Chapter 11 Barrier Options
Barrier options are financial derivatives whose payoffs depend on the crossing of a certain predefined barrier level by the underlying asset price (St)t∈[0,T ]. In this chapter we consider barrier options whose payoffs depend on an ex- tremum of (St)t∈[0,T ], in addition to the terminal value ST . Barrier options are priced by computing the discounted expected values of their claim payoffs, or by PDE arguments.
11.1 Options on Extrema ...... 393 11.2 Knock-Out Barrier Options ...... 398 11.3 Knock-In Barrier Options ...... 410 11.4 PDE Method ...... 414 Exercises ...... 419
11.1 Options on Extrema
Vanilla options with payoff C = φ(ST ) can be priced as
−rT ∗ −rT ∞ e E [φ(ST )] = e φ(y)ϕS (y)dy w0 T where ϕST (y) is the (one parameter) probability density function of ST , which satisfies y P(ST y) = ϕS (v)dv, y ∈ R. 6 w0 T Recall that typically we have x − K if x > K, φ(x) = (x − K)+ = 0 if x < K, for the European call option with strike price K, and
393
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$1 if x > K, 1 φ(x) = [K,∞)(x) = 0 if x < K, for the binary call option with strike price K. On the other hand, exotic options, also called path-dependent options, are options whose payoff C may depend on the whole path
{St : 0 6 t 6 T } of the underlying asset price process via a “complex” operation such as aver- aging or computing a maximum. They are opposed to vanilla options whose payoff C = φ(ST ), depend only using the terminal value ST of the price process via a payoff function φ, and can be priced by the computation of path integrals, see Sec- tion 17.2.
For example, the payoff of an option on extrema may take the form
T C := φ M0 , ST , where T M0 = Max St t∈[0,T ] is the maximum of (St)t∈R+ over the time interval [0, T ]. In such situations the option price at time t = 0 can be expressed as
−rT ∗ T −rT ∞ ∞ e E φ M0 , ST = e φ(x, y)ϕ T (x, y)dxdy w0 w0 M0 ,ST
T where ϕ T is the joint probability density function of (M , ST ), which M0 ,ST 0 satisfies
T x y P(M0 x and ST y) = ϕ T (u, v)dudv, x, y 0. 6 6 w0 w0 M0 ,ST >
General case
Using the joint probability density function of WfT = WT + µT and
T Xb0 = Max Wf = Max (Wt + µt), t∈[0,T ] t∈[0,T ]
T we are able to price any exotic option with payoff φ(WfT , Xb0 ), as
−(T −t)r ∗ T e E φ Xb0 , WfT Ft , 394 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance with in particular, letting a ∨ b := Max(a, b),
−rT ∗ T −rT ∞ ∞ ∗ T e E φ Xb0 , WfT = e φ(x, y)dP Xb0 x, WfT y . w−∞ wy∨0 6 6
In this chapter we work in a (continuous) geometric Brownian model in which the asset price (St)t∈[0,T ] has the dynamics
dSt = rStdt + σStdWt, t > 0, where (Wt)t∈R+ is a standard Brownian motion under the risk-neutral prob- ∗ ability measure P . In particular, by Lemma 5.15 the value Vt of a self- financing portfolio satisfies
−rT T −rt VT e = V0 + σ ξtSt e dWt, t ∈ [0, T ]. w0 In order to price barrier∗ options by the above probabilistic method, we will use the probability density function of the maximum
T M0 = Max St t∈[0,T ] of geometric Brownian motion (St)t∈R+ over a given time interval [0, T ] and the joint probability density function ϕ T (u, v) derived in Chapter 10 by M0 ,ST the reflection principle. Proposition 11.1. An exotic option with integrable claim payoff of the form T C = φ M0 , ST = φ Max St, ST t∈[0,T ] can be priced at time 0 as
e−rT E∗[C] −rT r ∞ ∞ e 2 σy σx −µ2T /2+µy−(2x−y)2/(2T ) = φ S0 e , S0 e (2x − y) e dxdy T 3/2 π w0 wy −rT r 0 ∞ e 2 σy σx −µ2T /2+µy−(2x−y)2/(2T ) + φ S0 e , S0 e (2x − y) e dxdy. T 3/2 π w−∞ w0 Proof. We have
2 σWT −σ T /2+rT (WT +µT )σ σWT ST = S0 e = S0 e = S0 e e , with
∗ “A former MBA student in finance told me on March 26, 2004, that she did not understand why I covered barrier options until she started working in a bank” Lyuu (2021).
395 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault
σ r µ := − + and WfT = WT + µT , 2 σ and
2 T σWt−σ t/2+rt M0 = Max St = S0 Max e t∈[0,T ] t∈[0,T ]
σWt σ Max Wt = S0 Max e e = S0 e t∈[0,T ] e t∈[0,T ] σXT = S0 e b0 , we have
2 T T σWT −σ T /2+rT T σWT σX C = φ ST , M0 = φ S0 e , M0 = φ S0 e e , S0 e b0 , hence
−rT ∗ −rT ∗ σW σXT e E [C] = e E φ S0 e eT , S0 e b0 −rT ∞ ∞ σy σx T = e φ S0 e , S0 e dP Xb0 x, WfT y w−∞ wy∨0 6 6 −rT r ∞ ∞ e 2 σy σx −µ2T /2+µy−(2x−y)2/(2T ) = φ S0 e , S0 e (2x − y) e dxdy T 3/2 π w−∞ wy∨0 −rT r ∞ ∞ e 2 σy σx −µ2T /2+µy−(2x−y)2/(2T ) = φ S0 e , S0 e (2x − y) e dxdy T 3/2 π w0 wy −rT r 0 ∞ e 1 2 σy σx −µ2T /2+µy−(2x−y)2/(2T ) + φ S0 e , S0 e (2x − y) e dxdy. T 3/2 π w−∞ w0
Pricing barrier options
The payoff of an up-and-out barrier put option on the underlying asset price St with exercise date T , strike price K and barrier level (or call level) B is
+ (K − ST ) if Max St < B, 06t6T + C = (K − ST ) 1n o = Max St < B 0 t T 0 if Max St > B. 6 6 06t6T This option is also called a Callable Bear Contract, or Bear CBBC with no residual value, or turbo warrant with no rebate, in which the call level B usually satisfies B 6 K.
396 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance
The payoff of a down-and-out barrier call option on the underlying asset price St with exercise date T , strike price K and barrier level B is
+ (ST − K) if min St > B, 06t6T + C = (ST − K) 1n o = min St > B 0 t T 0 if min St 6 B. 6 6 06t6T This option is also called a Callable Bull Contract, or Bull CBBC with no residual value, or turbo warrant with no rebate, in which B denotes the call level. ∗
Category ’R’ Callable Bull/Bear Contracts, or CBBCs, also called turbo warrants, involve a rebate or residual value computed as the payoff of a down-and-in lookback option. Category ’N’ Callable Bull/Bear Contracts do not involve a residual value or rebate, and they usually satisfy B = K. See Eriksson and Persson(2006), Wong and Chan(2008) and Exercise 11.2 for the pricing of Category ’R’ CBBCs with rebate.
Option type CBBC Behavior Payoff Price Figure + 1 down-and-out (ST − K) n o B 6 K (11.10) 11.4a Bull min St > B (knock-out) 06t6T B > K (11.11) 11.4b + 1 down-and-in (ST − K) n o B 6 K (11.13) 11.7a min St < B (knock-in) 06t6T B > K (11.14) 11.7b Barrier call + 1 up-and-out (ST − K) n o B 6 K 0 N.A. Max St < B (knock-out) 06t6T B > K (11.5) 11.1 + 1 up-and-in (ST − K) n o B 6 K BSCall 6.4 Max St > B (knock-in) 06t6T B > K (11.15) 11.8 + 1 down-and-out (K − ST ) n o B 6 K (11.12) 11.6 min St > B (knock-out) 06t6T B > K 0 N.A. + 1 down-and-in (K − ST ) n o B 6 K (11.16) 11.9 min St < B (knock-in) 06t6T B > K BSPut 6.11 Barrier put + 1 up-and-out (K − ST ) n o B 6 K (11.8) 11.2a Bear Max St < B (knock-out) 06t6T B > K (11.9) 11.2b + 1 up-and-in (K − ST ) n o B 6 K (11.17) 11.10a Max St > B (knock-in) 06t6T B > K (11.18) 11.10b
Table 11.1: Barrier option types.
∗ Download the corresponding for the pricing of Bull CBBCs (down-and-out barrier call options) with B > K.
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We can distinguish between eight different variations on barrier options, ac- cording to Table 11.1.
In-out parity
We have the following parity relations between the prices of barrier options and vanilla call and put options:
−(T −t)r ∗ + Cup-in(t) + Cup-out(t) = e E [(ST − K) | Ft], (11.1) −(T −t)r ∗ + Cdown-in(t) + Cdown-out(t) = e E [(S − K) | F ], (11.2) T t
−(T −t)r ∗ + Pup-in(t) + Pup-out(t) = e E [(K − ST ) | Ft], (11.3) −(T −t)r ∗ + Pdown-in(t) + Pdown-out(t) = e E [(K − ST ) | Ft], (11.4) where the price of the European call, resp. put option with strike price K are obtained from the Black-Scholes formula. Consequently, in the sequel we will only compute the prices of the up-and-out barrier call and put options and of the down-and-out barrier call and put options.
t t Note that all knock-out barrier option prices vanish when M0 > B or m0 < B, while the barrier up-and-out call, resp. the down-and-out barrier put option prices require B > K, resp. B < K, in order not to vanish.
11.2 Knock-Out Barrier Options
Up-and-out barrier call option
Let us consider an up-and-out barrier call option with maturity T , strike price K, barrier (or call level) B, and payoff ST − K if Max St 6 B, 06t6T + C = (ST − K) 1n o = Max St < B 0 t T 0 if Max St > B, 6 6 06t6T with B > K. Proposition 11.2. When K 6 B, the price + ST −t e−(T −t)r1 E∗ x − K 1 M t
h i −(T −t)r ∗ + e E (ST − K) 1 T Ft (11.5) M0
1+2r/σ2 2 ) B T −t B T −t B − Φ δ+ − Φ δ+ St KSt St ( −(T −t)r 1 T −t St T −t St − e K t Φ δ− − Φ δ− M0
1−2r/σ2 2 ) St T −t B T −t B − Φ δ− − Φ δ− B KSt St 1 1 T −t St = t Bl(St, r, T − t, σ, K) − St t Φ δ+ M0
where 1 σ2 δτ (s) = √ log s + r ± τ , s > 0. (11.6) ± σ τ 2 The price of the up-and-out barrier call option is zero when B 6 K. The following R code implements the up and out pricing formula (11.5).
399 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault
1 dp <- function( T , r, v, s ) { ( log(s) + ( r + v*v/2.0)*T)/v/sqrt(T) } dm <- function( T , r , v, s ) { ( log(s) + ( r - v*v/2.0)*T)/v/sqrt(T) } 3 ind<-function(condition) ifelse(condition,1,0) CBBC <- function(S,K,B,T,r,sig){ S*ind(S Note that taking B = +∞ in the above identity (11.5) recovers the Black- Scholes formula i St St e−(T −t)rE∗[(S − K)+ | F = S Φ δT −t − e−(T −t)rKΦ δT −t T t t + K − K for the price of European call options. The graph of Figure 11.1 represents the up-and-out barrier call option price given the value St of the underlying asset and the time t ∈ [0, T ] with T = 220 days. 16 14 12 10 8 6 4 200 2 160 0 Time in days 50 60 70 120 80 90 Underlying Fig. 11.1: Graph of the up-and-out call option price with B = 80 > K = 65.∗ T Proof of Proposition 11.2. We have C = φ ST , M0 with (x − K)+ if y < B, + φ(x, y) = (x − K) 1{y ∗ Right-click on the figure for interaction and “Full Screen Multimedia” view. 400 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance h i −(T −t)r ∗ +1 1 = e E (ST − K) t T Ft M0 B x=S t6r6T St t " # S + −(T −t)r1 E∗ x T −t − K 1 = e t n Sr o M0K S0 e b0 K} {S eσx b b (2x − y) e2x(y−x)/T dx = (2x − y) e2x(y−x)/T dx wa wa T h ix=b = − e2x(y−x)/T 2 x=a 401 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault T = ( e2a(y−a)/T − e2b(y−b)/T ) 2 T = ( e2(y∨0)(y−y∨0)/T − e2b(y−b)/T ) 2 T = (1 − e2b(y−b)/T ), 2 −1 hence, letting c = σ log(K/S0), we have h i ∗ +1 E (ST − K) T M0 402 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance 2 −(r+µ2/2)T −2b2/T +(σ+µ+2b/T )2T /2 T B T B −S0 e Φ δ+ − Φ δ+ KS0 S0 S0 S0 −K e−rT Φ δT − Φ δT − K − B 2 −(r+µ2/2)T −2b2T +(µ+2b/T )2T /2 B B +K e Φ δ− − Φ δ− , KS0 S0 T 0 6 x 6 B, where δ±(s) is defined in (11.6). Given the relations µ2 b2 T 2b2 r σ 2r B −T r + − 2 + σ + µ + = 2b + = 1 + 2 log , 2 T 2 T σ 2 σ S0 and µ2 b2 T 2b2 2r B −T r + − 2 + µ + = −rT + 2µb = −rT + −1 + 2 log , 2 T 2 T σ S0 this yields h i −rT ∗ +1 e E (ST − K) T (11.7) M0 403 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault Up-and-out barrier put option This option is also called a Callable Bear Contract, or Bear CBBC with no residual value, or turbo warrant with no rebate, in which B denotes the call level∗. The price + ST −t e−(T −t)r1 E∗ K − x 1 M t K, by h i −(T −t)r ∗ +1 e E (K − ST ) T Ft M0 ∗ Download the corresponding for the pricing of Bear CBBCs (up-and-out barrier put options) with B 6 K. 404 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance 1+2r/σ2 2 ! 1 T −t St B T −t B = St t −Φ −δ+ + Φ −δ+ M0 50 12 40 10 8 30 6 20 4 10 200 2 200 160 160 0 Time in days 0 Time in days 50 50 60 70 120 60 70 120 80 90 80 90 Underlying Underlying (a) Case K = 100 > B = 80. (b) Case B = 80 > K = 60. Fig. 11.2: Graphs of the up-and-out put option prices (11.8)-(11.9). The following Figure 11.3 shows the market pricing data of an up-and-out barrier put option on BHP Billiton Limited ASX:BHP with B = K = $28 for half a share, priced at 1.79. 405 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault Fig. 11.3: Pricing data for an up-and-out put option with K = B = $28. The attached performs an implied volatility calculation for up-and- out barrier put option (or Bear CBBC) prices with B < K, based on this set. Down-and-out barrier call option Let us now consider a down-and-out barrier call option on the underlying asset price St with exercise date T , strike price K, barrier level B, and payoff ST − K if min St > B, 06t6T + C = (ST − K) 1n o = min St > B 0 t T 0 if min St 6 B, 6 6 06t6T with 0 6 B 6 K. The down-and-out barrier call option is also called a Callable Bull Contract, or Bull CBBC with no residual value, or turbo war- ∗ rant with no rebate, in which B denotes the call level. When B 6 K, we have −(T −t)r ∗ + e E (ST − K) 1n o Ft (11.10) min St > B 06t6T 1 T −t St −(T −t)r 1 T −t St = St t Φ δ+ − e K t Φ δ− m0>B K m0>B K ∗ Download the corresponding for Bull CBBC pricing with B > K. 406 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance 2r/σ2 2 1 B T −t B −B t Φ δ+ m0>B St KSt 1−2r/σ2 2 −(T −t)r 1 St T −t B + e K t Φ δ− m0>B B KSt 1 = t Bl(St, r, T − t, σ, K) m0>B 2r/σ2 2 1 B T −t B −B t Φ δ+ m0>B St KSt 1−2r/σ2 2 −(T −t)r 1 St T −t B + e K t Φ δ− m0>B B KSt 1 = t Bl(St, r, T − t, σ, K) m0>B 2r/σ2 1 B B K −St t Bl , r, T − t, σ, , m0>B St St B 0 6 t 6 T . When B > K, we find −(T −t)r ∗ + 1 e E (ST − K) n o Ft (11.11) min St > B 06t6T 1 T −t St −(T −t)r 1 T −t St = St t Φ δ+ − e K t Φ δ− m0>B B m0>B B 2r/σ2 1 B T −t B −B t Φ δ+ m0>B St St 1−2r/σ2 −(T −t)r 1 St T −t B + e K t Φ δ− , m0>B B St St > B, 0 6 t 6 T , see Exercise 11.1 below. 60 50 16 40 14 30 12 10 20 8 10 6 4 0 2 200 0 120 90 Time in days 160 80 160 70 Time in days 120 200 90 Underlying 60 50 60 70 80 50 Underlying (a) Case B = 60 < K = 80. (b) Case K = 40 < B = 60. Fig. 11.4: Graphs of the down-and-out call option price (11.10)-(11.11). 407 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault In the next Figure 11.5 we plot∗ the down-and-out barrier call option price (11.11) as a function of volatility with B = 349.2 > K = 346.4, r = 0.03, T = 99/365, and S0 = 360. 0.20 Bull CBBC Market Price Bull CBBC Pricing Formula −rT S0 − Ke S0 − B 0.18 0.16 0.14 Bull CBBC Price 0.12 0.10 0.0 0.2 0.4 0.6 0.8 1.0 σ Fig. 11.5: Down-and-out call option price as a function of σ. We note that with such parameters, the down-and-out barrier call option −rT price (11.11) is upper bounded by the forward contract price S0 − K e in the limit as σ tends to zero, and that it decreases to S0 − B in the limit as σ tends to infinity. Down-and-out barrier put option When K > B, the price + ST −t e−(T −t)r1 E∗ K − x 1 mt >B n o 0 S0 x min Sr/S0 > B 0 r T −t 6 6 x=St of the down-and-out barrier put option with maturity T , strike price K and barrier level B is given by −(T −t)r ∗ +1 e E (K − ST ) T Ft m0 >B 1 T −t St T −t St = St t Φ δ+ − Φ δ+ m0>B K B 1+2r/σ2 2 ) B T −t B T −t B − Φ δ+ − Φ δ+ St KSt St ( −(T −t)r 1 T −t St T −t St − e K t Φ δ− − Φ δ− m0>B K B ∗ Download the corresponding . 408 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance 1−2r/σ2 2 ) St T −t B T −t B − Φ δ− − Φ δ− B KSt St 1 T −t St T −t St = St t Φ −δ+ − Φ −δ+ m0>B B K 1+2r/σ2 2 ) B T −t B T −t B − Φ δ+ − Φ δ+ St KSt St ( −(T −t)r 1 T −t St T −t St − e K t Φ −δ− − Φ −δ− m0>B B K 1−2r/σ2 2 ) St T −t B T −t B − Φ δ− − Φ δ− B KSt St 1 1 T −t St = t Blput(St, r, T − t, σ, K) + St t Φ −δ+ (11.12) m0>B m0>B B 2r/σ2 2 1 B T −t B T −t B −B t Φ δ+ − Φ δ+ m0>B St KSt St −(T −t)r 1 T −t St − e K t Φ −δ− m0>B B 1−2r/σ2 2 −(T −t)r 1 St T −t B T −t B + e K t Φ δ− − Φ δ− , m0>B B KSt St while the corresponding price vanishes when K 6 B. 14 12 10 8 6 4 200 2 160 0 Time in days 50 60 120 70 80 Underlying 90 Fig. 11.6: Graph of the down-and-out put option price (11.12) with K = 80 > B = 65. Note that although Figures 11.2b and 11.4a, resp. 11.2a and 11.4b, appear to share some symmetry property, the functions themselves are not exactly symmetric. Regarding Figures 11.1 and 11.6, the pricing function is actually the same, but the conditions B < K and B > K play opposite roles. 409 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault 11.3 Knock-In Barrier Options Down-and-in barrier call option When B 6 K, the price of the down-and-in barrier call option is given from the down-and-out barrier call option price (11.10) and the down-in-out call parity relation (11.2) as −(T −t)r ∗ +1 e E (ST − K) T Ft (11.13) m0 B St KSt 1−2r/σ2 2 −(T −t)r 1 St T −t B − e K t Φ δ− . m0>B B KSt 20 15 1 10 0.8 0.6 5 0.4 0 0.2 160 180 0 180 90 200 Time in days 80 200 70 220 Underlying 30 220Time in days 60 50 40 60 90 80 70 50 Underlying (a) Case K = 80 > B = 65. (b) Case K = 40 < B = 60. Fig. 11.7: Graphs of the down-and-in call option price (11.13)-(11.14). When B > K, the price of the down-and-in barrier call option is given from the down-and-out barrier call option price (11.11) and the down-in-out call parity relation (11.2) as h i −(T −t)r ∗ +1 e E (ST − K) T Ft (11.14) mt B B m0>B B 1+2r/σ2 1 B T −t B + t St Φ δ+ m0>B St St 410 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance 1−2r/σ2 −(T −t)r 1 St T −t B − e K t Φ δ− , 0 6 t 6 T . m0>B B St Up-and-in barrier call option When B > K, the price of the up-and-in barrier call option is given from (11.5) and the up-in-out call parity relation (11.1) as −(T −t)r ∗ +1 e E (ST − K) T Ft (11.15) M0 >B 1 1 T −t St = t Bl(St, r, T − t, σ, K) + St t Φ δ+ M0>B M0 25 20 15 10 5 180 200 0 90 Time in days 80 70 220 60 Underlying 50 Fig. 11.8: Graph of the up-and-in call option price (11.15) with B = 80 > K = 65. When B 6 K, the price of the up-and-in barrier call option is given from the Black-Scholes formula and the up-in-out call parity relation (11.1) as h i −(T −t)r ∗ +1 e E (ST − K) T Ft = Bl(St, r, T − t, σ, K). M0 >B 411 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault Down-and-in barrier put option When B 6 K, the price of the down-and-in barrier put option is given from (11.12) and the down-in-out put parity relation (11.4) as −(T −t)r ∗ +1 e E (K − ST ) T Ft (11.16) mt B B 2r/σ2 2 1 B T −t B T −t B + B t Φ δ+ − Φ δ+ m0>B St KSt St −(T −t)r 1 T −t St + e K t Φ −δ− m0>B B 1−2r/σ2 2 −(T −t)r 1 St T −t B T −t B − e K t Φ δ− − Φ δ− , m0>B B KSt St 0 6 t 6 T . 30 25 20 15 10 5 0 160 50 200 60 Time in days 70 80 Underlying 90 Fig. 11.9: Graph of the down-and-in put option price (11.16) with K = 80 > B = 65. When B > K, the price of the down-and-in barrier put option is given from the Black-Scholes put function and the down-in-out put parity relation (11.4) as −(T −t)r ∗ +1 e E (K − ST ) T Ft = Blput(St, r, T − t, σ, K), mt 0 6 t 6 T . 412 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance Up-and-in barrier put option When B 6 K, the price of the down-and-in barrier put option is given from (11.8) and the up-in-out put parity relation (11.3) as −(T −t)r ∗ +1 e E (K − ST ) T Ft (11.17) M0 >B = Blput(St, r, T − t, σ, K) 1+2r/σ2 ! 1 B T −t B T −t St −St t Φ −δ+ − Φ −δ+ M0 0 6 t 6 T . 1.2 10 1 8 0.8 6 0.6 180 180 4 0.4 2 200 0.2 200 Time in days Time in days 0 0 220 90 220 90 80 70 80 70 Underlying 60 Underlying 60 (a) K = 80 > B = 70. (b) Case K = 70 < B = 80. Fig. 11.10: Graphs of the up-and-in put option price (11.17)-(11.18). By (11.9) and the up-in-out put parity relation (11.3), the price of the up- and-in barrier put option is given when B > K by h i −(T −t)r ∗ +1 e E (K − ST ) T Ft (11.18) M0 >B 1 = t Blput(St, r, T − t, σ, K) M0>B 1+2r/σ2 2 1 B T −t B −St t Φ −δ+ M0 413 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault 11.4 PDE Method The up-and-out barrier call option price has been evaluated by probabilistic arguments in the previous sections. In this section we complement this ap- proach with the derivation of a Partial Differential Equation (PDE) for this option price function. The up-and-out barrier call option price can be written as h i −(T −t)r ∗ +1 e E (ST − K) T Ft M0 −(T −t)r ∗ + = e 1n oE (ST − K) 1n o Ft Max Sr < B Max Sr < B 06r6t t6r6T 1 = t g(t, St), M0 −(T −t)r ∗ + g(t, x) = e E (ST − K) 1n o St = x . (11.19) Max Sr < B t6r6T Next, by the same argument as in the proof of Proposition 6.1 we derive the Black-Scholes partial differential equation (PDE) satisfied by g(t, x), and written for the value of a self-financing portfolio. Proposition 11.3. Let (ηt, ξt)t∈R+ be a portfolio strategy such that (i) (ηt, ξt)t∈R+ is self-financing, (ii) the portfolio value Vt := ηtAt + ξtSt, t > 0, is given as in (11.19) by 1 Vt = t g(t, St), t > 0. M0 Then, the function g(t, x) satisfies the Black-Scholes PDE ∂g ∂g 1 ∂2g rg(t, x) = (t, x) + rx (t, x) + x2σ2 (t, x), (11.20) ∂t ∂x 2 ∂x2 t > 0, 0 < x < B, under the boundary condition g(t, B) = 0, 0 6 t 6 T , and ξt is given by ∂g ξ = (t, S ), t ∈ [0, T ], (11.21) t ∂x t 414 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance t provided that M0 < B. Proof. By (11.19) the price at time t of the down-and-out barrier call option discounted to time 0 is given by −rt1 e t g(t, St) M0 −rT ∗ + = e 1 E (S − K) 1 Ft M t −rT ∗ + = e E (S − K) 1 1 Ft T M t −rT ∗ + = e E (ST − K) 1n o St , Max Sr < B 06r6T which is a martingale indexed by t > 0. Next, applying the Itô formula to −rt t t 7−→ e g(t, St) “on {M0 6 B, 0 6 t 6 T }”, we have −rt −rt −rt d( e g(t, St)) = − r e g(t, St)dt + e dg(t, St) ∂g = −r e−rtg(t, S )dt + e−rt (t, S )dt t ∂t t 2 −rt ∂g 1 −rt 2 2 ∂ g + r e St (t, St)dt + e σ S (t, St)dt ∂x 2 t ∂x2 ∂g + e−rtσS (t, S )dW . (11.22) t ∂x t t In order to derive (11.21) we note that, as in the proof of Proposition 6.1, the self-financing condition (5.8) implies −rt −rt −rt d( e Vt) = −r e Vtdt + e dVt −rt −rt −rt = −r e Vtdt + ηt e dAt + ξt e dSt −rt −rt −rt −rt = −r(ηtAt + ξtSt) e dt + rηtAt e dt + rξtSt e dt + σξtSt e dWt −rt = σξtSt e dWt, t > 0, (11.23) and (11.21) follows by identification of (11.22) with (11.23) which shows that the sum of components in factor of dt have to vanish, hence 2 2 ∂g ∂g σ 2 ∂ g −rg(t, St) + (t, St) + rSt (t, St) + S (t, St) = 0. ∂t ∂x 2 t ∂x2 415 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault In the next proposition we add a boundary condition to the Black-Scholes PDE (11.20) in order to hedge the up-and-out barrier call option with ma- turity T , strike price K, barrier (or call level) B, and payoff ST − K if Max St 6 B, 06t6T + C = (ST − K) 1n o = Max St < B 0 t T 0 if Max St > B, 6 6 06t6T with B > K. 1 Proposition 11.4. The value Vt = t g(t, St) of the self-financing M0 ∂g ∂g 1 ∂2g rg t x t x rx t x x2σ2 t x ( , ) = ( , ) + ( , ) + 2 ( , ), (11.24a) ∂t ∂x 2 ∂x g(t, x) = 0, x > B, t ∈ [0, T ], (11.24b) + g(T , x) = (x − K) 1{x + g(T , x) = (x − K) 1{x g(t, x) = 0, x > B. (11.25) Condition (11.25) holds since the price of the claim at time t is 0 whenever St = B. When K 6 B, the closed-form solution of the PDE (11.24a) under the boundary conditions (11.24b)-(11.24c) is given from (11.5) in Proposi- tion 11.2 as 416 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance x x g(t, x) = x Φ δT −t − Φ δT −t (11.26) + K + B x −1−2r/σ2 B2 B −x Φ δT −t − Φ δT −t B + Kx + x x x −K e−(T −t)r Φ δT −t − Φ δT −t − K − B x 1−2r/σ2 B2 B +K e−(T −t)r Φ δT −t − Φ δT −t , B − Kx − x 0 < x 6 B, 0 6 t 6 T , see Figure 11.1. We note that the expression (11.26) can be rewritten using the standard Black-Scholes formula S S Bl(S, r, T , σ, K) = SΦ δT − K e−rT Φ δT + K − K for the price of the European call option, as x x g(t, x) = Bl(x, r, T − t, σ, K) − xΦ δT −t + e−(T −t)rKΦ δT −t + B − B 2 B 2r/σ B2 B −B Φ δT −t − Φ δT −t x + Kx + x x 1−2r/σ2 B2 B + e−(T −t)rK Φ δT −t − Φ δT −t , B − Kx − x 0 < x 6 B, 0 6 t 6 T . Table 11.2 summarizes the boundary conditions satisfied for barrier option pricing in the Black-Scholes PDE. 417 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault Boundary conditions Option type CBBC Behavior Maturity T Barrier B + down-and-out B 6 K (x − K) 0 Bull +1 (knock-out) B > K (x − K) {x>B} 0 down-and-in B 6 K 0 Bl(B, r, T − t, σ, K) +1 (knock-in) B > K (x − K) {x Table 11.2: Boundary conditions for barrier option prices. Hedging barrier options Figure 11.11 represents the value of Delta obtained from (11.21) for the up- and-out barrier call option in Exercise 11.1-(a). Fig. 11.11: Delta of the up-and-out barrier call with B = 80 > K = 55.∗ 418 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance Down-and-out barrier call option Similarly, the price g(t, St) at time t of the down-and-out barrier call option satisfies the Black-Scholes PDE 2 ∂g ∂g 1 2 2 ∂ g rg(t, x) = (t, x) + rx (t, x) + x σ (t, x), ∂t ∂x 2 ∂x2 g(t, B) = 0, t ∈ [0, T ], + g(T , x) = (x − K) 1{x>B}, on the time-space domain [0, T ] × [0, B] with terminal condition g(T , x) = + (x − K) 1{x>B} and the additional boundary condition g(t, x) = 0, x 6 B, since the price of the claim at time t is 0 whenever St 6 B, see (11.10) and Figure 11.4a when B 6 K, and (11.11) and Figure 11.4b when B > K. Exercises Exercise 11.1 Barrier options. a) Compute the hedging strategy of the up-and-out barrier call option on the underlying asset price St with exercise date T , strike price K and barrier level B, with B > K. b) Compute the joint probability density function dP(YT a and WT b) ϕ (a, b) = 6 6 , a, b ∈ R, YT ,WT dadb of standard Brownian motion WT and its minimum YT = min Wt. t∈[0,T ] c) Compute the joint probability density function b b dP(Y T a and WfT b) ϕ (a, b) = 6 6 , a, b ∈ R, Y T ,WeT dadb drifted W = W + µT minimum of Brownianb motion fT T and its Y T = min Wft = min (Wt + µt). t∈[0,T ] t∈[0,T ] ∗ The animation works in Acrobat Reader on the entire pdf file. 419 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault d) Compute the price at time t ∈ [0, T ] of the down-and-out barrier call option on the underlying asset price St with exercise date T , strike price K, barrier level B, and payoff ST − K if min St > B, 06t6T + C = (ST − K) 1n o = min St > B 0 t T 0 if min St 6 B, 6 6 06t6T in cases 0 < B < K and B > K. Exercise 11.2 Pricing Category ’R’ CBBC rebates. Given τ > 0, consider an asset price (St)t∈[τ,∞), given by 2 rt+σWt−σ t/2 Sτ+t = Sτ e , t > 0, where (Wt)t∈R+ is a standard Brownian motion, with r > 0 and σ > 0. In the sequel, ∆τ is the deterministic length of the Mandatory Call Event (MCE) valuation period which commences from the time upon which a MCE occurs up to the end of the following trading session. h + i a) Compute the expected rebate (or residual) E min Sτ+s − K Fτ s∈[0,∆τ] of a Category ’R’ CBBC Bull Contract having expired at a given time τ < T , knowing that Sτ = B > K > 0, with r > 0. h + i b) Compute the expected rebate E min Sτ+s − K Fτ of a Cat- s∈[0,∆τ] egory ’R’ CBBC Bull Contract having expired at a given time τ < T , knowing that Sτ = B > 0, with r = 0. c) Find the expression of the probability density function of the first hitting time τB = inf{t > 0 : St = B} of the level B > 0 by the process (St)t∈R+ . d) Price the CBBC rebate h +i −∆τ −τ 1 e E e [0,T ](τ) min St − K t∈[τ,τ+∆τ] h h + ii −∆τ −rτ 1 = e E e [0,T ](τ)E min St − K Fτ . t∈[τ,τ+∆τ] Exercise 11.3 Barrier forward contracts. Compute the price at time t of the following barrier forward contracts on the underlying asset price St with exercise date T , strike price K, barrier level B, and the following payoffs. In addition, compute the corresponding hedging strategies. 420 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance a) Up-and-in barrier long forward contract. Take ST − K if Max St > B, 06t6T C = (ST − K) 1n o = Max St > B 0 t T 0 if Max St 6 B. 6 6 06t6T b) Up-and-out barrier long forward contract. Take ST − K if Max St < B, 06t6T C = (ST − K) 1n o = Max St < B 0 t T 0 if Max St > B. 6 6 06t6T c) Down-and-in barrier long forward contract. Take ST − K if min St < B, 06t6T C = (ST − K) 1n o = min St < B 0 t T 0 if min St > B. 6 6 06t6T d) Down-and-out barrier long forward contract. Take ST − K if min St > B, 06t6T C = (ST − K) 1n o = min St > B 0 t T 0 if min St 6 B. 6 6 06t6T e) Up-and-in barrier short forward contract. Take K − ST if Max St > B, 06t6T C = (K − ST ) 1n o = Max St > B 0 t T 0 if Max St 6 B. 6 6 06t6T f) Up-and-out barrier short forward contract. Take K − ST if Max St < B, 06t6T C = (K − ST ) 1n o = Max St < B 0 t T 0 if Max St > B. 6 6 06t6T g) Down-and-in barrier short forward contract. Take 421 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault K − ST if min St < B, 06t6T C = (K − ST ) 1n o = min St < B 0 t T 0 if min St > B. 6 6 06t6T h) Down-and-out barrier short forward contract. Take K − ST if min St > B, 06t6T C = (K − ST ) 1n o = min St > B 0 t T 0 if min St 6 B. 6 6 06t6T Exercise 11.4 Compute the Vega of the down-and-out and down-and-in barrier call option prices, i.e. compute the sensitivity of down-and-out and down-and-in barrier option prices with respect to the volatility parameter σ. Exercise 11.5 Stability warrants. Price the up-and-out binary barrier option with payoff 1 1 1 C := {ST >K} T = T M0 K and M0 6B at time t = 0, with K 6 B. Exercise 11.6 Check that the function g(t, x) in (11.26) satisfies the bound- ary conditions g t B t ∈ T ( , ) = 0, [0, ], g(T , x) = 0, x K < B, 6 g(T , x) = x − K, K 6 x < B, g(T , x) = 0, x > B. Exercise 11.7 European knock-in/knock-out barrier options. Price the fol- lowing vanilla options by computing their conditional discounted expected payoffs: S − K +1 a) European knock-out barrier call option with payoff ( T ) {ST 6B}, K − S +1 b) European knock-in barrier put option with payoff ( T ) {ST 6B}, S − K +1 c) European knock-in barrier call option with payoff ( T ) {ST >B}, K − S +1 d) European knock-out barrier put option with payoff ( T ) {ST >B}, 422 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html