7. Barrier options, lookback options and Asian options
Path dependent options: payouts are related to the underlying asset price path history during the whole or part of the life of the option.
The barrier option is either nullified, activated or exercised when • the underlying asset price breaches a barrier during the life of the option.
The payoff of a lookback option depends on the minimum or • maximum price of the underlying asset attained during certain period of the life of the option
1 The payoff of an average option (usually called an Asian option) • depends on the average asset price over some period within the life of the option.
An interesting example is the Russian option, which is in fact a • perpetual American lookback option. The owner of a Russian option on a stock receives the historical maximum value of the asset price when the option is exercised and the option has no pre-set expiration date.
2 Discrete and continuous monitoring of asset price process
Due to the path dependent nature of these options, the asset • price process is monitored over the life of the option contract either for breaching of a barrier level, observation of new ex- tremum value or sampling of asset prices for computing average value.
In actual implementation, these monitoring procedures can only • be performed at discrete time instants rather than continuously at all times.
When the asset price path is monitored at discrete time instants, • the analytic forms of the price formulas become quite daunting since they involve multi-dimensional cumulative normal distri- bution functions and the dimension is equal to the number of monitoring instants.
3 Barrier options
1. An out-barrier option (or knock-out option) is one where the option is nullified prior to expiration if the underlying asset price touches the barrier. The holder of the option may be compen- sated by a rebate payment for the cancellation of the option. An in-barrier option (or knock-in option) is one where the option only comes in existence if the asset price crosses the in-barrier, though the holder has paid the option premium up front.
2. When the barrier is upstream with respect to the asset price, the barrier option is called an up-option; otherwise, it is called a down-option.
One can identify eight types of European barrier options, such as down-and-out calls, up-and-out calls, down-and-in puts, down- and-out puts, etc.
4 How both buyer and writer benefit from the up-and-out call?
With an appropriate rebate paid upon breaching the upside barrier, this type of barrier options provide the upside exposure for option buyer but at a lower cost.
The option writer is not exposed to unlimited liabilities when • the asset price rises acutely. Barrier options are attractive since they give investors more flex- • ibility to express their view on the asset price movement in the option contract design.
5 Discontinuity at the barrier (circuit breaker effect upon knock-out)
Pitched battles often erupt around popular knock-out barriers in • currency barrier options and these add much unwanted volatility to the markets. George Soros once said “knock-out options relate to ordinary • options the way crack relates to cocaine.”
6 European down-and-out call
∂c σ2 ∂2c ∂c = S2 + rS rc, S > B and τ (0, T ], ∂τ 2 ∂S2 ∂S − ∈ subject to
knock-out condition: c(B,τ)= R(τ) terminal payoff: c(S, 0) = max(S X, 0), − Set y = ln S, the barrier becomes the line y = ln B. ∂c σ2 ∂2c σ2 ∂c = + r rc ∂τ 2 ∂y2 − 2 ! ∂y − defined in the semi-infinite domain: y > ln B and τ (0, T ]. ∈
7 The auxiliary conditions become
c(ln B,τ)= R(τ) and c(y, 0) = max(ey X, 0). −
σ2 Write µ = r , the Green function in the infinite domain: < − 2 −∞ y < is given by ∞ rτ 2 e− (y + µτ ξ) G0(y,τ; ξ)= exp − , σ√2πτ − 2σ2τ ! where G0(y,τ; ξ) satisfies the initial condition:
lim G0(y,τ; ξ)= δ(y ξ). τ 0+ − →
8 Method of images
Assuming that the Green function in the semi-infinite domain takes the form G(y,τ; ξ)= G (y,τ; ξ) H(ξ)G (y,τ; η), 0 − 0 we are required to determine H(ξ) and η (in terms of ξ) such that the zero Dirichlet boundary condition G(ln B,τ; ξ) = 0 is satisfied.
Note that both G0(y,τ; ξ) and H(ξ)G0(y,τ; η) satisfy the differential equation. Also, provided that η (ln B, ), then 6∈ ∞ lim G0(y,τ; η)=0 for all y > ln B. τ 0+ →
By imposing the boundary condition, H(ξ) has to satisfy G (ln B,τ; ξ) (ξ η)[2(ln B + µτ) (ξ + η)] ( )= 0 = exp H ξ − 2 − . G0(ln B,τ; η) 2σ τ !
9 The assumed form of G(y,τ; ξ) is feasible only if the right hand side becomes a function of ξ only. This can be achieved by the judicious choice of η = 2ln B ξ, − so that 2µ ( ) = exp ( ln ) H ξ 2 ξ B . σ −
This method works only if µ/σ2 is a constant, independent of τ. • The parameter η can be visualized as the mirror image of ξ with • respect to the barrier y = ln B.
10 H(ξ)G0(y,τ; η) 2µ e rτ [y + µτ (2 ln B ξ)]2 = exp ( ln ) − exp 2 ξ B − 2 − σ − σ√2πτ − 2σ τ ! 2 B 2µ/σ e rτ [(y ξ)+ µτ 2(y ln B)]2 = − exp − −2 − . S σ√2πτ − 2σ τ !
The Green function in the specified semi-infinite domain: ln B 11 Pictorial representation of the method of image. The mirror is placed at y = ln B. 12 We consider the barrier option with zero rebate, where R(τ) = 0, and let K = max(B, X). The price of the zero-rebate European down-and-out call can be expressed as ∞ ξ cdo(y,τ) = max(e X, 0)G(y,τ; ξ) dξ Zln B − = ∞ (eξ X)G(y,τ; ξ) dξ Zln K − e rτ (u µτ)2 = − ∞ (Seu X) exp − √ 2 σ 2πτ Zln K/S − " − 2σ τ ! 2 B 2µ/σ (u 2β µτ)2 exp − − du. 2 − S − 2σ τ ! 13 The direct evaluation of the integral gives B δ+1 cdo(S, τ) = S N(d1) N(d3) " − S # δ 1 rτ B − Xe− N(d2) N(d4) , − " − S # where S σ2 ln K + r + 2 τ d1 = , d2 = d1 σ√τ, σ√τ − 2 B 2 B 2r d3 = d1 + ln , d4 = d2 + ln , δ = . σ√τ S σ√τ S σ2 14 Suppose we define c (S, τ; X, K)= SN(d ) Xe rτ N(d ), E 1 − − 2 then cdo(S, τ; X,e B) can be expressed in the following succinct form δ 1 2 B − B cdo(S, τ; X, B)= cE(S, τ; X, K) cE ,τ; X, K . − S S ! e e δ 1 2 B − B One can show by direct calculation that the function cE ,τ S S ! satisfies the Black-Scholes equation identically. The above form al- e lows us to observe readily the satisfaction of the boundary condition: cdo(B,τ) = 0, and terminal payoff condition. 15 Remarks 1. Closed form analytic price formulas for barrier options with ex- γτ ponential time dependent barrier, B(τ) = Be− , can also be derived. However, when the barrier level has arbitrary time de- pendence, the search for analytic price formula for the barrier option fails. Roberts and Shortland (1997) show how to derive the analytic approximation formula by estimating diffusion pro- cess boundary hitting times via the Brownian bridge technique. 2. Closed form price formulas for barrier options can also be ob- tained for other types of diffusion processes followed by the underlying asset price; for example square root constant elas- ticity of variance process the double exponential jump diffusion process. 16 3. Assuming B < X, the price of a down-and-in call option can be deduced to be δ 1 2 B − B cdi(S, τ; X, B)= cE ,τ; X . S S ! 4. The method of images can be extended to derive the density functions of restricted multi-state diffusion processes where bar- rier occurs in only one of the state variables. 5. The monitoring period for breaching of the barrier may be lim- ited to only part of the life of the option. 17 Transition density function and first passage time density The realized maximum and minimum value of the asset price from time zero to time t (under continuous monitoring) are defined by t m0 = min Su 0 u t t ≤ ≤ M0 = max Su, 0 u t ≤ ≤ respectively. The terminal payoffs of various types of barrier options T T can be expressed in terms of m0 and M0 . For example, consider the down-and-out call and up-and-out put, their respective terminal payoff can be expressed as ( ; ) = max( 0) cdo ST , T X, B ST X, 1 mT >B − { 0 } ( ; ) = max( 0) puo ST , T X, B X ST , 1 M T 18 Suppose B is the down-barrier, we define τB to be the stopping time at which the underlying asset price crosses the barrier for the first time: τ = inf t S B , S = S. B { | t ≤ } 0 Assume S > B and due to path continuity, we may express τB (commonly called the first passage time) as τ = inf t S = B . B { | t } In a similar manner, if B is the up-barrier and S < B, we have τ = inf t S B = inf t S = B . B { | t ≥ } { | t } 19 It is easily seen that τ > T and mT > B are equivalent • { B } { 0 } events if B is a down-barrier. By virtue of the risk neutral valuation principle, the price of a down-and-out call at time zero is given by rT cdo(S, 0; X, B) = e− EQ[max(ST X, 0)1 T ] − m0 >B rT { } = e− EQ[(ST X)1 S >max(X,B) 1 τ >T ]. − { T } { B } The determination of the price function c (S, 0; X, B) requires • d0 the determination of the joint distribution function of ST and T m0 . 20 Reflection principle 0 µ Let Wt (Wt ) denote the Brownian motion that starts at zero, with constant volatility σ and zero drift rate (constant drift rate µ). We T µ would like to find P [m0 < m,WT > x], where x m and m 0. ≥ 0 ≤ First, we consider the zero-drift Brownian motion Wt . T Given that the minimum value m0 falls below m, then there exists some time instant ξ, 0 <ξ 21 Suppose we define a random process W 0 for t < ξ W 0 = t t 2m W 0 for ξ t T, ( − t ≤ ≤ 0 f 0 that is, Wt is the mirror reflection of Wt at the level m within the time interval between ξ and T . It is then obvious that W 0 > x is f { T } equivalent to W 0 < 2m x . Also, the reflection of the Brownian { T − } path dictates that f W 0 W 0 = (W 0 W 0), u> 0. ξ+u − ξ − ξ+u − ξ f f 22 The stopping time ξ only depends on the path history W 0 : 0 • { t ≤ t ξ and it will not affect the Brownian motion at later times. ≤ } By the strong Markov property of Brownian motions, we argue • that the two Brownian increments have the same distribution, and the distribution has zero mean and variance σ2u. For every Brownian path that starts at 0, travels at least m units (down- ward, m 0) before T and later travels at least x m units ≤ − (upward, x m), there is an equally likely path that starts at ≥ 0, travels m units (downward, m 0) some time before T and ≤ travels at least m x units (further downward, m x). − ≤ Suppose W 0 >x, then W 0 < 2m x, • T T − P [W 0 > x,mT < m] =f P [W 0 < 2m x]= P [W 0 < 2m x] T 0 T − T − 2m x = N f − , m min(x, 0). σ√T ! ≤ 23 Pictorial representation of the reflection principle of the Brownian 0 motion Wt . The dotted path after time ξ is the mirror reflection of 0 the Brownian path at the level m. Suppose WT ends up at a value higher than x, then the reflected path at time T has a value lower than 2m x. − 24 Next, we apply the Girsanov Theorem to effect the change of mea- sure for finding the above joint distribution when the Brownian mo- µ tion has non-zero drift. Suppose under the measure Q, Wt is a Brownian motion with drift rate µ. We change the measure from µ Q to Q such that Wt becomes a Brownian motion with zero drift under Q. Consider the following joint distribution e e µ T P [WT > x,m0 < m] = EQ[1 µ 1 T ] WT >x m0 25 µ By applying the reflection principle and observing that WT is a zero- drift Brownian motion under Q, we obtain µ T e P [WT > x,m0 < m] 2 µ µ µ T = E 1 µ exp (2m W ) Q 2m W >x 2 T 2 " { − T } σ − − 2σ !# 2µm 2 µ µ µ T = e σe2 E 1 µ exp W Q W <2m x 2 T 2 " { T − } −σ − 2σ !# 2µm 2m x 1 z2 µz µ2T = e σ2 e − e−2σ2T e−σ2− 2σ2 dz √2 2 Z−∞ πσ T 2µm 2m x 1 (z + µT )2 = e σ2 − exp dz √2 2 − 2σ2T ! Z−∞ πσ T 2µm 2m x + µT = e σ2 N − , m min(x, 0). σ√T ! ≤ 26 µ Suppose the Brownian motion Wt has a downstream barrier m over T the period [0, T ] so that m0 > m, we would like to derive the joint distribution P [W µ > x,mT > m], and m min(x, 0). T 0 ≤ By applying the law of total probabilities, we obtain µ T P [WT > x,m0 > m] = P [W µ >x] P [W µ > x,mT < m] T − T 0 x + µT 2µm 2m x + µT = N − e σ2 N − , m min(x, 0). σ√T ! − σ√T ! ≤ µ Under the special case m = x, since WT > m is implicitly implied T from m0 > m, we have 2µm T m + µT 2 m + µT P [m0 > m]= N − e σ N . σ√T ! − σ√T ! 27 Extension to upstream barrier µ When the Brownian motion Wt has an upstream barrier M over the T µ period [0, T ] so that M0 28 By swapping µ for µ, M for m and y for x, we obtain − − − 2µM µ y 2M µT T σ2 P [WT < y,M0 >M]= e N − − ,M max(y, 0). σ√T ! ≥ In a similar manner, we obtain µ T P [WT < y,M0 2µM T M µT 2 M + µT P [M0 29 Density functions of restricted Brownian processes µ We define fdown(x, m, T ) to be the density function of WT with the downstream barrier m, where m min(x, 0), that is, ≤ f (x, m, T ) dx = P [W µ dx, mT > m]. down T ∈ 0 By differentiating with respect to x and swapping the sign, we obtain fdown(x, m, T ) 2µm 1 x µT 2 x 2m µT = n − e σ n − − 1 m min(x,0) . σ√T " σ√T ! − σ√T !# { ≤ } 30 µ Similarly, we define fup(x,M,T ) to be the density function of WT with the upstream barrier M, where M > max(y, 0), then P [W µ dy,M T 31 Suppose the asset price St follows the lognormal process under the S risk neutral measure such that ln t = W µ, where S is the asset price S t σ2 at time zero and the drift rate µ = r . Let ψ(S ; S, B) denote − 2 T the transition density of the asset price ST at time T given the asset price S at time zero and conditional on S > B for 0 t T . Here, t ≤ ≤ B is the downstream barrier. By Eq. (4.1.27b), we deduce that ψ(ST ; S, B) is given by 2 ln ST r σ T 1 S − − 2 ψ(ST ; S, B) = n σ√TST σ√T S B σ2 2r 1 ln T 2 ln r T B σ2− S − S − − 2 n . − S σ√T 32 First passage time density functions Let Q(u; m) denote the density function of the first passage time at which the downstream barrier m is first hit by the Brownian µ path W , that is, Q(u; m) du = P [τm du]. First, we determine t ∈ the distribution function P [τm > u] by observing that τm > u and { } mu > m are equivalent events. { 0 } u P [τm >u] = P [m0 > m] m + µu 2µm m + µu = N − e σ2 N . σ√u ! − σ√u ! 33 The density function Q(u; m) is then given by Q(u; m) du = P [τm du] ∈ 2µm ∂ m + µu 2 m + µu = N − e σ N du 1 m<0 −∂u " σ√u ! − σ√u !# { } m (m µu)2 = exp − −2 du 1 m<0 . √2πσ2u3 − 2σ u ! { } Let Q(u; M) denote the first passage time density associated with the upstream barrier M. 2µM ∂ M µu 2 M + µu Q(u; M) = N − e σ N 1 M>0 −∂u " σ√u ! − − σ√u !# { } M (M µu)2 = exp −2 1 M>0 . √2πσ2u3 − 2σ u ! { } 34 We write B as the barrier, either upstream or downstream. When B B the barrier is downstream (upstream), we have ln < 0 ln > 0 . S S 2 2 B B σ ln ln S r 2 u Q(u; B)= S exp − − . √ 2 3 − 2σ2u 2 πσ u Suppose a rebate R(t) is paid to the option holder upon breaching the barrier at level B by the asset price path at time t. Since the expected rebate payment over the time interval [u,u + du] is given by R(u)Q(u; B) du, then the expected present value of the rebate is given by T ru rebate value = e− R(u)Q(u; B) du. Z0 35 When R(t) = R0, a constant value, direct integration of the above integral gives B α+ ln B + βT rebate value = R N δ S 0 S σ√T B α ln B βT + − N δ S − , S σ√T where 2 2 σ2 σ 2 r 2 β β = v r + 2rσ , α = − ± u − 2 ! ± σ2 u t S and δ = sign ln . B Here, δ is a binary variable indicating whether the barrier is down- stream (δ = 1) or upstream (δ = 1). − 36 Correction formula for discretely monitored barrier options Let V (B; m) be the price of a discretely monitored knock-in or knock-out down call or up put option with constant barrier B. Let V (B) be the price of the corresponding continuously monitored bar- rier option. We have βσ√δt 1 V (B; m)= V (Be± )+ o , √m! 1 where β = ξ √2π 0.5826, ξ is the Riemann zeta function, − 2 , ≈ σ is the volatility. The “+” sign is chosen when B>S, while the “ ” sign is chosen − when B One observes that the correction shifts the barrier away from the current underlying asset price by a factor of eβσ√δt. 37 Lookback options Let T denote the time of expiration of the option and [T0, T ] be the lookback period. We denote the minimum value and maximum value of the asset price realized from T to the current time t (T t T ) 0 0 ≤ ≤ by t mT = min Sξ 0 T ξ t 0≤ ≤ and t M T = max Sξ 0 T ξ t 0≤ ≤ 38 A floating strike lookback call gives the holder the right to buy • at the lowest realized price while a floating strike lookback put allows the holder to sell at the highest realized price over the lookback period. Since S mT and M T S so that the T ≥ T0 T0 ≥ T holder of a floating strike lookback option always exercise the option. Hence, the respective terminal payoff of the lookback call and put are given by S mT and M T S . T − T0 T0 − T 39 A fixed strike lookback call (put) is a call (put) option on the • maximum (minimum) realized price. The respective terminal payoff of the fixed strike lookback call and put are max(M T T0 − X, 0) and max(X mT , 0), where X is the strike price. − T0 Under the risk neutral measure, the process for the stochastic • S variable U = ln ξ is a Brownian process with drift rate µ = ξ S σ2 r and variance rate σ2, where r is the riskless interest rate − 2 and S is the asset price at current time t (dropping subscript t for brevity). 40 We define the following stochastic variables mT y = ln t = min U , ξ [t, T ] T S { ξ ∈ } M T Y = ln t = max U , ξ [t, T ] , T S { ξ ∈ } and write τ = T t. For y 0 and y u, we can deduce the following − ≤ ≤ joint distribution function of UT and yT from the transition density function of the Brownian process with the presence of a downstream barrier 2µy u + µτ 2 u + 2y + µτ P [UT u,yT y]= N − e σ N − . ≥ ≥ σ√τ ! − σ√τ ! Here, Uξ is visualized as a restricted Brownian process with drift rate µ and downstream absorbing barrier y. 41 Similarly, for y 0 and y u, the corresponding joint distribution ≥ ≥ function of UT and YT is given by 2µy u µτ 2 u 2y µτ P [UT u,YT y]= N − e σ N − − . ≤ ≤ σ√τ ! − σ√τ ! By taking y = u in the above two joint distribution functions, we obtain the distribution functions for yT and YT 2µy y + µτ 2 y + µτ P (yT y) = N − e σ N , y 0, ≥ σ√τ ! − σ√τ ! ≤ 2µy y µτ 2 y µτ P (YT y) = N − e σ N − − , y 0. ≤ σ√τ ! − σ√τ ! ≥ The density functions of yT and YT can be obtained by differentiating the above distribution functions. 42 European fixed strike lookback options Consider a European fixed strike lookback call option whose terminal payoff is max(M T X, 0). The value of this lookback call option at T0 − the current time t is given by c (S, M, t)= e rτ E max(max(M,M T ) X, 0) , fix − t − t h i where St = S, M = M and τ = T t, and the expectation is T0 − taken under the risk neutral measure. The payoff function can be simplified into the following forms, depending on M X or M > X: ≤ (i) M X ≤ max(max(M,M T ) X, 0) = max(M T X, 0) t − t − (ii) M > X max(max(M,M T ) X, 0) = (M X) + max(M T M, 0). t − − t − 43 Define the function H by H(S, τ; K)= e rτ E[max(M T K, 0)], − t − where K is a positive constant. Once H(S, τ; K) is determined, then H(S, τ; X) if M X c (S,M,τ) = ≤ fix e rτ (M X)+ H(S, τ; M) if M > X ( − − = e rτ max(M X, 0) + H(S, τ; max(M, X)). − − c (S,M,τ) is independent of M when M X because the • fix ≤ terminal payoff is independent of M when M X. ≤ When M > X, the terminal payoff is guaranteed to have the • floor value M X. If we subtract the present value of this − guaranteed floor value, then the remaining value of the fixed strike call option is equal to a new fixed strike call but with the strike being increased from X to M. 44 Since max(M T K, 0) is a non-negative random variable, its ex- t − pected value is given by the integral of the tail probabilities where e rτ E[max(M T K, 0)] − t − rτ ∞ T = e− P [Mt K x] dx 0 − ≥ Z T rτ ∞ Mt z = e− P ln ln dz z = x + K ZK " S ≥ S # rτ ∞ y z = e− Se P [YT y] dy y = ln ln K ≥ Z S S + 2µy rτ ∞ y y µτ 2 y µτ = e− Se N − + e σ N − − dy, ln K Z S " σ√τ ! σ√τ !# 45 rτ H(S, τ; K) = SN(d) e− KN(d σ√τ) − − 2r σ2 S 2 2r +e rτ S erτ N(d) −σ N d √τ , − 2r − K − σ where S σ2 ln K + r + 2 τ d = . σ√τ The European fixed strike lookback put option with terminal payoff max(X mT , 0) can be priced in a similar manner. Write m = mt − T0 T0 and define the function h(S, τ; K)= e rτ E[max(K mT , 0)]. − − t 46 The value of this lookback put can be expressed as p (S, m, τ)= e rτ max(X m, 0) + h(S, τ; min(m, X)), fix − − where rτ ∞ T h(S, τ; K) = e− P [max(K mt , 0) x] dx 0 − ≥ Z K rτ T T = e− P [K mt x] dx since 0 max(K mt , 0) K 0 − ≥ ≤ − ≤ Z K rτ T = e− P [mt z] dz z = K x Z0 ≤ − ln K rτ S y y = e− Se P [yT y] dy y = ln Z0 ≤ S ln K 2µy rτ S y y µτ 2 y + µτ = e− Se N − + e σ N dy Z0 " σ√τ ! σ√τ !# 2 rτ rτ σ = e− KN( d + σ√τ) SN( d)+ e− S − − − 2r 2 S 2r/σ 2r − N d + √τ erτ N( d) . K − σ − − 47 European floating strike lookback options By exploring the pricing relations between the fixed and floating lookback options, we can deduce the price functions of floating strike lookback options from those of fixed strike options. Consider a European floating strike lookback call option whose terminal payoff is S mT , the present value of this call option is given by T − T0 rτ T cfℓ(S, m, τ) = e− E[ST min(m, mt )] rτ − T = e− E[(ST m) + max(m mt , 0)] rτ − − = S me− + h(S, τ; m) − 2 rτ rτ σ = SN(dm) e− mN(dm σ√τ)+ e− S − − 2r 2r S −σ2 2r rτ N dm + √τ e N( dm) , m − σ − − where S σ2 ln m + r + 2 τ dm = . σ√τ 48 In a similar manner, consider a European floating strike lookback put option whose terminal payoff is M T S , the present value of T0 − T this put option is given by rτ T pfℓ(S,M,τ) = e− E[max(M,Mt ) ST ] rτ T − = e− E[max(Mt M, 0) (ST M)] − rτ− − = H(S, τ; M) (S Me− ) − − 2 rτ rτ σ = e− MN( d + σ√τ) SN( d )+ e− S − M − − M 2r 2r S 2 2r erτ N(d ) −σ N d √τ , M M − M − σ where S σ2 ln M + r + 2 τ d = . M σ√τ 49 Boundary condition at S = m Consider the particular situation when S = m, that is, the cur- • rent asset price happens to be at the minimum value realized so far. The probability that the current minimum value remains to be the realized minimum value at expiration is expected to be zero. We can argue that the value of the floating strike lookback • call should be insensitive to infinitesimal changes in m since the change in option value with respect to marginal changes in m is proportional to the probability that m will be the realized minimum at expiry ∂c fℓ(S, m, τ) = 0. ∂m S=m 50 Rollover strategy and strike bonus premium The sum of the first two terms in c can be seen as the price • fℓ function of a European vanilla call with strike price m, while the third term can be interpreted as the strike bonus premium. We consider the hedging of the floating strike lookback call by • the following rollover strategy. At any time, we hold a European vanilla call with the strike price set at the current realized mini- mum asset value. In order to replicate the payoff of the floating strike lookback call at expiry, whenever a new realized minimum value of the asset price is established at a later time, one should sell the original call option and buy a new call with the same expiration date but with the strike price set equal to the newly established minimum value. 51 Since the call with a lower strike is always more expensive, an • extra premium is required to adopt the rollover strategy. The sum of these expected costs of rollover is termed the strike bonus premium. We would like to show how the strike bonus premium can be • obtained by integrating a joint probability distribution function T involving mt and ST . First, we observe that strike bonus premium = h(S, τ; m)+ S me rτ c (S, τ; m) − − − E = h(S, τ; m) p (S, τ; m), − E where cE(S, τ, m) and pE(S, τ; m) are the price functions of Eu- ropean vanilla call and put, respectively. The last result is due to put-call parity relation. 52 Recall m rτ T h(S, τ; m)= e− P [mt ξ] dξ Z0 ≤ and rτ ∞ pE(S, τ; m) = e− P [max(m ST , 0) x] dx 0 − ≥ Z m rτ = e− P [ST ξ] dξ. Z0 ≤ Since the two stochastic state variables satisfies 0 mT S , we ≤ t ≤ T have P [mT ξ] P [S ξ]= P [mT ξ 53 Differential equation formulation We would like to illustrate how to derive the governing partial dif- ferential equation and the associated auxiliary conditions for the European floating strike lookback put option. First, we define the quantity t 1/n n Mn = (Sξ) dξ , t>T0, "ZT0 # the derivative of which is given by 1 Sn dM = dt n n 1 n(Mn) − so that dMn is deterministic. Taking the limit n , we obtain →∞ M = lim Mn = max Sξ, n T ξ t →∞ 0≤ ≤ giving the realized maximum value of the asset price process over the lookback period [T0, t]. 54 We attempt to construct a hedged portfolio which contains one • unit of a put option whose payoff depends on Mn and units of −△ the underlying asset. Again, we choose so that the stochastic △ components associated with the option and the underlying asset cancel. Let p(S, Mn, t) denote the value of the lookback put option and • let Π denote the value of the above portfolio. We then have Π= p(S, Mn, t) S. − △ The dynamics of the portfolio value is given by • ∂p 1 Sn ∂p ∂p σ2 ∂2p dΠ= dt + dt + dS + S2 dt dS n 1 2 ∂t n(Mn) − ∂Mn ∂S 2 ∂S − △ ∂p by virtue of Ito’s lemma. Again, we choose = so that the △ ∂S stochastic terms cancel. 55 Using usual no-arbitrage argument, the non-stochastic portfolio • should earn the riskless interest rate so that dΠ= rΠ dt, where r is the riskless interest rate. Putting all equations to- gether, ∂p 1 Sn ∂p σ2 ∂2p ∂p + + S2 + rS rp = 0. n 1 2 ∂t n(Mn) − ∂Mn 2 ∂S ∂S − Now, we take the limit n and note that S M. When • 1 Sn → ∞ ∂p ≤ S 56 The domain of the pricing model has an upper bound M on S. The variable M does not appear in the equation, though M appears as a parameter in the auxiliary conditions. The final condition is p(S, M, T )= M S. − In this European floating strike lookback put option, the boundary conditions are applied at S = 0 and S = M. Once S becomes zero, it stays at the zero value at all subsequent times and the payoff at expiry is certain to be M. Discounting at the riskless interest rate, the lookback put value at the current time t is r(T t) p(0,M,t)= e− − M. The boundary condition at the other end S = M is given by ∂p =0 at S = M. ∂M 57 Asian options Asian options are averaging options whose terminal payoff de- • pends on some form of averaging of the price of the underlying asset over a part or the whole of option’s life. There are frequent situations where traders may be interested • to hedge against the average price of a commodity over a period rather than, say, end-of-period price. Averaging options are particularly useful for business involved in • trading on thinly-traded commodities. The use of such financial instruments may avoid the price manipulation near the end of the period. 58 The most common averaging procedures are the discrete arithmetic averaging defined by 1 n A = St T n i iX=1 and the discrete geometric averaging defined by n 1/n A = S . T ti iY=1 Here, S is the asset price at discrete time t ,i = 1, 2, ,n. ti i ··· In the limit n , the discrete sampled averages become the → ∞ continuous sampled averages. The continuous arithmetic average is given by 1 T AT = St dt, T Z0 while the continuous geometric average is defined to be 1 T AT = exp ln St dt . T Z0 ! 59 Partial differential equation formulation Suppose we write the average of the asset price as t A = f(S, u) du, Z0 where f(S, t) is chosen according to the type of average adopted in 1 the Asian option. For example, f(S, t) = S corresponds to con- t 1 n tinuous arithmetic average, f(S, t) = exp δ(t t ) ln S corre- n − i iX=1 sponds to discrete geometric average, etc. 60 Suppose f(S, t) is a continuous time function, then by the mean value theorem t+∆t dA = lim f(S, u) du = lim f(S, u∗) dt = f(S, t) dt, ∆t 0 t ∆t 0 → Z → t Consider a portfolio which contains one unit of the Asian option and ∆ units of the underlying asset. We then choose ∆ such − that the stochastic components associated with the option and the underlying asset cancel off each other. 61 Assume the asset price dynamics to be given by dS = [µS D(S, t)] dt + σS dZ, − where Z is the standard Wiener process, D(S, t) is the dividend yield on the asset, µ and σ are the expected rate of return and volatility of the asset price, respectively. Let V (S, A, t) denote the value of the Asian option and let Π denote the value of the above portfolio. The portfolio value is given by Π= V (S, A, t) ∆S, − and its differential is found to be ∂V ∂V ∂V σ2 ∂2V dΠ= dt+f(S, t) dt+ dS + S2 dt ∆ dS ∆D(S, t) dt. ∂t ∂A ∂S 2 ∂S2 − − 62 The last term in the above equation corresponds to the contribution of the dividend from the asset to the portfolio’s value. As usual, we ∂V choose ∆ = so that the stochastic terms containing dS cancel. ∂S The absence of arbitrage dictates dΠ= rΠ dt, where r is the riskless interest rate. Putting the results together, we obtain ∂V σ2 ∂2V ∂V ∂V + S2 + [rS D(S, t)] + f(S, t) rV = 0. ∂t 2 ∂S2 − ∂S ∂A − The equation is a degenerate diffusion equation since it contains diffusion term corresponding to S only but not for A. The auxiliary conditions in the pricing model depend on the specific details of the Asian option contract. 63 Continuously monitored geometric averaging options We take time zero to be the time of initiation of the averaging • period, t is the current time and T denotes the expiration time. We define the continuously monitored geometric averaging of • the asset price Su over the time period [0, t] by 1 t Gt = exp ln Su du . t Z0 The terminal payoff of the fixed strike call option and floating strike call option are, respectively, given by c (S ,G , T ; X) = max(G X, 0) fix T T T − c (S ,G , T ) = max(S G , 0), fℓ T T T − T where X is the fixed strike price. 64 European fixed strike Asian call option We assume the existence of a risk neutral pricing measure Q under which discounted asset prices are martingales, implying the absence of arbitrage. Under the measure Q, the asset price follows dSt = (r q) dt + σ dZt, St − where Zt is a Q-Brownian motion. For 0 By integrating ln St, we obtain t 1 σ2 (T t)2 ln GT = ln Gt + (T t) ln St + r q − T T " − − − 2 ! 2 # σ T + (Zu Zt) du. T Zt − 65 σ T The stochastic term (Zu Zt) du can be shown to be Gaus- T Zt − σ2 (T t)3 sian with zero mean and variance − . By the risk neutral T 2 3 valuation principle, the value of the European fixed strike Asian call option is given by c (S ,G , t)= e r(T t)E[max(G X, 0)], fix t t − − T − where E is the expectation under Q conditional on St = S, Gt = G. We assume the current time t to be within the averaging period. By defining σ2 (T t)2 σ (T t)3 µ = r q − and σ = − , − − 2 ! 2T T s 3 GT can be written as t/T (T t)/T GT = Gt St − exp(µ + σZ), where Z is the standard normal random variable.b b 66 Recall E[max(F exp(µ + σZ) X, 0] F − 2 F 2 ln + µ + σ ln + µ = F eµ+σ /2N X b XN X , σ − σ we then deduce that r(T t) t/T (T t)/T µ+σ2/2 cfix(S, G, t)= e− − G S − e N(d1) XN(d2) , − where t T t d2 = ln G + − ln S + µ ln X σ, T T − , d1 = d1 + σ. 67 European floating strike Asian call option Since the terminal payoff of the floating strike Asian call option involves ST and GT , pricing by the risk neutral expectation approach would require the joint distribution of ST and GT . For floating strike Asian options, the partial differential equation method provides the more effective approach to derive the price formula for cfℓ(S, G, t). This is because the similarity reduction technique can be applied to reduce the dimension of the differential equation. When continuously monitored geometric averaging is adopted, the governing equation for cfℓ(S, G, t) can be expressed as ∂c σ2 ∂2c ∂c G S ∂c fℓ + S2 fℓ + (r q)S fℓ + ln fℓ rc = 0, ∂t 2 ∂S2 − ∂S t G ∂G − fℓ 0 68 Next, we define the similarity variables G c (S, G, t) y = t ln and W (y, t)= fℓ . S S This is equivalent to choose S as the numeraire. In terms of the similarity variables, the governing equation for cfℓ(S, G, t) becomes ∂W σ2t2 ∂2W σ2 ∂W + r q + t qW = 0, 0 69 We write τ = T t and let F (y,τ; η) denote the fundamental solution − to the following parabolic equation with time dependent coefficients ∂F σ2(T τ)2 ∂2F σ2 ∂F = − r q + (T τ) , τ> 0, ∂τ 2 ∂y2 − − 2 ! − ∂y with initial condition at τ = 0 (corresponding to t = T ) given as F (y, 0; η)= δ(y η). − 70 Though the differential equation has time dependent coefficients, the fundamental solution is readily found to be σ2 τ y η r q + 2 0 (T u) du ( ; )= − − − − F y,τ η n R . σ τ (T u)2 du 0 − qR The solution to W (y,τ) is then given by W (y,τ)= e qτ ∞ max(1 eη/T , 0)F (y,τ; η) dη. − − Z−∞ 71 The direct integration of the above integral gives c (S, G, t)= Se q(T t)N(d ) Gt/T S(T t)/T e q(T t)e QN(d ), fℓ − − 1 − − − − − 2 where b b b S σ2 T 2 t2 t ln + r q + 3 3 G − 2 2− σ T t d1 = , d2 = d1 − , T 3 t3 − T s 3 σ − b 3 b b 2 r r q + σ T 2 t2 σ2 T 3 t3 Q = − 2 − − . 2 T − 6 T 2 b 72 Continuously monitored arithmetic averaging options We consider a European fixed strike European Asian call based on continuously monitored arithmetic averaging. The terminal payoff is defined by c (S ,A , T ; X) = max (A X, 0) . fix T T T − To motivate the choice of variable transformation, we consider the following expectation representation of the price of the Asian call at time t c (S ,A , t) = e r(T t)E [max (A X, 0)] fix t t − − T − t T r(T t) 1 1 = e− − E max Su du X + Su du, 0 " T Z0 − T Zt !# T St r(T t) Su = e− − E max xt + du, 0 , T " Zt St !# where the state variable xt is defined by 1 t xt = (It XT ), It = Su du = tAt. St − Z0 73 In subsequent exposition, it is more convenient to use It instead of At as the averaging state variable. Since Su/St,u>t, is independent of the history of the asset price up to time t, one argues that the conditional expectation is a function of xt only. We then deduce that cfix(St, It, t)= Stf(xt, t) for some function of f. In other words, f(xt, t) is given by r(T t) T e− − Su f(xt, t)= E max xt + du, 0 . T " Zt St !# If we write the price function of the fixed strike call as cfix(S, I, t), then the governing equation for cfix(S, I, t) is given by ∂c σ2 ∂2c ∂c ∂c fix + S2 fix + (r q)S fix + S fix rc = 0. ∂t 2 ∂S2 − ∂S ∂I − fix 74 Suppose we define the following transformation of variables: 1 c (S, I, t) x = (I XT ) and f(x, t)= fix , S − S then the governing differential equation for f(x, t) becomes ∂f σ2 ∂2f ∂f + x2 + [1 (r q)x] qf = 0, 75 1 t When xt 0, which corresponds to Su du X, it is possible to ≥ T Z0 ≥ find closed form analytic solution to f(x, t). Since xt is an increasing function of t so that x 0, the terminal condition f(x, T ) reduces T ≥ to x/T . In this case, f(x, t) admits solution of the form f(x, t)= a(t)x + b(t). By substituting the assumed form of solution into the governing equation, we obtain the following pair of governing equations for a(t) and b(t): da(t) 1 ra(t) = 0, a(T )= , dt − T db(t) a(t) qb(t) = 0, b(T ) = 0. dt − − 76 When r = q, a(t) and b(t) are found to be 6 e r(T t) e q(T t) e r(T t) a(t)= − − and b(t)= − − − − − . T T (r q) − Hence, the option value for I XT is given by ≥ q(T t) r(T t) I r(T t) e− − e− − c (S, I, t)= X e− − + − S. fix T − T (r q) − Though the volatility σ does not appear explicitly in the above price formula, it appears implicitly in S and A. The gamma is easily seen to be zero while the delta is a function of t and T t but not S or − A. For I 77 Put-call parity relation Let cfix(S, I, t) and pfix(S, I, t) denote the price function of the fixed strike arithmetic averaging Asian call option and put option, respec- tively. Their terminal payoff functions are given by c (S, I, T )= max I X, 0 fix T − I pfix(S, I, T ) = max X , 0 , − T T where I = Su du. Let D(S, I, t) denote the difference of cfix and Z0 pfix. Since both cfix and pfix are governed by the same equation so does D(S, I, t). 78 The terminal condition of D(S, I, t) is given by I I I D(S, I, T ) = max X, 0 max X , 0 = X. T − − − T T − The terminal condition D(S, I, T ) is the same as that of the contin- uously monitored arithmetic averaging option with I XT . Hence, ≥ when r = q, the put-call parity relation between the prices of fixed 6 strike Asian options under continuously monitored arithmetic aver- aging is given by cfix(S, I, t) pfix(S, I, t) − q(T t) r(T t) I r(T t) e− − e− − = X e− − + − S. T − T (r q) − Similar techniques can be used to derive the put-call parity relations between other types of Asian options (floating/fixed strike and ge- ometric/arithmetic averaging). 79 Fixed-floating symmetry relations By applying a change of measure and identifying a time-reversal of a Brownian motion, it is possible to establish the symmetry rela- tions between the prices of floating strike and fixed strike arithmetic averaging Asian options at the start of the averaging period. Suppose we write the price functions of various continuously moni- tored arithmetic averaging option at the start of the averaging pe- riod (taken to be time zero) as c (S , λ, r, q, T )= e rT E [max (λS A , 0)] fl 0 − T − T p (S , λ, r, q, T )= e rT E [max (A λS , 0)] fl 0 − T − T c (X,S , r, q, T )= e rT E [max (A X, 0)] fix 0 − T − p (X,S , r, q, T )= e rT E [max (X A , 0)] . fix 0 − − T 80 Assume that the asset price St follows the Geometric Brownian process under the risk neutral measure Q, where dSt = (r q) dt + σ dZt. St − Here, Zt is a Q-Brownian process. Suppose the asset price is used as the numeraire, then rT cfl e− cfl∗ = = E [max (λST AT , 0)] S0 S0 − S e rT max (λS A , 0) = E T − T − T . " S0 ST # To effect the change of numeraire, we define the measure Q∗ by 2 rT dQ∗ σ + ST e− = e 2 T σZT = . − qT dQ S0e− 81 By virtue of the Girsanov Theorem, Z = Z σt is a Q -Brownian t∗ t − ∗ process. If we write AT∗ = AT /ST , then c = e qT E max λ A , 0 , fl∗ − ∗ − T∗ where E∗ denotes the expectation under Q∗. Now, we consider 1 T Su 1 T AT∗ = du = Su∗(T ) du, T Z0 ST T Z0 where σ2 Su∗(T ) = exp r q (T u) σ(ZT Zu) . − − − 2 ! − − − ! In terms of the Q -Brownian process Z , where Z Zu = σ(T ∗ t∗ T − − u)+ Z Z , we can write T∗ − u∗ σ2 Su∗(T ) = exp r q + (u T )+ σ(Zu∗ ZT∗ ) . − 2 ! − − ! 82 Furthermore, we define a reflected Q∗-Brownian process starting at zero by Zt, where Zt = Zt∗, then ZT u equals in law to Zu∗ ZT∗ due − − − to the stationary increment property of a Brownian process. Hence, b b b we establish T σ2 law 1 σZT u+ r q+ 2 (u T ) AT∗ = AT = e − − − du, T 0 Z b b and via time-reversal of ZT u, we obtain − b T σ2 1 σZξ+ q r 2 ξ AT = e − − dξ. T 0 Z b b Note that AT S0 is the arithmetic average of the price process with drift rate q r. b− 83 Summing the results together, we have c = S c = e qT E max λS A S , 0 , fl 0 fl∗ − ∗ 0 − T 0 h i and from which we deduce the following fixed-floatingb symmetry relation cfl(S0, λ, r, q, T )= pfix(λS0,S0, q, r, T ). By combining the put-call parity relations for floating and fixed Asian options and the above symmetry relation, we can derive the follow- ing fixed-floating symmetry relation between cfix and pfl. X cfix(X,S0, r, q, T )= pfl S0, , q, r, T . S0 ! 84