Closed-Form Solutions to the Valuation and Hedging of an Arithmetic Asian Option ∗ Zhaojun Yang† Lihong Huang School of Mathematics and Econometrics, Hunan University, Changsha, 410082 P.R. China Chaoqun Ma School of Business Administration, Hunan University, Changsha, 410082 P.R. China
Abstract This paper first shows one property of the Gradient operator on Banach space D1,1 , which plays a particular role in deriving hedging strategies of some options. Utilizing a result in Yor (1992), the paper then acquires the density of a valuable random variable in Finance by Girsanov’s transformation. Following the above results, the paper presents the closed-form solutions to both the valuation and hedging of an arithmetic Asian call option by Gradient operator method. In the end, the paper provides the corresponding results for the arithmetic Asian put option and some comparative statics. KEY WORDS: an arithmetic Asian option, Gradient operator method, valuation and hedging, closed-form solutions. AMS SUBJECT CLASSIFICATION(1991): 93E20, 90A09
1
Introduction
It is critically important to price options and derive their hedging strategies in financial economics. Unfortunately most of options don’t possess any (perfect) hedging strategy, thus, they don’t possess a unique fair price, either. Furthermore, only very few options can be obtained a closed-form solution to their valuation or hedging strategy even though the hedging strategy does exist. Asian option is an exotic option whose terminal payoff is based on average values of the fundamental asset prices during some period within the option’s lifetime. Therefore, it is almost impossible for an investor to increase his wealth at random by manipulating the ∗ A project supported by scientific research fund of Hunan Provincial Education Department (No. 01B031). The results of this paper have partially been drawn from the first author’s doctoral dissertation at Central South University. The authors are indebted to Prof. J.Z. Zou , Z.T. Hou & Z.Z. Li at Central South University and Prof. G.F. Zhou at Washington University. Especially, our greatest single debt of gratitude goes to Academician J.A. Yan, who read carefully the manuscript and offered substantial comments including correction of some defects on it. Any remaining errors are the authors’ own responsibility. † E-mail: zhaojun
[email protected]
1
asset prices during the near maturity dates. As a result, the Asian options are better than the European options in this respect. The Asian options fall into two types, i.e., the arithmetic and the geometric. For the geometric Asian option we have already acquired closed-form solutions to both the valuation and hedging. This is because the geometric averaging value of log-normal random variables is still log-normal. However, the arithmetic averaging value doesn’t possess such excellent property. Therefore, the task of finding a closed-form solution for the valuation of an arithmetic Asian option appeared to be surprisingly involved, let alone to derive its hedging strategy. Rogers & Shi (1995) pointed that the existence of the explicit solution to its valuation would be impossible. Consequently it would be also impossible for the existence of the closed-form solution to its hedging. Actually many studies of the arithmetic Asian option were based either on approximations or on numerical methods. For example, the Monte-Carlo simulation were employed by Kemna & Vorst (1990); an algorithm based on Wilkinson estimates for pricing Asians were employed by Turnbull & Wakeman (1991); a lower bound by conditioning on appropriate Gaussian variables were determined by Rogers & Shi (1995); using some results from risk theory on stop-loss order and comonotone risks, an upper bound by the price of a portfolio of European call options were derived recently by Simon, Goovaerts & Dhaene (2000); and finally the most efficient analytical tools leading to quasiexplicit pricing formulae were developed by Geman & Yor (1993,1995). Furthermore, all of them didn’t deal with the problem of how to hedge the option. In the next section, this paper first shows one property of a gradient operator on Banach space D1,1 , which plays a particular role in deriving the hedging strategies of some options. Then, utilizing a result in Yor (1992), the paper acquires the density of a valuable random variable in Finance by Girsanov’s transformation. Following the above results, Section 3 presents the closed-form solutions to both the valuation and hedging of an arithmetic Asian call option by Gradient operator method. In the end, Section 4 provides the corresponding results for the arithmetic Asian put option and some comparative statics.
2
The model and some lemmas
Consider a complete probability space (Ω, F, P ) and a standard 1-dimension Brownian motion W = (W (t), 0 ≤ t ≤ T ) defined on it. We shall denote by {Ft } the P-augmentation of the natural filtration FtW = σ(W (s); 0 ≤ s ≤ t), 0 ≤ t ≤ T. There are two assets on a frictionless market. One of the assets is risk-free bond with a constant deterministic interest rate r. The other is risky security (stock) on the space (Ω, F, P ) with a measurable, adapted price process S = (S(t), Ft ; 0 ≤ t ≤ T ). The dynamics of the price process is determined by stochastic differential equation (SDE) dS(t) = µS(t)dt + σS(t)dW (t), S(0) = s0 , 0 ≤ t ≤ T.
(2.1)
It is well-known that such market is complete. According to the context of no-arbitrage pricing in a complete market, we may assume µ = r without loss of generality. Consequently (2.1) can be equivalent to the following equation, ³
´
S(t) = s0 exp σW (t) − σ 2 t/2 + rt , 0 ≤ t ≤ T,
(2.2)
where constants r, σ, T > 0. All of us know that the fair price of every integrable contingent claim fT ∈ FT is given by V (t) ≡ exp (−r(T − t)) E ( fT | Ft ) , 0 ≤ t ≤ T. 2
(2.3)
The payoff at maturity from the arithmetic Asian call option is Ã
fT =
!+
Z T
1 T
S(t)dt − q
0
,
(2.4)
where constant q > 0. In order to derive both the valuation and hedging of such option we now show two lemmas in the following. Lemma 2.1 Assume F ∈ D1,1 , then we have F + ∈ D1,1 , and DF + = 1(F >0) (ω)DF . Proof Let ψ(x) = x+ , we define the infinitely differentiable function ρ(x) by µ
¶
1 ρ(x) = c1(0,2) (x) exp , [(x − 1)2 − 1] where c is a constant satisfying
R
< ρ(x)dx
= 1. Define Z
ρn (x) ≡ nρ(nx), ψn (x) ≡ then, we have
0) D(F ), a.s. ,
we obtain then n¯
¯
°
°o
° ° limn→∞ E ¯ψn (F ) − F + ¯ + °Dψn (F ) − 1(F >0) D(F )° = 0,
by dominated convergence. Consequently the desired result follows from that D is a closed operator on D1,1 . 2 Remark 2.1 All of the new notations are defined in Ocone & Karatzas (1991). Therefore, we don’t explain here. 3
Lemma 2.2 Denote the probability density function of then for x > 0 we have p(t, x; a, b) = M (x) where
Z +∞ 0
N (v)
·Z +∞ 0
y
2a b2
Rt 0
exp (au + bW (u)) du by p(t, x; a, b), ¶
µ
Ã
√ 4π 2 − (at)2 M (x) = 8(πb x 2πt)−1 exp 2b2 t Ã
4πv 2v 2 N (v) = sin( 2 )sinh(v) exp − 2 b t b t
(2.5)
!
3 2
and
¸
2 exp − 2 (y 2 + 2ycosh(v) + 1) dy dv, b x
,
!
.
Furthermore, it is obvious that p(t, x; a, b) = 0 when x < 0. Proof Let ft (x, y) be the joint probability density function of random vector Then, if x > 0, it can be derived from (6.c) in Yor (1992) that ³
ft (x, y) =
exp
2xyt+π 2 x−t−t exp(2y) 2xt
´
√ x2 2π 3 t
Z ∞ 0
Ut (x; a, b) = P
Z
b2 t 4
0
t 0 exp(2Wu )du, Wt
!
µ
t 0 exp(au
+ bWu )du ≥ x .
Ã
¶
πz z 2 exp(y) cosh(z) sinh(z) sin ; dz exp − − 2t x t
In addition, ft (x, y) = 0 (x ≤ 0) is obvious. Denote Ut (x; a, b) ≡ P According to Lemma 9.4 in Karatzas & Shreve (1988), we obtain
³R
µ µ
exp 2 Wu +
2au b2
³R
´
¶¶
du ≥
b2 x 4
.
e such that Define an equivalent probability measure P, ¯
!
Ã
e ¯¯ dP 2a2 2a ¯ = exp − 2 Wt − 4 t . dP ¯F b b t
Obviously, there exists such probability measure. Then we conclude by Girsanov theorem that Ã
Z
2a2 2a − Ut (x; a, b) = IA exp W 2 b t b2 b4 4 Ω
Ã
b2 t 4
!!
dP,
( ¯ 2 ) ¯R b t ¯ 4 b2 x where set A ≡ ω ¯ 0 exp(2Wu )du ≥ 4 . Therefore we derive that ¯ Ã
a2 t Ut (x; a, b) = exp − 2 2b
!Z
∞
Z ∞
µ
¶
2ay exp f b2 t (u, y)dudy. b2 4 −∞
b2 x 4
In the end, substituting the joint probability density function ft (x, y) into this equality and properly arranging it, (2.5) can be derived. 2 4
´
.
3
Closed-form solutions to the valuation and hedging of Asian call option ³
1 Let Ξ ≡ S(t) qT − theorem.
´
Rt
0 S(u)du . Now, we establish the main results of the paper in the following
Theorem 3.1 At any time t (0 ≤ t ≤ T ), the fair price of the arithmetic Asian call option fT is given by 1 V (t) = exp (−r(T − t)) S(t) T
Z +∞ Ξ
Ã
!
σ2 (x − Ξ)p T − t, x; r − , σ dx. 2
(3.1)
As for its hedging strategy, the amount invested in the stock is 1 π(t) = V (t) + ΞS(t) exp (−r(T − t)) T
Z +∞ Ξ
p(T − t, u; r −
σ2 , σ)du; t(0 ≤ t ≤ T ) 2
(3.2)
The amount invested in the bond is simply determined by V (t) − π(t). Proof Obviously, it suffices to prove (3.1) and (3.2). We notice from (2.4) that "Z
1 fT = S(t) T
t
T
¶
µ
#+
1 exp σ(W (u) − W (t)) + (r − σ 2 )(u − t) du − Ξ 2
.
(3.3)
Hence, fT must be integrable. According to the properties of Brownian motion and Lemma 2.2, the density conditioned on Ft of the random variable, Z T t
³
2
µ
¶
1 exp σ(W (u) − W (t)) + (r − σ 2 )(u − t) du 2
´
is p T − t, x; r − σ2 , σ . Thus, synthesizing (2.3) and (3.3), we obtain (3.1) at once. On the other hand, by virtue of Lemma 2.1 and the properties of operator D, we conclude fT exp (−rT ) ∈ D1,1 and ·
1 Dt [fT exp (−rT )] = fT + 1(gT >0) (ω)(q − T
Z t 0
¸
S(u)du) σ exp (−rT ) .
(3.4)
R
where gt ≡ T1 0T S(u)du − q. According to the extended Clark’s formula ( please see Karatzas ,Ocone & Li 1991), we obtain fT exp (−rT ) = E [fT exp (−rT )] +
Z T 0
E [Dt (fT exp (−rT ))|Ft ] dW (t).
(3.5)
Furthermore, by the definition of hedging strategy the process of the amount invested in the stock (π(t), Ft ; 0 ≤ t ≤ T ) satisfies fT exp (−rT ) = E [fT exp (−rT )] +
Z T 0
exp (−rt) π(t)dW (t).
(3.6)
Comparing (3.5) and (3.6), we arrive at π(t) =
1 exp (rt) E [Dt (fT exp (−rT ))|Ft ] . σ 5
(3.7)
By this means, we derive that ·
1 T
π(t) = V (t) + exp (−r(T − t)) q −
¸
Z t
S(u)du P (gT > 0|Ft ).
0
(3.8)
Hence, according to the properties of Brownian motion and Lemma 2.2, (3.2) can be easily established from (3.8). 2 Remark 3.1 From (3.1) and (3.2), we are able to verify that π(t) = S(t)
∂V (t) , ∂S(t)
(3.9)
which is just such one as we expect. Hence, the Gradient operator method isn’t necessary to derive the hedging strategy of such Asian option. But in many other cases of optimal portfolios associated with option pricing, maximizing utility from terminal wealth, and maximizing utility from consumption (please see Yang & Ma 2001 and Ocone & Karatzas 1991), the Gradient operator method plays a very important role and is even indispensable. Corollary 3.1 The amount invested in the stock is always positive. Especially, when qT we have S(t) π(t) = (1 − exp (−r(T − t))) . rT
Rt 0
S(u)du > (3.10)
Proof Obviously, it is sufficient to prove (3.10). In fact, it follows from (3.1) and (3.2) that 1 π(t) = S(t) exp (−r(T − t)) T
Z +∞ Ξ
Ã
!
σ2 xp T − t, x; r − , σ dx. 2
Moreover, we notice that Z +∞ 0
Ã
!
"Z
σ2 xp T − t, x; r − , σ dx = E 2
T −t 0
!
Ã
#
σ2 exp (r − )u + σdWu du . 2
Synthesizing the above results, (3.10) can be easily verified. 2
4
The Put Option and some Comparative Statics
In this section, we only present the closed-form solution to the put option and some comparative statics and omit their proofs because of ³the simplicity. The payoff at maturity from the ´+ RT 1 arithmetic Asian put option is given by f¯T ≡ q − S(t)dt . Let T
U (T − t, Ξ) ≡
0
Z +∞ Ã Ξ
!
σ2 p T − t, x; r − , σ dx. 2
(4.1)
Theorem 4.1 For the arithmetic Asian put option, at any time t (0 ≤ t ≤ T ), its fair price is determined by 1 V¯ (t) = S(t) exp (−r(T − t)) T
Z Ξ 0
Ã
!
σ2 (Ξ − x)p T − t, x; r − , σ dx. 2
6
(4.2)
The hedging strategy is as follows. The amount invested in the stock is 1 π ¯ (t) = V¯ (t) + ΞS(t) exp (−r(T − t)) [1 − U (T − t, Ξ)] , T and the amount invested in the bond is V¯ (t) − π ¯ (t). Theorem 4.2 For some comparative statics, we have ·
(4.3)
¸
∂V (t) 1 1 ∆≡ = V (t) + S(t)ΞU (T − t, Ξ) exp (−r(T − t)) ; ∂S(t) S(t) T Γ≡
∂ 2 V (t) 1 σ2 2 = Ξ p(T − t, Ξ; r − , σ) exp (−r(T − t)) ; ∂ 2 S(t) T S(t) 2 ∂V (t) = −U (T − t, Ξ) exp (−r(T − t)) . ∂q
(4.4) (4.5) (4.6)
Remark 4.1 For the other comparative statics, it is tedious to write, therefore, we don’t show here.
References [1] Geman, H. and M. Yor, Bessel processes, Asian options and perpetuities, Mathematical Finance 4(1993)345-371. [2] Geman, H. and M. Yor, The valuation of double-barrier: A probabilitic approach, Working paper(1995). [3] Karatzas, I. and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York(1988). [4] Karatzas, I., D. l. Ocone, and J. Li, An extension of Clark’s formula, Stochastics and Stochastic Reports, 37(1991)127-131. [5] Kemna, A. G. Z. And A. C. F. Vorst, A pricing method for options based on average asset values, Journal of Banking and Finance 14(1990)113-129. [6] Ocone, D. L. and I. Karatzas, A generalized Clark representation formula, with application to optimal portfolios, Stochastics and Stochastic Reports 34(1991)187-220. [7] Rogers, L. C. G. and Z. Shi, The value of an Asian option, Journal of Applied Probability 32(1995)1077-1088. [8] Simon, S., M. J. Goovaerts, and J. Dhaene , An easy computable upper bound for the price of an arithmetic Asian option, Insurance: Mathematics and Economics 26(2000)175-183. [9] Turnbull,S. M. and L. M. Wakeman, A quick algorithm for pricing European average options, Journal of Financial and Quantitative Analysis 26(1991)377-389. [10] Yang, Z. J., Problems on Options,Investment and Consumption, Doctoral Dissertation, Central South University (2000). [11] Yang, Z.J. and Ma, C.Q., Optimal trading strategy with partial information and the value of information: the simplified and generalized models, International Journal of Theoretical and Applied Finance,4(2001)759-772. [12] Yor, M., On some exponential functionals of Brownian Motion, Adv. Appl. Prob. 24(1992) 509-531. 7
