AMERICAN OPTIONS REVIEW OF STOPPING TIMES. Important example: the first passage time for continuous process X:

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1 AMERICAN OPTIONS REVIEW OF STOPPING TIMES τ is stopping time if {τ t} F t for all t Important example: the first passage time for continuous process X: τ m = min{t 0 : X(t) = m} (τ m = if X(t) never takes the value m This is a stopping time (see Sect 8.2 of Shreve) Optional sampling theorem: Let τ be a stopping time. If (M t ) is a martingale (submartingale, supermartingele), then (M τ t ) is a martingale (submartingale, supermartingele) Winter Per A. Mykland

2 THE PERPETUAL AMERICAN PUT Process: ds t = rs t dt + σs t dw t T = all stopping times that take value in [0, ] Value of perpetual put: v (x) = max τ T E [e rτ (K S τ ) + I {τ< } S 0 = x] = max τ T E [e rτ (K S τ )I {τ< } S 0 = x] The last equality: no point in exercising if S t > K Reasonable to exercise when S t hits some level L What is L? What is the price v (x)? Winter Per A. Mykland

3 INTERMEDIATE PROBLEM Hitting time: τ L = min{t 0 : S(t) = L} v L (x) = E [e rτ L (K S τl )I {τl < } S 0 = x] Program: find v L (x), then find value L which maximizes v L (x) This is simpler, since v L (x) does not involve any maximization over stopping times Since S τl = L on {τ L < }: v L (x) = E [e rτ L (K L)I {τl < } S 0 = x] which depends only on distribution of τ L One derivation for this price: Shreve Section (more elegant) We shall use: Winter Lecture 1 (brute force) Winter Per A. Mykland

4 Set X t = log S t log S 0 and ν = r 1 2 σ2 Then X t = νt + σw t Hitting time τ = min{t : X t = b} From Lecture 1 (p. 16): density of τ: f τ (t) = Since S 0 = x: ( b 2πσ2 t exp b2 3 2σ 2 t + ν σ 2 b 1 2 ν 2 ) σ 2 t S τl = L <=> X τl = log(l) log(x) And so τ = τ L if b = log(l) log(x) = log(x/l) Winter Per A. Mykland

5 For x L: BACK TO ORIGINAL PROBLEM v L (x) = E [e rτ L (K L)I {τl < } S 0 = x] = e rt (K L)f(t)dt 0 ( ) 2rb = (K L) exp = (K L) σ 2 ( x L) 2r σ 2 Obviously, for x L: v L (x) = (K x) (exercise immediately) Winter Per A. Mykland

6 OPTIMAL VALUE OF L { 2r v L (x) = (K L)L σ 2 x 2r σ 2 for x L (K x) otherwise For fixed x: maximize with respect to L: where 2r (K L)L σ L 2 = L 2r σ 2 = 2r (K L)L σ L 2 L = 2r + σ2 σ 2 + 2r 2r (K L)L σ σ2 2 1 L 2r σ 2 + 2r 2r KL σ σ2 2 1 = 0 <=> L = L 2r 2r + σ 2 K This corresponds to a maximum of (K L)L 2r σ 2 since only stationary point and since (K L)L 2r σ 2 = 0 for L = 0 and (K L)L 2r σ 2 as L. Winter Per A. Mykland

7 ANALYTIC CHARACTERIZATION OF PUT PRICE v L (x) = { (K L )(x/l ) 2r σ 2 (K x) otherwise for x L Hence v L (x) = { (K L ) 2r σ 2 x (x/l ) 2r σ 2 1 otherwise for x L Right derivative at L : v L (L +) = (K L ) 2r σ 2 L = 1 v L (x) is continuous at L : smooth pasting Winter Per A. Mykland

8 CAN VERIFY DIRECTLY THAT (i) v L (x) (K x) + for all x 0 (ii) rv L (x) rxv L (x) 1 2 σ2 x 2 v L (x) 0 for all x 0 (iii) for each x 0, one of (i) or (ii) is an equality (i)-(iii) (complementarity conditions) determine v L (x) TRADING INTERPRETATION d[e rt v L (S t )] = e rt [ rv L (S t )dt + v L (S t )ds t v L (S t )d[s, S] t ] = v L (S t )d S t d D t where, since d S t = d[e rt S t ] = re rt S t dt + e rt ds t : d D t = e rt [ rv L (S t )dt rv L (S t )S t 1 2 v L (S t )d[s, S] t ] = e [rv rt L (S t )dt rv L (S t )S t dt 1 ] 2 v L (S t )σ 2 St 2 dt 0 by (ii) (ii) (i) + (ii) means that v L (S t ) is a superreplication of the American option Winter Per A. Mykland

9 STRUCTURE OF THE DIVIDEND Precise form of (ii) (for all x 0 rv L (x) rxv L (x) 1 2 σ2 x 2 v L (x) = { 0 if x > L rk if x < L Hence d D t = e rt [rv L (S t ) rv L (S t )S t 1 2 v L (S t )σ 2 S 2 t ] dt = e rt rki {St <L }dt FINANCIAL INTERPRETATION The hedging strategy pays a dividend of rk $ for when S t < L. This is arbitrage profit if the owner of the option does not exercise at time τ L PROBABILISTIC INTERPRETATION M t = e rt v L (S t ) is a supermartingale with Doob-Meyer decomposition v L (S t )d S t d D t M τl t is a martingale Winter Per A. Mykland

10 FINALLY: v L (x) = v (x) Proof: Since e rt v L (S t ) is a supermartingale: For any stopping time τ T and so v L (x) = v L (S 0 ) E [e r(τ t) v L (S τ t )] E [e rτ v L (S τ )] (as t ) v L (x) max τ T E [e rτ v L (S τ )] = v (x) On the other hand, since τ L T The equality follows v (x) = max τ T E [e rτ v L (S τ )] E [e rτ L vl (S τl )] = v L (x) Winter Per A. Mykland

11 THE REGULAR AMERICAN PUT Exercise time τ must be T v(t, x) = max τ T t,t E [e r(τ t) (K S τ ) + S t = x] where T t,t is the set of all stopping times taking values in [t, T ] Winter Per A. Mykland

12 ANALYTIC CRITERIA FROM TRADING INTERPRETATION Solvency requires: v(t, S t ) (K S t ) +, or (i) v(x, t) (K x) + Replication considerations. Ito s formula: d[e rt v(t, S t )] = v x (t, S t )d S t d D t Where d D t = e [rv(t, rt S t ) v t (t, S t ) rs t v x (t, S t ) 1 ] 2 σ2 St 2 v xx (t, S t ) dt Superreplication requires: d D t 0, or (ii) rv(t, x) v t (t, x) rxv x (t, x) 1 2 σ2 x 2 v xx (t, x) 0 Getting the lowest price: (iii) for each x 0, one of (i) or (ii) is an equality (otherwise one could lower v(x,t) and still have a solvent superreplication) (i)-(iii): the complementarity conditions again Winter Per A. Mykland

13 STOPPING AND CONTINUATION REGIONS Stopping region: S = {(t, x) : v(t, x) = (K x) + } Continuation region: C = {(t, x) : v(t, x) > (K x) + } Rationale: If (t, S t ) C: exercise value. Keep it. option is worth more than On the other hand, if v(t, x) = (K x) + : rv(t, x) v t (t, x) rxv x (t, x) 1 2 σ2 x 2 v xx (t, x) = rk Hence if (t, S t ) S: d D t = e [rv(t, rt S t ) v t (t, S t ) rs t v x (t, S t ) 1 ] 2 σ2 St 2 v xx (t, S t ) dt = rkdt You re being arbitraged. Get rid of it. Winter Per A. Mykland

14 STRUCTURE OF THE DIVIDEND rv(t, x) v t (t, x) rxv x (t, x) 1 2 σ2 x 2 v xx (t, x) = { 0 for (t, x) C rk for (t, x) S Hence, as before: d D t = rki {(t,st ) S}dt STOPPING RULE τ = min{t [0, T ] : (t, S t ) S} M t = e rt v(s t, t) is a supermartingale with Doob-Meyer decomposition v x (S t, t)d S t d D t M τ t is a martingale Winter Per A. Mykland

15 STOPPING BOUNDARY Stopping region: S = {(t, x) : v(t, x) = (K x) + } Continuation region: C = {(t, x) : v(t, x) > (K x) + } Boundary: x = L(T t): (t, x) S iff x L(T t) (t, x) C iff x > L(T t) smooth pasting continues to hold: v x (x+, t) = v x (x, t) = 1 for x = L(T t), for t < T Winter Per A. Mykland

16 AMERICAN CALL OPTIONS CASE OF NO DIVIDEND The calculations from the discrete case (Autumn Lecture 5) carry over. The value is the same as for European options WITH DIVIDEND AT DISCRETE TIMES Reduces to a discrete time problem. Between dividend times, reduces to a European options problem Winter Per A. Mykland