CHAPTER 4 Path Dependent Options

The competition pressure for financial products push financial institutions to design and develop more innovative financial derivatives, many of which are tended toward specific needs of customers. Recently, there has been a growing popularity for path dependent options, so called since their 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 the most popular path dependent option that is either nullified, activated or exercised when the underlying asset price breaches a barrier during the life of the option. The capped stock-index options, based on Standard and Poor’s (S & P) 100 and 500 Indexes, are barrier options now traded in option exchanges (they were launched in the Chicago Board of Exchange in 1991). These capped options will be exercised automatically when the index value exceeds the cap at the close of the day. 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, while 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. In this chapter, we present details of the product nature of barrier options, lookback options and Asian options, together with the analytic procedures for their valuation. 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 extremum value or sam- pling 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. However, most pricing models of path dependent options assume continuous monitoring of asset price in order to achieve good analytic tractability. We derive analytic price formulas for most common types of continuously monitored barrier and lookback options, and geometric averaged Asian options. For arithmetic averaged Asian options, we manage to obtain analytic approximation valuation formulas. Under the Black-Scholes pricing paradigm, we assume that the uncertainty in the finan- 166 4 Path Dependent Options cial market over the time horizon [0, T ] is modeled by a filtered probability space (Ω, , t t [0,T ], Q), where Q is the unique risk neutral (equivalent F {F } ∈ martingale) probability measure and the filtration t is generated by the standard Brownian process Z(u) : 0 u t . UnderF Q, the asset price { ≤ ≤ } process St follows the Geometric Brownian process with riskless interest rate r as the drift rate and constant volatility σ. In addition, all discounted prices of securities are martingales under Q. 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 distribution functions and the dimension is equal to the number of monitoring instants. We discuss briefly some effective analytic approximation techniques for estimating the prices of discretely monitored path dependent options.

4.1 Barrier options

Options with the barrier feature are considered to be the simplest types of path dependent options. The distinctive feature is that the payoff depends not only on the final price of the underlying asset but also whether the asset price has breached (one-touch) some barrier level during the life of the option. 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 compensated 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. 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. Also, we may have two-sided barrier options that have both upside and downside barriers. Nullification or activation of the contract occurs either when one of the barriers is touched or only when the two barriers are breached in a pre-specified sequential order. The later type of options are called sequential barrier options. Suppose the knock-in or knock-out feature is activated only when the asset price breaches the barrier for a pre-specified length of time (rather than one touch of the barrier), we name this special type of barrier options as Parisian options (Chesney et al., 1997). Why barrier options are popular? From the perspective of the buyer of an option contract, he can achieve option premium reduction through the barrier provision by not paying premium to cover those unlikely scenarios as viewed by himself. For example, the buyer of a down-and-out call believes that the asset price would never fall below some floor value, so he can reduce option 4.1 Barrier options 167

premium by allowing the option to be nullified when the asset price does fall below the perceived floor value. How both buyer and writer may 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. On the other hand, the option writer is not exposed to unlimited liabilities when the asset price rises acutely. In general, barrier options are attractive since they give investors more flexibility to express their view on the asset price movement in the option contract design. The very nature of discontinuity at the barrier (circuit breaker effect upon knock-out) creates hedging problems with the barrier options. It is extremely difficult for option writers to hedge barrier options when the asset price is around the barrier level. 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.” More details on the discussion of hedging problems of barrier options can be found in Linetsky’s paper (1999). Also, Hsu (1997) discusses the difficulty of market implementation of criteria for determining barrier events. In order to avoid unpleasant surprises of being knocked out, the criteria used should be impartial, objective and consistent. Consider a portfolio of one European in-option and one European out- option, both have the same barrier, strike price and date of expiration. The sum of their values is simply the same as that of a corresponding European option with the same strike price and date of expiration. From financial intuition, this is obvious since only one of the two barrier options survives at expiry and either payoff is the same as that of the European option. Hence, provided there is no rebate payment, we have

cordinary = cdown-and-out + cdown-and-in (4.1.1a)

pordinary = pup-and-out + pup-and-in (4.1.1b) where c and p denote call and put values, respectively. That is, the value of an out-option can be found easily once the value of the corresponding in-option is available, or vice versa. In this section, we derive analytic price formulas for European options with either one-sided barrier or two-sided barriers based on continuous monitoring of the asset price process. Under the Black-Scholes pricing paradigm, we can solve the pricing models using both the partial differential equation approach and martingale pricing approach. We derive the Green function (fundamental solution) of the governing Black-Scholes equation in a restricted domain using the method of images. When the martingale approach is used, we obtain the transition density function using the reflection principle in the Brownian motion literature. To compute the expected present value of the rebate payment, we derive the density function of the first passage time to the barrier. We also extend our derivation of price formulas to options with 168 4 Path Dependent Options double barriers. The effects of discrete monitoring of the barrier on option prices will be discussed at the end of the section. Barrier option models can also be used to model the nullifying effect on a bond when its issuer defaults. We model default occurrence of a bond issuer by the fall of the issuer’s firm value to some liability level. In Sec. 9.1, we show how to apply the barrier option models to devise the pricing models of defaultable bonds.

4.1.1 European down-and-out call options

The down-and-out call options have been available in the U.S. market since 1967 and the analytic price formula ﬁrst appears in the pioneering paper by Merton (1973; Chap. 1). A down-and-out call has similar features as an ordinary call option, except that it becomes nulliﬁed when the asset price S falls below the knock-out level B.

Partial diﬀerential equation formulation Let B denote the constant down-and-out barrier. The domain of deﬁnition for the barrier option model now becomes [B, ) [0, T ] in the S τ plane. Let R(τ) denote the time-dependent rebate paid∞ to×the holder when− the barrier is hit. Taking the usual Black-Scholes assumptions (frictionless market, continuous trading, etc.), the governing equation of the down-and-out barrier call option is given by ∂c σ2 ∂2c ∂c = S2 + rS rc, S > B and τ (0, T ], (4.1.2) ∂τ 2 ∂S2 ∂S − ∈ subject to

knock-out condition: c(B, τ) = R(τ) (4.1.3a) terminal payoff: c(S, 0) = max(S X, 0), (4.1.3b) − where c = c(S, τ) is the barrier option value, r and σ are the constant riskless interest rate and volatility, respectively. The down-barrier is normally set to be below the strike price X, otherwise the down-and-out call may be knocked out even it expires in-the-money. The partial differential equation formulation implies that knock-out occurs when the barrier is breached at any time during the life of the option. Suppose we apply the transformation of the independent variable: y = ln S, the barrier becomes the line y = ln B. Now, the Black-Scholes equation (4.1.2) is reduced to the following constant coefficient equation for c(y, τ)

∂c σ2 ∂2c σ2 ∂c = + r rc (4.1.4a) ∂τ 2 ∂y2 − 2 ∂y − deﬁned in the semi-inﬁnite domain: y > ln B and τ (0, T ]. The auxiliary conditions become ∈ 4.1 Barrier options 169

c(ln B, τ) = R(τ) and c(y, 0) = max(ey X, 0). (4.1.4b) − σ2 Write µ = r and recall that the Green function of Eq. (4.1.4a) in the − 2 infinite domain: < y < is given by [see Eq. (3.3.5)] −∞ ∞ rτ 2 e− (y + µτ ξ) G (y, τ; ξ) = exp − , (4.1.5) 0 √ − 2σ2τ σ 2πτ where G0(y, τ; ξ) satisfies the initial condition: lim G0(y, τ; ξ) = δ(y ξ). τ 0+ − → Method of images We would like to seek the Green function of Eqs. (4.1.4a,b) in the semi-infinite domain: ln B < y < with zero Dirichlet boundary condition at y = ln B. Assuming that the Green∞ function takes the form G(y, τ; ξ) = G (y, τ; ξ) H(ξ)G (y, τ; η), (4.1.6) 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 G(y, τ; ξ) satisfies Eq. (4.1.4a) since 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. By imposing the boundary condition, τ 0+ H→(ξ) has to satisfy G (ln B, τ; ξ) (ξ η)[2(ln B + µτ) (ξ + η)] H(ξ) = 0 = exp − − .(4.1.7) G (ln B, τ; η) 2σ2τ 0 The assumed form of G(y, τ; ξ) is feasible only if the right hand side of Eq. (4.1.7) becomes a function of ξ only. This can be achieved by the judicious choice of η = 2 ln B ξ, (4.1.8a) − so that 2µ H(ξ) = exp (ξ ln B) . (4.1.8b) σ2 − The parameter η can be visualized as the mirror image of ξ with respect to the barrier y = ln B. This is how the name of this method is derived (see Fig. 4.1). Now, by grouping the terms involving exponentials, the second term in Eq. (4.1.6) can be expressed as

H(ξ)G0(y, τ; η) rτ 2 2µ e− [y + µτ (2 ln B ξ)] = exp (ξ ln B) exp − − σ2 − σ√2πτ − 2σ2τ 2 2µ/σ rτ 2 B e− [(y ξ) + µτ 2(y ln B)] = exp − − − . (4.1.9) S √ − 2σ2τ σ 2πτ 170 4 Path Dependent Options

Collecting all the terms together, the Green function in the speciﬁed semi- inﬁnite domain: ln B < y < becomes ∞ rτ 2 e− (u µτ) G(y, τ; ξ) = exp − σ√2πτ − 2σ2τ 2 B 2µ/σ (u 2β µτ)2 exp − − ,(4.1.10) − S − 2σ2τ ) B where u = ξ y and β = ln B y = ln . − − S

Fig. 4.1 Pictorial representation of the method of image. The mirror is placed at y = ln B.

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

∞ c (y, τ) = max(eξ X, 0)G(y, τ; ξ) dξ do − Zln B ∞ = (eξ X)G(y, τ; ξ) dξ ln K − Z rτ 2 e− ∞ (u µτ) = (Seu X) exp − σ√2πτ − − 2σ2τ Zln K/S 2 B 2µ/σ (u 2β µτ)2 exp − − du. (4.1.11) − S − 2σ2τ # The direct evaluation of the integral gives 4.1 Barrier options 171

B δ+1 c (S, τ) = S N(d ) N(d ) do 1 − S 3 " # δ 1 rτ B − Xe− N(d ) N(d ) , (4.1.12a) − 2 − S 4 " # where

S σ2 ln K + r + 2 τ d1 = , d2 = d1 σ√τ, σ√τ − (4.1.12b) 2 B 2 B 2r d = d + ln , d = d + ln , δ = . 3 1 σ√τ S 4 2 σ√τ S σ2 Suppose we deﬁne

rτ c (S, τ; X, K) = SN(d ) Xe− N(d ), (4.1.13a) E 1 − 2 then c (S, τ; X, B) can be expressed in the following succinct form do e δ 1 B − B2 c (S, τ; X, B) = c (S, τ; X, K) c , τ; X, K . (4.1.13b) do E − S E S δ 1 e e B − B2 One can show by direct calculation that the function c , τ S E S satisﬁes the Black-Scholes equation identically (see Problem 4.1). The above form allows us to observe readily the satisfaction of the boundarye condition: cdo(B, τ) = 0, and terminal payoﬀ condition. The barrier option price formula (4.1.13b) indicates that cdo(S, τ; X, B) < cE (S, τ; X, K) < cE (S, τ; X), that is, a down-and-out call is less expensive than the corresponding vanilla European call. This is obvious since the in- eherent nullifying property lowers the option premium.

Remarks 1. Closed form analytic price formulas for barrier options with exponential γτ time dependent barrier, B(τ) = Be− , can also be derived [see Problem 4.2]. However, when the barrier level has arbitrary time dependence, 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 process boundary hitting times via the Brownian bridge technique. 2. Closed form price formulas for barrier options can also be obtained for other types of diffusion processes followed by the underlying asset price. Lo et al. (2002) derive barrier option price formulas under the square root constant elasticity of variance process while Sepp (2004) uses Laplace transform method to obtain price formulas under the double exponential jump diffusion process. 172 4 Path Dependent Options

3. By the relation (4.1.1a) and assuming B < X, the price of a down-and-in call option can be deduced to be

δ 1 B − B2 c (S, τ; X, B) = c , τ; X . (4.1.14) di S E S 4. The barrier option with rebate payment will be considered later when the density function of the first passage time is available. Alternatively, one can use the known solution for the diffusion equation in the semi-infinite domain with time dependent boundary condition to derive the additional option premium due to the rebate (see Problem 4.3). 5. The method of images can be extended to derive the density functions of restricted multi-state diffusion processes where barrier occurs in only one of the state variables (Wong and Kwok, 2003a). The pricing of a two-asset down-and-out call is illustrated in Problem 4.6. 6. The monitoring period for breaching of the barrier may be limited to only part of the life of the option. The price formulas for these partial barrier options have been obtained by Heynen and Kat (1994a) (see Problem 4.7).

4.1.2 Transition density function and ﬁrst passage time density

We may formulate the pricing models of barrier options using the martingale pricing approach and derive the price formulas by computing the discounted expectation of the terminal payoﬀ (subject to knock-out or knock-in provision) under the risk neutral measure Q. Let the time interval [0, T ] denote the life of the barrier option, that is, the option is initiated at time zero and will expire at time T . 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, (4.1.15) 0 u t ≤ ≤ respectively. The terminal payoﬀs of various types of barrier options can be T T 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

cdo(ST , T ; X, B) = max(ST X, 0)1 mT >B − { 0 } puo(ST , T ; X, B) = max(X ST , 0)1 M T

respectively. 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. (4.1.17a) B { | t ≤ } 0 4.1 Barrier options 173

Assume S > B and due to path continuity, we may express τB (commonly called the ﬁrst passage time) as

τ = inf t S = B . (4.1.17b) 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 . (4.1.17c) B { | t ≥ } { | t } T It is easily seen that τB > T and m0 > B are equivalent 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 mT >B ] − { 0 } rT = e− EQ[(ST X)1 ST >max(X,B) 1 τB >T ].(4.1.18) − { } { }

The determination of the price function cd0 (S, 0; X, B) requires the determi- T nation of the joint distribution function of ST and m0 . We illustrate how to obtain the joint distribution function step by step via the reﬂection principle.

Reflection principle We illustrate how to apply the reflection principle to derive the joint law of the minimum value over [0, T ] and terminal value of a Brownian motion. Let 0 µ Wt (Wt ) denote the Brownian motion that starts at zero, with volatility σ T µ and zero drift rate (drift rate µ). We would like to find P [m0 < m, WT > x], where x m and m < 0. First, we consider the zero-drift Brownian motion 0 ≥ T Wt . Given that the minimum value m0 falls below m, then there exists some 0 time instant ξ, 0 < ξ < T , such that ξ is the first time that Wξ equals m. As Brownian paths are continuous, there exist some times during which 0 0 Wt < m. In other words, Wt decreases at least below m and then increases at least up to level x (higher than m) at time T . Suppose we define a random process W 0 for t < ξ W 0 = t (4.1.19) t 2m W 0 for ξ t T , − t ≤ ≤ 0 0 that is, Wt is the mirrorf reflection of Wt at the level m within the time 0 interval between ξ and T (see Fig. 4.2). It is then obvious that WT > x 0 { } is equivalenf t to WT < 2m x . Also, the reflection of the Brownian path dictates that { − } f W 0 W 0 = (W 0 W 0), u > 0. (4.1.20) ξ+u − ξ − ξ+u − ξ 0 The stopping timef ξ onlyf depends on the path history Wt : 0 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 in Eq. (4.1.20) have the same distribution, and the distribution has zero mean and variance σ2u. For every Brownian path that starts at 0, travels 174 4 Path Dependent Options at least m units (downward, 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, tra−vels m units (downward,≥ m < 0) some time before T and travels at least m x 0 T − units (further downward, m x). Suppose WT > x, then W0 < 2m x, and together with relation (4.1.20),≤ we obtain − f P [W 0 > x, mT < m] = P [W 0 < 2m x] = P [W 0 < 2m x] T 0 T − T − 2m x = N − , m min(x, 0). (4.1.21) f √ ≤ σ T

Fig. 4.2 Pictorial representation of the reflection princi- 0 ple of the Brownian motion Wt . The dotted path after time ξ is the mirror reflection of the Brownian path at the 0 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. − Next, we apply the Girsanov Theorem to effect the change of measure for finding the above joint distribution when the Brownian motion has non-zero µ drift. Suppose under the measure Q, Wt is a Brownian motion with drift rate µ µ. We change the measure from Q to Q such that Wt becomes a Brownian motion with zero drift under Q. Consider the following joint distribution e P [W µ > x, mT < m] T 0e = EQ[1 W µ>x 1 mT x 1 mT

e 4.1 Barrier options 175

µ T P [WT > x, m0 < m] 2 µ µ µ T µ = E 1 2m W >x exp (2m WT ) Q { − T } σ2 − − 2σ2 2 2µm µ µ µ T σ2 µ = e e E 1 W <2m x exp WT Q { T − } −σ2 − 2σ2 2m x 2µm − z2 µz µ2T = e σ2 e e− 2σ2T e− σ2 − 2σ2 dz Z−∞ 2m x 2 2µm − 1 (z + µT ) = e σ2 exp dz √2πσ2T − 2σ2T Z−∞ 2µm 2m x + µT = e σ2 N − , m min(x, 0). (4.1.22b) √ ≤ σ T µ Suppose the Brownian motion Wt has a downstream barrier m over the T period [0, T ] so that m0 > m, we would like to derive the joint distribution P [W µ > x, mT > m], where 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). (4.1.23a) √ − √ ≤ σ T σ T Under the special case m = x, we have

µm T m + µT 2 m + µT P [m > m] = N − e σ2 N . (4.1.23b) 0 √ − √ σ T σ T µ When the Brownian motion Wt has an upstream barrier M over the T µ period [0, T ] so that M0 < M, the joint distribution function of WT and T T T M0 can be deduced using the following relation between M0 and m0 : T M0 = max (σZt + µt) = min ( σZt µt), (4.1.24) 0 t T − 0 t T − − ≤ ≤ ≤ ≤

where Zt is the standard Brownian motion. Since Zt has the same distribu- − µ tion as Zt, the distribution of the maximum value of Wt is the same as that µ of the negative of the minimum value of Wt− . By swapping µ for µ, M for m and y for x in Eq. (4.1.22b), we obtain − − −

µM µ T 2 y 2M µT P [W < y, M > M] = e σ2 N − − , M max(y, 0). (4.1.25) T 0 √ ≥ σ T In a similar manner, we obtain 176 4 Path Dependent Options

µ T P [WT < y, M0 < M] = P [W µ < y] P [W µ < y, M T > M] T − T 0 y µT 2µM y 2M µT = N − e σ2 N − − , M max(y, 0), (4.1.26a) √ − √ ≥ σ T σ T and by setting y = M, we obtain

µM T M µT 2 M + µT P [M < M] = N − e σ2 N . (4.1.26b) 0 √ − − √ σ T σ T 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]. (4.1.27a) down T ∈ 0 By diﬀerentiating Eq. (4.1.23a) with respect to x and swapping the sign, we obtain

fdown(x, m, T ) 1 x µT 2µm x 2m µT = n − e σ2 n − − √ √ − √ σ T σ T σ T 1 m min(x,0) . (4.1.27b) { ≤ } µ In a similar manner, we deﬁne fup(x, M, T ) be the density function of WT with the upstream barrier M, where M > max(y, 0), then P [W µ dy, M T < M] T ∈ 0 = fup(y, M, T ) dy 1 y µT 2µM y 2M µT = n − e σ2 n − − dy √ √ − √ σ T σ T σ T 1 M>max(y,0) . (4.1.27c) { }

Suppose the asset price St follows the lognormal process under the risk S neutral measure such that ln t = W µ, where S is the asset price at time S t σ2 zero and the drift rate µ = r . Let ψ(S ; S, B) denote the transition − 2 T 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, B is the downstream barrier. t ≤ ≤ 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√T ST   σ√T 

   2 2r 1 ST B σ B σ2 − ln S 2 ln S r 2 T n − − − . (4.1.28) − S  √  σ T   4.1 Barrier options 177

First passage time density functions Let Q(u; m) denote the density function of the ﬁrst passage time at which µ the downstream barrier m is ﬁrst bit by the Brownian process Wt , that is, Q(u; m) du = P [τm du]. First, we determine the distribution function ∈ u P [τm > u] by observing that τm > u and m0 > m are equivalent events. By Eq. (4.1.23b), we obtain { } { }

u P [τm > u] = P [m0 > m] m + µu 2µm m + µu = N − e σ2 N . (4.1.29) σ√u − σ√u The density function Q(u; m) is then given by Q(u; m) du = P [τ du] m ∈ ∂ m + µu 2µm m + µu 2 = N − e σ N du 1 m<0 −∂u σ√u − σ√u { } m (m µu)2 = − exp − du 1 m<0 . (4.1.30a) √ 2 3 − 2σ2u { } 2πσ u In a similar manner, let Q(u; M) denote the ﬁrst passage time density associated with the upstream barrier M. Using the result in Eq. (4.1.26b), we have

∂ M µu 2µM M + µu 2 Q(u; M) = N − e σ N 1 M>0 −∂u σ√u − − σ√u { } M (M µu)2 = exp − 1 M>0 . (4.1.30b) √ 2 3 − 2σ2u { } 2πσ u We write B as the barrier, either upstream or downstream. When the B B barrier is downstream (upstream), we have ln < 0 ln > 0 . We may S S combine Eqs. (4.1.30a,b) into one equation as follows

2 2 B B σ ln ln S r 2 u Q(u; B) = S exp − − . (4.1.31) 2 3  h 2 i  √ 2πσ u − 2σ u     Suppose a rebate R(t) is paid to the option holder upon breaching the barrier at level B 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. (4.1.32) Z0 When R(t) = R0, a constant value, direct integration of the above integral gives 178 4 Path Dependent Options

B α+ ln B + βT rebate value = R N δ S 0 S √ " σ T ! B α− ln B βT + N δ S − , (4.1.33a) S √ σ T !# where

2 σ2 2 r σ β β = r + 2rσ2, α = − 2 s − 2 σ2 S and δ = sign ln . (4.1.33b) B Here, δ is a binary variable indicating whether the barrier is downstream (δ = 1) or upstream (δ = 1). − Fokker-Planck equation formulation We would like to ﬁnd the partial diﬀerential equation formulation of the transition density function ψB (x, t; x0, t0) for the restricted Brownian process with upstream absorbing barrier B. The absorbing condition resembles the knock-out feature in barrier options. The appropriate boundary condition for an absorbing barrier is given by (Cox and Miller, 1965)

ψB(x, t; x0, t0) = 0. (4.1.34) x=B

The forward Fokker-Planck equation that governs ψB is known to be [see Eq. (2.3.12)]

∂ψ ∂ψ σ2 ∂2ψ B = µ B + B , < x < B, t > t , (4.1.35) ∂t − ∂x 2 ∂x2 −∞ 0 with boundary condition: ψ (B, t) = 0. Since x x as t t so that B → 0 → 0

lim ψB (x, t; x0, t0) = δ(x x0). (4.1.36) t t → 0 −

As deduced from the density function in Eq. (4.1.27c), ψB is found to be 1 x x µ(t t ) ψ (x, t; x , t ) = n − 0 − − 0 B 0 0 σ√t t σ√t t − 0 − 0 − 2µ(B x0 ) (x x0) 2(B x0) µ(t t0) e σ2 n − − − − − , − σ√t t − 0 x < B, t > t0, x0 < B. (4.1.37)

µ The probability that Wt never crosses the barrier over [t0, t] is given by 4.1 Barrier options 179

P [τ > t] = P W µ B, M t B W µ = x B t ≤ t0 ≤ t0 0 B

= ψB (x, t; x0, t0) dx. (4.1.38) Z−∞

Price formula for European up-and-out call Consider a European up-and-out call with strike X and upstream barrier B. Since the writer uses the knock-out feature to cap the upside liability, the option structure makes sense only if we choose X < B. As the option is always in-the-money upon knock-out, some form of rebate (say, B X) should be paid upon breaching the barrier. Suppose X B, the up-and-out− call is knocked out before it becomes in-the-money. Hence,≥ the up-and-out call is indeed worthless. We would like to compute the non-rebate portion of the value of the up- and-out call. By the risk neutral valuation principle, the non-rebate call value is given by

rT T e− EQ (ST X)1 X

rT ∞ x e− (Se X)f (x, B, T ) dx − down Zln X/S δ 1 B − B2 = c (S, T ; X) c , T ; X , (4.1.40) E − S E S 

where cE (S, τ; X) is the price function of a European vanilla call option with time to expiry τ. Since fdown(x, B, T ) and fup(y, B, T ) have the same analytic form, the integral in Eq. (4.1.39) can be related to the integral in Eq. (4.1.40). Hence, we deduce that

up-and-out call value δ 1 B − B2 = c (S, T ; X) c , T ; X E − S E S " # δ 1 B − B2 c (S, T ; B) c , T ; B . (4.1.41) − E − S E S " # The corresponding up-and-in call value is given by 180 4 Path Dependent Options

δ 1 B − B2 up-and-in call value = c , T ; X + c (S, T ; B) S E S E δ 1 B − B2 c , T ; B . (4.1.42) − S E S All other price formulas for European out/in put options with either upstream or downstream barrier can be derived in a similar manner (see Prob- lem 4.5). The paper by Rich (1994) contains a comprehensive list of price formulas of European options with one-sided barrier.

4.1.3 Options with double barriers

A double barrier option has two barriers, upstream barrier at U and downstream barrier at L. In its simplest form, when the asset price reaches one of the two barriers, either nullification (knock-out) or activation (knock-in) is triggered. More complicated payoff structure may include rebate payment upon hitting either one of the knock-out barriers as a compensation to the option holder. In sequential barrier options, knock-out is triggered only when the two barriers are hit at a pre-specified sequential order (see Problem 4.12). The papers by Luo (2001) and Kolkiewicz (2002) contain comprehensive dis- cussions on various types of double barrier options that can be structured. With the presence of two barriers, the following first passage times of the asset price process St can be defined τ = inf t S = U and τ = inf t S = L . (4.1.43) U { | t } L { | t } During the life of the option [0, T ], we can distinguish the following three mutually exclusive events: (i) the upper barrier is first reached, (ii) the lower barrier is first reached, (iii) neither of the two barriers is reached. We take the usual assumption that St is a Geometric Brownian process St with volatility σ under the risk neutral measure Q. Let Xt = ln so that S0 σ2 X = 0 and X is a Brownian process with drift rate r and variance 0 t − 2 rate σ2. We are interested to find the following density functions:

g(x, T ) dx = P [X dx, min(τ , τ ) > T ] (4.1.44a) T ∈ L U g+(x, T ) dx = P [X dx, min(τ , τ ) T, τ < τ ] (4.1.44b) T ∈ L U ≤ U L g−(x, T ) dx = P [X dx, min(τ , τ ) T, τ < τ ]. (4.1.44c) T ∈ L U ≤ L U Once the above densities are available, most double barrier options can be priced easily. Some examples are listed below. 1. Double knock-out call option (call payoﬀ received at maturity if none of the barriers is breached) 4.1 Barrier options 181

o rT cLU = e− E (ST X)1 ST >X 1 min(τL ,τU )>T − { } { } ln U/S rT x = e− (Se X)g(x, T ) dx, X (L, U), (4.1.45) − ∈ Zln X/S

where S0 = S and the expectation E is taken under the risk neutral measure Q. 2. Upper-barrier knock-in call option (a vanilla call comes into life if the upper barrier is breached before the lower barrier is breached during the option life, that is, τ < τ and τ T ) U L U ≤ i rT cU = e− E (ST X)1 ST >X 1 τU <τL 1 min(τL ,τU ) T − { } { } { ≤ } ln U/S rT x + = e− (Se X)g (x, T ) dx, X (L, U). (4.1.46) − ∈ Zln X/S The analytic form for g+(x, T ) can be found in Problem 4.11. 3. Lower-barrier knock-out call option (call payoﬀ received at maturity if the lower barrier is never breached or the upper barrier is breached before the lower barrier, that is, τL > T or τU < τL). Since the sum of a lower-barrier knock-out call and a lower-barrier knock-in call equals a vanilla call, we have

co = c ci L E − L ln U/S rT x = c e− (Se X)g−(x, T ) dx, X (L, U), (4.1.47) E − − ∈ Zln X/S

where cE is the price of a European vanilla call option.

Density functions of Brownian processes with two-sided barriers All the density functions defined in Eqs. (4.1.44a,b,c) satisfy the forward Fokker-Planck equation. Their complete partial differential equation formu- lations require the prescription of appropriate auxiliary conditions. We take the initial position X(0) = x0 = 0. Let g(x, t; `, u) denote the density function of the restricted Brownian process Xt with two-sided absorbing barriers at x = ` and x = u, where the barriers are positioned such that St ` < 0 < u. Recall that Xt = ln , and if L and U are the absorbing barriers S0 L U of the asset price process S , respectively, then ` = ln and u = ln with t S S S0 = S. The partial differential equation formulation for g(x, t; `, u) is given by ∂g ∂g σ2 ∂2g = µ + , ` < x < u, t > 0, (4.1.48a) ∂t − ∂x 2 ∂x2 with auxiliary conditions: 182 4 Path Dependent Options

g(`, t) = g(u, t) = 0 and g(x, 0+) = δ(x). (4.1.48b) By deﬁning the transformation

µx µ2t g(x, t) = e σ2 − 2σ2 g(x, t), (4.1.49) then g(x, t) satisfies the forward Fokker-Planck equation with zero drift: b ∂g σ2 ∂2g b (x, t) = (x, t). (4.1.50) ∂t 2 ∂x2 The auxiliary conditions forb g(x, t) remainbthe same as those given in Eq. (4.1.48b). Without the barriers, the infinite-domain fundamental solution to Eq. (4.1.50) is known to be b 1 x2 φ(x, t) = exp . (4.1.51) √ 2 −2σ2t 2πσ t Like the one-sided barrier case, we try to add extra terms to the above solution such that the homogeneous boundary conditions at x = ` and x = u are satisfied. The following procedure is an extension of the method of images to two-sided barriers. First, we attempt to add the pair of negative terms φ(x 2`, t) and φ(x 2u, t) whereby − − − − [φ(x, t) φ(x 2`, t)] = 0 (4.1.52a) − − x=`

and [φ(x, t) φ(x 2u, t)] = 0. (4.1.52b) − − x=u

Note that φ(x 2`, t) and φ(x 2u, t) corresp ond to the fundamental soluton with initial condition− δ(x 2`)−and δ(x 2u), respectively. Writing the above partial sum with three terms− as − g (x, t) = φ(x, t) φ(x 2`, t) φ(x 2u, t), (4.1.53) 3 − − − − we observe that the homogeneous boundary conditions are not yet satisﬁed since b

g (`, t) = φ(x 2u, t) = 0 (4.1.54a) 3 − − 6 x=`

gb3(u, t) = φ(2 2`, t) = 0. (4.1.54b) − − 6 x=u

b To nullify the non-zero value of φ(x 2u, t) and φ(x 2`, t) , we − − − − x=` x=u add a new pair of positive terms φ(x 2(u `), t) and φ(x + 2(u `), t). − − − Similarly, we write the partial sum with ﬁve terms as 4.1 Barrier options 183

g (x, t) = g (x, t) + φ(x 2(u `), t) + φ(x + 2(u `), t), (4.1.55) 5 3 − − − and observe that b b g (`, t) = φ(x 2(u `), t) = 0 (4.1.56a) 5 − − 6 x=`

gb5(u, t) = φ(x + 2(u `), t) = 0. (4.1.56b) − 6 x=u

Fig. 4.3 A pictorial representation of the inﬁnite number of sources and sinks due to a pair of absorbing barriers (mirrors) with the object placed at the origin. The po- sitions of the sources and sinks are α = 2(u `)j, j = j − 0, 1, 2, ; βj = 2u + 2(u `)(j 1) if j > 0 and β = 2` + ·2(· ·u `)(j 1) if j <−0. − j − − Whenever a new pair of positive terms or negative terms are added, the value of the partial sum at x = ` and x = u becomes closer to zero. In a recursive manner, we add successive pairs of positive and negative terms so as to come closer to the satisfaction of the homogeneous boundary conditions at x = ` and x = u. Apparently, the two absorbing barriers may be visualized as a pair of mirrors with the object placed at the origin (see Fig. 4.3 for a pictorial representation). The source at the origin generates a sink at x = 2` due to the mirror at x = ` and another sink at x = 2u due to the mirror at x = u. To continue, the sink at x = 2` (x = 2u) generates a source at x = 2(u `) [x = 2(` u)] due to the mirror at x = u (x = `). As the procedure−continues, this− leads to the sum of an inﬁnite number of positive and negative terms. The solution to g(x, t) is deduced to be

µx µ2t g(x, t) = e σ2 − 2σ2 g(x, t)

µx µ2t ∞ = e σ2 − 2σ2 [φ(x 2n(u `, t), t) φ(x 2` 2n(u `), t)] b − − − − − − n= X−∞ µx µ2t 2 e σ2 − 2σ2 ∞ [x 2n(u `)] = exp − − √ 2 − 2σ2t 2πσ t n= X−∞ [(x 2`) 2n(u `)]2 exp − − − . (4.1.57) − − 2σ2t 184 4 Path Dependent Options

The double-mirror analogy provides the intuitive argument showing why g(x, t) involves an inﬁnite number of terms [see the text by Karatzas and Shreve (1991) for a rigorous proof]. Once g(x, t) is known, it becomes quite o straightforward to derive the price formula of cLU (see Problem 4.8).

Next, we would like to derive the density function of the ﬁrst passage time to either barrier, which is deﬁned by

q(t; `, u) dt = P [min(τ , τ ) dt], (4.1.58) ` u ∈

where τ` = inf t Xt = ` and τu = inf t Xt = u . First, we consider the corresponding distribution{ | } function { | }

P [min(τ`, τu) t] = 1 P [min(τ`, τu) > t] ≤ − u = 1 g(x, t) dx (4.1.59a) − Z` so that ∂ u q(t; `, u) = g(x, t) dx. (4.1.59b) −∂t Z` After some tedious manipulation, we manage to obtain 1 q(t; `, u) = √2πσ2t3 ∞ µ` µ2t (2n(u `) `]2 [2n(u `) `] exp exp − − − − σ2 − 2σ2 − 2σ2t n= X−∞ µu µ2t [2n(u `) + u]2 + [2n(u `) + u] exp exp − .(4.1.60) − σ2 − 2σ2 − 2σ2t We may also be interested to ﬁnd the exit time to a particular barrier. The density function of the exit time to the lower and upper barriers are deﬁned by

q−(t; `, u) dt = P [τ dt, τ < τ ] (4.1.61a) ` ∈ ` u q+(t; `, u) dt = P [τ dt, τ < τ ]. (4.1.61b) u ∈ u `

Since τ` dt, τ` < τu τu dt, τu < τ` = min(τ`, τu) dt , we deduce that { ∈ } ∪ { ∈ } { ∈ }

+ q(t; `, u) = q−(t; `, u) + q (t; `, u). (4.1.62)

A judicious decomposition of q(t; `, u) in Eq. (4.1.60) into its two components would suggest (see also Problem 4.10) 4.1 Barrier options 185

1 ∞ q−(t; `, u) = [2n(u `) `] √ 2 3 2πσ t n= − − X−∞ µ` µ2t [2n(u `) `]2 exp exp − − (4.1.63a) σ2 − 2σ2 − 2σ2t 1 ∞ q+(t; `, u) = [2n(u `) + u] √ 2 3 − 2πσ t n= X−∞ µu µ2t [2n(u `) + u]2 exp exp − . (4.1.63b) σ2 − 2σ2 − 2σ2t To show the claim, we define the probability flow by σ2 ∂g J(x, t) = µg(x, t) (x, t) (4.1.64) − 2 ∂x and observe that ∂ u u ∂g q(t; `, u) = g(x, t) dx = dx. (4.1.65) −∂t − ∂t Z` Z` Since g satisfies the forward Fokker-Planck equation, we have u ∂g σ2 ∂2g q(t; `, u) = µ dx = J(u, t) J(`, t). (4.1.66) ∂x − 2 ∂x2 − Z` One may visualize the probability flow across x = ` and x = u as

J(`, t) = P [τ dt, τ < τ ] (4.1.67a) − ` ∈ ` u J(u, t) = P [τ dt, τ < τ ]. (4.1.67b) u ∈ u ` + The exit time densities q−(t; `, u) and q (t; `, u) are seen to satisfy σ2 ∂g q−(t; `, u) = J(`, t) = µg(x, t) (x, t) (4.1.68a) − − − 2 ∂x x=` 2 + σ ∂g q (t; `, u) = J(u, t) = µg(x, t) (x, t) . (4.1.68b) − 2 ∂x x=u

An alternative proof to Eqs. (4.1.68a,b) is given by Kolkiewicz (2002). Sup- + pose rebate R−(t) [R (t)] is paid when the lower (upper) barrier is ﬁrst breached during the life of the option, then the value of the rebate portion of the double-barrier option is given by

T rξ + + rebate value = e− [R−(ξ)q−(ξ; `, u) + R (ξ)q (ξ; `, u)] dξ. (4.1.69) Z0 186 4 Path Dependent Options

4.1.4 Discretely monitored barrier options

The barrier option pricing models have good analytic tractability when the barrier is monitored continuously (knock-in or knock-out is presumed to oc- cur if the barrier is breached at any instant). In ﬁnancial markets, practitioners necessarily have to specify a discrete monitoring frequency. Kat and Verdonk (1995) show that the price diﬀerences between discrete and continuous barrier options can be quite substantial, even under daily monitoring of the barrier. We would expect that discrete monitoring would lower the cost of knock-in options but raise the cost of knock-out options, compared to their counterparts with continuous monitoring. The analytic price formulas of discrete monitoring barrier options can be derived, but their numerical valuation would be much tedious. The analytic representation involves the multi-variate normal distribution functions (see Problem 4.13). Broadie et al. (1997) obtain an approximation formula for discretely monitored barrier options, which requires only a simple continuity correction to the continuous barrier option formulas. Let δt denote the uniform time interval between monitoring instants and there are m monitoring instants prior to expiration. The precise statement of their analytic approximation formula is given below.

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 barrier option. We have

βσ√δt 1 V (B; m) = V (Be ) + o , (4.1.70) √m 1 where β = ξ √2π 0.5826, ξ is the Riemann zeta function, σ is the − 2 ≈ volatility. The“+” sign is chosen when B > S, while the “ ” sign is chosen when B < S. One observes that the correction shifts the barrier− away from the current underlying asset price by a factor of eβσ√δt. The extensive numerical experiments performed by Broadie et al. (1997) reveal the remarkably good accuracy of the above approximation formula.

4.2 Lookback options

Lookback options are path dependent options whose payoﬀs depend on the maximum or the minimum of the underlying asset price attained over a certain period of time (called the lookback period). We ﬁrst consider lookback options where the lookback period is taken to be the whole life of the option. Let T denote the time of expiration of the option and [T0, T ] be the lookback 4.2 Lookback options 187

period. We denote the minimum value and maximum value of the asset price realized from T to the current time t (T t T ) by 0 0 ≤ ≤ t mT = min Sξ (4.2.1a) 0 T ξ t 0≤ ≤ and t M T = max Sξ (4.2.1b) 0 T ξ t 0≤ ≤ respectively. The above formulas implicitly imply continuous monitoring of the asset price, though discrete monitoring for the extremum value is normally adopted in practical implementation. Lookback options can be classified into two types: fixed strike and floating strike. 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 holder T T0 T0 T of a floating strike lookback option≥ always exercise≥ the option. Hence, the T respective terminal payoff of the lookback call and put are given by ST m − T0 and M T S . In a sense, floating strike lookback options are not options. T0 T A fixed strik− e lookback call (put) is a call (put) option on the maximum (minimum) realized price. The respective terminal payoff of the fixed strike T T lookback call and put are max(M T0 X, 0) and max(X mT0 , 0), where X is the strike price. Lookback options−guarantee “no-regret”− outcome for the holders, and thus the holders are relieved from making the difficult decision on the optimal timing for entry into or exit from the market. Generally speaking, lookback options are most desirable for investors who have confidence in the view of the range of the asset price movement over a certain period. One would expect that the prices of lookback options are much more sensitive to volatility. Also, the writer would charge a much higher premium in view of the favorable payoff to the holder. In this section, we derive the price formulas for various types of lookback options, including those with exotic forms of lookback specification. The analytic formulas are limited to options which are European style, and continuous monitoring of the asset price for the extremum value is assumed. We adopt the usual Black-Scholes pricing framework and the underlying asset price is assumed to follow the Geometric Brownian process. Under the S risk neutral measure, the process for the stochastic variable U = ln ξ is a ξ S σ2 Brownian process with drift rate µ = r and variance rate σ2, where r is − 2 the riskless interest rate and S is the asset price at current time t (dropping subscript t for brevity). We define the following stochastic variables mT y = ln t = min U , ξ [t, T ] (4.2.2a) T S { ξ ∈ } M T Y = ln t = max U , ξ [t, T ] , (4.2.2b) T S { ξ ∈ } 188 4 Path Dependent Options 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 [see Eq. (4.1.23a)]

u + µτ 2µy u + 2y + µτ P [U u, y y] = N − e σ2 N − . (4.2.3a) T ≥ T ≥ σ√τ − σ√τ Here, Uξ is visualized as a restricted Brownian process with drift rate µ and downstream absorbing barrier y. Similarly, for y 0 and y u, the ≥ ≥ corresponding joint distribution function of UT and YT is given by [see Eq. (4.1.26a)]

u µτ 2µy u 2y µτ P [U u, Y y] = N − e σ2 N − − . (4.2.3b) T ≤ T ≤ σ√τ − σ√τ By taking y = u in the above two joint distribution functions, we obtain the distribution functions for yT and YT

y + µτ 2µy y + µτ P (y y) = N − e σ2 N , y 0, (4.2.4a) T ≥ σ√τ − σ√τ ≤ y µτ 2µy y µτ P (Y y) = N − e σ2 N − − , y 0. (4.2.4b) T ≤ σ√τ − σ√τ ≥ The density functions of yT and YT can be obtained by diﬀerentiating the above distribution functions (see Problem 4.14).

4.2.1 European ﬁxed strike lookback options

Consider a European ﬁxed strike lookback call option whose terminal payoﬀ T is max(M T0 X, 0). The value of this lookback call option at the current time t is given− by

rτ T c (S, M, t) = e− E max(max(M, M ) X, 0) , (4.2.5) fix t − t where St = S, MT0 = M and τ = T t, and the expectation is 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) (4.2.6a) t − t − (ii) M > X

max(max(M, M T ) X, 0) = (M X) + max(M T M, 0). (4.2.6b) t − − t − Deﬁne the function H by 4.2 Lookback options 189

rτ T H(S, τ; K) = e− E[max(M K, 0)], (4.2.7) t − where K is a positive constant. Once H(S, τ; K) is determined, then H(S, τ; X) if M X cfix(S, M, τ) = rτ ≤ e− (M X) + H(S, τ; M) if M > X − rτ = e− max(M X, 0) + H(S, τ; max(M, X)). (4.2.8) − Interesting, cfix(S, M, τ) is independent of M when M X. This is obvious because the terminal payoff is independent of M when M≤ X. On the other hand, when M > X, the terminal payoff is guaranteed to ha≤ve 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. T Since max(Mt K, 0) is a non-negative random variable, its expected value is given by the−integral of the tail probabilities where

rτ T e− E[max(M 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 S ≥ S ZK rτ ∞ y z = e− Se P [YT y] dy y = ln ln K ≥ S Z S µy rτ ∞ y y + µτ 2 y µτ = e− Se N − + e σ2 N − − dy, (4.2.9) ln K σ√τ σ√τ Z S where the last integral is obtained by using the distribution function in Eq. (4.2.4b). By performing straightforward integration, we obtain

rτ H(S, τ; K) = SN(d) e− KN(d σ√τ) − − 2r 2 σ2 rτ σ rτ S − 2r + e− S e N(d) N d √τ ,(4.2.10a) 2r − K − σ " # where

S σ2 ln K + r + 2 τ d = . (4.2.10b) σ√τ The European fixed strike lookback put option with terminal payoff T t max(X mT0 , 0) can be priced in a similar manner. Write m = mT0 and define the− function

rτ T h(S, τ; K) = e− E[max(K m , 0)]. (4.2.11) − t The value of this lookback put can be expressed as 190 4 Path Dependent Options

rτ p (S, m, τ) = e− max(X m, 0) + h(S, τ; min(m, X)), (4.2.12) fix − where

rτ ∞ T h(S, τ; K) = e− P [max(K m , 0) x] dx − t ≥ Z0 K rτ T T = e− P [K m x] dx since 0 max(K m , 0) K − t ≥ ≤ − t ≤ Z0 K rτ T = e− P [mt z] dz z = K x 0 ≤ − Z K ln S rτ y y = e− Se P [yT y] dy y = ln 0 ≤ S Z K ln S rτ y y µτ 2µy y + µτ = e− Se N − + e σ2 N dy σ√τ σ√τ Z0 2 rτ rτ σ = e− KN( d + σ√τ) SN( d) + e− S − − − 2r 2r/σ2 S − 2r N d + √τ erτ N( d) . (4.2.13) K − σ − − " #

4.2.2 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 T T0 call option is given by − rτ T cf`(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(d ) e− mN(d σ√τ) + e− S m − m − 2r 2r S − σ2 2r N d + √τ erτ N( d ) ,(4.2.14a) m − m σ − − m " # where

S σ2 ln m + r + 2 τ d = . (4.2.14b) m σ√τ In a similar manner, consider a European ﬂoating strike lookback put option whose terminal payoﬀ is M T S , the present value of this put T0 T option is given by − 4.2 Lookback options 191

rτ T pf`(S, M, τ) = e− E[max(M, Mt ) ST ] rτ T − = e− E[max(Mt M, 0) (ST M)] − r−τ − = H(S, τ; M) (S Me− ) − − 2 rτ rτ σ = e− MN( d + σ√τ) SN( d ) + e− S − M − − M 2r 2r S − σ2 2r erτ N(d ) N d √τ , (4.2.15a) M − M M − σ " # where

S σ2 ln M + r + 2 τ d = . (4.2.15b) M σ√τ 

Boundary condition at S = m Consider the particular situation when S = m, that is, the current 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 (see Problem 4.16). Now, 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 (Goldman et al., 1979). Mathematically, this is represented by ∂c f` (S, m, τ) = 0. (4.2.16) ∂m S=m

The above property can be veriﬁed by direct diﬀerentiation of the call price formula (4.2.14a).

Rollover strategy and strike bonus premium The sum of the first two terms in cf` can be seen as the price function of a European vanilla call with strike price m, while the third term can be interpreted as the strike bonus premium (Garman, 1992). To interpret 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 minimum 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. 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. 192 4 Path Dependent Options

We would like to show how the strike bonus premium can be obtained T by integrating a joint probability distribution function involving mt and ST . First, we observe that

rτ strike bonus premium = h(S, τ; m) + S me− c (S, τ; m) − − E = h(S, τ; m) p (S, τ; m), (4.2.17) − E where cE (S, τ, m) and pE (S, τ; m) are the price functions of European vanilla call and put, respectively. The last result is due to put-call parity relation. Recall from Eq. (4.2.13) that m rτ T h(S, τ; m) = e− P [m ξ] dξ (4.2.18a) t ≤ Z0 and in a similar manner

rτ ∞ pE (S, τ; m) = e− P [max(m ST , 0) x] dx 0 − ≥ Z m rτ = e− P [S ξ] dξ. (4.2.18b) T ≤ Z0 T Since the two stochastic state variables always satisﬁes 0 mt ST , we have ≤ ≤ P [mT ξ] P [S ξ] = P [mT ξ < S ] (4.2.19a) t ≤ − T ≤ t ≤ T so that (Wong and Kwok, 2003b) m rτ T strike bonus premium = e− P [m ξ S ] dξ. (4.2.19b) t ≤ ≤ T Z0

4.2.3 More exotic forms of European lookback options

The lookback options discussed above are the most basic types where the payoff functions at expiry are of standard forms and the lookback period spans the whole life of the option. In the financial market, a wide variety of options have been structured with properties similar to the above prototype lookback options but being less expensive. Some examples are: partial T lookback call option with terminal payoff max(ST λmT0 , 0), λ > 1, partial −T lookback put option with terminal payoff max(λM T0 ST , 0), 0 < λ < 1 (see Problem 4.19). Also, the price of a lookback option− will be lowered if the lookback period is spanning only part of the life of the option. When an investor is faced with the problem of deciding the optimal timing for market entry (market exit), he may be interested to purchase an option whose lookback period covers the early part (period near expiration) of the life of the option. In what follows, we discuss the properties of a “limited period” floating strike lookback call option which is designed for optimal market entry. 4.2 Lookback options 193

The discussion of the corresponding ﬁxed strike lookback options for optimal market exit is relegated to an exercise Problem 4.21.

Lookback options for market entry Suppose an investor has a view that the asset price will rise substantially in the next 12 months and he buys a call option on the asset with the strike price equal to the current asset price. Suppose the asset price drops a few percent within a few weeks after the purchase, though it does rise up strongly to a high level at expiration, the investor should have a better return if he had bought the option a few weeks later. The optimal timing for market entry is always difficult to be decided. The investor could have avoided the above difficulty if he has purchased a “limited period” floating strike lookback call option whose lookback period only covers the early part of the option’s life. It would cause the investor too much if a full period floating strike lookback call were purchased instead. Let [T0, T1] denote the lookback period where T1 < T , T is the expiration time, and let the current time t [T0, T1]. The terminal payoff function of the “limited period” lookback call is∈max(S mT1 , 0). We write S = S, mt = m T T0 t T0 and τ = T t. The value of this lookbac−k call is given by − rτ T1 c(S, m, τ) = e− E[max(ST m , 0)] − T0 rτ 1 T = e− E[max(ST m, 0) m m 1 ] − { ≤ t } rτ T1 1 T + e− E[max(ST mt , 0) m>m 1 ] − { t } rτ = e− E[ST 1 T1 ] ST >m,m m (4.2.20) { ≤ t } rτ e− mE 1 T1 ST >m,m m − { ≤ t } rτ h i + e− E[ST 1 T1 T1 ] ST >m ,m>m { t t } rτ T1 e− E[m 1 T1 T1 ], t < T1, t ST >m ,m>m − { t t } where E is the expectation under the risk neutral measure. The solution for the call value requires the derivation of the appropriate distribution functions. For the first term, the expectation can be expressed as

∞ ∞ ∞ xz E[ST 1 T1 = Se k(z)h(x, y) dz dx dy, ST >m,m m { ≤ t } ln m y ln m x Z S Z Z S − (4.2.21a) S where k(z) is the density function for z = ln T and h(x, y) is the bivariate ST1 S mT1 density function for x = ln T1 and y = ln t [see Eq. (4.2.3b)]. Similarly, S S the third and fourth terms can be expressed as 194 4 Path Dependent Options

m ln S ∞ ∞ xz E[ST 1 T1 T1 = Se k(z)h(x, y) dz dx dy ST >m ,m>m { t t } y y x Z−∞ Z Z − (4.2.21b) and

m ln S T1 ∞ ∞ y E[m 1 T1 T1 ] = Se k(z)h(x, y) dzdxdy. t ST >m ,m>m { t t } y y x Z−∞ Z Z − (4.2.21c)

After performing tedious integration procedures in the above discounted expectation calculations, the price formula of the present “limited-period” lookback call is found to be (Heynen and Kat, 1994b)

c(S, m, τ)

rτ T T1 = SN(d ) me− N(d ) + SN d , e ; − 1 − 2 2 − 1 1 − T t r − !

rτ T1 t + e− mN f , d ; − 2 − 2 2 − T t r − !

2r 2 σ2 rτ σ S − 2r 2r T1 t + e− S N2 f1 + T1 t, d1 + √τ; − 2r " m − σ − − σ r T t ! p − T T erτ N d , e ; − 1 − 2 − 1 1 − T t r − !# 2 r(T T ) σ + e− − 1 1 + SN(e )N( f ), t < T , (4.2.22) 2r 2 − 1 1 where

S σ2 ln m + r + 2 τ d = , d = d σ√τ, 1 σ√τ 2 1 − 2 r + σ (T T ) 2 − 1 (4.2.23) e1 = , e2 = e1 σ T T1, σ√T T − − − 1 2 p ln S + r + σ (T t) m 2 1 − f1 = , f2 = f1 σ T1 t. σ√T1 t − − − p One can check easily that when T1 = T (full lookback period), the above price formula reduces to price formula (4.2.14a). Suppose the current time passes beyond the lookback period, t > T , the realized minimum value mT1 1 T0 is now a known quantity. This “limited period” lookback call option then becomes identical to a vanilla call option with the known strike price mT1 . T0 4.2 Lookback options 195

4.2.4 Diﬀerential equation formulation

We would like to illustrate how to derive the governing partial differential 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, (4.2.24) ZT0 the derivative of which is given by 1 Sn dMn = n 1 dt (4.2.25) n (Mn) − so that dM is deterministic. Taking the limit n , we obtain n → ∞

M = lim Mn = max Sξ , (4.2.26) n T ξ t →∞ 0≤ ≤ giving the realized maximum value of the asset price process over the lookback period [T0, t]. 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−4components 4 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, M , t) S. (4.2.27) n − 4 The dynamics of the portfolio value is given by

n 2 2 ∂p 1 S ∂p ∂p σ 2 ∂ p dΠ = dt + n 1 dt + dS + S 2 dt dS (4.2.28) ∂t n (Mn) − ∂Mn ∂S 2 ∂S − 4 ∂p by virtue of Ito’s lemma. Again, we choose = so that the stochastic 4 ∂S terms cancel. Using usual no-arbitrage argument, the non-stochastic portfolio should earn the riskless interest rate so that

dΠ = rΠ dt, (4.2.29)

where r is the riskless interest rate. Putting all equations together, we obtain

n 2 2 ∂p 1 S ∂p σ 2 ∂ p ∂p + n 1 + S 2 + rS rp = 0. (4.2.30) ∂t n (Mn) − ∂Mn 2 ∂S ∂S − Now, we take the limit n and note that S M. When S < M, 1 Sn → ∞ ∂p ≤ lim = 0; and when S = M, = 0. Hence, the second term in n n 1 →∞ n (Mn) − ∂M 196 4 Path Dependent Options

Eq. (4.2.30) becomes zero as n . In conclusion, the governing equation for the floating strike lookback put→ ∞is given by ∂p σ2 ∂2p ∂p + S2 + rS rp = 0, 0 < S < M, t > T , (4.2.31) ∂t 2 ∂S2 ∂S − 0 which is identical to the usual Black-Scholes equation (Goldman et al., 1979), except that the domain of the pricing model has an upper bound M on S. It is interesting to observe that the variable M does not appear in the equation, though M appears as a parameter in the auxiliary conditions. The final condition is the terminal payoff function, namely, p(S, M, T ) = M S. (4.2.32a) − 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. (4.2.32b) The boundary condition at the other end S = M is given by ∂p = 0 at S = M. (4.2.32c) ∂M

Remarks 1. The satisfaction of the differential equation by the price function implies implicitly that the lookback option is hedgeable. 2. One can show by direct differentiation that the put price formula given in Eq. (4.2.15a,b) satisfies Eq. (4.2.31) and the auxiliary conditions (4.2.32a,b,c). T 3. When the terminal payoff assumes the more general form f(ST , MT0 ), Xu and Kwok (2005) manage to derive an integral representation of the lookback option price formula using the partial differential equation approach.

4.2.5 Discretely monitored lookback options

In real market situations, practitioners necessarily specify a discrete monitoring frequency since continuous monitoring of the asset price movement is almost impractical. We would expect discrete monitoring causes the price of the lookback options to go lower since a new extremum value may be missed out in discrete monitoring. Heynen and Kat (1995) show in their numerical experiments that monitoring the asset price discretely instead of continuously may have a significant effect on the values of lookback options. The analytic price formulas for lookback options with discrete monitoring (Heynen and Kat, 1995) involve the n-variate normal distribution functions, 4.3 Asian options 197 where n is the number of monitoring instants within the remaining life of the option. Levy and Mantion (1997) propose a simple but effective analytic approximation method to price discretely monitored lookback options. The method assumes a second order Taylor expansion of the option value in powers of √δt, where δt is the time between successive monitoring instants (assumed to be at regular intervals). The two coefficients of the Taylor expansion are determined by fitting the approximate formula with option values corresponding to δt = τ and δt = τ/2, where τ is the time to expiry. The construction and implementation of the method are quite straightforward, the details of which are presented in Problem 4.23. As demonstrated by their numerical experiments, the accuracy of this analytic approximation method is quite remarkable. Similar to the discretely monitored barrier options, Broadie et al. (1999) derive analytic approximation formulas for the price functions of discretely monitored lookback options on a monitoring date in terms of the price functions of the continuously monitored counterparts.

4.3 Asian options

Asian options are averaging options whose terminal payoff depends 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. For example, suppose a manufacturer expects to make a stream of copper purchases for his factory over some fixed time horizon. He would be interested to acquire price protection which is linked to the average price over the period. The hedging of risk against the average price may be achieved through the purchase of an appropriate averaging option. 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. Most Asian options are of European-type since an Asian option of American-type may be redeemed as early as the start of the averaging period and lose the intent of protection from averaging. There are two main classes of Asian options, the fixed strike (average rate) and the floating strike (average strike) options, whose terminal call payoff are max(AT X, 0) and max(S A , 0), respectively. Here, S is the asset price at expiry− , X is the T − T T strike price and AT denotes some form of average of the price of the underlying asset over the averaging period [0, T ]. The value of AT depends on the realization of the asset price path. The most common averaging procedures are the discrete arithmetic averaging defined by n 1 A = S (4.3.1a) T n ti Xi=1 198 4 Path Dependent Options and the discrete geometric averaging defined by

n 1/n

AT = Sti . (4.3.1b) "i=1 # Y

Here, Sti is the asset price at discrete time ti, i = 1, 2, , n. In the limit n , the discrete sampled averages become the continuous· · · sampled averages.→ ∞The continuous arithmetic average is given by 1 T A = S dt, (4.3.2a) T T t Z0 while the continuous geometric average is deﬁned to be

1 T A = exp ln S dt . (4.3.2b) T T t Z0 ! A wide variety of averaging options have been proposed. The comprehensive summaries of them can be found in Boyle’s (1993) and Zhang’s (1994) papers. The most commonly used sampled average is the discrete arithmetic average. If the Geometric Brownian motion is assumed for the underlying asset price process, the pricing of this type of Asian option is in general analytically intractable since the sum of lognormal densities has no explicit representation. On the other hand, the analytic derivation of the price formula of a European Asian option with geometric averaging is feasible since the product of lognormal prices remains lognormal. In this section, we first derive the general partial differential equation formulation for pricing Asian options. We then consider the pricing of continuously monitored Asian options with geometric or arithmetic averaging. Put-call parity relations and fixed-floating symmetry relations between the prices of continuously monitored Asian options are deduced. For discretely monitored Asian options, we derive closed form price formulas for geometric averaging options and deduce analytic approximation formulas for arithmetic averaging options using the Edgeworth expansion technique.

4.3.1 Partial diﬀerential equation formulation

Suppose we write the average of the asset price as t A = f(S, u) du, (4.3.3) Z0 where f(S, t) is chosen according to the type of average adopted in the Asian 1 option. For example, f(S, t) = S corresponds to continuous arithmetic av- t n 1 erage, f(S, t) = exp δ(t ti) ln S corresponds to discrete geometric n − ! Xi=1 4.3 Asian options 199 average, etc. The price of an Asian option is a function of time to expiry and the two state variables, S and A. 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 0 → Zt → t < u∗ < t + ∆t, (4.3.4)

so dA is deterministic. Hence, a riskless hedge for the Asian option requires only eliminating the asset-induced risk. 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. Assume the asset price dynamics to be given by

dS = [µS D(S, t)] dt + σS dZ, (4.3.5) − 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, (4.3.6) − 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 − − (4.3.7) The last term in the above equation corresponds to the contribution of the ∂V dividend from the asset to the portfolio’s value. As usual, we choose ∆ = ∂S so that the stochastic terms containing dS cancel. The absence of arbitrage dictates dΠ = rΠ dt, (4.3.8) where r is the riskless interest rate. Putting Eqs. (4.3.7) and (4.3.8) together, we obtain the following governing differential equation for V (S, A, t) ∂V σ2 ∂2V ∂V ∂V + S2 + [rS D(S, t)] + f(S, t) rV = 0. (4.3.9) ∂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. 200 4 Path Dependent Options

4.3.2 Continuously monitored geometric averaging options

We would like to derive analytic price formulas for the European Asian options whose terminal payoff depends on the continuously monitored geometric averaging of the underlying asset price. 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 G = exp ln S du . (4.3.10) t t u Z0 The terminal payoffs 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), (4.3.11) f` T T T − T where X is the fixed strike price. We illustrate how to use the risk neutral expectation approach to derive the price formula of the European fixed strike Asian call option. On the other hand, the partial differential equation method is used to derive the price formula of the floating strike counterpart.

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, (4.3.12) St −

where Zt is a Q-Brownian motion. For 0 < t < T , the solution of the above stochastic diﬀerential equation is given by [see Eqs. (2.4.18a,b)] σ2 ln S = ln S + r q (T t) + σ(Z Z ). (4.3.13) T t − − 2 − T − t By integrating ln St as required by Eq. (4.3.10), we obtain t 1 σ2 (T t)2 ln G = ln G + (T t) ln S + r q − T T t T − t − − 2 2 σ T + (Z Z ) du. (4.3.14) T u − t Zt σ T The stochastic term (Zu Zt) du can be shown to be Gaussian with T t − Zσ2 (T t)3 zero mean and variance − (see Problem 2.32). By the risk neutral T 2 3 4.3 Asian options 201 valuation principle, the value of the European ﬁxed strike Asian call option is given by

r(T t) c (S , G , t) = e− − E[max(G X, 0)], (4.3.15) fix t t T − where E is the expectation under Q conditional on the ﬁltration generated by the Q-Brownian motion. 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 = − , (4.3.16a) − − 2 2T T 3 r GT can be written as t/T (T t)/T GT = Gt St − exp(µ + σZ), (4.3.16b)

where Z is the standard normal random variable. Recallb from our usual expectation calculations with call payoﬀ that b 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 , (4.3.17) σ ! − σ ! we then deduce that

r(T t) t/T (T t)/T µ+σ2/2 c (S , G , t) = e− − G S − e N(d ) XN(d ) , (4.3.18) fix t t t t 1 − 2 h i where t T t d = ln G + − ln S + µ ln X σ, 2 T t T t − d1 = d1 + σ. (4.3.19)

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 cf`(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 cf`(S, G, t) can be expressed as ∂c σ2 ∂2c ∂c G S ∂c f` + S2 f` + (r q)S f` + ln f` rc = 0, ∂t 2 ∂S2 − ∂S t G ∂G − f` 0 < t < T. (4.3.20) 202 4 Path Dependent Options

Next, we deﬁne the similarity variables G c (S, G, t) y = t ln and W (y, t) = f` . (4.3.21) S S This is equivalent to choose S as the numeraire. In terms of the similarity variables, the governing equation for cf`(S, G, t) becomes ∂W σ2t2 ∂2W σ2 ∂W + r q + t qW = 0, 0 < t < T, (4.3.22) ∂t 2 ∂y2 − − 2 ∂y − with terminal condition: W (y, T ) = max(1 ey/T , 0). 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, (4.3.23a) ∂τ 2 ∂y2 − − 2 − ∂y with initial condition at τ = 0 (corresponding to t = T ) given as

F (y, 0; η) = δ(y η). (4.3.23b) − Though the diﬀerential equation has time dependent coeﬃcients, the fundamental solution is readily found to be [see Eq. (3.3.15)]

σ2 τ y η r q + 2 0 (T u) du F (y, τ; η) = n − − − − . (4.2.24) τ  σ (T u)R2 du  0 −  q  The solution to W (y, τ) is then given byR

qτ ∞ η/T W (y, τ) = e− max(1 e , 0)F (y, τ; η) dη. (4.3.25) − Z−∞ The direct integration of the above integral gives (Wu et al., 1999)

q(T t) t/T (T t)/T q(T t) Q c (S, G, t) = Se− − N(d ) G S − e− − e− N(d ), (4.3.26) f` 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 3 σ 3− r b σ2 q b b r q + T 2 t2 σ2 T 3 t3 Q = − 2 − − . (4.3.27) 2 T − 6 T 2 b 4.3 Asian options 203

4.3.3 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) . (4.3.28) fix T T T − He and Takahashi (2000) propose a variable reduction method which reduces the dimension of the governing diﬀerential equation by one. To motivate the choice of variable transformation, we consider the following expectation representation of the price of the Asian call at time t

r(T t) c (S , A , t) = e− − E [max (A X, 0)] fix t t T − t T r(T t) 1 1 = e− − E max S du X + S du, 0 T u − T u " Z0 Zt !# T St r(T t) Su = e− − E max x + du, 0 , (4.3.29) T t S " Zt t !#

where the state variable xt is deﬁned by 1 t x = (I XT ), I = S du = tA . (4.3.30) t S t − t u t t Z0 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 in Eq. (4.3.29) is a function of xt only. We then deduce that

cfix(St, It, t) = Stf(xt, t) (4.3.31) for some function of f. In other words, f(xt, t) is given by

e r(T t) T S f(x , t) = − − E max x + u du, 0 . (4.3.32) t T t S " Zt t !#

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. (4.3.33) ∂t 2 ∂S2 − ∂S ∂I − fix Suppose we define the following transformation of variables: 1 c (S, I, t) x = (I XT ) and f(x, t) = fix , (4.3.34) S − S then the governing differential equation for f(x, t) becomes 204 4 Path Dependent Options

∂f σ2 ∂2f ∂f + x2 + [1 (r q)x] qf = 0, < x < , t > 0.(4.3.35) ∂t 2 ∂x2 − − ∂x − −∞ ∞ The terminal condition is given by 1 f(x, T ) = max(x, 0). (4.3.36) T 1 t When x 0, which corresponds to S du X, it is possible to ﬁnd t ≥ T u ≥ Z0 closed form analytic solution to f(x, t). Since xt is an increasing function of t so that xT 0, the terminal condition f(x, T ) reduces to x/T . In this case, f(x, t) admits≥solution of the form

f(x, t) = a(t)x + b(t). (4.3.37)

By substituting the assumed form of solution into Eq. (4.3.35), 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. (4.3.38a) dt − − When r = q, a(t) and b(t) are found to be 6 r(T t) q(T t) r(T t) e− − e− − e− − a(t) = and b(t) = − . (4.3.38b) 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. (4.3.39) 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 < XT , there is no closed− form analytic solution available. Curran (1994) and Rogers and Shi (1995) propose the conditioning method to ﬁnd a lower bound on the Asian option price. They both use the approach of projecting the averaging state variable AT on a T -measurable Gaussian random variable Z, where F

E[max(A X, 0)] = E [E [max(A X, 0) Z]] T − T − | E [max (E[A X Z], 0)] . (4.3.40) ≥ T − | The resulting expectation involving Z may be solvable in closed form. The choice of Z by Curran is the geometric average while that by Rogers and Shi is the logarithm of the geometric average. The approximation error would 4.3 Asian options 205 be small since the correlation coeﬃcient between the geometric average and arithmetic average is close to one. The analytic approximation approach has also been applied to continuously monitored ﬂoating strike Asian options. Bouaziz et al. (1994) use the linear approximation technique to approximate the law of AT , ST by a joint lognormal distribution [see also the extension to quadratic{ approximation} by Chung et al. (2003)]. Several other analytic approximation methods for pricing Asian options can be found in the papers by Henderson et al. (2004), Milevsky and Posner (1998), Nielsen and Sandmann (2003) and Tsao et al. (2003).

4.3.4 Put-call parity and ﬁxed-ﬂoating symmetry relations

It is well known that the difference of the prices of European vanilla call and put options is equal to a European forward. Do we have similar put-call parity relations for European Asian options? Also, can we establish symmetry relations between the prices of fixed strike and floating strike Asian options, like those of lookback options? In this section, we derive these parity and symmetry relations for continuously monitored Asian options under the Black-Scholes framework. Some of these relations can be extended to more general stochastic price dynamics (Hoogland and Neumann, 2000).

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, respectively. Their terminal payoﬀ functions are given by I c (S, I, T ) = max X, 0 (4.3.41a) fix T − I p (S, I, T ) = max X , 0 . (4.3.41b) fix − T

Let D(S, I, t) denote the diﬀerence of cfix and pfix. Since both cfix and pfix are governed by the same equation [see Eq. (4.3.33)], so does D(S, I, t). 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. (4.3.42) T − − − T T − The terminal condition D(S, I, T ) is the same as that of the continuously monitored arithmetic averaging option with I XT . Hence, when r = q, the put-call parity relation between the prices of≥ ﬁxed strike Asian options6 under continuously monitored arithmetic averaging is given by [see also Eq. (4.3.39)] 206 4 Path Dependent Options

c (S, I, t) p (S, I, t) fix − fix q(T t) r(T t) I r(T t) e− − e− − = X e− − + − S. (4.3.43) 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 geometric/arithmetic averaging) (see Problems 4.28 and 4.29).

Fixed-floating symmetry relations By applying a change of measure and identifying a time-reversal of a Brow- nian motion (Henderson and Wojakowski, 2002), it is possible to establish the symmetry relations 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 monitored arithmetic averaging option at the start of the averaging period (time zero) as

rT cf`(S0, λ, r, q, T ) = e− E [max (λST AT , 0)] (4.3.44a) rT − pf`(S0, λ, r, q, T ) = e− E [max (AT λST , 0)] (4.3.44b) rT − cfix(X, S0, r, q, T ) = e− E [max (AT X, 0)] (4.3.44c) rT − p (X, S , r, q, T ) = e− E [max (X A , 0)] . (4.3.44d) fix 0 − T

Assume that the asset price St follows the Geometric Brownian process under the risk neutral measure Q, where

dSt = (r q) dt + σ dZt. (4.3.45) St −

Here, Zt is a Q-Brownian process. Suppose the asset price is used as the numeraire, then

rT cf` e− cf∗` = = E [max (λST AT , 0)] S0 S0 − rT S e− max (λS A , 0) = E T T − T . (4.3.46) S S 0 T To eﬀect the change of numeraire, we deﬁne the measure Q∗ by

rT dQ∗ σ2 ST e− 2 T +σZT = e− = qT . (4.3.47) dQ S0e−

By virtue of the Girsanov Theorem, Z∗ = Z σt is a Q∗-Brownian process t t − [see Problem 3.9]. If we write AT∗ = AT /ST , then qT c∗ = e− E∗ [max (λ A∗ , 0)] , (4.3.48) f` − T where E∗ denotes the expectation under Q∗. Now, we consider 4.3 Asian options 207

T T 1 Su 1 A∗ = du = S∗(T ) du, (4.3.49a) T T S T u Z0 T Z0 where σ2 S∗(T ) = exp r q (T u) σ(Z Z ) . (4.3.49b) u − − − 2 − − T − u 

In terms of the Q∗-Brownian process Zt∗, where ZT Zu = σ(T u)+ZT∗ Zu∗, we can write − − − σ2 S∗(T ) = exp r q + (u T ) + σ(Z∗ Z∗ ) . (4.3.50) u − 2 − u − T 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 propert− y of a Brownian− process. Hence, −we establish b b b T σ2 law 1 σZT −u+ r q+ (u T ) A∗ = A = e − 2 − du, (4.3.51a) T T T Z0 b b and via time-reversal of ZT u, we obtain −

T σ2 1 σZξ+ q r ξ A b= e − − 2 dξ. (4.3.51b) T T Z0 b b Note that AT S0 is the arithmetic average of the price process with drift rate q r. Summing the results together, we have − b qT c = S c∗ = e− E∗ max λS A S , 0 (4.3.52) f` 0 f` 0 − T 0 h i and from which we deduce the following ﬁxed-ﬂoatingb symmetry relation

cf`(S0, λ, r, q, T ) = pfix(λS0, S0, q, r, T ). (4.3.53)

By combining the put-call parity relations for floating and fixed Asian options and the above symmetry relation, we can derive the fixed-floating symmetry relation between cfix and pf` (see Problem 4.30) X c (X, S , r, q, T ) = p S , , q, r, T . (4.3.54) fix 0 f` 0 S 0 208 4 Path Dependent Options

4.3.5 Fixed strike options with discrete geometric averaging

Consider the discrete geometric averaging of the asset prices at evenly distributed discrete times t = i∆t, i = 1, 2, , n, where ∆t is the uniform time i · · · interval between ﬁxings and tn = T is the time of expiration. Deﬁne the running geometric averaging by

k 1/k G = S , k = 1, 2, , n. (4.3.55) k ti · · · "i=1 # Y The terminal payoﬀ of a European average value call option with discrete geometric averaging is given by max(Gn X, 0), where X is the strike price. Suppose the asset price is lognormally distributed,− then the asset price ratio

Sti Ri = , i = 1, 2, , n is also lognormally distributed. Assume that under Sti−1 · · · the risk neutral measure σ2 ln R N r ∆t, σ2∆t , i = 1, 2, , n, (4.3.56) i ∼ − 2 · · · where r is the riskless interest rate and N(µ, σ2) represents a normal distribution with mean µ and variance σ2.

European ﬁxed strike call option The price formula of the European ﬁxed strike call depends on whether the current time t is prior to or after time t0. First, we consider t < t0 and write

2 n 1/n G S S St − S n = t0 tn n 1 t1 , (4.3.57a) S S S − S − · · · S t t ( tn 1 tn 2 t0 ) so that

Gn St0 1 ln = ln + [ln Rn + 2 ln Rn 1 + + n ln R1] , t < t0. (4.3.57b) St St n − · · ·

St0 Since ln Ri, i = 1, 2, n and ln represent independent Brownian incre- · · · St ments over non-overlapping time intervals, they are normally distributed and G independent. Observe that ln n is a linear combination of these independent St Brownian increments, so it remains to be normally distributed with mean

σ2 1 σ2 n r (t t) + r ∆t i − 2 0 − n − 2 Xi=1 σ2 n + 1 = r (t t) + (T t ) , (4.3.58a) − 2 0 − 2n − 0 4.3 Asian options 209 and variance n 1 (n + 1)(2n + 1) σ2(t t) + σ2∆t i2 = σ2 (t t) + (T t ) . 0 − n2 0 − 6n2 − 0 i=1 X (4.3.58b) Let τ = T t, where τ is the time to expiry. Suppose we write − (n + 1)(2n + 1) σ2 τ = σ2 τ 1 (T t ) (4.3.59a) G − − 6n2 − 0 σ2 σ2 n 1 µ G τ = r τ − (T t ) , (4.3.59b) G − 2 − 2 − 2n − 0 then the transition density function of Gn at time T , given the asset price St at an earlier time t < t0, can be expressed as

2 2 σG 1 ln Gn ln St + µG 2 τ ψ(G ; S ) = exp − − . n t 2  n h 2σ2τ io  Gn 2πσGτ − G   p  (4.3.60) By the risk neutral valuation approach, the price of the European ﬁxed strike call with discrete geometric averaging is given by

rτ cG(St, t) = e− E[max(Gn X, 0)] − (4.3.61a) rτ µGτ = e− [S e N(d ) XN(d )] , t < t t 1 − 2 0 where 2 St σG ln X + µG + 2 τ d1 = , d2 = d1 σG√τ. (4.3.61b) σG√τ − We consider the two extreme cases where n = 1 and n . When σ2 σ→2 ∞ n = 1, σ2 τ and µ G τ reduce to σ2τ and r τ, respec- G G − 2 − 2 tively, so that thecall pricereduces to that of a Europ ean vanilla call 2 option. We observe that σGτ is a decreasing function of n, which is consistent with the intuition that the more frequent we take the averaging, σ2 the lower the volatility. When n , σ2 τ and µ G τ tend to → ∞ G G − 2 2 σ2 T t σ2 τ (T t ) and r τ − 0 , respectively. Correspond- − 3 − 0 − 2 − 2 ingly , discrete geometric averaging becomes itscontinuous analog. In particular, the price of a European ﬁxed strike call with continuous geometric averaging at t = t0 is found to be [see also Eq. (4.3.18)]

1 σ2 r+ (T t0) r(T T ) c (S , t ) = S e− 2 6 − N(d ) Xe− − 0 N(d ), (4.3.62a) G t0 0 t0 1 − 2 where b b 210 4 Path Dependent Options

2 St0 1 σ ln + r + (T t0) X 2 6 − T t0 d1 = , d2 = d1 σ − . (4.3.62b) T t0 − 3 σ −3 r b q b b Next, we consider the in-progress option where the current time t is within the averaging period, that is, t t0. Here, t = tk +ξ∆t for some integer k, 0 k n 1 and 0 ξ < 1. Now,≥S , S S , S are known quantitites; and≤ ≤ − ≤ t1 t2 · · · tk t St St S the price ratios k+1 , k+2 , , tn are independent lognormal random St Stk+1 · · · Stn−1 variables. We may write

1/n (n k)/n G = [S S ] S − n t1 · · · tk t 2 n k 1/n S St − St − tn n 1 k+1 (4.3.63) S − S − · · · S ( tn 1 tn 2 t ) so that

Gn 1 ln = [ln Rn + 2 ln Rn 1 + + (n k 1) ln Rk+2 + (n k) ln Rt] − St n · · · − − − (4.3.64a) wheree 1/n (n k)/n k/n (n k)/n S = [S S ] S − = G S − and R = S /S . (4.3.64b) t t1 · · · tk t k t t tk+1 t G σ2 Lete the variance and mean of ln n be denoted by σ2 τ and µ G τ, G G − 2 St respectively. They are found to be e e e (n k)2 e (n k 1)(n k)(2n 2k 1) σ2 τ = σ2∆t − (1 ξ) + − − − − − G n2 − 6n2 (4.3.65a) and e σ2 σ2 n k (n k 1)(n k) µ G τ = r ∆t − (1 ξ) + − − − . G − 2 − 2 n − 2n e (4.3.65b) Similare to formula (4.3.61a), the price formula of the in-progress European ﬁxed strike call option takes the form

rτ µG τ c (S , τ) = e− S e N(d ) XN(d ) , t t , (4.3.66a) G t t 1 − 2 ≥ 0 h i where e e 2 e e St σG ln X + µG + 2 τ d = , d = d σ √τ. (4.3.66b) 1 2 1 G σG√τ e − e e σ˜2 Again, by takinge the limit n , the limitinge e valuese of σ˜2 , µ˜ G and e → ∞ G G − 2 S˜(t) become 4.3 Asian options 211

2 2 2 2 2 T t σ σG σ T t lim σG = − , lim µG = r − , n T t 3 n − 2 − 2 2(T t ) →∞ − 0 →∞ − 0 e (4.3.67a) and e e

T −t t − T t0 1 lim St = St Gt where Gt = exp ln Su du . (4.3.67b) n T t →∞ − 0 Zt0 The priceeof the correspe ondingecontinuous geometric averaging call option can be obtained by substituting these limiting values into the price formula (4.3.66a).

European ﬁxed strike put option Using a similar derivation procedure, the price of the corresponding European ﬁxed strike put option with discrete geometric averaging can be found to be

rτ µG τ e− [XN( d ) Se N( d )] , t < t − 2 − − 1 0 pG(S, τ) = rτ µG τ (4.3.68) e− XN( d ) Se N( d ) , t t , ( − 2 − − 1 ≥ 0 h i where d1 and d2 are given by Eq.e (4.3.61b),e e and ed˜1 and d˜2 are given by Eq. (4.3.66b). The put-call parity relation for the European ﬁxed strike options with discrete geometric averaging can be deduced to be

rτ µG τ rτ e− Se Xe− , t < t0 cG(S, τ) pG(S, τ) = − (4.3.69) rτ µG τ rτ − e− Se Xe− , t t . − ≥ 0 Additional analytic price formulas for Asian options with geometric aver- e e aging can be found in Boyle’s paper (1993).

4.3.6 Fixed strike options with discrete arithmetic averaging

The most common type of Asian options are the fixed strike options whose terminal payoff is determined by the discrete arithmetic average of past prices. The valuation of these options is made difficult by the choice of Geometric Brownian motion for the underlying asset price since the sum of lognormal components has no closed form representation. First, we give a formal description of this type of option contracts. Suppose the average of asset prices is calculated over the time interval [t0, tn] and at t t discrete points t = t + i t, t = 0, 1, , n, t = n − 0 . The running i 0 4 · · · 4 n average A(t) is defined for the current time t, t t < t , by m ≤ m+1 1 m A(t) = S , 0 m n, (4.3.70) m + 1 ti ≤ ≤ Xi=0 and A(t) = 0 for t < t0. Let tn be the time of expiration so that the payoff function at expiry is given by max(A(t ) X, 0) for the fixed strike call, where n − 212 4 Path Dependent Options

X is the strike price. It may be more convenient to consider the terminal payoﬀ in terms of A˜(tn; t) as deﬁned by

m + 1 1 n A(t ; t) = A(t ) A(t) = S , (4.3.71a) n n − n + 1 n + 1 ti i=m+1 X which is thee average of the unknown stochastic components beyond the current time. For example, the terminal payoff of the fixed strike call can be rewritten as max(A(t ; t) X∗, 0), where n − m + 1 e X∗ = X A(t) (4.3.71b) − n + 1 is the effective strike price of the option. It is easily seen that if X ∗ becomes negative, then the option is surely exercised at expiration. The probability distribution of either A(tn) or A(tn; t) has no available explicit representation. The best approach for deriving approximate analytic price formulas is to approximate the distribution ofe A(tn) [or A(tn; t)] by an approximate lognormal distribution through the method of generalized Edgeworth series expansion. The Edgeworth series expansion is quitee similar to the Taylor series expansion for analytic functions in complex function theory. A brief discussion of the Edgeworth series expansion is presented below (Jarrow and Rudd, 1982).

Edgeworth series expansion Given a probability distribution Ft(s), called the true distribution, we would like to approximate Ft(s) by an approximating distribution Fa(s). The dis- dF (s) tributions considered are restricted to the class where t = f (s) and ds t dF (s) a = f (s) exist, that is, distributions which have continuous density ds a functions. First, we deﬁne the following quantities: (i) jth moment of distribution F :

∞ j αj(F ) = s f(s) ds (4.3.72a) Z−∞ (ii) jth central moment of distribution F :

∞ µ (F ) = [s α (F )]jf(s) ds (4.3.72b) j − 1 Z−∞ (iii) characteristic function of F :

∞ φ(F, t) = eitsf(s) ds, i = √ 1. (4.3.72c) − Z−∞ Here, it is assumed that the moments α (F ) exist for j n. Next, the j ≤ cumulants kj(F ) are deﬁned by 4.3 Asian options 213

n 1 − j (it) n 1 ln φ(F, t) = k (F ) + o(t − ). (4.3.73) j j! Xj=1 It can be shown by theoretical analysis that the ﬁrst n 1 cumulants exist, − provided that αn(F ) exists. The ﬁrst four cumulants are

2 k1(F ) = α1(F ), k2(F ) = µ2(F ), k3(F ) = µ3(F ), k4(F ) = µ4(F ) 3[µ2(F )] . − (4.3.74) djF (s) Also, we assume the existence of the derivatives: a , j m. Let N = dsj ≤ min(n, m), the diﬀerence of ln φ(Ft, t) and ln φ(Fa, t) can be represented by

N 1 − j (it) N 1 ln φ(F , t) = [k (F ) k (F )] + ln φ(F , t) + o(t − ). (4.3.75) t j t − j a j! a j=1 X N−1 Taking the exponentials of the above equation [note that eo(t ) = 1 + N 1 o(t − )], we obtain

N 1 − j (it) n 1 φ(F , t) = exp [k (F ) k (F )] φ(F , t) + o(t − ). (4.3.76) t  j t − j a j!  a j=1 X   Suppose the above exponential term is expanded in polynomial in it, we have

N 1 N 1 − j − j (it) (it) N 1 exp [k (F ) k (F )] = E + o(t − ), (4.3.77)  j t − j a j!  j j! j=1 j=0 X X   where the ﬁrst few coeﬃcients are 2 E0 = 1, E1 = k1(Ft) k1(Fa), E2 = [k2(Ft) k2(Fa)] + E , − − 1 (4.3.78) E = [k (F ) k (F )] + 3E [k (F ) k (F )] + E3, etc. 3 3 t − 3 a 1 2 t − 2 a 1 In terms of cumulants, Eq. (4.3.75) can be rewritten as

N 1 − j (it) N 1 φ(F , t) = E φ(F , t) + o(t − ). (4.3.79) t  j j!  a Xj=0   Lastly, we take the inverse Fourier transform of the above equation. Using the following relations

1 ∞ its f (s) = e− φ(F , t) dt, (4.3.80a) t 2π t Z−∞ j j d fa(s) 1 ∞ its j ( 1) = e− (it) φ(F , t) dt, − dsj 2π a Z−∞ j = 0, 1, , N 1, (4.3.80b) · · · − 214 4 Path Dependent Options we obtain the following representation of the Edgeworth series expansion

N 1 − ( 1)j djf (s) f (s) = f (s) + E − a + (s, N) (4.3.81a) t a j j! dsj j=1 X where 1 ∞ its N 1 (s, N) = e o(t − ) dt. (4.3.81b) 2π Z−∞ In order that (s, N) exists for all s, it is necessary to observe

lim (s, N) = 0 for all s. (4.3.82) N →∞ | | Suppose all moments of the true and approximating distributions can be calculated, one may claim theoretically that a given distribution can be ap- proximated by another distribution to any desired level of accuracy. It is most convenient to use the lognormal distribution as the approximating distribution in valuation problems for arithmetic averaging Asian options. Suppose we choose the parameters of the approximating lognormal distribution such that its ﬁrst two moments match with the ﬁrst two moments of the true distribution, that is, α1(Ft) = α1(Fa) and µ2(Ft) = µ2(Fa), then the corresponding two-term Edgeworth series expansion becomes

ft(s) = fa(s) + (s, 3), (4.3.83) since E = α (F ) α (F ) = 0 and E = µ (F ) µ (F ) + E2 = 0. 1 1 t − 1 a 2 2 t − 2 a 1 Fixed strike call option Consider the fixed strike call option with discrete arithmetic averaging whose terminal payoff is max(A(tn) X, 0). By the risk neutral valuation approach, the price of the fixed strike call− is given by

rτ c(S, A, t) = e− E[max(A(t ) X, 0)] n − rτ = e− E[max(A(t ; t) X∗, 0)] (4.3.84) n − where E is the expectation under the riske neutral measure conditional on St = S and A(t) = A, and A(tn; t) and X∗ are defined by Eqs. (4.3.71a,b). We would like to approximate the distribution of A(tn; t) by a lognormal distribution. More specificallye , we approximate the distribution of ln A(tn; t) by a normal distribution whose mean and variance aree µ(t) and σ(t)2, respectively. The first and the second moments of the approximating lognormale distribution are then given by (see Problem 2.24)

σ(t)2 α (F ) = µ(t) + (4.3.85a) 1 a 2 2 α2(Fa) = 2µ(t) + 2σ(t) (4.3.85b) 4.3 Asian options 215 respectively. Suppose we adopt the two-term Edgeworth approximation for A(tn; t), which can be achieved by setting the ﬁrst two moments of the approximating lognormal distribution and the distribution of A(tn; t) to be equal,e that is, e σ(t)2 µ(t) + = ln E[A(t ; t)] (4.3.86a) 2 n 2 2 2µ(t) + 2σ(t) = ln E[Ae(tn; t) ]. (4.3.86b) 2 Solving for µ(t) and σ(t) , we obtain e 1 µ(t) = 2 ln E[A(t ; t] ln E[A(t ; t)2] (4.3.87a) n − 2 n σ(t)2 = ln E[A(t ; t)2] 2 ln E(A[t ; t)]. (4.3.87b) en − e n Suppose ln A(tn; t) were normallye distributed withe mean µ(t) and variance σ(t)2, then the value of the ﬁxed strike call would become

e r(T t) c(S, A, t) = e− − E[A˜(t ; t)]N(d ) X∗N(d ) (4.3.88) { n 1 − 2 } where 2 µ(t) + σ(t) ln X∗ d = − , d = d σ(t). (4.3.89) 1 σ(t) 2 1 − The above price formula provides a good approximation to the true price of the ﬁxed strike call when the approximating lognormal distribution is a good approximation to the true distribution of A(tn; t). The remaining procedure 2 amounts to the determination of E[A(tn; t)] and E[A(tn; t) ]. Let St denote the spot asset price at thee current time t, where t = tm + ξ t, 0 ξ < 1. For t t , we have e e 4 ≤ ≥ 0 n 1 E[A˜(t ; t)] = E[S ]. (4.3.90) n n + 1 ti i=m+1 X The asset price dynamics under the risk neutral measure Q is assumed to be dS t = r dt + σ dZ (4.3.91) St where r is the constant riskless interest rate and σ2 is the constant variance rate. We then have r(i m ξ) t E[Sti ] = Ste − − 4 , (4.3.92) so that r(n m) t St r(1 ξ) t 1 e − 4 E[A˜(t ; t)] = e − 4 − , t t . (4.3.93) n n + 1 1 er t ≥ 0 − 4 For t < t0, it can be shown similarly that 216 4 Path Dependent Options

r(n+1) t St r(t t) 1 e 4 E[A˜(t ; t)] = e 0− − , t < t . (4.3.94) n n + 1 1 er t 0 − 4 Next, we consider E[A(t ; t)2]. For t t , we have n ≥ 0 n n 1 E[A(t ; t)2] = e E[S S ] n (n + 1)2 ti tj i=m+1 j=m+1 X X e n n 1 r(i+j 2ξ) t+σ2(min(i,j) ξ) t 2 = e − 4 − 4 S , (n + 1)2 t i=Xm+1 j=Xm+1 (4.3.95) and after some tedious manipulation, we obtain (Levy, 1992)

2 σ2 2 St 2ξ r+ t E[A(t ; t) ] = e− 2 4 (A A +A A ), t t , (4.3.96a) n (n + 1)2 1 − 2 3 − 4 ≥ 0 wheree (2r+σ2 ) t (2r+σ2 )(N m+1) t e 4 e − 4 A = − , 1 (1 er t)[1 e(2r+σ2 ) t] − 4 − 4 [r(N m+2)+σ2 ] t (2r+σ2 )(N m+1) t e − 4 e − 4 A2 = − , (1 er t)[1 e(r+σ2 ) t] − 4 − 4 (4.3.96b) (3r+σ2 ) t [r(N m+2)+σ2 ] t e 4 e − 4 A = − , 3 (1 er t)[1 e(r+σ2 ) t] − 4 − 4 2(2r+σ2 ) t (2r+σ2 )(N m+1) t e 4 e − 4 A = − . 4 [1 e(r+σ2 ) t][1 e(2r+σ2 ) t] − 4 − 4 2 The calculation of E[A(tn; t) ] for t < t0 can be performed similarly [see Problem (4.32)]. e

4.4 Problems

4.1 Consider the function S λ B2 f(S, τ) = c , τ , B E S 

where cE (S, τ) is the price of a vanilla European call option, λ is some constant parameter. Show that f(S, τ) satisﬁes the Black-Scholes equation ∂f σ2 ∂2f ∂f = S2 + rS rf ∂τ 2 ∂S2 ∂S − r when λ is chosen to be 2 + 1. − σ2 4.4 Problems 217

Hint: Substitution of f(S, τ) into the Black-Scholes equation gives ∂f σ2 ∂2f ∂f S2 + rS rf ∂τ − 2 ∂S2 ∂S − S λ ∂c σ2 ∂2c = E ξ2 E B ∂τ − 2 ∂ξ2 ∂c σ2 ∂c + (λ 1)σ2ξ E λ(λ 1) c rλc + rξ E + rc − ∂ξ − − 2 E − E ∂ξ E B2 where c = c (ξ, τ), ξ = . E E S 4.2 Consider the European zero-rebate up-and-out put option with expo- γτ nential barrier function B(τ) = Be− , where B(τ) > X for all τ. Show that the price of this barrier option is given by

δ 1 B(τ) − B(τ)2 2(r γ) p(S, τ) = p (S, τ) p , τ , δ = − , E − S E S σ2 

where pE (S, τ) is the price of the corresponding European vanilla put option. Deduce the price of the corresponding European up-and-in put option with the same barrier. S Hint: Let y = ln , show that p(y, τ) satisﬁes B(τ)

∂p σ2 ∂2p σ2 ∂p = + r γ rp. ∂τ 2 ∂y2 − 2 − ∂y − 4.3 By applying the following transformation on the dependent variable in the Black-Scholes equation

c = eαy+βτ w, 1 r α2σ2 where α = , β = r, show that Eq. (4.1.4a) is reduced 2 − σ2 − 2 − to the prototype diﬀusion equation ∂w σ2 ∂2w = , ∂τ 2 ∂y2 while the auxiliary conditions are transformed to become

βτ αy y w(0, τ) = e− R(τ) and w(y, 0) = max(e (e X), 0). − Consider the following diﬀusion equation with semi-inﬁnite domain ∂v ∂2v = a2 , x > 0 and t > 0, a is a positive constant, ∂t ∂x2 218 4 Path Dependent Options

with initial condition: v(x, 0) = f(x) and boundary condition: v(0, t) = g(t), the solution to the above problem is given by (Kevorkian, 1990)

1 ∞ x ξ)2 /4a2t (x+ξ)2/4a2t v(x, t) = f(ξ)[e− − e− ] dξ 2a√πt 0 − Z 2 2 x t e x /4a ω + − g(t ω) dω. (i) 2a√π ω3/2 − Z0 Using the form of solution given by Eq. (i), show that the price of the European down-and-out option is given by

αy+βτ 1 ∞ αξ ξ c(y, τ) = e max(e− (e X), 0) √2πτσ − Z0 (y ξ)2/2σ2τ (y+ξ)2/2σ2τ e− − e− dξ − h 2 2i y τ e β(τ ω)e y /2σ ω + − − − R(τ ω) dω . √2πσ ω3/2 − Z0 ) Assuming B < X, show that the price of the European down-and-out call option is given by [see Eqs. (4.1.13a,b)]

δ 1 B − B2 c(S, τ) = c (S, τ) c , τ E − S E S 2 2 ln S + r σ ω exp − B − 2 τ S 2σ2w rω ln B ! + e− R(τ ω) dω. √2πσ ω3/2 − Z0 The last term represents the additional option premium due to the rebate payment. 4.4 Suppose the asset price follows the lognormal distribution with drift rate r, volatility σ, and barrier B. Find the density function of the asset price ST at expiration time T , with the current asset price S starting below the barrier then breaching the barrier but ending below the barrier at expiration. 4.5 We deﬁne

S σ2 ln K + r + 2 τ d = , d = d σ√τ, 1 σ√τ 2 1 − 2 ln B 2r d = S + d , d = d σ√τ, δ = , 3 σ√τ 1 4 3 − σ2 4.4 Problems 219

rτ V (S, τ; K) = [Xe− N( d ) SN( d )] b − 2 − − 1 δ 1 2 B − rτ B Xe− N( d ) N( d ) . − S − 4 − S − 3 Show that the price functions of various barrier put options are given by

puo(S, τ; X, B) = Vb(S, τ; min(X, B)) p (S, τ; X, B) = max(V (S, τ; X) V (S, τ; B), 0) do b − b p (S, τ; X, B) = p (S, τ; X) p (S, τ; X, B) ui E − uo p (S, τ; X, B) = p (S, τ; X) p (S, τ; X, B), di E − do where pE (S, τ; X) is the price function of the European put option. 4.6 Consider a European two-asset down-and-out call option where the terminal payoﬀ depends on the payoﬀ state variable S1,t and knock-out occurs when the barrier state variable S2,t breaches the downstream barrier B2. Assume that under the risk neutral measure Q.

dSi,t = r dt + σi dZi,t, i = 1, 2 and dZ1 dZ2 = ρ dt. Si,t

Let X1 denote the strike price. Show that the price of this down-and-out call with an external barrier is given by (Kwok et al., 1998) call price rτ T = e− EQ[(S1,T X)1 S1,T >X 1 m >B ] − { } { 2,0 2} δ2 1+2γ12 B2 − = S N (d , e ; ρ) N(d0 , e0 ; ρ) 1,0 2 1 1 − S 1 1 " 2,0 # δ2 1 rT B2 − e− X N(d , e ; ρ) N(d0 , e0 ; ρ) , − 1 2 2 − S 2 2 " 2,0 # where S σ2 ln 1,0 + r + 1 T X1 2 d1 = , d2 = d1 σ1√T , σ1√T − 2γ ln B2 12 S2,0 d10 = d1 + , d20 = d10 σ1√T , σ1√T − 2 S2,0 σ1 ln + r + ρσ1σ2 T B2 − 2 e1 = , e2 = e1 ρσ1√T , σ2√T − 2 ln B2 S2,0 e10 = e1 + , e20 = e10 ρσ1√T , σ2√T − 2r σ1 δ2 = 2 , γ12 = ρ . σ2 σ2 220 4 Path Dependent Options

4.7 Consider a European down-and-out partial barrier call where the barrier provision is activated only between option starting date (time 0) and t1. Here, t1 is some time earlier than the expiration date T so that 0 < t1 < T . Let B and X denote the down-barrier and strike, respec- St tively, where B < X. Let ln = r dt + σZt under the risk neutral S0 measure Q. Assuming S0 > B, show that the down-and-out call price is given by (Heynen and Kat, 1994a) call price rT 1 1 t = e− EQ (ST X) ST >X m 1 >B − { } { 0 } h δ+1 i t1 B t1 = S N d , e ; N d0 , e0 ; 0 1 1 T − S 1 1 T " r ! r !# δ 1 rT t1 B − t1 e− X N d , e ; N d0 , e0 ; , − 2 2 T − S 2 2 T " r ! r !# where

2 S0 σ ln X + r + 2 T d1 = , d2 = d1 σ√T , σ√T − 2 ln B S0 d10 = d1 + , d20 = d10 σ√T , σ√T − 2 S0 σ ln B + r + 2 t1 e1 = , e2 = e1 σ√t1, σ√t1 − 2 ln B S0 e10 = e1 + , e20 = e10 σ√t1, σ√t1 − 2r δ = . σ2 Show that the above price formula reduces to that in Eq.(4.1.12a,b) when t1 is set equal T . t Hint: Modify the price formula in Problem 4.6 by setting ρ = 1 , r T t1 σ1 = σ and σ2 = σ so that γ12 = 1. r T 4.8 Consider a typical term in g(x, t)

µx µ2 t exp σ2 2σ2 (x ξ)2 − exp − , √ 2 − 2σ2t 2πσ t where ξ can be either 2n(u `) or 2` + 2n(u `). Show that the above term can be rewritten as − − 4.4 Problems 221

µξ 2 e σ2 (x ξ µt) exp − − − . √ 2 2σ2t 2πσ t Hence, show that u g(x, t) dx Z` ∞ 2µn(u `) = exp − σ2 n= X−∞ u µt 2u(u `) ` µt 2n(u `) N − − − N − − − σ√t − σ√t 2µ[` + n(u `)] exp − − σ2 u µt 2` 2n(u `) ` µt 2` 2n(u `) N − − − − N − − − − . σ√t − σ√t Use the above result to derive the price formula of the double knock-out call option [see Eq. (4.1.45)]. Analytic tractability of the price formula remains even when the barriers are exponential functions in time. Suppose the upper and lower δ1t δ2 t barriers become Ue and Le , t [0, T ]. Here, δ1 and δ2 are constant and the barriers do not intersect ov∈er [0, T ]. Show that the price formula for the double knock-out call option can be expressed as (Kunitomo and Ikeda, 1992)

µ µ ∞ U n 1 L 2 co = S [N(d ) N(d )] LU Ln S 1 − 2 n= X−∞ Ln+1 µ3 [N(d ) N(d )] − U nS 3 − 4 µ 2 µ ∞ n 1 2 rT U − L Xe− N(d σ√T ) N(d σ√T ) − Ln S 1 − − 2 − n= ( ) X−∞ µ 2 Ln+1 3 − [N(d σ√T ) N(d σ√T )] , − U nS 3 − − 4 − ) where

SU 2n σ2 SU 2n σ2 ln XL2n + r + 2 T ln F L2n + r + 2 T d1 = , d2 = , σ√T σ√T L2n+2 σ2 L2n+2 σ2 ln XSU 2n + r + 2 T ln F SU 2n + r + 2 T d3 = , d4 = , σ√T σ√T 222 4 Path Dependent Options

2[r δ n(δ δ )] δ δ µ = − 2 − 1 − 2 + 1, µ = 2n 1 − 2 , 1 σ2 2 γ2 2[r δ2 + n(δ1 δ2)] µ = − − + 1, F = Ueδ1 T . 3 σ2

4.9 Let P (x, t; x0, t0) denote the transition density function of the restricted µ Brownian process Wt = µt + σZt with two absorbing barriers at x = 0 and x = `. Using the method of separation of variables (Kevorkian, 1990), show that the solution to P (x, t; x0, t0) admits the following eigen-function expansion [which diﬀers drastically in analytic form from that in Eq. (4.1.57)]

µ ∞ (x x0) 2 λk(t t ) kπx kπx0 P (x, t; x , t ) = e σ2 − e− − 0 sin sin 0 0 ` ` ` Xk=1 where the eigenvalues are given by 1 µ2 k2π2σ2 λ = + . k 2 σ2 `2 

Hint: P (x, t; x0, t0) satisﬁes the forward Fokker-Planck equation with + auxiliary conditions: P (0, t) = P (`, t) = 0 and P (x, t0 ; x0, t0) = δ(x x0). Pelsser (2000) derives the above solution by performing the−Laplace inversion using Bromwich contour integration. + 4.10 Let the exit time density q (t; x0, t0) have dependence on the initial + + state X(t0) = x0. We write τ = t t0 so that q (t; x0, t0) = q (x0, τ). Show that the partial diﬀerential −equation formulation is given by ∂q+ ∂q+ σ2 ∂2q+ = µ + 2 , ` < x0 < u, τ > 0, ∂τ ∂x0 2 ∂x0 [with auxiliary conditions:

+ + + q (u, τ) = 0, q (`, τ) = 0 and q (x0, 0) = δ(x0).

+ Solve for q (x0, τ) using the partial diﬀerential equation approach and compare the solution with that given in Eq. (4.1.63b). Also, show that

t t u + q (s; x0, t0) ds + q−(s; x0, t0) ds + P (x, t; x0, t0) dx = 1, Zt0 Zt0 Z`

where P (x, t; x0, t0) is the transition density function deﬁned in Prob- lem 4.9. 4.11 By using the method of path counting, Sidenius (1998) shows that g+(x, T ) deﬁned in Eq. (4.1.44b) has the following analytic solution 4.4 Problems 223

µx µ2 g+(x, T ) = exp T σ2 − σ2 ∞ φ(x; α+, σ√T ) φ(x, β+, σ√T ) n − n "n=1 # X where 1 (x λ)2 φ(x; λ, ν) = exp − √ 2 − 2ν2 2πν α+ = 2n(u `) + 2` and β+ = 2n(u `). n − n − Find the closed form price formula for the upper-barrier knock-in call option [see Eq. (4.1.46) and Luo’s paper (2001)]. 4.12 A sequential barrier option is a barrier option with two-sided barriers where nullification occurs only when the barriers are breached at a pre-specified order (say, up then down). Show that the price formula of the sequential up-then-down out call option can be inferred from those of the double knock-out call option (see Problem 4.8) except that the infinite summation over n is replaced by summation over only two terms (Li, 1999). Hint: Show that the price formula obtained by summing the two specific terms satisfies the condition that the sequential up-then- down out call option becomes the corresponding down-and-out call when the asset price hits the upper barrier. 4.13 Consider a discretely monitored down-and-out call option with strike price X and barrier level Bi at discrete time ti, i = 1, 2, , n. Show that the price of this barrier call is given by (Heynen and ·Kat,· · 1996)

cd0 (S0, T ; X, B1, B2, , Bn) 1 2 ·n·+1· rT 1 2 n+1 = S N (d , d , , d ; Γ ) e− N (d , d , , d ; Γ ) 0 n+1 1 1 · · · 1 − n+1 2 2 · · · 2 where

2 S0 σ ln + r + ti i Bti 2 i i d1 = , i = 1, 2, , n, d2 = d1 σ√ti, σ√ti · · · − 2 S0 σ ln X + r + 2 T dn+1 = , dn+1 = dn+1 σ√T . 1 σ√T 2 1 − Also, Γ is the (n + 1) (n + 1) correlation matrix whose entries are given by ×

min(tj, tk) tj ρjk = , 1 j, k n; ρj,n+1 = , j = 1, 2, , n. √tj√tk ≤ ≤ r T · · · 224 4 Path Dependent Options

4.14 Let fmin(y) and fmax(y) denote the density function of yT and YT , respectively. Show that

1 y + µτ 2µ 2µy y + µτ f (y) = n − + e σ2 N min σ√τ σ√τ σ2 σ√τ 2µy 1 y + µτ + e σ2 n , σ√τ σ√τ 1 y µτ 2µ 2µy y µτ f (y) = n − e σ2 N − − max σ√τ σ√τ − σ2 σ√τ 2µy 1 y µτ + e σ2 n − − . σ√τ σ√τ

Compare with fdown(x, m, T ) and fup(y, M, T ) in Eqs. (4.1.27b,c). 4.15 As an alternative approach to derive the value of a European ﬂoating strike lookback call, we consider

rτ T cf`(S, m, τ) = e− E[ST min(m, mt )] rτ − T = S e− E[min(m, m )], − t t where St = S, mT0 = m and τ = T t. We may decompose the above expectation calculation into two terms:−

T T T E[min(m, mt )] = mP [m mt ] + E[mt 1 m>mT ]. ≤ { t } Show that the ﬁrst term is given by mT m mP ln t ln S ≥ S m 1 2r m ln + µτ S − σ2 ln + µτ = m N − S N S . σ√τ − m σ√τ " # Now, the second term can be expressed as

m ln S T y E[mt 1 mT >m ] = Se fmin(y) dy. { t } Z−∞ By performing the tedious integration procedure, show that the same price function for cf`(S, m, τ) [see Eq. (4.2.14a,b)] is obtained. T 4.16 From the following form of the distribution function of mt [see Eq. (4.2.4a)]

m 1 2r m ln + µτ S − σ2 ln + µτ P [m mT ] = N − S N S , ≤ t σ√τ − m σ√τ Show that P [m mT ] becomes zero when S = m. ≤ t 4.4 Problems 225

4.17 Suppose we use a straddle (combination of call and put with the same strike m) in the rollover strategy for hedging the ﬂoating strike lookback call and we write

cf`(S, m, τ) = cE (S, τ; m) + pE (S, τ; m) + strike bonus premium.

Find an integral representation of the strike bonus premium in terms T of distribution functions of ST and mt . 4.18 Prove the following put-call parity relation between the prices of the ﬁxed strike lookback call and ﬂoating strike lookback put:

rτ c (S, M, τ; X) = p (S, max(M, X), τ) + S Xe− . fix f` − Deduce that ∂c fix = 0 for M < X. ∂M

Give a ﬁnancial interpretation why cfix is insensitive to M when M < X [see also Eq. 4.2.8)]. 4.19 Suppose the terminal payoﬀs of the partial lookback call and put options are max(S λmT , 0), λ > 1 and max λM T S , 0 , 0 < λ < T T0 T0 T 1, respectively. Sho−w that the price formulas of these −lookback options are, respectively (Conze and Viswanathan, 1991)

ln λ rτ ln λ c (S, m, τ) = SN d λme− N d σ√τ m − σ√τ − m − σ√τ − 2 2 2r/σ rτ σ S − 2r ln λ + e− λS N d + σ 2r m − m σ − σ√τ " 2 ln λ erτ λ2r/σ N d − − m − σ√τ ln λ p (S, M, τ) = SN d + − − M σ√τ rτ ln λ + λMe− N d + + σ√τ − M σ√τ 2 2 2r/σ rτ σ S − 2r ln λ e− λS N d √τ + − 2r M M − σ σ√τ " 2 ln λ erτ λ2r/σ N d + , − M σ√τ

where dm and dM are given by Eqs. (4.2.14b) and (4.2.15b), respectively. 4.20 The terminal payoﬀ of the lookback spread option is given by 226 4 Path Dependent Options

c (S, m, M, 0) = max(M T mT X, 0). sp T0 − T0 − Show that the current value of the European lookback spread option can be expressed as (Wong and Kwok, 2003b) (i) currently at- or in-the-money, that is, M m X 0 − − ≥ rτ c (S, m, M, τ) = c (S, m, τ) + p (S, M, τ) Xe− ; sp f` f` −

(ii) currently out-of-the-money, that is, M m X < 0 − − rτ c (S, m, M, τ) = c (S, m, τ) + p (S, M, τ) Xe− sp f` f` − m+X rτ T T + e− P [M < ξ m + X] dξ. t ≤ t ZM 4.21 An investor holding a European in-the-money call option may suffer loss in profits if the asset price drops substantially just before expiration. The “limited period” fixed strike lookback feature may help remedy the above market exit problem. To achieve optimal timing for market exit, it is therefore sensible to choose the lookback period starting some time after the option’s starting date and ending at expiration. Let [T1, T ] denote the lookback period and consider the pricing of this fixed strike lookback call at time t before the lookback period, whose terminal payoff is max(M T X, 0). Note that when S > X, the call T1 T1 − T is guaranteed to be in-the-money at expiration since M T1 ST1 . Show that the value of the “limited-period” lookback call is given≥ by

rτ T c(S, τ) = e− E max M X, 0 T1 − M T rτ T1 T = e− E ST1 X 1 m >X,ST X , t < T1, { 1 } "( ST1 ! − ) # where X is the strike price. Assuming the usual lognormal diﬀusion process for the asset price under the risk neutral measure, show that 4.4 Problems 227

rτ T T1 c(S, τ) = SN(d ) e− XN(d ) SN e , d ; − 1 − 2 − 2 − 1 1 − T t r − ! 2r 2 σ2 rτ T1 t rτ σ S − e− XN f , d ; − + e− S − 2 2 − 2 − T t 2r − X r − ! " 

2r 2r T1 t N2 d1 √τ, f1 + T1 t; − − σ − σ − −r T t ! p − T T + erτ N e , d ; − 1 2 1 1 T t r − !# 2 r(T T ) σ + e− − 1 1 SN(f )N( e ), t < T , − 2r 1 − 2 1 where d1, e1, f1, , etc. are the same as those defined in Eq. (4.2.23) except that m is·replaced· · by X accordingly (Heynen and Kat, 1994b). Deduce the price formula for the corresponding “limited period” fixed strike lookback put option whose terminal payoff function is max(X T − mT1 , 0). 4.22 Use Eq. (4.2.31) to derive the following partial differential equation for the floating strike lookback put option ∂V σ2 ∂2V σ2 ∂V = r + , 0 < ξ < , τ > 0, ∂τ 2 ∂ξ2 − 2 ∂ξ ∞ M where V (ξ, τ) = p (S, M, t)/S and τ = T t, ξ = ln . The auxiliary f` − S conditions are ∂V V (ξ, 0) = eξ 1 and (0, τ) = 0. − ∂ξ Solve the above Neumann boundary value problem and check the result with the put price formula given in Eqs. (4.2.15a,b). ∂V Hint: Define W = so that W satisfies the same governing differen- ∂ξ tial equation but the boundary condition becomes W (0, τ) = 0. Solve for W (ξ, τ), then integrate W with respect to ξ to obtain V . Be aware that an arbitrary function φ(t) is generated upon integration with respect to ξ. Obtain an ordinary differential equation for φ(t) by substituting the solution for V into the original differential equation. 4.23 Let p(S, t; δt) denote the value of a floating strike lookback put option with discrete monitoring of the realized maximum value of the asset price, where δt is the regular interval between monitoring instants. Sup- pose we assume the following two-term Taylor expansion of p(S, t; δt) in powers of √δt (Levy and Mantion, 1997) 228 4 Path Dependent Options

p(S, t; δt) p(S, t; 0) + α√δt + βδt. ≈ With δt = 0, p(S, t; 0) represents the lookback put value corresponding to continuous monitoring. Let τ denote the time to expiry. By setting δt = τ and δt = τ/2, we deduce the following pair of linear equations for α and β:

τ τ τ α + β = p S, t; p(S, t; 0) 2 2 2 − r α√τ + βτ = p(S, t; τ) p(S, t; 0). − Hence, p(S, t; τ) is simply the vanilla put value with strike price equal to the current realized maximum asset price M. With only one monitoring instant at the mid-point of the remaining option’s life, show that τ p S, t; 2 is given by τ p S, t; 2 qτ rτ τ τ 1 = e− S + e− MN2 dM + σ , dM (τ) + σ√τ; − − 2 2 − √2 r (r q)τ τ 1 + e − SN2 dM (τ), d ; 2 √2 (r q) τ τ τ τ + e − 2 SN d , d + σ ; 0 , 2 M 2 − 2 2 r where

S σ2 σ2 ln M + r q + 2 τ r q + 2 √τ d (τ) = − and d(τ) = − . M σ√τ σ Once α and β are determined, we then obtain an approximate price formula of the discretely monitored ﬂoating strike lookback put. 4.24 Explain why

call Asian call Asian call European ≥ arithmetic ≥ geometric Hint: An average price is less volatile than the series of prices from which it is computed. 4.25 Consider the exchange option which entities the holder the right but not the obligation to exchange risky asset S2 for another risky asset S1. Let the price dynamics of S1 and S2 under the risk neutral measure be governed by

dSi = (r qi) dt + σi dZi, i = 1, 2, Si − 4.4 Problems 229

where dZ1 dZ2 = ρ dt. Let V (S1, S2, τ) denote the price function of the exchange option, whose terminal payoﬀ takes the form

V (S , S , 0) = max(S S , 0). 1 2 1 − 2

Show that the governing equation for V (S1, S2, τ) is given by

2 2 2 2 2 ∂V σ1 ∂ V ∂ V σ2 2 ∂ V = S1 2 + ρσ1σ2S1S2 + S2 2 ∂τ 2 ∂S1 ∂S1∂S2 2 ∂S2 ∂V ∂V + (r q2)S1 + (r q2)S2 rV. − ∂S2 − ∂S2 −

By taking S2 as the numeraire and deﬁning the similarity variables S V (S , S , τ) x = 1 and W (x, τ) = 1 2 , S2 S2 show that the governing equation for W (x, τ) becomes

∂W σ2 ∂2W ∂W = x2 + (q q )x . ∂τ 2 ∂x2 1 − 2 ∂x Verify that the solution to W (x, τ) is given by

q τ q τ W (x, τ) = e− 1 xN(d ) e− 2 N(d ) 1 − 2 or

q τ q τ V (S , S , τ) = S e− 1 N(d ) S e− 2 N(d ) 1 2 1 1 − 2 2 where

2 S1 σ ln S + q2 q1 + 2 τ d = 2 − , d = d σ√τ, 1 σ√τ 2 1 − σ2 = σ2 2ρσ σ + σ2. 1 − 1 2 2 Apply the above exchange option price formula to price the floating strike Asian call option based on the knowledge of the price formula of the fixed strike Asian call option. σ2(T t)2 Hint: The covariance between S and G is equal to − . T T 2 4.26 We define the geometric average of the price path of asset i, i = 1, 2, during the time interval [t, t + T ] by

1 T G (t + T ) = exp ln S (t + u) du . i T i Z0 ! 230 4 Path Dependent Options

Consider an Asian option involving two assets whose terminal payoﬀ is given by max(G1(t + T ) G2(t + T ), 0). Show that the price formula of this European Asian option− at time t is given by (Boyle, 1993)

V (S , S , t; T ) = S N(d ) S N(d ) 1 2 1 1 − 2 2 where e e r2 σ2 σ2 σ2 2 S = S exp + T , i = 1, 2, σ2 = 1 + 2 ρσ σ , i i − 2 12 3 3 − 3 1 2 2 S1 σ e ln + 2 T d = S2 , d = d σ√T . 1 √ 2 1 eσ T − e Hint: Apply the price formula of the exchange option.

4.27 Suppose we deﬁne the ﬂexible geometric average GF (n) of asset prices at n evenly spaced time instants by

n ωi GF (n) = Si , i=1 Y iα where ω = , i = 1, 2, , n and S is the asset price at time t . i n · · · i i iα Xi=1 Here, ωi is the weighting factor associated with Si with the property that the larger the value of α, the heavier are the weights allocated to the more recent asset price. Under the risk neutral measure, the asset price is assumed to follow the lognormal process dS = r dt + σ dZ. S We consider the ﬁxed strike Asian option whose terminal payoﬀ function is

V (S, G , T ) = max(φ(G (n) X), 0), F F − where X is the strike price, and φ is the binary variable which is set to 1 for a call or 1 for a put. Show that the Asian option value is given by (Zhang, 1994)−

f f f rτ f V (S, GF , t) = φ SAj N φ dn j + σ T n j Xe− N φdn j , − − − − q where 4.4 Problems 231

2 f f σ f f f Aj = exp r τ T µ,n j T µ,n j T n j Bj , − − − − 2 − − − j ωi f f Sn i B = 1, B = − , 1 j n, 0 j S ≤ ≤ i=1 Y S σ2 f f ln X + r 2 T µ,n j + ln Bj f − − dn j = , − f σ T n j − n q f T µ,n j = ωi[τ (n i)∆t], − − − i=Xj+1 n n j i 1 − − f 2 T n j = ωi [τ (n i)∆t] + 2 ωiωk[τ (n k)∆t], − − − − − i=j+1 i=2 X X kX=1 n is the number of asset prices taken for averaging, ∆t is the time interval between successive observational instants, j is the number of f observations already passed, Bj can be considered as the weighted average of the returns of those observations that have already passed. 4.28 Deduce the following put-call parity relation between the prices of Eu- ropean ﬁxed strike Asian call and put options under continuously monitored geometric averaging

c(S, G, t) p(S, G, t) − 2 2 r(T t) t/T (T t)/T σ T t = e− − G S − exp (T t) − − 6 T ( " 2 r q σ T t + − − 2 − X . 2 T # − !)

4.29 Suppose continuous arithmetic averaging of the asset price is taken from t = 0 to T , T is the expiration time. The terminal payoﬀ functions for the ﬂoating strike call and put options are, respectively,

1 T 1 T max S S du, 0 and max S du S , 0 . T − T u T u − T Z0 ! Z0 ! Show that the put-call parity relation for the above pair of European ﬂoating strike options is given by

q(T t) S r(T t) q(T t) r(T t) c p = Se− − + [e− − e− − ] e− − A , − (r q)T − − t − where 232 4 Path Dependent Options

1 t A = S du. t T u Z0 Suppose continuous geometric averaging of the asset price is taken, show that the corresponding put-call parity relation is given by

q(T t) t/T (T t)/T c p = Se− − G S − − − σ2 2 2 3 σ (T t) r q 2 (T t) exp − + − − − r(T t) ,  6T 2 2T − −    where 1 t G = exp ln S du . t t u Z0 4.30 Show that the put-call parity relations between the prices of ﬂoating strike and ﬁxed strike Asian options at the start of the averaging period are given by

qT rT S(e− e− ) p (S , λ, r, q, T ) c (S , λ, r, q, T ) = − λS f` 0 − f` 0 (r q)T − 0 −qT rT S(e− e− ) rT c (X, S , r, q, T ) p (X, S , r, q, T ) = − e− X. fix 0 − fix 0 (r q)T − − By combining the above put-call parity relations with the fixed-floating symmetry relation between cf` and pfix, deduce the following symmetry relation between cfix and pf` X c (X, S , r, q, T ) = p S , , q, r, T . fix 0 f` 0 S 0 4.31 Consider the European continuously monitored arithmetic average Asian option with terminal payoff max(A X S X , 0), where T − 1 T − 2 1 T A = S du. T T u Z0 At the current time t > 0, the average value over the time period [0, t] has been realized. Let V (S, τ; X1, X2) denote the price function of the Asian option at the start of the averaging period. Show that the value of the in-progress Asian option is given by T t X T X T A t − V S, T t; 1 , 2 t − . T − T t T t − T t − − −

2 4.32 Consider E A(tn; t) deﬁned in Eq. (4.3.95). Show that when t < t0, we have (Levyh , 1992)i e 4.4 Problems 233

2 2 St (2r+σ)(t t) E[A(t ; t) ] = e 0− (B B + B B ) n (n + 1)2 1 − 2 3 − 4 where e

2 2 1 e(2r+σ )(n+1)∆t er(n+1)∆t e(2r+σ )(n+1)∆t B = − , B = − , 1 (1 er∆t)[1 e(2r+σ2 )∆t] 2 (1 er∆t)[1 e(r+σ2 )∆t] − − − 2 − 2 er∆t er(n+1)∆t e(2r+σ )∆t e(2r+σ )(n+1)∆t B = − , B = − . 3 (1 er∆t)[1 e(r+σ2 )∆t] 4 [1 e(r+σ2 )∆t][1 e(2r+σ2 )∆t] − − − −