal shouting boundary exists for all times. 2. S n+1 (τ) < S n (τ) The holder should choose to shout at a higher critical stock price with less allowable reset rights outstanding. 3. The shouting boundary starts at X, Sn (0+ ) = X, and Sn (τ) is an increasing function of τ with a finite asymptotic limit at τ. Sn+1 ( ) < S n ( ), n = 1,2,.

20 S 1 * 1 shout 1.4 critical asset price 1.3 S 2 * S 3 * 2 shout shout time to expiry Plot of Sn (τ) against τ, n = 1,2,3; r < q. S n (τ) exists for all values of τ.

21 Behaviors of optimal reset policies r > q 1. Sn (τ) retains the monotonic properties in both n and τ. 2. S n (τ) always starts at S n (0+ ) = X. 3. S n (τ) is defined only for τ [0, τ n ), where τ n is the unique solution to d dτ [eqτ P n (τ)] = 0. That is, it is never optimal to shout the n-reset put whenever τ > τ n.

22 shout critical asset price shout 2 shout τ 1 τ τ time to expiry Plot of S n (τ) against τ, n = 1,2,3; r > q. S n (τ) exists only for τ < τ n (τ).

23 Infinite number of shout/reset rights Let S (τ) denote the critical stock price for infinite-reset put. 1. S (τ) is a non-decreasing function of τ, S (τ) = lim n S n ( ) = X. 2. S (τ) X. We then have S (τ) = X for all τ 0. The holder should exercise the infinite-reset put whenever the option becomes in-the-money.

24 Infinite-reset rights and lookback options Shout call and reset put share the same optimal shout/reset policies. The infinite-shout call becomes the fixed strike lookback call with terminal payoff = max(m T X,0), where M T = realized maximum stock value = max 0 u T S(u). The infinite-reset put becomes a lookback option with terminal payoff max(m T S T, X S T ). When X = 0, the terminal payoff becomes a floating strike lookback call option.

25 Closed form solution for n-reset shout floor Let R n (S, τ) denote the price function of the n-reset shout floor. (i) If r q, (ii) If r > q, R n (S, τ) = SP n (τ) τ (0, ). (11a) where τ n R n (S, τ) = { SPn (τ) τ (0, τ n ] e q(τ τ n) SP n (τ n ), τ (τ n, ), is the unique solution to d dτ [eqτ P n (τ)] = 0. (11b) Optimal shouting policies of shouting floors. 1. r q. Since we have R n (S, τ) = SP n (τ) for all values of τ, the first shouting right will be utilized at once at any asset price level. 2. r > q. It will not be shouted at any asset price when τ > τ n. However, it will be shouted at once at any asset price level whenever τ τ n.

26 Since there is no strike price appearing in the linear complementarity formulation, one deduces that R n (S, τ) is linearly homogeneous in S so that R n (S, τ) = Sg n (τ). { d dτ [eqτ g n (τ)] d dτ [eqτ g n (τ)] 0, } g n (τ) P n (τ), [g n (τ) P n (τ)] = 0 and g n (0) = 0. Within the time interval where d dτ [eqτ P n (τ)] 0, the equations are automatically satisfied by g n (τ) = P n (τ). However, at those times where d dτ [eqτ P n (τ)] < 0, g n (τ) must satisfy d dτ [eqτ g n (τ)] = 0. (i) r q. Since d dτ [eqτ P n (τ)] is strictly positive for all τ > 0 and P n (0) = 0, we then have g n (τ) = P n (τ), τ (0, ). (ii) r > q. For τ (0, τ n], we deduce similarly that g n (τ) = P n (τ). However, when τ > τ n, we have eqτ g n (τ) = e qτ np n (τ n ), so that g n(τ) = e q(τ τ n) P n (τ n ) for τ (τ n, ).

27 Nature of employee stock options as employee compensation package Non-transferable; may be exercised early (say, when the employee plans to leave) while an unconstrained investor ordinarily would sell the option. No short selling of firm s stock for hedging the risk of holding the option. Other embedded features, like reload and (time and/or performance) vesting requirement. Distinguish between value to holder and cost to option grantee. Market value of option Upper bound on the cost to option grantee.

28 How reloads work? Example X original = $100, S ξ = $150; $100/$150 = 2/3 units of owned share tendered to pay as strike upon exercise. Under the reload provision, the holder will be granted 2/3 unit of new option, with the same expiration date as the original option and the strike price set at $150. Remark The upper bound on a reload option is the underlying stock price, no matter how many reloads are possible and how long is the maturity.

29 Upper bound on the value of a reload option On the first exercise (at S 1 ), the holder receives 1 X S 1 shares and X S 1 new reload option with strike S 1. On the second exercise (at S 2 ), the employee nets an additional X ( 1 S 1 S 1 shares for a total of 1 X + X ( 1 S ) 1 = 1 X ( X shares and S 1 S 1 S 2 S 2 S 1 S 2 = X S 2 new reload options with strike S 2. ) S ) ( 2 ) S1 After the i th exercise (at S i ), the employee will hold 1 X S i shares and X S i new options with strike S i. Note that the value of the reload option is further reduced since the employee will not receive the early dividends.

30 Upon exercising, the employee receives one unit of stock and pays the strike X. Assumed that the employee uses X/S units of owned stock for the strike payment so that number of units of new stock received = 1 X S. The employee is assumed to keep these new stocks so that continuous dividend yields will be received. The employee receives X/S units of new call option with strike price set at the prevailing stock price at the exercise moment and same maturity date T.

31 Value of the reload option = S X + X c(s, τ; S, r, q). S By the linear homogeneity property of the call price function with respect to S, we obtain where c(s, τ; S, r, q) = Sĉ(τ; r, q), ĉ(τ; r, q) = e qτ N( d 1 ) e rτ N( d 2 ) and Similarly, d 1 = r q + σ2 2 σ τ and d 2 = r q σ2 2 σ τ. p(s, τ; S, r, q) = S p(τ; r, q) where p(τ; r, q) = e rτ N( d 2 ) e qτ N( d 1 ).

32 Linear complementarity formulation for option pricing The payoff function upon exercise of the single-reload option is S X + Xĉ(τ; r, q). V 1 τ L r,qv 1 0, V 1 (S, τ) S X + Xĉ(τ), [ ] V1 τ L r,qv 1 {V 1 (S, τ) [S X + Xĉ(τ)]} = 0, S (0, ), τ (0, T], where the differential operator L r,q is given by L r,q = σ2 2 S2 + (r q)s 2 S2 S V 1 (S,0) = (S X) +. r, r > 0 and q 0.

33 Zero dividend yield The price function of a single-reload option can be expressed in terms of the price functions of a forward contract and a shout call option. By defining U 1 (S, τ; X, r,0) = V 1 (S, τ; X, r,0) (S Xe rτ ) and observing the put-call parity Xĉ(τ; r,0) (X Xe rτ ) = X p(τ; r,0), one can show that U 1 (S, τ; X, r,0) is governed by U 1 τ L r,0u 1 0, U 1 (S, τ) X p(τ; r,0) [ ] U1 τ L r,0u 1 [U 1 (S, τ) X p(τ; r,0)] = 0, S (0, ), τ (0, T], U 1 (S,0) = (X S) +.

34 Suppose we use S as the numeraire, and accordingly, we define x = X S and W 1 (x, τ) = 1 S U 1(S, τ). By observing the put-call symmetry relation: p(τ; r,0) = ĉ(τ;0, r), then W 1 (x, τ) observes the following linear complementarity formulation W 1 τ L 0,rW 1 0, W 1 (x, τ) xĉ(τ;0, r), [ ] W1 τ L 0,rW 1 [W 1 (x, τ) xĉ(τ;0, r)] = 0, x (0, ), τ (0, T], W 1 (x,0) = (x 1) +.

35 The payoff upon exercise is xĉ(τ;0, r) = c(x, τ; x,0, r), which is a call option with strike price set at the prevailing stock price. Note that W 1 (x, τ) is the price function of the one-shout call option with zero interest rate, unit strike price, dividend yield r and strike price reset to the prevailing stock price upon shouting. In summary, we obtain V 1 (S, τ; X, r,0) = S Xe rτ + c shout,1 (X, τ; S,0, r), where c shout,1 (S, τ; X, r, q) is the price function of the one-shout call option.

36 Multi-reload options When the stock pays no dividend, the price function of a n-reload option can be expressed as the sum of the price functions of a forward contract and a n-shout call option. From financial intuition, one would expect that the holder will exercise their reload right at a lower critical stock price when there are more reload rights outstanding. For q > 0, one can derive the recursive relation that relates the critical stock prices for perpetual multi-reload options.

37 Linear complementarity formulation V n τ L r,qv n 0, V n (S, τ; X) S X + XV n 1 (1, τ;1) ( ) Vn τ L r,qv n {V n (S, τ; X) [S X + XV n 1 (1, τ;1)]} = 0 S (0, ), τ (0, T], V n (S,0) = (S X) +. The formulation is valid for n 1; and for notational convenience, we assume V 0 (S, τ; X, r, q) to be c(s, τ; X, r, q). From financial intuition, it is obvious that V n+1 (S, τ; X) > V n (S, τ; X) for all S > 0 and τ > 0.

38 Relation with n-shout call option When q = 0, we apply the transformation to obtain W n (x, τ) = 1 S [V n(s, τ; X, r,0) (S Xe rτ )] and x = X S W n τ L 0,rW n 0, W n (x, τ) xw n 1 (1, τ), [ ] Wn τ L 0,rW n [W n (x, τ) xw n 1 (1, τ)] = 0, x (0, ), τ (0, T], W n (x,0) = (x 1) +. Like the single-reload option, we can establish V n (S, τ; X, r,0) = S Xe rτ + c shout,n (X, τ; S,0, r), where c shout,n (S, τ; X, r, q) is the price function of a n-shout call option.

39 Properties of the critical stock price Sn (τ; r, q) The holder should never exercise at S < X and Sn (τ; r, q) starts at X as τ 0 +. Monotonic property: Sn+1 (τ; r, q) < S n (τ; r, q), an obvious fact from financial intuition. When q = 0, the optimal exercise policy of a n-reload option can be related directly with that of the n-shout call counterpart. When q > 0, we obtain the recursive relation: S n ( ) X = µ + µ µ + 1 [ S ] n 1 ( ) 1 µ+, n > 1. X Sn ( ) is monotonically decreasing with respect to n and lim n S n ( ) = X.

40 Employee stock options with infinite reloads Consider the function τ F(S, τ) = Se qτ Xe rτ + q 0 (Se qu Xe ru ) du, which satisfies the equation We define the transformation F τ L r,qf = q(s X) F(S,0) = S X. U (S, τ; X, r, q) = V (S, τ; X, r, q) F(S, τ), then U (S, τ; X, r, q) satisfies linear complementarity formulation U τ L r,qu q(x S), [ U τ L r,qu q(x S) S (0, ), U (S,0) = (X S) +. ] U (S, τ; X) XU (1, τ;1), [U (S, τ; X) XU (1, τ;1)] = 0 τ (0, T],

41 We define W (x, τ) = 1 S U (S, τ) and x = X S. The corresponding governing equation for W (x, τ) can be expressed as W L q,r W q(x 1) W (x, τ) xw (1, τ) [ τ ] W L q,r W q(x 1) [W (x, τ) xw (1, τ)] = 0, τ x (0, ), τ (0, T], W (x,0) = (x 1) +.

42 Let c float (S, m, τ; r, q) denote the price function of a floating strike lookback call option with terminal payoff S m, where m is the realized minimum value of the stock price over the life of the option. Interestingly, W (x, τ) is related to c float (S, m, τ; r, q) through the following relation: W (x, τ) = c float (X,min(S, X), τ; q, r) + q τ 0 c float(x,min(s, X), u; q, r) du. V (S, τ; X, r, q) = SW ( X S, τ ) + F(S, τ) = c float (S,min(S, X), τ; q, r) + (Se qτ Xe rτ ) + q τ 0 [c float(s,min(s, X), u; q, r) + (Se qu Xe ru )] du.

43 Proof of the price formula for infinite-reload option We take the limit n and define the infinite-shout call by c shout (S, τ; X, r, q) = lim n c shout,n(s, τ; X, r, q). The governing equation for c shout (S, τ) is given by c shout L r,q c shout τ = 0, S > S shout, (τ), τ > 0, c shout (S shout, (τ), τ) = S shout, (τ)c shout (1, τ;1), c shout S (S shout, (τ), τ) = c shout (1, τ;1), c shout (S,0) = (S X)+.

44 Exercise policy of infinite-shout calls The infinite-shout call has a simple exercise policy: the holder shouts when the option becomes at-the-money. We then have Sshout, (τ) = X for all τ > 0. Since the exercise boundary is known, the pricing model is no longer a free boundary value problem. Remark Mathematically, the free boundary value problem associated with the n-reload option reduces to a problem with fixed domain as n. However, the dimension of the problem increases by one.

45 Pricing model of lookback call option We consider the following pricing model of a floating strike lookback call option, whose price function is represented by c float (S, m, τ; r, q): c float L r,q c float = 0, S > m, τ > 0, τ c float m S=m = 0, c float (S, m,0) = S m.

46 We apply the transformations c float (y, τ) = c float(s, m, τ), y = S m m, so that the governing equations of the pricing model of the lookback option can be transformed into c float τ c float L r,q c float = 0, y > 1, τ > 0, y y=1 = c float (1, τ), c float (y,0) = y 1.

47 Comparing the pricing formulations of the infinite-shout call and the floating strike lookback call option, we then conclude that (a) when S > X, c shout (S, τ; X) = Xc shout ( ) ( ) S SX X, τ;1 = Xc float, τ = c float (S, X, τ). (b) when S X, c shout (S, τ; X) = Sc shout (1, τ;1) = Sc float(1, τ) = c float (S, S, τ). Combining the above results, we obtain c shout (S, τ; X) = c float(s,min(x, S), τ).

48 We rewrite the pricing formulation of W (x, τ) into the following alternative form: W L q,r W = q(x 1), x > 1, τ > 0 τ W x W x=1 = 0, W (x,0) = x 1. Comparing with the formulation of c float (y, τ), we then deduce that W (x, τ) = c float (S,min(S, X), τ; q, r) + q c float(s,min(s, X), u; q, r) du. 0 Lastly, the price function of the infinite-reload option is given by τ V (S, τ; X, r, q) = SW ( X S, τ ) + F(S, τ) = c float (S,min(S, X), τ; q, r) + (Se qτ Xe rτ ) + q τ 0 [c float(s,min(s, X), u; q, r) + (Se qu Xe ru )] du.

49 S q N = 500 N = 1000 N = 2000 Extrapolation Analytic formula Option values for the infinite-reload options obtained using binomial calculations and analytic formulas.

50 8 7 6 three reload critical stock price one reload two reload time to expiry Plots of critical stock price against time to expiry ( for employee ) options with one, two and three reloads q = 0, r > σ2. 2 The parameter values used in the calculations are: q = 0, r = 0.1, σ = 0.3 and X = 1. The critical stock price S n (τ; r, q) is defined only for τ < τ n, n = 1,2,3. These threshold values on time to expiry are found to be: τ 1 = 6.78, τ 2 = and τ 3 = 17.86; and they observe the monotonic property: τ 1 < τ 2 < τ 3.

51 30 25 critical stock price one reload 10 two reload 5 three reload time to expiry Plots of critical stock price against time to expiry ( for employee ) options with one, two and three reloads q = 0, r σ2. 2 The parameter values used in the calculations are: q = 0, r = 0.04, σ = 0.3 and X = 1. Apparently, the critical stock price is defined for all values of time to expiry and there is no asymptotic value for the critical stock price at infinite time to expiry.

52 Stochastic movement of the primary fund S 2 (t) and upgraded fund S 1 (t) A primary fund with value process S2 t, which is protected with reference to another (guaranteed) fund with value process S1 t. The holder has the right to reset the value of the fund to that of the guaranteed fund upon exercising the reset rights.

53 n(t) = number of units at time t F(t) = value of the profected fund

54 Pricing model of the perpetual protection fund Let V n (S 1, S 2 ) denote the value of the perpetual protection fund with n reset rights and withdrawal right. We take advantage of the linear homogeneity property of V n (S 1, S 2 ) and define W n (x) = V n(s 1, S 2 ), x = S 1. S 2 S 2 This corresponds to the choice of S 2 as the numeraire. In the continuation region, the governing equation for W n (x) takes the form σ 2 2 x2d2 W n dx 2 + (q 1 q 2 )x dw n dx q 2W n = 0, x w n < x < x r n, where σ 2 = σ1 2 2ρσ 1σ 2 + σ2 2, xw m and xr n are the threshold values for x at which the holder should optimally withdraw and reset, respectively.

55 Boundary conditions at withdrawal and reset thresholds W n (x w n ) = 1 and W n (xw n ) = 0, W n (x r n ) = xr n W n 1(1) and W n (xr n ) = W n 1(1). Upon withdrawal, V n (S 1, S 2 ) becomes S 2 and so we have W n (x w n) = 1. When the holder resets at x = x r n, the option writer has to supply enough funding to increase the number of units of the primary fund such that the new fund value equals S 1. The corresponding number of units equals x r n, which is the ratio of the fund values at the reset threshold x r n. Subsequently, the protection fund has one reset right less and x becomes 1 since the values of the new upgraded fund and guaranteed fund are identical upon reset.

56 Limiting case: infinite number of reloads Consider the limiting case n, we have W (x r ) = xr W (1). The equation is seen to be satisfied by x r = 1. This represents immediate reset whenever S2 t falls to St 1, given that the holder has infinite number of reset rights. The corresponding derivative condition becomes W (1) = W (1). This is because the value of V (S 1, S 2 ) is unaffected by marginal changes in S 2 when S 2 is close to S 1. number of units, n(t) = max ( 1, max 0 u t ( S1 (u) S 2 (u) )) ; F(t) = n(t)s 2 (t).

57 Price function: W (x) σ 2 2 x2dw2 dx 2 + (q 1 q 2 )x dw dx q 2W = 0, x w < x < 1, subject to the auxiliary conditions: W (x w ) = 1 and W (xw ) = 0, W (1) = W (1). The solution to W (x) is easily seen to be where W (x) = h(x) h(x w ), xw < x < 1, h(x) = (θ 2 1)x θ 1 (θ 1 1)x θ 2, x > 0.

58 The boundary condition W (x w ) = 1 is satisfied by the inclusion of the multiplicative factor 1/h(x w ) in W (x). The optimality condition, W (xw ) = 0, gives the following algebraic equation for x w : h (x w ) = θ 1(θ 2 1)(x w )θ 1 θ 2 (θ 1 1)(x w )θ 2 = 0. In summary, W (x) = h(x) h(x w ), xw < x < 1.