Option Pricing. Chapter 9 - Barrier Options - Stefan Ankirchner. University of Bonn. last update: 9th December 2013

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1 Option Pricing Chapter 9 - Barrier Options - Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1

2 Standard barrier option Agenda What is a barrier option? Deriving pricing PDEs Valuation using the distribution of the maximum of a BM Double barrier options Further reading: S. Shreve: Stochastic Calculus for Finance II, Chapter 7.3. Stefan Ankirchner Option Pricing 2

3 Standard barrier option What is a barrier option? Definition: A barrier option is an option where the right to exercise depends on whether the underlying crosses a certain barrier level before expiration. Two cases: knock-out options: the right to exercise is lost if the barrier is crossed. The option becomes worthless. knock-in options: the right to exercise is obtained if the barrier is crossed. Why barrier options? smaller premiums: barrier options are cheaper! Stefan Ankirchner Option Pricing 3

4 Standard barrier option Example: Down and Out Call European call that is knocked out (you also say deactivated ) if the underlying crosses a barrier L before expiration. L is smaller than the present asset value S 0. Payoff of the D&O Call { CT D&O (ST K) = +, if S t > L for all t T, 0, if S t L for at least one t T. Stefan Ankirchner Option Pricing 4

5 Standard barrier option Example: Down and Out Call Special case K = L Asset evolution K L S 0 Payoff call Standard D&O S T K S T K S T K Time T Stefan Ankirchner Option Pricing 5

6 Standard barrier option Standard barrier options Categorizing barrier options standard option if active: call or put barrier level in relation to current asset price: down or up knock-in or knock-out 2 3 = 8 standard types Call Put Up Down Up Down In Out In Out In Out In Out Stefan Ankirchner Option Pricing 6

7 D&O call D&O call U&O call By following our 4 step recipe one can derive pricing PDEs for barrier options. We will do so first for D&O calls. Notation: S t = price of the underlying (think of a stock) m t = min 0 u t S u Payoff of the D&O call: C D&O T minimum price between 0 and t = (S T K) + 1 {mt >L}. The time t value of the D&O call depends on S t and m t. However, under the assumption that there has been no knock out prior to t, the value depends only on S t! Stefan Ankirchner Option Pricing 7

8 D&O call U&O call D&O call: Rolling the 4 steps 1) Assume that the D&O call is replicable. Denote by v(t, x) the time t option value / replicating portfolio value under the assumption that the barrier has not been attained before t and that S t = x. 2) v(t, S t ) is an Ito process. Ito s formula implies [ dv(t, S t) = v x(t, S t)σs tdw t + v x(t, S t)µs t + 1 ] 2 vxx(t, St)S t 2 σ 2 + v t(t, S t) dt. The self-financing condition yields dv(t, S t) = (t)ds t + (v(t, S t) (t)s t)rdt = (t)σs tdw t + (t)µs tdt + (v(t, S t) (t)s t)rdt. Stefan Ankirchner Option Pricing 8

9 D&O 4 steps cont d D&O call U&O call [ dv(t, S t) = v x(t, S t)σs tdw t + v x(t, S t)µs t + 1 ] 2 vxx(t, St)S t 2 σ 2 + v t(t, S t) dt, dv(t, S t) = (t)σs tdw t + (t)µs tdt + (v(t, S t) (t)s t)rdt. 3) Matching the coefficients: = (t) = v x (t, S t ) and v(t, x) has to satisfy the PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0. Stefan Ankirchner Option Pricing 9

10 D&O 4 steps cont d D&O call U&O call Boundary conditions: at knock out the D&O call is worthless, i.e. v(t, L) = 0, 0 t T. payoff if no knock-out prior to expiration: (S T K) +. Thus v(t, x) = (x K) +, x > L. Stefan Ankirchner Option Pricing 10

11 D&O 4 steps cont d D&O call U&O call 4) Solving the PDE: Feynman-Kac (applies only up to knock-out), numerical solution: straightforward Boundary conditions for solving the PDE with a finite difference scheme: v(t, L) = 0, 0 t T, v(t, x) = (x K) +, x > L, v(t, S max ) S max e r(t t) K, 0 t T. Stefan Ankirchner Option Pricing 11

12 D&O call D&O call U&O call Stefan Ankirchner Option Pricing 12

13 D&O call: summary D&O call U&O call Theorem Let v(t, x) be the time t D&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, L) = 0, 0 t T, v(t, x) = (x K) +, x > L. Remark: formulation similar to Thm in Shreve: Stochastic Calculus for Finance II Stefan Ankirchner Option Pricing 13

14 U&O call D&O call U&O call Similarly we can derive pricing PDEs for U&O calls. Assumptions and notation: upper barrier U > K S t = price of the underlying (think of a stock) M t = max 0 u t S u Payoff of the U&O call: C U&O T maximal price between 0 and t = (S T K) + 1 {MT <U}. Stefan Ankirchner Option Pricing 14

15 D&O call U&O call Pricing PDE for U&O calls Theorem (see Thm in Shreve) Let v(t, x) be the time t U&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, U) = 0, 0 t T, v(t, 0) = 0, 0 t T, v(t, x) = (x K) +, x < U. Caution: v(t, x) is not continuous in (T, U)! Stefan Ankirchner Option Pricing 15

16 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls U&O call revisited Payoff of an U&O call: where U > K, C U&O T M t = max 0 u t S u. The value at time t = 0: = (S T K) + 1 {MT <U}, C U&O 0 = e rt E Q [(S T K) + 1 {MT <U}], where Q is the risk neutral measure. Next: the joint Q-distribution of (M T, S T ) is known. Therefore the value of the U&O call can be calculated explicitly. Stefan Ankirchner Option Pricing 16

17 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls The maximum of a Brownian motion with drift Payoff of an U&O call: Dynamics of S: C U&O T = (S T K) + 1 {MT <U}. ds t = rs t dt + σs t dw Q t, where W Q is a Brownian motion under the risk neutral measure Q. Definition: Let α = 1 σ2 σ (r 2 ) and Note that Ŵ t = αt + W Q t. S T = S 0 e σw Q T = S 0 e σŵt, and S T K iff ŴT k := 1 σ log ( K S 0 ). +(r σ2 2 )T Stefan Ankirchner Option Pricing 17

18 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls The maximum of a Brownian motion with drift Definition: Let M T denote the maximum Then M T = max Ŵ u. 0 u T max S u = max S 0e σŵu = S 0 e σ M T, 0 u T 0 u T and the barrier U is hit iff M T b := 1 σ log ( U S 0 ). The time t = 0 value of an U&O call is given by C U&O 0 = e rt E Q [(S 0 e σŵt K) + 1 { MT <b} ] = e rt E Q [(S 0 e σŵt K) 1 { MT <b,ŵt k} ]. Stefan Ankirchner Option Pricing 18

19 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls Distribution of the maximum of a BM with drift Theorem (see Thm in Shreve) The density of the joint distribution of ( M T, ŴT ) under Q is given by f (m, w) = 2(2m w) T 2πT eαw 1 2 α2 T 1 2T (2m w)2, w m, m 0, and f (m, w) = 0 for other values of m and w. Stefan Ankirchner Option Pricing 19

20 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls Price of an U&O call Recall: The time t = 0 value of an U&O call is given by C U&O 0 = e rt E Q [(S 0 e σŵt K) 1 { MT <b,ŵt k} ]. Using the joint density of ( M T, ŴT ) we obtain that the price of an U&O call with barrier U and strike K is given by C U&O 0 = e rt b k b (S 0 e σw 2(2m w) K) w + T 2πT eαw 1 2 α2 T 1 2T (2m w)2 dmdw Stefan Ankirchner Option Pricing 20

21 Maximum of a Brownian motion with drift Explicit pricing formula for U&O calls Price of an U&O call cont d b b C U&O 0 = e rt (S 0e σw K) k w + 2(2m w) T 2πT 1 eαw α 2 T 1 (2m w) 2 2 2T dmdw. With some tedious calculations one can show that ( C0 U&O = S 0 [Φ (δ + T, S0 K e rt K [ Φ )) ( Φ (δ + T, S0 U ( (δ T, S0 K )) Φ ( ) 2r [ S0 σ U 2 Φ (δ + (T, U2 U KS 0 ( ) 1 2r [ +e rt S0 σ K 2 Φ (δ (T, U2 U KS 0 )) ] ( (δ T, S0 U )) Φ ))] (δ + (T, US0 ))] )) ))] Φ (δ (T, US0, where δ ±(T, s) = 1 [ σ log s + (r ± 12 ) ] T σ2 T. Stefan Ankirchner Option Pricing 21

22 Pricing PDEs In-Out Parity U&I call and D&I call Assumptions and notation: L = lower barrier and U = upper barrier M t = max 0 u t S u maximum price between 0 and t m t = min 0 u t S u minimum price between 0 and t Payoff of an U&I call with strike K: C U&I T = (S T K) + 1 {MT U}. Payoff of an D&I call with strike K: C D&I T = (S T K) + 1 {mt L}. Stefan Ankirchner Option Pricing 22

23 Pricing PDEs In-Out Parity Pricing PDE for U&I calls Recall that BS call(s, K, T t, σ, r) denotes the price of a Plain Vanilla Call, where S is the current price of the underlying,... Theorem Let v(t, x) be the time t U&I call value under the assumption that it has not been activated before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, (1) with boundary conditions v(t, U) = BS call(u, K, T t, σ, r) 0 t T, v(t, 0) = 0 0 t T, v(t, x) = 0 x < U. Stefan Ankirchner Option Pricing 23

24 Pricing PDEs In-Out Parity Pricing PDE for D&I calls Theorem Let v(t, x) be the time t D&I call value under the assumption that it has not been activated before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE (1) with boundary conditions v(t, L) = BS call(l, K, T t, σ, r) 0 t T, v(t, x) = 0 x > L. Stefan Ankirchner Option Pricing 24

25 In-Out Parity Pricing PDEs In-Out Parity Let C T = (S T K) + be the payoff of a Plain Vanilla Call. Let CT U&O and CT U&I be the payoff of barrier calls with strike K and barrier U > S 0. Notice that C T = C U&O T + C U&I T. Thus the price of the U&I call at time 0 satisfies C U&I 0 = C 0 C U&O 0. Similarly, we have an In-Out parity for Down-options: C D&I 0 = C 0 C D&O 0. Stefan Ankirchner Option Pricing 25

26 Double barrier options Parisian barrier options Second generation barrier options First Generation Barrier Options: one barrier knock-in resp. knock-out if barrier is crossed once Second Generation Barrier Options: Double barrier options: additional barrier Parisian barrier options: knock-in resp. knock-out only if a certain amount of time is spent beyond the barrier Stefan Ankirchner Option Pricing 26

27 Double barrier options Parisian barrier options Double knock out barrier options Barriers: lower barrier L upper barrier U Double knock out call and put: Type DKOC DKOP Payoff (S T K) + 1 {min0 t T S t>l, max 0 t T S t<u} (K S T ) + 1 {min0 t T S t>l, max 0 t T S t<u} Stefan Ankirchner Option Pricing 27

28 Double barrier options Parisian barrier options Double knock out barrier options Asset evolution B u S 0 B L Time T Stefan Ankirchner Option Pricing 28

29 Double barrier options Parisian barrier options Double knock out call Theorem Let v(t, x) be the time t DKOC value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, L) = 0, 0 t T, v(t, U) = 0, 0 t T, v(t, x) = (x K) +, L < x < U. Stefan Ankirchner Option Pricing 29

30 Parisian Option Double barrier options Parisian barrier options Standard barrier options: the option trigger only depends on a single touching of the barrier by the underlying price process. The counterparty may manipulate the underlying for a short time such that the barrier option is knocked-out resp. knocked in Parisian barrier options require the knock-out / knock-in condition to be satisfied for a certain time, and thus prevent the counterparty to influence prices. Stefan Ankirchner Option Pricing 30

31 Double barrier options Parisian barrier options Standard and cumulative Parisian options 2 types of Parisian barrier options: Standard Parisian barrier option: option is knocked out if the underlying asset value stays consecutively below the barrier for a time longer than some pre specified time window d before the maturity date. Cumulative Parisian barrier option: option is knocked out if the underlying asset value spends until maturity in total d units of time below the barrier. Pricing of Parisian barrier options: Monte-Carlo method (see next Chapter) Stefan Ankirchner Option Pricing 31

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