Asian Options

Chapter 13 Asian Options

Asian options are special cases of average value options, whose claim payoffs are determined by the difference between the average underlying asset price over a certain time interval and a strike price K. This chapter covers several probabilistic and PDE techniques for the pricing and hedging of Asian options. Due to their dependence on averaged asset prices. Asian options are less volatile than plain vanilla options whose claim payoffs depend only on the terminal value of the underlying asset.

13.1 Bounds on Asian Option Prices ...... 449 13.2 Pricing by the Hartman-Watson distribution . . . . . 456 13.3 Laplace Transform Method ...... 459 13.4 Moment Matching Approximations ...... 460 13.5 PDE Method ...... 466 Exercises ...... 477

13.1 Bounds on Asian Option Prices

Asian options were ﬁrst traded in Tokyo in 1987, and have become particu- larly popular in commodities trading.

Arithmetic Asian options

Given an underlying asset price process (St)t∈[0,T ], the payoﬀ of the Asian call option on (St)t∈[0,T ] with exercise date T and strike price K is given by

1 T + C = Stdt − K . T w0

Similarly, the payoﬀ of the Asian put option on (St)t∈[0,T ] with exercise date T and strike price K is 449

" This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

1 T + C = K − Stdt . T w0

Due to their dependence on averaged asset prices, Asian options are less volatile than plain vanilla options whose payoﬀs depend only on the terminal value of the underlying asset.

As an example, Figure 13.1 presents a graph of Brownian motion and its moving average process 1 t Xt := Bsds, t > 0. t w0

∗ Fig. 13.1: Brownian motion Bt and its moving average Xt.

Related exotic options include the Asian-American options, or Hawaiian options, that combine an Asian claim payoﬀ with American style exercise, and can be priced by variational PDEs, cf. § 8.6.3.2 of Crépey(2013).

An option on average is an option whose payoﬀ has the form

C = φ(ΛT , ST ), where T 2 T σBu+ru−σ u/2 ΛT = S0 e du = Sudu, T 0. w0 w0 > • For example when φ(y, x) = (y/T − K)+ this yields the Asian call option with payoﬀ + + 1 T ΛT Sudu − K = − K , (13.1) T w0 T

∗ The animation works in Acrobat Reader on the entire pdf ﬁle.

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which is a path-dependent option whose price at time t ∈ [0, T ] is given by " # T + −(T −t)r ∗ 1 e E Sudu − K Ft . (13.2) T w0

• As another example, when φ(y, x) := e−y this yields the price

h T i ∗ − Sudu ∗ −Λ P (0, T ) = E e r0 = E e T

at time 0 of a bond with underlying short-term rate process St.

Using the time homogeneity of the process (St)t∈R+ , the option with payoﬀ C = φ(ΛT , ST ) can be priced as

T −(T −t)r ∗ −(T −t)r ∗ e E [φ(ΛT , ST ) | Ft] = e E φ Λt + Sudu, ST Ft wt T Su ST = e−(T −t)rE∗ φ y + x du, x wt S S t t y=Λt,x=St T −t Su ST −t = e−(T −t)rE∗ φ y + x du, x . (13.3) w0 S S 0 0 y=Λt,x=St

Using the Markov property of the process (St, Λt)t∈R+ , we can write down the option price as a function

−(T −t)r ∗ f(t, St, Λt) = e E [φ(ΛT , ST ) | Ft] −(T −t)r ∗ = e E [φ(ΛT , ST ) | St, Λt] of (t, St, Λt), where the function f(t, x, y) is given by

T −t −(T −t)r ∗ Su ST −t f(t, x, y) = e E φ y + x w du, x . 0 S0 S0

As we will see below there exists no easily tractable closed-form solution for the price of an arithmetically averaged Asian option.

Geometric Asian options

On the other hand, replacing the arithmetic average

n 1 X 1 T St (tk − tk−1) ' Sudu T k T w0 k=1 with the geometric average

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n n ! Y (t −t )/T Y (t −t )/T S k k−1 S k k−1 tk = exp log tk k=1 k=1 n ! 1 X t −t = exp log S k k−1 T tk k=1 n ! 1 X = exp (t − t ) log S T k k−1 tk k=1 1 T ' exp log Sudu T w0 leads to closed-form solutions using the Black Scholes formula, see Exer- cise 13.5.

Pricing by probability density functions

We note that the prices of option on averages can be estimated numerically using the joint probability density function ψΛT −t,BT −t of (ΛT −t, BT −t), as follows:

T −t −(T −t)r ∗ Su ST −t f(t, x, y) = e E φ y + x w du, x 0 S0 S0 ∞ ∞ −(T −t)r σu+(T −t)r−(T −t)σ2/2 e φ y + xz, x e ψ ,B (z, u)dzdu, w0 w−∞ ΛT −t T −t see Section 13.2 for details.

Bounds on Asian option prices

We note (see Lemma 1 of Kemna and Vorst(1990) and Exercise 13.7 below for the discrete-time version of that result), that the Asian call option price can be upper bounded by the corresponding European call option price using convexity arguments. Proposition 13.1. Assume that r > 0, and let φ be a convex and non- decreasing payoﬀ function. We have the bound −rT ∗ 1 T −rT ∗ e E φ Sudu − K e E [φ(ST − K)]. T w0 6 Proof. By Jensen’s inequality for the uniform measure with probability den- 1 ∗ sity function (1/T ) [0,T ] on [0, T ] and for the probability measure P , we have

−rT ∗ T du −rT ∗ T du e E φ Su − K = e E φ (Su − K) w0 T w0 T 452 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance

−rT ∗ T du e E φ(Su − K) 6 w0 T T −rT ∗ −(T −u)r ∗ du = e E φ e E [ST | Fu] − K w0 T T −rT ∗ ∗ −(T −u)r du = e E φ E [ e ST − K | Fu] w0 T T −rT ∗ ∗ −(T −u)r du e E E φ e ST − K Fu (13.4) 6 w0 T

−rT T ∗ ∗ du e E E φ(ST − K) Fu (13.5) 6 w0 T −rT T ∗ du = e E [φ(ST − K)] w0 T −rT ∗ = e E [φ(ST − K)], where from (13.4) to (13.5) we used the facts that r > 0 and φ is non- decreasing. In particular, taking φ(x) : (x − K)+, Proposition 13.1 shows that Asian option prices are upper bounded by European call option prices, as " # T + −rT ∗ 1 −rT ∗ + e E Sudu − K e E [(ST − K) ], T w0 6 see Figure 13.2 for an illustration, as the averaging feature of Asian options reduces their underlying volatility.

nSim=99999;N=1000;t <- 1:N; dt <- 1.0/N; sigma=2;r=0.5; european=0;asian=0;K=1.5 2 dev.new(width=16,height=7); par(oma=c(0,5,0,0)) for (j in 1:nSim){S<-exp(sigma*cumsum(rnorm( N, 0,sqrt(dt)))+r*t/N-sigma**2*t/2/N);color="blue" 4 A<-sum(c(1,S))/(N+1);if (S[N]>=K) {european=european+S[N]-K} if (A>=K) {asian=asian+A-K};if (S[N]>A) {color="darkred"} else {color="darkgreen"} 6 plot(c(0,t/N),c(1,S), xlab ="Time", type= 'l', lwd = 3, ylab ="", ylim =c(0,exp(4*r)), col= color,main=paste("Asian Price=",format(round(asian,2)),"/", j,"=",format(round(asian/j,2)),"European Price=",format(round(european,2)), "/",j,"=",format(round(european/j,2))), xaxs='i',xaxt='n',yaxt='n', yaxs='i', yaxp = c(0,10,10), cex.lab=2, cex.main=2) text(0.3,6,paste("A-Payoﬀ=",format(round(max(A-K,0),2)),",E-Payoﬀ=", format(round(max(S[N]-K,0),2))), col=color,cex=2) 8 axis(1, las=1, cex.axis=2) axis(2, at=c(0,K,A,1,2,3,4,5,6,7,8,9,10), labels=c(0,"K","Average",1,2,3,4,5,6,7,8,9,10), las=2, cex.axis=2) 10 lines(c(0,t/N),rep(K,N+1),col="red",lty = 1, lwd = 4); lines(c(0,t/N),rep(A,N+1),col="darkgreen",lty = 2, lwd = 4); Sys.sleep(0.1) 12 if (S[N]>K || A>K) {readline(prompt="Pause. Press to continue...")}}

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Fig. 13.2: Asian option price vs European option price.∗

In the case of Asian call options we have the following result. Proposition 13.2. We have the conditional bound " # T + −(T −t)r ∗ 1 e E Sudu − K Ft (13.6) T w0 " # t + −(T −t)r ∗ 1 T − t 6 e E Sudu + ST − K Ft T w0 T on Asian option prices, t ∈ [0, T ]. Proof. Let the function f(t, x, y) be deﬁned as

" + # ∗ 1 T f(t, St, Λt) = E Sudu − K Ft , T w0 i.e., from Proposition 13.1,

" +# ∗ 1 T −t Su f(t, x, y) = E y + x w du − K T 0 S0 " +# ∗ 1 x = E y + ΛT −t − K T S0 " +# ∗ y x = E − K + ΛT −t T TS0

∗ The animation works in Acrobat Reader on the entire pdf ﬁle. 454 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance

" # (T − t)x yS KTS Λ + = E∗ 0 − 0 + T −t TS0 (T − t)x (T − t)x T − t " +# (T − t)x ∗ yS0 KTS0 6 E − + ST −t TS0 (T − t)x (T − t)x " +# y (T − t)xST −t = E∗ − K + , x, y > 0, T TS0 which yields (13.6). See also Proposition 3.2-(ii) of Geman and Yor(1993) for lower bounds when r takes negative values. We also have the following bound which yields the behavior of Asian call option prices in large time. Proposition 13.3. The Asian call option price satisﬁes the bound " # T + −(T −t)r t −(T −t)r −(T −t)r ∗ 1 e 1 − e e E Sudu − K Ft 6 Sudu + St , T w0 T w0 rT t ∈ [0, T ], and tends to zero (almost surely) as time to maturity T tends to inﬁnity: " #! T + −(T −t)r ∗ 1 lim e E Sudu − K Ft = 0, t > 0. T →∞ T w0 Proof. We have the bound

" # T + −(T −t)r ∗ 1 0 6 e E Sudu − K Ft T w0 T −(T −t)r ∗ 1 e E Sudu Ft 6 T w0 t T −(T −t)r ∗ 1 −(T −t)r ∗ 1 = e E Sudu Ft + e E Sudu Ft T w0 T wt t T −(T −t)r 1 ∗ 1 −(T −t)r ∗ = e E Su | Ft du + e E Su | Ft du T w0 T wt t T −(T −t)r 1 1 −(T −t)r (u−t)r = e Sudu + e e Stdu T w0 T wt t T 1 −(T −t)r St −(T −u)r = e Sudu + e du T w0 T wt t −(T −t)r 1 −(T −t)r 1 − e = e Sudu + St . T w0 rT

455 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

Note that as T tends to inﬁnity the Black-Scholes European call price tends to St, i.e., we have

−(T −t)r ∗ + lim e E [(ST − K) | Ft] = St, t 0, T →∞ > see Exercise 6.4-aiii).

13.2 Pricing by the Hartman-Watson distribution

First, we note that the numerical computation of Asian call option prices can be done using the probability density function of

T ΛT = Stdt. w0 In Yor(1992), Proposition 2, the joint probability density function of

t 2 σBs−pσ s/2 (Λt, Bt) = e ds, Bt − pσt/2 , t > 0, w0 has been computed in the case σ = 2, cf. also Dufresne(2001) and Matsumoto and Yor(2005). In the next proposition, we restate this result for an arbitrary variance parameter σ after rescaling. Let θ(v, τ) denote the function deﬁned as

2 π /(2τ) ∞ v e −ξ2/(2τ) −v cosh ξ θ(v, τ) = √ w e e sinh(ξ) sin (πξ/τ) dξ, v, τ > 0. 2π3τ 0 (13.7) Proposition 13.4. For all t > 0 we have

t 2 σBs−pσ s/2 σt P e ds ∈ dy, Bt − p ∈ dz w0 2 σz σz/2 2 ! σ 2 2 1 + e 4 e σ t dy = e−pσz/2−p σ t/8 exp −2 θ , dz, 2 σ2y σ2y 4 y y > 0, z ∈ R. The expression of this probability density function can then be used for the pricing of options on average such as (13.3), as

T −t −(T −t)r ∗ Sv ST −t f(t, x, y) = e E φ y + x w dv, x 0 S0 S0 = e−(T −t)r ∞ T −t σu+(T −t)r−(T −t)σ2/2 Sv × w φ y + xz, x e P w dv ∈ dz, BT −t ∈ du 0 0 S0 456 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance

σ 2 2 ∞ ∞ 2 = e−(T −t)r+(T −t)p σ /8 φ y + xz, x eσu+(T −t)r−(T −t)(1+p)σ /2 2 w0 w−∞ 2 ! 2 ! 1 + eσu−(T −t)pσ /2 p 4 eσu/2−(T −t)pσ /4 (T − t)σ2 dz × exp −2 − σu θ , du σ2z 2 σ2z 4 z

2 2 ∞ ∞ 2 = e−(T −t)r−(T −t)p σ /8 φ y + x/z, xv2 e(T −t)r−(T −t)σ /2 w0 w0 1 + v2 4vz (T − t)σ2 dz ×v−1−p exp −2z θ , dv , σ2 σ2 4 z which actually stands as a triple integral due to the deﬁnition (13.7) of θ(v, τ). Note that here the order of integration between du and dz cannot be exchanged without particular precautions, at the risk of wrong computations.

By repeating the argument of (13.3) for φ(x, y) := (x − K)+, the Asian call option can be priced as " # T + −(T −t)r ∗ 1 e E Sudu − K Ft T w0 " # T + −(T −t)r ∗ 1 = e E Λt + Sudu − K Ft T wt " # T + −(T −t)r ∗ 1 Su = e E y + x w du − K Ft T t St x=St, y=Λt " # T −t + −(T −t)r ∗ 1 Su = e E y + x w du − K . T 0 S0 x=St, y=Λt

Hence the Asian call option can be priced as " # T + −(T −t)r ∗ 1 f(t, St, Λt) = e E Sudu − K Ft , T w0 where the function f(t, x, y) is given by " # T −t + −(T −t)r ∗ 1 Su f(t, x, y) = e E y + x w du − K T 0 S0 " +# −(T −t)r ∗ 1 x = e E y + ΛT −t − K , x, y > 0. (13.8) T S0

From Proposition 13.4, we deduce the marginal probability density function of ΛT , also called the Hartman-Watson distribution see e.g. Barrieu et al. (2004).

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Proposition 13.5. The probability density function of

T 2 σBt−pσ t/2 ΛT := e dt, w0 is given by

T 2 P eσBt−pσ t/2dt ∈ du w0 σv−pσ2T /2 ! σv/2−pσ2T /4 2 ! σ 2 2 ∞ 1 + e p 4 e σ T = ep σ T /8 exp −2 − σv θ , dvdu 2u w−∞ σ2u 2 σ2u 4 2 2 2 2 ∞ 1 + v 4v σ T du = e−p σ T /8 v−1−p exp −2 θ , dv , w0 σ2u σ2u 4 u u > 0. From Proposition 13.5, we get

T P(ΛT /S0 ∈ du) = P Stdt ∈ du (13.9) w0 2 2 2 2 ∞ 1 + v 4v σ T du = e−p σ T /8 v−1−p exp −2 θ , dv , w0 σ2u σ2u 4 u

σB −pσ2t/2 2 where St = S0 e t and p = 1 − 2r/σ . By (13.8), this probability density function can then be used for the pricing of Asian options, as

" +# −(T −t)r ∗ 1 x f(t, x, y) = e E y + ΛT −t − K (13.10) T S0

∞ + −(T −t)r y + xz = e − K P(ΛT −t/S0 ∈ dz) w0 T + σ 2 2 ∞ ∞ y + xz = e−(T −t)r e−(T −t)p σ /8 − K 2 w0 w0 T 1 + v2 4v σ2 dz ×v−1−p exp −2 θ , (T − t) dv σ2z σ2z 4 z 1 2 2 ∞ ∞ = e−(T −t)r−(T −t)p σ /8 (xz + y − KT ) T w0∨(KT −y)/x w0 1 + v2 4v σ2 dz × exp −2 θ , (T − t) dv σ2z σ2z 4 z 2 + 4x 2 2 ∞ ∞ 1 (KT − y)σ = e−(T −t)r−(T −t)p σ /8 − σ2T w0 w0 z 4x 1 + v2 σ2 dz ×v−1−p exp −z θ vz, (T − t) dv , 2 4 z

458 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance cf. Theorem in § 5 of Carr and Schröder(2004), which is actually a triple integral due to the deﬁnition (13.7) of θ(v, t). Note that since the integrals are not absolutely convergent, here the order of integration between dv and dz cannot be exchanged without particular precautions, at the risk of wrong computations.

13.3 Laplace Transform Method

The time Laplace transform of the rescaled option price

" +# ∗ 1 t C(t) := E Sudu − K , t > 0, t w0 as √ √ K/2 2 2 ∞ e−xx−2+(p+ 2λ+p )/2(1 − 2Kx)2+( 2λ+p −p)/2dx −λt r0 w e C(t)dt = p , 0 λ(λ − 2 + 2p)Γ − 1 + p + 2λ + p2 /2 with here σ := 2, and Γ(z) denotes the gamma function, see Relation (3.10) in Geman and Yor(1993). This expression can be used for pricing by numerical inversion of the Laplace transform using e.g. the Widder method, the Gaver- Stehfest method, the Durbin-Crump method, or the Papoulis method. The following Figure 13.3 represents Asian call option prices computed by the Geman and Yor(1993) method.

30 25 20 15 10 5 0

Underlying 90

85 350 250 300 150 200 80 50 100 0 Time in months

Fig. 13.3: Graph of Asian call option prices with σ = 1, r = 0.1 and K = 90.

We refer to e.g. Carr and Schröder(2004), Dufresne(2000), and references therein for more results on Asian option pricing using the probability density function of the averaged geometric Brownian motion.

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Figure 9.6 presents a graph of implied volatility surface for Asian options on light sweet crude oil futures.

13.4 Moment Matching Approximations

Lognormal approximation

Other numerical approaches to the pricing of Asian options include Levy (1992), Turnbull and Wakeman(1992) which rely on approximations of the average price distribution based on the lognormal distribution. The lognormal distribution has the probability density function

1 2 2 dx g(x) = √ e−(µ−log x) /(2η ) , x > 0, η 2π x where µ ∈ R, η > 0, with moments

µ+η2 2 2 2µ+2η2 E[X] = e / and E X = e . (13.11)

The approximation is implemented by matching the above ﬁrst two moments to those of time integral T ΛT := Stdt w0 of geometric Brownian motion

2 σBt+(r−σ /2)t St = S0 e , 0 6 t 6 T , as computed in the next proposition, cf. also (7) and (8) page 480 of Levy (1992). Proposition 13.6. We have

erT − 1 E∗[Λ ] = S , T 0 r and (σ2+2r)T 2 rT 2 ∗ 2 2 r e − (σ + 2r) e + (σ + r) E (ΛT ) = 2S . 0 (σ2 + r)(σ2 + 2r)r Proof. The computation of the ﬁrst moment is straightforward: we have

∗ ∗h T i E ΛT = E Sudu w0 T ∗ = E [Su]du w0 T ru = S0 e du w0

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erT − 1 = S . 0 r

For the second moment we have, letting p := 1 − 2r/σ2,

T T 2 2 ∗ 2 2 −pσ a/2−pσ b/2 ∗ σBa σBb E ΛT = S0 e E e e dbda w0 w0 T a 2 −pσ2a/2−pσ2b/2 (a+b)σ2/2 bσ2 = 2S0 e e e dbda w0 w0 T a 2 −(p−1)σ2a/2 −(p−3)σ2b/2 = 2S0 e e dbda w0 w0 2 4S T 2 2 = 0 e−(p−1)σ a/2(1 − e−(p−3)σ a/2)da (p − 3)σ2 w0 2 2 4S T 2 4S T 2 2 = 0 e−(p−1)σ a/2da − 0 e−(p−1)σ a/2 e−(p−3)σ a/2da (p − 3)σ2 w0 (p − 3)σ2 w0 2 2 8S 2 4S T 2 = 0 (1 − e−(p−1)σ T /2) − 0 e−(2p−4)σ a/2da (p − 3)(p − 1)σ4 (p − 3)σ2 w0 2 2 8S 2 4S 2 = 0 (1 − e−(p−1)σ T /2) − 0 (1 − e−(p−2)σ T ) (p − 3)(p − 1)σ4 (p − 3)(p − 2)σ4 2 r e(σ +2r)T − (σ2 + 2r) erT + (σ2 + r) = 2S2 , 0 (σ2 + r)(σ2 + 2r)r

2 2 since r − σ /2 = −pσ /2. By matching the ﬁrst and second moments

2 2 µˆ T +ηˆ T /2 2 2(µˆ T +ηˆ T ) E[ΛT ] ' e T and E ΛT ' e T of the lognormal distribution with the moments of Proposition 13.6 we estimate µbT and ηbT as 2 ! 2 1 E ΛT 1 ∗ 1 2 ηbT = log ∗ 2 and µbT = log E [ΛT ] − ηbT . T (E [ΛT ]) T 2

Under this approximation, the probability density function ϕΛT of ΛT = T S dt is approximated by the lognormal probability density function r0 t 1 (µ − log x)2 ϕ x ≈ − bT x > ΛT ( ) p exp 2 , 0. xσt,T 2(T − t)π 2(T − t)ηbT

461 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

2.5 Exact expression Lognormal approximation 2

1.5

1 Probability density 0.5

0 0 0.5 1 1.5 2 2.5 3 x

Fig. 13.4: Lognormal approximation for the probability density function of ΛT . As a consequence of Lemma 7.8 we ﬁnd the approximation " # + + −rT ∗ 1 T −rT ∞ x e E Stdt − K = e − K ϕ (x)dx T w0 w0 T ΛT −rT 2 e ∞ x + (µT − log x) dx ' − K exp − b p w0 T 2 x σt,T 2(T − t)π 2(T − t)ηbT 1 2 = e(bµ+bη /2)T Φ(d ) − KΦ(d ), (13.12) T 1 2 where √ ∗ 2 log(E [ΛT ]/(KT )) T µTb + ηb T − log(KT ) d1 = √ + ηb = √ ηb T 2 ηb T and √ ∗ √ log(E [ΛT ]/(KT )) T d2 = d1 − ηb T = √ − ηb . ηb T 2 The next Figure 13.5 compares the lognormal approximation to a Monte Carlo estimate of Asian call option prices with σ = 0.5, r = 0.05 and K/St = 1.1.

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1 Lognormal approximation 0.9 Monte Carlo estimate Stratifed lognormal approximation 0.8

0.7

0.6

0.5

0.4 Asian option price 0.3

0.2

0.1 2 4 6 8 10 Time t

Fig. 13.5: Lognormal approximation to the Asian call option price.

Figure 13.5 also includes the stratiﬁed approximation

" +# −rT 1 T e E Stdt − K (13.13) T w0 + −rT ∞ h x i = e E − K ST = y ϕ |S y(x)dP(ST y)dx w0 T ΛT T = 6 −rT ∞ e −p(y/x)σ2(y/x)T /2+σ2(y/x)T /2 ' e Φ(d+(K, y, x)) − KT Φ(d−(K, y, x)) T w0 × dP(ST 6 y)dx, cf. Privault and Yu(2016), where √ 1 2x(bT (y/x) − (1 + y/x)aT (y/x)) σ(y/x) T d±(K, y, x) := √ log ± 2σ(y/x) T σ2K2T 2 2 and  1 2 b (z)  σ2 z T − − z  ( ) := log 2 1 ,  T σ aT (z) aT (z)    1 log z 1√ log z 1√ √ 2 √ 2 aT (z) := 2 Φ + σ T − Φ − σ T ,  σ p(z) σ2T 2 σ2T 2    √ √  1 log z 2 log z 2  bT (z) := Φ √ + σ T − Φ √ − σ T ,  σ2q(z) σ2T σ2T and

1 2 2 2 1 2 2 2 p(z) := √ e−(σ T /2+log z) /(2σ T ), q(z) := √ e−(σ T +log z) /(2σ T ). 2πσ2T 2πσ2T

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Conditioning on the geometric mean price

Asian options on the arithmetic average 1 T Stdt T w0 have been priced by conditioning on the geometric mean price

1 T 1 T 1 T G := exp log Stdt exp log Stdt = Stdt T w0 6 T w0 T w0 in Curran(1994), as

" +# −rT ∗ 1 T e E Sudu − K T w0 " + # −rT ∞ ∗ 1 T = e E Sudu − K G = x dP(G 6 x) w0 T w0 " + # −rT K ∗ 1 T = e E Sudu − K G = x dP(G 6 x) w0 T w0 " + # −rT ∞ ∗ 1 T + e E Sudu − K G = x dP(G 6 x) wK T w0

= C1 + C2, where " + # −rT K ∗ 1 T C1 := e E Sudu − K G = x dP(G 6 x), w0 T w0 and " + # −rT ∞ ∗ 1 T C2 := e E Sudu − K G = x dP(G 6 x) wK T w0 −rT ∞ ∗ 1 T = e E Sudu − K G = x dP(G x) wK T w0 6 −rT e ∞ ∗ T −rT ∞ = E Sudu G = x dP(G x) − K e dP(G x) T wK w0 6 wK 6 −rT e ∗ T −rT = E Sudu1{G K} − K e P(G K). T w0 > >

The term C1 can be estimated by a lognormal approximation given that G = x. As for C2, we note that

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1 T G = exp log Stdt T w0 1 T σ2t = exp µt + σBt − dt T w0 2 T σ2 σ T = exp µ − + Btdt , 2 2 T w0 hence T 2 σ T log G = (µ − σ /2) + Btdt 2 T w0 2 2 has the Gaussian distribution N (µ − σ /2)T /2, σ T /3 with mean (µ − σ2/2)T /2, and variance " # T 2 T T E Btdt = E BsBtdsdt w0 w0 w0

T T = E[BsBt]dsdt w0 w0 T t = 2 sdsdt w0 w0 T = t2dt w0 T 3 = . 3 Hence, we have

P(G > K) = P(log G > log K) T σ2 σ T = P µ − + Btdt > log K 2 2 T w0 T T T σ2 = P Btdt > − µ − + log K w0 σ 2 2 √ 3 T σ2 = Φ √ µ − − log K . σ T 2 2

Basket options

Basket options on the portfolio

N X (k) AT := αkST k=1

465 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

have also been priced in Milevsky(1998) by approximating AT by a lognormal or a reciprocal gamma random variable, see also Deelstra et al.(2004) for additional conditioning on the geometric average of asset prices.

Asian basket options

Moment matching techniques combined with conditioning have been applied to Asian basket options in Deelstra et al.(2010). See also Dahl and Benth (2002) for the pricing of Asian basket options using quasi Monte Carlo sim- ulation.

13.5 PDE Method

Two variables

The price at time t of the Asian call option with payoff (13.1) can be written as " # T + −(T −t)r ∗ 1 f(t, St, Λt) = e E Sudu − K Ft , 0 6 t 6 T . T w0 (13.14) Next, we derive the Black-Scholes partial differential equation (PDE) for the value of a self-financing portfolio. Until the end of this chapter we model the asset price (St)t∈[0,T ] as

dSt = µStdt + σStdBt, t > 0, (13.15) where (Bt)t∈R+ is a standard Brownian motion under the historical probability measure P.

Proposition 13.7. Let (ηt, ξt)t∈R+ be a self-ﬁnancing portfolio strategy whose value Vt := ηtAt + ξtSt, t > 0, takes the form

Vt = f(t, St, Λt), t > 0, where f ∈ C1,2,1((0, T ) × (0, ∞)2) is given by (13.14). Then, the function f(t, x, y) satisﬁes the PDE

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∂f ∂f ∂f 1 ∂2f rf(t, x, y) = (t, x, y) + x (t, x, y) + rx (t, x, y) + x2σ2 (t, x, y), ∂t ∂y ∂x 2 ∂x2 (13.16) 0 6 t 6 T , x > 0, under the boundary conditions  y +  + −(T −t)r f(t, 0 , y) = lim f(t, x, y) = e − K , (13.17a)  x&0 T  f(t, x, 0+) = lim f(t, x, y) = 0, (13.17b) y&0   y + f(T , x, y) = − K , (13.17c)  T

0 6 t 6 T , x > 0, y > 0, and ξt is given by ∂f ξ = (t, S , Λ ), 0 t T . (13.18) t ∂x t t 6 6 Proof. We note that the self-ﬁnancing condition (5.8) implies

dVt = ηtdAt + ξtdSt = rηtAtdt + µξtStdt + σξtStdBt (13.19) = rVtdt + (µ − r)ξtStdt + σξtStdBt, t > 0.

Since dΛt = Stdt, an application of Itô’s formula to f(t, x, y) leads to ∂f ∂f dV = f(t, S , Λ ) = (t, S , Λ )dt + (t, S , Λ )dΛ t t t ∂t t t ∂y t t t 2 ∂f 1 2 2 ∂ f ∂f +µSt (t, St, Λt)dt + S σ (t, St, Λt)dt + σSt (t, St, Λt)dBt ∂x 2 t ∂x2 ∂x ∂f ∂f = (t, S , Λ )dt + S (t, S , Λ )dt ∂t t t t ∂y t t 2 ∂f 1 2 2 ∂ f ∂f +µSt (t, St, Λt)dt + S σ (t, St, Λt)dt + σSt (t, St, Λt)dBt. ∂x 2 t ∂x2 ∂x (13.20)

By respective identiﬁcation of components in dBt and dt in (13.19) and (13.20), we get

467 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

 ∂f ∂f ∂f  rηtAtdt + µξtStdt = (t, St, Λt)dt + St (t, St, Λt)dt + µSt (t, St, Λt)dt  ∂t ∂y ∂x  2  1 2 2 ∂ f + S σ (t, St, Λt)dt, 2 t ∂x2    ∂f ξ S σdB = S σ (t, S , Λ )dB ,  t t t t ∂x t t t hence  2 ∂f ∂f 1 2 2 ∂ f  rVt − rξtSt = (t, St, Λt) + St (t, St, Λt)dt + St σ (t, St, Λt),  ∂t ∂y 2 ∂x2

 ∂f  ξ = (t, S , Λ ), t ∂x t t i.e.  ∂f ∂f ∂f  rf(t, St, Λt) = (t, St, Λt) + St (t, St, Λt) + rSt (t, St, Λt)  ∂t ∂y ∂x  2  1 2 2 ∂ f + S σ (t, St, Λt), 2 t ∂x2    ∂f  ξ = (t, S , Λ ).  t ∂x t t

Remarks.

i) We have ξT = 0 at maturity from (13.17c) and (13.18), which is consis- tent with the fact that the Asian option is cash-settled at maturity and, + close to maturity, its payoﬀ (ΛT /T − K) becomes less dependent on the underlying asset price ST . ii) If Λt/T > K, by Exercise 13.8 we have " # T + −(T −t)r ∗ 1 f(t, St, Λt) = e E Sudu − K Ft T w0 −(T −t)r Λt 1 − e = e−(T −t)r − K + S , (13.21) T t rT

0 6 t 6 T . In particular, the function

y 1 − e−(T −t)r f(t, x, y) = e−(T −t)r − K + x , T rT

0 6 t 6 T , x > 0, y > 0, solves the PDE (13.16). iii) When Λt/T > K, the Delta ξt is given by

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∂f 1 − e−(T −t)r ξ = (t, S , Λ ) = , 0 t T . (13.22) t ∂x t t rT 6 6 Next, we examine two methods which allow one to reduce the Asian option pricing PDE from three variables (t, x, y) to two variables (t, z). Reduction of dimensionality can be of crucial importance when applying discretization scheme whose complexity are of the form N d where N is the number of discretization steps and d is the dimension of the problem (curse of dimensionality).

(1) One variable with time-independent coeﬃcients

Following Lamberton and Lapeyre(1996), page 91, we deﬁne the auxiliary process 1 1 t 1 Λt Zt := w Sudu − K = − K , 0 6 t 6 T . St T 0 St T

With this notation, the price of the Asian call option at time t becomes

T + −(T −t)r ∗h 1 i −(T −t)r ∗ + e E Sudu − K Ft = e E ST (ZT ) Ft . T w0

Lemma 13.8. The price (13.2) at time t of the Asian call option with payoﬀ (13.1) can be written as

f(t, St, Λt) = Stg(t, Zt) (13.23) " # T + −(T −t)r ∗ 1 = e E Sudu − K Ft , t ∈ [0, T ], T w0 with the relation 1 y f(t, x, y) = xg t, − K , x > 0, y 0, 0 t T , x T > 6 6 where " # T −t + −(T −t)r ∗ 1 Su g(t, z) = e E z + w du (13.24) T 0 S0 " +# ΛT −t = e−(T −t)rE∗ z + , S0T with the boundary condition

g(T , z) = z+, z ∈ R.

469 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

Proof. For 0 6 s 6 t 6 T , we have 1 t St d(StZt) = d Sudu − K = dt, T w0 T hence t t Su StZt = SsZs + d(SuZu) = SsZs + du, ws ws T and therefore

StZt 1 t Su = Zs + w du, 0 6 s 6 t 6 T . Ss T s Ss

Since for any t ∈ [0, T ], St is positive and Ft-measurable, and Su/St is independent of Ft, u > t, we have: " + # −(T −t)r ∗ + −(T −t)r ∗ ST e E ST (ZT ) Ft = e StE ZT Ft St " # T + −(T −t)r ∗ 1 Su = e StE Zt + w du Ft T t St " # T + −(T −t)r ∗ 1 Su = e StE z + w du T t St z=Zt " # T −t + −(T −t)r ∗ 1 Su = e StE z + w du T 0 S0 z=Zt " +# −(T −t)r ∗ ΛT −t = e StE z + S0T z=Zt = Stg(t, Zt), which proves (13.24). When Λt/T > K we have Zt > 0, hence in this case by (13.21) and (13.23) we ﬁnd

1 − e−(T −t)r g(t, Z ) = e−(T −t)rZ + , 0 t T . (13.25) t t rT 6 6 Note that as in (13.10), g(t, z) can be computed from the probability density function (13.9) of ΛT −t, as

" +# ΛT −t g(t, z) = E∗ z + S0T ∞ u + Λt = w z + dP 6 u 0 T S0 470 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance

2 2 = e−p σ t/8 ∞ u + ∞ 1 + v2 4v σ2 du × z + v−1−p exp −2 θ , (T − t) dv w0 T w0 σ2 σ2u 4 u 2 2 = e−p σ t/8 ∞ u ∞ 1 + v2 4v σ2 du × z + v−1−p exp −2 θ , (T − t) dv w(−zT )∨0 T w0 σ2 σ2u 4 u 2 2 2 2 ∞ ∞ 1 + v 4v σ du = z e−p σ t/8 v−1−p exp −2 θ , (T − t) dv w(−zT )∨0 w0 σ2 σ2u 4 u 2 2 1 2 2 ∞ ∞ 1 + v 4v σ + e−p σ t/8 v−1−p exp −2 θ , (T − t) dvdu. T w(−zT )∨0 w0 σ2 σ2u 4

The next proposition gives a replicating hedging strategy for Asian options.

Proposition 13.9. (Rogers and Shi(1995)). Let (ηt, ξt)t∈R+ be a self- ﬁnancing portfolio strategy whose value Vt := ηtAt + ξtSt, t ∈ [0, T ], is given by 1 Λt Vt = Stg(t, Zt) = Stg t, − K , 0 6 t 6 T , St T where g ∈ C1,2((0, T ) × (0, ∞)) is given by (13.24). Then, the function g(t, z) satisﬁes the PDE

∂g 1 ∂g 1 ∂2g (t, z) + − rz (t, z) + σ2z2 (t, z) = 0, (13.26) ∂t T ∂z 2 ∂z2 0 6 t 6 T , under the terminal condition

g(T , z) = z+, z ∈ R, (13.27) and the corresponding replicating portfolio Delta is given by ∂g ξ = g(t, Z ) − Z (t, Z ), 0 t T . (13.28) t t t ∂z t 6 6

Proof. By (13.15) and the Itô formula applied to 1/St, we have

2 1 dSt 2 (dSt) d = − 2 + 3 St (St) 2 (St) 1 2 = −µ + σ dt − σdBt , St hence 1 Λt dZt = d − K St T

471 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

Λ K = d t − TSt St 1 Λ 1 = d t − Kd T St St 1 dΛ Λ 1 = t + t − K d T St T St dt 1 = + StZtd T St dt 2 = + Z −µ + σ dt − Z σdB . T t t t By the self-ﬁnancing condition (5.8) we have

dVt = ηtdAt + ξtdSt = rηtAtdt + µξtStdt + σξtStdBt, t > 0. (13.29)

Another application of Itô’s formula to f(t, St, Zt) = Stg(t, Zt) leads to d(Stg(t, Zt)) = g(t, Zt)dSt + Stdg(t, Zt) + dSt • dg(t, Zt) ∂g ∂g = g(t, Z )dS + S (t, Z )dt + S (t, Z )dZ t t t ∂t t t ∂z t t 2 1 ∂ g 2 + St (t, Zt)(dZt) + dSt • dg(t, Zt) 2 ∂z2 ∂g = µS g(t, Z )dt + σS g(t, Z )dB + S (t, Z )dt t t t t t t ∂t t 2 ∂g 1 ∂g ∂g +S Z −µ + σ (t, Z )dt + S (t, Z )dt − σS Z (t, Z )dB t t ∂z t T t ∂z t t t ∂z t t 2 1 2 2 ∂ g 2 ∂g + σ Z St (t, Zt)dt − σ StZt (t, Zt)dt 2 t ∂z2 ∂z ∂g 2 ∂g 1 ∂g = µS g(t, Z )dt + S (t, Z )dt + S Z −µ + σ (t, Z )dt + S (t, Z )dt t t t ∂t t t t ∂z t T t ∂z t 2 1 2 2 ∂ g 2 ∂g + σ Z St (t, Zt)dt − σ StZt (t, Zt)dt 2 t ∂z2 ∂z ∂g +σS g(t, Z )dB − σS Z (t, Z )dB . (13.30) t t t t t ∂z t t

By respective identiﬁcation of components in dBt and dt in (13.29) and (13.30), we get

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 ∂g ∂g  rη A + µξ S = µS g(t, Z ) + S (t, Z ) − µS Z (t, Z )  t t t t t t t ∂t t t t ∂z t  2  1 ∂g 1 2 2 ∂ g  + St (t, Zt) + σ Z St (t, Zt),  T ∂z 2 t ∂z2     ∂g  ξ S σ = σS g(t, Z ) − σS Z (t, Z ), t t t t t t ∂z t hence  2 ∂g 1 ∂g 1 2 2 ∂ g  rVt − rξtSt = St (t, Zt) + St (t, Zt) + σ Z St (t, Zt),  ∂t T ∂z 2 t ∂z2

 ∂g  ξt = g(t, Zt) − Zt (t, Zt), ∂z i.e.  2 ∂g 1 ∂g 1 2 2 ∂ g  (t, z) + − rz (t, z) + σ z (t, z) = 0,  ∂t T ∂z 2 ∂z2

 ∂g  ξ = g(t, Z ) − Z (t, Z ), t t t ∂z t under the terminal condition g(T , z) = z+, z ∈ R, which follows from (13.24). When Λt/T > K we have Zt > 0 and (13.25) and (13.28) show that ∂g ξ = g(t, Z ) − Z (t, Z ) t t t ∂z t 1 − e−(T −t)r = e−(T −t)rZ + − e−(T −t)rZ t rT t 1 − e−(T −t)r = , 0 t T , rT 6 6 which recovers (13.22). Similarly, from (13.27) we recover ∂g ξ = g(T , Z ) − Z (T , Z ) = Z 1 − Z 1 = 0 T T T ∂z T T {ZT >0} T {ZT >0} at maturity.

We also check that ∂f ∂f ξ = e−(T −t)rσS f(t, S , Z ) − σZ f(t, S , Z ) t t ∂x t t t ∂z t t ∂g = e−(T −t)r −Z (t, Z ) + g(t, Z ) t ∂z t t

473 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

! t −(T −t)r ∂g 1 1 = e St t, Sudu − K + g(t, Zt) ∂x x T w0 |x=St t ∂ −(T −t)r 1 1 = x e g t, Sudu − K , 0 t T . ∂x x T w0 6 6 |x=St

We also find that the amount invested on the riskless asset is given by ∂g η A = Z S (t, Z ). t t t t ∂z t Next we note that a PDE with no first order derivative term can be obtained using time-dependent coefficients.

(2) One variable with time-dependent coeﬃcients

Deﬁne now the auxiliary process

−(T −t)r t 1 − e −(T −t)r 1 1 Ut := + e w Sudu − K rT St T 0 1 −(T −t)r −(T −t)r = 1 − e + e Z , 0 t T , rT t 6 6 i.e. e(T −t)r − 1 Z = e(T −t)rU + , 0 t T . t t rT 6 6 We have 1 dU = − e−(T −t)rdt + r e−(T −t)rZ dt + e−(T −t)rdZ t T t t −(T −t)r 2 −(T −t)r −(T −t)r = e σ Ztdt − e σZtdBt − (µ − r) e Ztdt −(T −t)r = − e σZtdBbt, t > 0, where µ − r dBbt = dBt − σdt + dt = dBet − σdt σ is a standard Brownian motion under

2 ST dPb = eσBT −σ t/2dP∗ = e−rT dP∗. S0 Lemma 13.10. The Asian call option price can be written as " # T + −(T −t)r ∗ 1 Sth(t, Ut) = e E Sudu − K Ft , T w0 where the function h(t, y) is given by 474 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance

+ h(t, y) = Eb (UT ) Ut = y , 0 6 t 6 T . (13.31) Proof. We have

1 1 T UT = w Sudu − K = ZT , ST T 0 and −rT dPb 2 S |Ft (BT −Bt)σ−(T −t)σ /2 e T ∗ = e = −rt , dP e St |Ft hence the price of the Asian call option is

The next proposition gives a replicating hedging strategy for Asian options. See § 7.5.3 of Shreve(2004) and references therein for a diﬀerent derivation of the PDE (13.32).

Proposition 13.11. (Večeř(2001)). Let (ηt, ξt)t∈R+ be a self-ﬁnancing portfolio strategy whose value Vt := ηtAt + ξtSt, t > 0, is given by

Vt = Sth(t, Ut), t > 0, where h ∈ C1,2((0, T ) × (0, ∞)) is given by (13.31). Then, the function h(t, z) satisﬁes the PDE

!2 ∂h σ2 1 − e−(T −t)r ∂2h (t, y) + − y (t, y) = 0, (13.32) ∂t 2 rT ∂y2 under the terminal condition

h(T , z) = z+, and the corresponding replicating portfolio is given by ∂h ξ = h(t, U ) − Z (t, U ), 0 t T . t t t ∂y t 6 6 475 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

Proof. By the self-ﬁnancing condition (13.19) we have

dVt = rηtAtdt + µξtStdt + σξtStdBt (13.33) = rVtdt + (µ − r)ξtStdt + σξtStdBt, t > 0. By Itô’s formula we get

d(Sth(t, Ut)) = h(t, Ut)dSt + Stdh(t, Ut) + dSt • dh(t, Ut) = µSth(t, Ut)dt + σSth(t, Ut)dBt 2 ∂h ∂h 1 ∂ h 2 +St (t, Ut)dt + (t, Ut)dUt + (t, Ut)(dUt) ∂t ∂y 2 ∂y2 ∂h + (t, U )dS • dU ∂y t t t ∂h = µS h(t, U )dt + σS h(t, U )dB − S (µ − r) (t, U )Z dt t t t t t t ∂y t t 2 2 ∂h ∂h σ 2 ∂ h +St (t, Ut)dt − σ (t, Ut)ZtdBet + Z (t, Ut)dt ∂t ∂y 2 t ∂y2 ∂h −σ2S (t, U )Z dt t ∂y t t ∂h = µS h(t, U )dt + σS h(t, U )dB − S (µ − r) (t, U )Z dt t t t t t t ∂y t t 2 2 ∂h ∂h σ 2 ∂ h +St (t, Ut)dt − σ (t, Ut)Zt(dBt − σdt) + Z (t, Ut)dt ∂t ∂y 2 t ∂y2 ∂h −σ2S (t, U )Z dt. (13.34) t ∂y t t

By respective identiﬁcation of components in dBt and dt in (13.33) and (13.34), we get  ∂h ∂h  rη A + µξ S = µS h(t, U ) − (µ − r)S Z (t, U )dt + S (t, U )  t t t t t t t t ∂y t t ∂t t  2 2  σ 2 ∂ h  + StZ (t, Ut),  2 t ∂y2     ∂h ξ = h(t, U ) − Z (t, U ),  t t t ∂y t hence

476 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance

 2 2 ∂h σ 2 ∂ h  rηtAt = −rSt(ξt − h(t, Ut)) + St (t, Ut) + StZt (t, Ut),  ∂t 2 ∂y2

 ∂h  ξt = h(t, Ut) − Zt (t, Ut), ∂y and  !2  ∂h σ2 1 − e−(T −t)r ∂2h  (t, y) + − y (t, y) = 0,  ∂t rT ∂y2  2  !  1 − e−(T −t)r ∂h  ξ = h(t, U ) + − U (t, U ),  t t rT t ∂y t under the terminal condition

h(T , z) = z+.

We also ﬁnd the riskless portfolio allocation ! 1 − e−(T −t)r ∂h ∂h η A = e(T −t)rS U − (t, U ) = S Z (t, U ). t t t t rT ∂y t t t ∂y t

Exercises

Exercise 13.1 Compute the ﬁrst and second moments of the time integral T Stdt for τ ∈ [0, T ), where (St)t∈R is the geometric Brownian motion wτ + σB +rt−σ2t/2 St := S0 e t , t > 0.

Exercise 13.2 Consider the short rate process rt = σBt, where (Bt)t∈R+ is a standard Brownian motion. T a) Find the probability distribution of the time integral r ds. r0 s b) Compute the price

" +# −rT ∗ T e E rudu − K w0

T of a caplet on the forward rate r ds. r0 s

Exercise 13.3 Asian call option with negative strike price. Consider the asset price process 477 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault

2 rt+σBt−σ t/2 St = S0 e , t > 0, where (Bt)t∈R+ is a standard Brownian motion. Assuming that K 6 0, compute the price " # T + −(T −t)r ∗ 1 e E Sudu − K Ft T w0 of the Asian option at time t ∈ [0, T ].

Exercise 13.4 Consider the Asian forward contract with payoﬀ

T Sudu − K, (13.35) w0

2 σBu+ru−σ u/2 where Su = S0 e , u > 0, and (Bu)u∈R+ is a standard Brownian motion under the risk-neutral measure P∗. a) Price the Asian forward contract at any time t ∈ [0, T ]. (10 marks) b) Find the self-ﬁnancing portfolio strategy (ξt)t∈[0,T ] hedging the Asian forward contract with payoﬀ (13.35), where ξt denotes the quantity invested at time t ∈ [0, T ] in the risky asset St. (10 marks)

Exercise 13.5 Compute the price " # T + −(T −t)r ∗ 1 e E exp log Sudu − K Ft , 0 6 t 6 T . T w0 at time t of the geometric Asian option with maturity T , where St = rt+σB −σ2t/2 S0 e t , t ∈ [0, T ].

Hint: When X 'N (0, v2) we have

2 E∗[( em+X − K)+] = em+v /2Φ(v + (m − log K)/v) − KΦ((m − log K)/v).

Exercise 13.6 Consider a CIR process (rt)t∈R+ given by √ drt = −λ(rt − m)dt + σ rtdBt, (13.36) where (Bt)t∈R+ is a standard Brownian motion under the risk-neutral probability measure P∗, and let 1 t Λt := rsds, t ∈ [τ, T ]. T − τ wτ

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Compute the price at time t ∈ [τ, T ] of the Asian option with payoﬀ + (ΛT − K) , under the condition Λt > K.

Exercise 13.7 Consider an asset price (St)t∈R+ which is a submartingale under the risk-neutral probability measure P∗, in a market with risk-free in- terest rate r > 0, and let φ(x) = (x − K)+ be the (convex) payoﬀ function of the European call option.

Show that, for any sequence 0 < T1 < ··· < Tn, the price of the option on average with payoﬀ S + ··· + S φ T1 Tn n can be upper bounded by the price of the European call option with maturity Tn, i.e. show that ST + ··· + ST E∗ φ 1 n E∗[φ(S )]. n 6 Tn

Exercise 13.8 Let (St)t∈R+ denote a risky asset whose price St is given by

dSt = µStdt + σStdBt, where (Bt)t∈R+ is a standard Brownian motion under the risk-neutral probability measure P∗. Compute the price at time t ∈ [τ, T ] of the Asian option with payoﬀ 1 T + Sudu − K , T − τ wτ under the condition that 1 t At := Sudu K. T − τ wτ >

Exercise 13.9 Pricing Asian options by PDEs. Show that the functions g(t, z) and h(t, y) are linked by the relation ! 1 − e−(T −t)r g(t, z) = h t, + e−(T −t)rz , 0 t T , z > 0, rT 6 6 and that the PDE (1.35) for h(t, y) can be derived from the PDE (1.33) for g(t, z) and the above relation.

σB +rt−σ2t/2 Exercise 13.10(Brown et al.(2016)) Given St := S0 e t a geometric Brownian motion and letting

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−(T −t)r −(T −t)r e 1 t e Λt Zet := w Sudu − K = − K , 0 6 t 6 T , St T 0 St T

ﬁnd the PDE satisﬁed by the pricing function ge(t, z) such that " # T + −(T −t)r ∗ 1 Stge t, Zet = e E w Sudu − K Ft . T 0

Exercise 13.11 Hedging Asian options (Yang et al.(2011)). a) Compute the Asian option price f(t, St, Λt) when Λt/T > K. b) Compute the hedging portfolio allocation (ξt, ηt) when Λt/T > K. + c) At maturity we have f(T , ST , ΛT ) = (ΛT /T − K) , hence ξT = 0 and

−rT + e ΛT 1 ΛT ηT AT = AT − K {ΛT >KT } = − K . A0 T T

+ d) Show that the Asian option with payoﬀ (ΛT − K) can be hedged by the self-ﬁnancing portfolio 1 −(T −t)r Λt 1 Λt ξt = f(t, St, Λt) − e − K h t, − K St T St T

in the asset St and

−rT e Λt 1 Λt ηt = − K h t, − K , 0 6 t 6 T , A0 T St T

rt in the riskless asset At = A0 e , where h(t, z) is solution to a partial diﬀerential equation to be written explicitly.

Exercise 13.12 Asian options with dividends. Consider an underlying asset price process (St)t∈R+ modeled as dSt = (µ − δ)Stdt + σStdBt, where (Bt)t∈R+ is a standard Brownian motion and δ > 0 is a continuous-time dividend rate. a) Write down the self-ﬁnancing condition for the portfolio value Vt = ξtSt + rt ηtAt with At = A0 e , assuming that all dividends are reinvested. b) Derive the Black-Scholes PDE for the function gδ(t, x, y) such that Vt = gδ(t, St, Λt) at time t ∈ [0, T ].

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install.packages("quantmod") 2 library(quantmod) getDividends("Z74.SI",from="2018-01-01",to="2018-12-31",src="yahoo") 4 getSymbols("Z74.SI",from="2018-11-16",to="2018-12-19",src="yahoo") T <- chart_theme(); T$col$line.col <-"black" 6 chart_Series(Op(`Z74.SI`),name="Opening prices(black)- Closing prices (blue)",lty=4,theme=T) add_TA(Cl(`Z74.SI`),lwd=2,lty=5,legend='Diﬀerence',col="blue",on = 1)

Z74.SI.div 2018-07-26 0.107 2018-12-17 0.068 2018-12-18 0.068

Opening prices (purple) − Closing prices (blue) 2018−11−16 / 2018−12−18 3.12

3.10 3.10

3.08

3.06 3.05 3.04

3.02

3.00 3.00

2.98

Nov 16 Nov 20 Nov 22 Nov 26 Nov 28 Nov 30 Dec 04 Dec 06 Dec 10 Dec 12 Dec 14 Dec 18 2018 2018 2018 2018 2018 2018 2018 2018 2018 2018 2018 2018

Fig. 13.6: SGD0.068 dividend detached on 18 Dec 2018 on Z74.SI.

The diﬀerence between the closing price on Dec 17 ($3.06) and the opening price on Dec 18 ($2.99) is $3.06 − $2.99 = $0.07. The adjusted price on Dec 17 ($2.992) is the closing price ($3.06) minus the dividend ($0.068).

Z74.SI Open High Low Close Volume Adjusted (ex-dividend) 2018-12-17 3.05 3.08 3.05 3.06 17441000 2.992 2018-12-18 2.99 2.99 2.96 2.96 28456400 2.960

The dividend rate α is given by α = 0.068/3.06 = 2.22%.

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