Fast narrow bounds on the value of Asian options

Fast narrow bounds on the value of Asian options G. W. P. Thompson Centre for Financial Research, Judge Institute of Management, University of Cambridge Abstract We consider the problem of finding bounds on the value of fixed-strike and floatingstrike Asian options. A good lower bound for both types was derived in Rogers & Shi 995). We provide an alternative derivation, which leads to a simpler expression for the bound, and also to the bound given by Curran 992) for fixed-strike options; we derive an analogous bound for floating-strike options. Combining these results with a new upper bound allows the accurate valuation of fixed-strike and floating-strike Asian options for typical parameter values. Keywords: Brownian motion, Asian option, fixed-strike, floating-strike. Introduction A fixed-strike Asian call) option with strike K > and maturity T on an asset with price process {S t } is a contract with value T T S t dt K) + at time T. We assume that T = year and that the asset price follows a geometric Brownian motion S t = S expσb t + ct) where c is a constant, S and σ are positive constants and B t is a Brownian motion. The arbitragefree time- value of this option is then equal to exp ρ)e[ S expσb t + αt) dt K) + ], where ρ is the risk-free interest rate assumed constant) and α = ρ 2 σ2 see Baxter & Rennie 996) for an introduction to arbitrage-free valuation). Thus the problem of valuing the option boils down to calculating E[S expσb t +αt) dt K) + ]. Another type of Asian option, the floating-strike option, pays out S t dt S ) +, and leads to the consideration of E[ expσb t + αt) dt expσb + α)) + ]. Several approaches to the problem of valuing Asian options have been tried. Carverhill & Clewlow 99) use a convolution method to compute the distribution of S t dt, and work by Yor 992) and Geman & Yor 993) has lead to a formula for the price as a triple integral. Approximate formulae and bounds in the form of single and double integrals

by Levy 992), Levy & Turnbull 992), Curran 992) and Rogers & Shi 995) seem to be faster to evaluate however, and the methods of this paper fall into this category. Using intuition and simple optimization we derive bounds on the value of fixed-strike and floating-strike Asian options which can be computed quickly, and which are accurate for typical parameter values. The method generalizes to other options on sums of lognormal assets: discretely monitored Asians, currency basket options, and swaptions in Gaussian HJM models, for example. The plan of this paper is as follows: in Section 2 we state some useful facts about covariances related to Brownian motion, and in Section 3 present the impressive lower bound of Rogers & Shi 995) for fixed-strike and floating-strike options, and the approximation to the fixed-strike bound given by Curran 992). This approximation is notable since it is very close to the bound of Rogers & Shi 995) and is much easier to compute, involving only one-dimensional integrals rather than a troublesome two-dimensional integral. We give an alternative derivation of the bounds of Rogers & Shi 995), leading to an expression involving only one-dimensional integrals; we also provide a generalization Curran s approximation to floating-strike options. In Section 4 we derive upper bounds to complement the lower bounds, and in Section 5 we present a numerical comparison of the various bounds discussed here, using the parameter values from Curran 992). 2 Useful covariances Like most of the approximation formulae in the literature, we will exploit the high correlation between expσb t + αt) dt and B t dt for the values of σ and α met in practice, and the fact that the second integral is Gaussian. We will need the covariance matrix of the bivariate Gaussian random variable B t, B s ds): ) T E B t, B s ds B t, B s ds) ) ) t t t/2) =. t t/2) 3 Thus the conditional distribution of B t given B s ds = z is normal with mean 3t t/2)z and variance t 3t 2 t/2) 2, and the conditional distribution of B s ds given B t = x is normal with mean t/2)x and variance 3 t t/2)2. We will also need the covariance matrix of B t, B s ds B ): E B t, B s ds B ) T B t, B s ds B ) ) = t t 2 /2 t 2 /2 3 ), 2

so, given B s ds B = x, the conditional distribution of B t is normal with mean 3t 2 x/2 and variance t 3t 4 /4. 3 Lower bounds In this section we present two derivations of the bound of Rogers & Shi 995): their own, and an alternative, which yields a simpler expression for the bound. It also leads to the bound of Curran 992) for fixed-strike options. We show how a similar bound for floating-strike options may be derived. The derivation of Rogers & Shi 995) exploits the inequality: E [ A +] = E [ E A + C )] E [ E A C)) +] which holds for any random variables A and C. For fixed-strike Asian option they choose A = S t dt K and C = B t dt. Since the inner expectation is E S exp σb t + αt) ) B s ds dt K and, conditional B t dt = z, B t is normal with mean 3t t/2)z and variance t 3t 2 t/2) 2, we have the bound ) [ V fixed e ρ ] + 3φ 3z Se 3σt t/2)z+αt+ 2 σ2 t 3t 2 t/2) 2 ) dt K dz. 3.) To bound a floating-strike option they use A = S t dt S, C = B t dt B, and get ) [ V floating e ρ 3φ 3z S exp 3σt 2 z/2 + αt + 2 σ2 t 3t 4 /4 )) dt S exp 3σz/2 + α + σ 2 /8 ))] + dz. Both of these formulae are slightly tricky to evaluate since the outer integration has a non-smooth integrand. An alternative approach is to approximate the event that the option eventually ends in-the-money with something more tractable. Let A = {ω : S t dt > K}, and note that [ ) + ] E S exp σb t + αt) dt K = E [S exp σb t + αt) K) IA)] dt. 3.2) If we replace A by some other event A, we no longer have equality in 3.2); the right hand side is now a lower bound. We will use A = { B t dt > γ}. This Gaussian form allows the expectation to be written as a Black-Scholes type formula once γ has been determined, and just leaves us with a one-dimensional integral of a smooth integrand, which should be very fast. 3

To determine the optimal value of γ, let N t = σb t + αt + log S and note that for any random variable X with density f X x) γ E expn t ) K; X > γ) dt = Thus the optimal value of γ, γ satisfies E exp N t ) K X = γ) f X γ)) dt. With our choice of X = B t dt, we conclude that EexpN t ) X = γ ) dt = K. 3.3) S exp 3γ σt t/2) + αt + 2 σ2 t 3t 2 t/2) 2)) dt = K, 3.4) which determines γ uniquely. We now have the bound [ Se V fixed e ρ σb E t+αt K ) )] I B s ds > γ dt, and it remains to calculate the expectation. Fix t, ) and let N = σb t + αt + log S and N 2 = B s ds γ. Write µ i = EN i ), σi 2 = VarN i) and c = CovN, N 2 ), then using E [ e N K ) I N 2 > ) ] = e µ + 2 σ2 Φ µ2 + c σ 2 ) KΦ where Φ is the normal distribution function, and substituting µ = αt + log S, µ 2 = γ, σ 2 = σ2 t, σ2 2 = 3, c = σt t/2), we have [ V fixed e ρ Se αt+ γ 2 σ2t ) + σt t/2) Φ / 3 µ2 σ 2 ), )] γ dt KΦ /. 3 Integrating this numerically is significantly easier than integrating 3.). To see that this bound gives the same answers as that of Rogers & Shi 995), let Y = S t dt, Z = B t dt and note that E[EY K Z)) + ] = E[EY K Z))IEY K Z) > )], and since from 3.3) and 3.4), EY K Z) is strictly increasing in Z, we have: EY K Z) > if and only if Z > γ. Thus γ satisfies E[EY K Z)) + ] = E[EY K Z))IZ > γ )] which is just E[Y K)IZ > γ )]. The bound of Curran 992) arises from solving 3.4) approximately, using the following method: let fγ) = E S expσb t + αt) dt B s ds = γ), and note that a reasonable approximation to f is fγ) := S expγσ + α/2), obtained by interchanging the orders of integration and exponentiation. Recall that we seek γ = f K) and observe that if f f 4

then f x) f 2x f f x)), the approximation being exact if f f is constant. Thus γ f 2K f f K)), giving 2K γ σ [log S ) ] e 3logK/S) α/2)t t/2)+αt+ 2 σ2 t 3t 2 t/2) 2) dt α/2, which is the continuous limit of the bound given by Curran 992). For the floating-strike option, let A = {ω : S t dt > S } and use an approximation to A of the form A = { B t dt B > γ}. With this choice, γ, the optimal value of γ, satisfies [ ] E S exp σb t + αt) S exp σb + α)) dt B s ds B = γ = giving exp 3γ σ/2 α σ 2 /8 ) exp 3γ σt 2 /2 + αt + 2 σ2 t 3t 4 /4 )) dt =, 3.5) which has a unique solution. Our lower bound for the floating-strike case is thus [ S V floating e ρ E exp σbt + αt) S exp σb + α) ) )] I B s ds B > γ dt, which reduces to [ V floating e ρ Se αt+ γ 2 σ2t σt 2 ) /2 Φ / dt Se α+ γ )] /2 2 σ2 Φ 3 /. 3 Again this bound gives the same answers as that of Rogers & Shi 995). To see this, let Y = S t dt, Z = B t dt, and note first that γ E Y S Z B = γ) = 3σt 2 /2) expat, γ)) dt + 3σ/2) expa, γ)), where At, γ) = log S 3γσt 2 /2 + αt + 2 σ2 t 3t 4 /4 ). Since γ solves EY S Z B = γ ) =, we have expat, γ )) dt = expa, γ )), so γ E Y S Z B = γ) = γ=γ 3σ t 2 )/2) expat, γ )) dt >. Thus E Y S Z B = γ) > if and only if γ > γ. A similar argument to the fixedstrike case completes the proof. We can now generalize Curran s formula to the case of floating-strike options by solving 3.5) approximately. Let fγ) = e 3γσ/2 α σ2 /8 E eσbt+αt B t dt B = γ) and let 5

fγ) = expσγ α/2) be the approximation to f obtained by interchanging the orders of integration and exponentiation. Then the solution to 3.5) is given approximately by γ f 2 f f )), giving γ σ [α/2 + log 2 e α/4 σ2 /8 4 Upper bounds exp αt 3αt 2 /4 + 2 σ2 t 3t 4 /4 )) )] dt. In this section we derive a new upper bound on the value of fixed-strike and floating-strike Asian options in the form of a double integral. Rogers & Shi 995) obtained an upper bound by considering the error made by their lower bound see Section 3). As an indication of their relative accuracy, if σ =.3, ρ =.9, S =, and T = year and k =, the lower bounds of Section 3 are both 8.8275 and the upper bound of this section is 8.8333. By comparison, the upper bound given by Rogers & Shi 995) is 9.39. The inequality underlying the new bound is the following: let X be a random variable, and let f t ω) be a random function with f tω) dt = for all ω. Then [ ) + ] [ ) + ] E S expσb t + αt) dt X = E S expσb t + αt) Xf t ) dt [ ] E S expσb t + αt) Xf t ) + dt = E [ S exp σb t + αt) Xf t ) +] dt. 4.) For both the fixed-strike and floating-strike cases we will use f t = µ t + σb t B s ds), where µ t is a deterministic function satisfying µ t dt = ; and derive an expression for µ t which is approximately optimal in each case. As the bounds have a very similar derivation we will concentrate on the fixed-strike option and give the appropriate modifications for the floating-strike case at the end of the section. Take X = K in 4.) and first consider the choice f t = µ t. To choose µ t we will minimize the right hand side of 4.) over the set of deterministic functions f t such that f t dt =. Let Lλ, {f t }) = E[ S expσb t + αt) Kf t ) + dt λ f t dt )] be the Lagrangian, and consider stationarity with respect to {f t } for the unconstrained problem. This gives the condition KP S exp σb t + αt) Kf t ) λ) ɛ t dt =, where {ɛ t } is some small deterministic perturbation. Thus we see that PS expσb t + αt) Kf t ) must be independent of t. Equivalently we have logkf t /S) αt = γσ t for some 6

constant γ. Thus the optimal choice for f t is f t = S/K) exp σγ t + αt ), where the constant γ is chosen so that f t dt =. Since f t dt is monotone increasing in γ, the correct value for γ is easy to estimate numerically. If instead f t = µ t + σb t B s ds), the condition for stationarity with respect to small deterministic perturbations is [ P S expσb t + αt) K µ t + σ B t ))] B s ds = λ, for all t, 4.2) but this cannot easily be re-arranged to give the dependence of µ t on λ. Instead we will use the approximation expσb t ) + σb t which should be reasonable for small σ. This leads to the condition PS expαt) + σb t ) Kf t ) = λ for all t. Letting N t = S expαt) + S expαt)σ Kσ)B t + Kσ B s ds, we conclude that PN t Kµ t ) must be independent of t. Using the facts about the joint distribution of B t, B s ds) given in Section 2, we deduce that where Imposing µ t dt = gives µ t = K S exp αt) + γ v t ), 4.3) v t = VarN t ) = c 2 t t + 2Kσ)c t t t/2) + Kσ) 2 /3, 4.4) c t = S expαt)σ Kσ. 4.5) γ = ) / K Se α )/α vt dt. 4.6) We estimate the integral vt dt numerically. We now know the constant γ and hence the function µ t for our upper bound: [ V fixed e ρ E S exp σb t + αt) K µ t + σ B t ))) + ] B s ds dt. Conditioning on B t = x, this becomes ) V fixed e ρ x φ E [ at, x) + bt, x)n) +] dx dt, 4.7) t t 7

where N has a N, ) distribution, and the functions a and b are given by at, x) = S exp σx + αt) Kµ t + σx) + Kσ t/2)x, 4.8) bt, x) = Kσ 3 t t/2)2. 4.9) The calculation of E[a + bn) + ] is straightforward and gives aφa/b) + bφa/b). In the form of 4.7) the integrand is badly behaved near, ) so we perform the change of variables v = t, w = x/ t, giving V fixed e ρ [ 2vφw) at, x)φ ) at, x) + bt, x)φ bt, x) )] at, x) dw dv. bt, x) This expression, combined with 4.3), 4.4), 4.5), 4.6), 4.8) and 4.9), constitutes the upper bound in the fixed-strike case. For the case of a floating-strike option, we now take X = S. Setting Z = B t dt, the condition for stationarity with respect to small deterministic perturbations analogous to 4.2) is E [ S expσb + α); S expσb t + αt) µ t + σ B t Z))S expσb + α))] = λ, t. We approximate this by P[S expσb t + αt) µ t + σb t Z))S expσb + α)] = λ, t, and further, using the approximation expσb t B )) + σb t B ), by P[expαt )) + σb t B ) µ t + σb t Z))] = λ, t. This implies that for some γ, µ t = exp α t )) + γ v t where v t = Var[expαt )) + σb t B )) σb t Z)]. Since µ t dt = we must have γ = exp α))/α)/ vt dt. Our bound is thus [ Se V floating e ρ σb E t+αt µ t + σ B t Z)) Se σb +α ) ] + dt [ ) ] = e ρ E e Nt) N 2 t)e N + 3t) dt, where N t) = σb t + αt + log S, N 2 t) = µ t + σb t B s ds) and N 3 t) = σb + α + log S. If we condition on N 2 t) = x we can perform the remaining expectation analytically. Let 8

µ i t) = EN i t)), σ ij t) = CovN i t), N j t)) and denote by tildes the conditional distributions given N 2 t) = x: µ i t, x) = µ i + x µ 2 )σ i2 /σ 22 and σ ij = σ ij σ i2 σ j2 /σ 22. Finally let v 2 = VarN t) N 3 t) N 2 t) = x) = σ 2 σ 3 + σ 33. Our upper bound on the price of a floating-strike Asian option is then V floating e ρ where we take logx) = for x. x )[e µ ) µ2 φ + 2 σ µ µ 3 logx) + σ σ 3 Φ σ22 σ22 v )] xe µ 3+ 2 σ 33 µ µ 3 logx) + σ 3 σ 33 ) Φ dx dt, v 5 Numerical Results In Table we consider fixed-strike options and show the upper bound of Rogers & Shi 995), the upper and lower bounds derived in Sections 3 and 4, the approximation of Curran 992) and the Monte-Carlo results of Levy & Turnbull 992). All calculations assume ρ =.9, an initial stock price of S = and an expiry time of year. For the lower bound and the new upper bound, the approximate time taken on an HP 9/73) is parenthesized; for the Monte-Carlo studies, the estimated standard error is bracketed beneath. In Table 2 we show how the upper and lower bounds of Rogers & Shi 995) compare to the upper bound of Section 4 and the generalization to floating-strike options of Curran s lower bound, described in Section 3. The approximate time taken is parenthesized. References Baxter, M. W. & Rennie, A. 996), Financial Calculus, Cambridge University Press. Carverhill, A. & Clewlow, L. 99), Flexible convolution, Risk 34), 25 29. Curran, M. 992), Beyond average intelligence, Risk 5), 6. Geman, H. & Yor, M. 993), Bessel processes, Asian options and Perpetuities, Mathematical Finance 34), 349 375. Levy, E. 992), Pricing European average rate currency options, J. Intnl. Money and Finance, 474 49. Levy, E. & Turnbull, S. 992), Average Intelligence, Risk 52), 5 9. 9

Rogers, L. C. G. & Shi, Z. 995), The value of an Asian option, J. Appl. Prob. 32, 77 88. Yor, M. 992), Some aspects of Brownian motion, Part : Some Special Functionals, Birkäuser.

Volatility σ Strike K Curran lower R S lower M C result Upper bound R S upper 95 8.888 8.888 8.8 8.889 8.82.6).9) [.].3).5 4.382 4.382 4.3 4.384 4.38.2).) [.].9) 5.9583.9583.95.9585.968.2).) [.].9) 95 8.98 8.98 8.9 8.93 8.95.6).8) [.].9). 4.95 4.95 4.9 4.955 5..23).7) [.].2) 5 2.699 2.699 2.6 2.74 2.34.23).8) [.].2) 9 4.9827 4.9827 4.96 4.9929 5.94.23).9) [.].24).3 8.8275 8.8275 8.8 8.8333 9.39.23).9) [.].24) 4.6949 4.6949 4.68 4.727 4.96.23).8) [.].28) 9 8.829 8.829 8.4 8.228 8.57.23).9) [.3].28).5 3.225 3.225 2.98 3.569 3.69.23).8) [.3].63) 9.79 9.79 9. 9.56 9.97.23).8) [.3].64) Table Comparison of various bounds on fixed-strike Asian option prices for S =, ρ =.9, and an expiry time of year. Parenthesized numbers are computation times in seconds, bracketed numbers are estimates of standard errors from Curran 992)).

Volatility σ Interest Rate ρ Generalized R S lower Upper bound R S upper.5.2454.2454.2457.355.8).36).27)..9.6992.6992.6997.825.2).54).25).5.256.256.2525.45.2).78).24).5 3.444 3.444 3.464 3.83.7).24).3).2.9 2.626 2.626 2.6237 3.62.2).32).33).5.798.798.724 2.87.2).44).28).5 5.6246 5.6246 5.638 6.584.7).8).28).3.9 4.7382 4.7382 4.7456 5.76.2).22).33).5 3.685 3.685 3.666 4.64.2).32).33) Table 2 Comparison of bounds on floating-strike Asian option prices for S = with an expiry time of year. Parenthesized numbers are computation times in seconds. The parameter values are those used in Rogers & Shi 995). 2