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 2005 1 Per A. Mykland
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 2005 2 Per A. Mykland
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 8.3.1 (more elegant) We shall use: Winter Lecture 1 (brute force) Winter 2005 3 Per A. Mykland
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 2005 4 Per A. Mykland
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 2005 5 Per A. Mykland
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 2005 6 Per A. Mykland
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 2005 7 Per A. Mykland
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 + 1 2 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 2005 8 Per A. Mykland
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 2005 9 Per A. Mykland
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 2005 10 Per A. Mykland
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 2005 11 Per A. Mykland
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 2005 12 Per A. Mykland
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 2005 13 Per A. Mykland
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 2005 14 Per A. Mykland
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 2005 15 Per A. Mykland
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 2005 16 Per A. Mykland