Lecture 15: Exotic Options: Barriers

Transcription

1 Lecture 15: Exotic Options: Barriers Dr. Hanqing Jin Mathematical Institute University of Oxford Lecture 15: Exotic Options: Barriers p. 1/10

2 Barrier features For any options with payoff ξ at exercise time τ, barrier features can be applied Set one barrier B, (or multi-barrier) Option activated/disactivated when the underlying asset price hits the barrier B before expiry Knock-ins: activated when the barrier is hit. Up-and-in: the option is activated when the barrier is hit from below Down-and-in: activated when the barrier is hit from above. Knock-outs: Up-and-out; Down-and-out Lecture 15: Exotic Options: Barriers p. 2/10

3 Barrier call options Example 1: Given a European call option with K = 100, and a barrier B = 90. The terminal payoff is max{s T K, 0} if it is active at time T, and 0 if not. Up-and-in option: if S 0 = 80, the option will be activated when S t goes up to 90 before T. Down-and-in option: if S 0 = 95, the option will be activated when S t goes down to 90 before T. Payoffs of barriers: path dependent Prices of barriers: for a European option, price is a function of (t,s t ) before the barrier hitting. Weakly path dependent. Lecture 15: Exotic Options: Barriers p. 3/10

4 Why barriers Barrier options are cheaper than vanilla options Example 2: An investor holds 1000 shares of a stock with price S 0 = 100, he wants to hedge the risk of lower price at 6 months later by put option with strike K = 90. If the price goes up to 110, he may not need the option any more. He can buy a put option with knock-out barrier B = 110, which reduces the premium of the option. Reverse knock-out option: the option will be knocked-out at some barrier where the option is in-the-money In E.g. 2, when the knock-out barrier B = 80, the option is a reverse knock-out put option. Lecture 15: Exotic Options: Barriers p. 4/10

5 Other barrier options Multiple barriers: E.g., A double knock-out option has two barriers B 1,B 2, The option is deactivated when either of the barriers was hit. Time dependent barrier: the barrier B changes with time, e.g. B(t) = Be rt. American digital: the strike price K can be regarded as a barrier. Barrier with rebate: when the option knocked out, a rebate will be paid immediately. Lecture 15: Exotic Options: Barriers p. 5/10

6 Pricing of barriers V knock-in + V knock-out = V no-barrier. For a down-and-out European call option (d/o for short) with strike K, barrier B, denote its price by C d/o (t,s t ) when it is not knocked out. Suppose K > B. Black-Scholes PDE hold for S t > B, L BS C d/o (t,s t ) = 0 on (t,s t ) [0,T) (B, + ) Boundary condition: C d/o (T,S T ) = max{s T K, 0} C d/o (t,b) = 0 Lecture 15: Exotic Options: Barriers p. 6/10

7 Pricing of barriers Denote V 1 (t,s t ) = C(t,S t ) C d/o (t,s t ), then L BS V 1 = 0 and V 1 (T,S T ) = 0 when S T > B V 1 (t,b) = C(t,B) Check that U(t,S t ) = ( ) S t 1 2r/σ 2 B C(t, B 2 S t ) also satisfies Black-Scholes PDE Notice U(t,B) = C(t,B), and U(T,S T ) = 0 for S T > B. So V 1 (t,s t ) = U(t,S t ). C d/o (t,s t ) = C(t,S t ) ( St B ) 1 2r/σ 2 C(t, B2 S t ) Lecture 15: Exotic Options: Barriers p. 7/10

8 Pricing of barriers Suppose K < B. Then at time T, if not knocked out, then S T > B, the payoff max{s T K, 0}1 ST >B = max{s T B, 0}+(B K)1 ST >B a d/o on call option with strike B + (B K) d/o on cash-or-nothing call with strike B. By the same way, we can calculate the value of d/o on cash-or-nothing call with strike B is C c/n (t,s t ) ( S t B ) 1 2r/σ 2 C c/n (t, B2 S t ) Value of the d/o on call option with strike K is C d/o (t, S t ) = C(t, S t ; B)+(B K)C c/n (t, S t ; B) ( S t B )1 2r σ 2 [C(t, B2 S t ; B)+(B K)C c/n (t, B2 S t ; B)] Lecture 15: Exotic Options: Barriers p. 8/10

9 Risk-neutral pricing For a stochastic process X t, denote M X t 1,t 2 = sup t [t1,t 2 ] X(s), m X t 1,t 2 = inf t [t1,t 2 ] X(s) By risk neutral pricing formula, given m S 0,t > B, C d/o (t,s t ) = e r(t t) E[max{ S T K, 0}1 m St,T >B S t = S t ] = e r(t t) E[( S T K)1 ST >K,m S t,t >B S t = S t ] = e r(t t) E[(S t e X T t K)1 St e X T t>k,m S t ex 0,T t >B] = e r(t t) E[(S t e X T t K)1 XT t >ln(k/s t ),m X 0,T t >ln(b/s t)] where X t = (r σ 2 /2)t + σw t for some standard Brownian motion W. Lecture 15: Exotic Options: Barriers p. 9/10

10 Risk-neutral pricing Distributions of X t,m X 0,t For Y t = a + µt + σw t, Distribution of M Y t ( x a µt F M Y t (x) = Φ σ t is required. for any x a is ) e 2µ(x a) σ 2 Φ Distribution of m Y t for any x a is ( ) x a µt F m Y t (x) = Φ σ t + e 2µ(x a) σ 2 Joint density function of (Y t,m Y t ) at x y,y > a, f Yt,M Y t (x,y) = 2(2y (x + a)) 2πσ6 t 3 e (2y (x+a))2 2σ 2 t ( ) x + a µt σ t ( ) x a + µt Φ σ t + µ(x a) σ 2 µ2 t 2σ 2 Lecture 15: Exotic Options: Barriers p. 10/10