On Pricing of Discrete Barrier Options

Transcription

1 On Pricing of Discrete Barrier Options S. G. Kou Department of IEOR 312 Mudd Building Columbia University New York, NY This version: April 2001 Abstract A barrier option is a derivative contract that is activated or extinguished when the price of the underlying asset crosses a certain level. Most models assume continuous monitoring of the barrier. However, in practice, most, if not all, of the barrier options traded are discretely monitored. Unlike their continuous counterparts, there is essentially no closed form solution available, and even numerical pricing is di cult. This paper extends an approximation by Broadie, Glasserman, and Kou (1997) for discretely monitored barrier options by covering more cases and giving a simpler proof. The techniques used here come from sequential analysis, particularly Siegmund and Yuh (1982) and Siegmund (1985). Key words: Siegmund s corrected di usion approximation, level crossing probabilities, Girsanov theorm. 1. Introduction A barrier option is a nancial derivative contract that is activated (knocked in) or extinguished (knocked out) when the price of the underlying asset (could be a stock, index, exchange rate, interest rate, etc.) crosses a certain level (called barrier). For example, an up-and-out call option gives the option holder the payo of a European call option, if the price of the underlying asset does not reach a higher barrier level before the expiration date. More complicated barrier options may have two barriers (double barrier options), and may have the nal payo determined by one asset and the barrier level determined by the other asset (two dimensional barrier options). Taken together, they are among the most popular path-dependent options traded in exchanges worldwide and also in over-the-counter markets. This paper focus exclusively on one-dimensional, single barrier options, which include eight possible types: up (down)-and-in (out) call (put) options.

2 An important issue of pricing barrier options is whether the barrier crossing is monitored in continuous time. Most models assume continuous monitoring of the barrier. In other words, in the models a knock-in or knock-out occurs if the barrier is reached at any instant before the expiration date, mainly because it leads to analytical solutions; see, for example, Merton (1973), Heynen and Kat (1994a, 1994b), Kunitomo and Ikeda (1992) for various formulae for continuously monitored barrier options under the classical Brownian motion framework; and Kou and Wang (2001) for continuously monitored barrier options under a jump di usion framework. However, in practice, most, if not all, of the barrier options traded are discretely monitored; in other words, they specify xed times for monitoring of the barrier (typically daily closings). Besides practical implementation issues, there are some legal and nancial reasons why discretely monitored barrier options are preferred to continuously monitored barrier options. For example, some discussions in trader s literature ( Derivatives Week, May 29th, 1995) voice concern that, when the monitoring is continuous, extraneous barrier breached may occur in less liquid markets while the major western markets are closed, and may lead to certain arbitrage opportunities. Although discretely monitored barrier options are popular and important, pricing them is not as easy as that of their continuous counterparts. (1) Essentially there are no closed solutions, except using m-dimensional normal distribution functions (m is the number of monitoring points), which can hardly be computed easily if, for example, m>5; see Reiner (2000). (2) Even direct Monte Carlo simulation or standard binomial trees is di cult, and could take hours or even days to get accurate results; see Broadie, Glasserman, and Kou (1999). (3) Although the central limit theorem asserts that, as m!1, the di erence between the discretely and continuously monitored barrier options should be small, it is well known in the trader s literature that numerically the di erence is surprisingly big, even for large m. This suggests that approximation by using the central limit theorem is not helpful. To deal with these di culties, Broadie, Glasserman, and Kou (1997) propose a continuity correction for the discretely monitored barrier option, and justify the correction both theoretically and numerically (Chuang, 1996, also suggests independently the approximation in a heuristic way). The resulting approximation, which only relies on a simple correction to the Merton (1973) formula (thus trivial to implement), is nevertheless quite accurate, and has been used in practice; see, for example, the textbook by Hull (2000). The idea goes back to classical technique in sequential analysis, in which corrections to normal approximation are used to 2

3 adjust for the overshoot e ects when a discrete random walk crosses a barrier; see, for example, Cherno (1965), Siegmund (1985), and Woodroofe (1982). Therefore, from a statistical point of view, it is an interesting application of sequential analysis to a real life problem. The goals of the current paper are twofold. (1) A new and shorter proof of Broadie, Glasserman, Kou (1997) is given here, which also makes the link between the sequential analysis and the barrier correction formula more transparent. (2) The new proof covers all eight cases of the barrier options, while the proof in the original paper only covers four of them. These are made possible by the results of Siegmund and Yuh (1982) and Siegmund (1985, pp ), and a change of measure argument via a simple discrete Girsanov theorem. The mathematical formulation of the problem and the main result is stated in the next section, while the proof is deferred to Section 3. Discussion is given in the last section. 2. Main Result We assume the following Brownian dynamics for the price of the underlying asset S(t), t 0, S(t) =S(0) exp f¹t + B(t)g ; where under the risk-neutral probability P, the drift is ¹ = r 2 =2, r being the risk-free interest rate and B(t) is a standard Brownian motion under P. In the continuously monitoring case, standard nance theory implies that the price of a barrier option will be the expectation, takenwithrespecttotherisk-neutralmeasurep, of the discounted (with the discount factor being e rt with T the expiration date of the option) payo of the nal payo of the option. For example, the price of a continuous up-and-out call option is given by V (H) =E (e rt (S(T) K) + I( (H; S) >T)); where K 0 is the strike price, H>S(0) is the barrier, and for any process Y (t), the notation (x; Y ) means that (x; Y ):=infft 0:Y (t) xg; the price of a continuous down-and-in put option is given by V (H) =E (e rt (K S(T )) + I(~ (H; S) T)); where H<S(0) is the barrier, and ~ (x; Y )=infft 0:Y (t) xg: The other six types of the barrier options can be priced similarly. In the Brownian motion framework, all of the eight types of the barrier options can be priced in closed forms; see Merton (1973). 3

4 In the discretely monitoring case, under the risk neutral measure P,atthen-th monitoring point, n t, t = T=m, the asset price is given by ( S n = S(0) exp ¹n t + p ) nx t Z i = S(0) exp(w n p t); n =1; 2; :::; m; where the random walk W n is de ned by W n := nx Z i + ¹ p t ; the drift is given by ¹ = r 2 =2, andz i s are independent standard normal random variables. By analogy, the price of the discrete up-and-out-call option is given by V m (H) = E (e rt (S m K) + I( 0 (H; S) >m)) = E fe rt (S m K) + If 0 (a=( p T ); W) >mgg; where a := log(h=s(0)) > 0, 0 (H; S) = inffn 1:S n Hg; 0 (x; W) = inffn 1:W n x p mg: Note that in this case, we have a rst passage problem for the random walk W n,withsmall drifts ( ¹ p p p t! 0, asm!1), to cross a high boundary (a m=( T )!1,asm!1). All the other seven types of the discrete barrier options can be represented similarly. Since essentially there is no closed form solution for the discrete barrier options, the following result is useful in giving an approximation for the prices. Theorem 2.1. Let V (H) be the price of a continuous barrier option, and V m (H) be the price of an otherwise identical barrier option with m monitoring points. Then for any of the eight discrete monitored regular barrier options, we have the approximation: V m (H) =V (He pt=m )+o(1= p m); (2.1) with + for an up option and for a down option, where the constant = ³(1=2) p 2¼ ¼ 0:5826; with ³ the Riemann zeta function. 4

5 Corrected Relative error Continuous Barrier of (eq. 2.1) Barrier Barrier (eq. 2.1) True (in percent) Table 2.1: Up-and-Out Call Option Price Results, m =50(daily monitoring). This table is taken from Table 2.6 in Broadie, Glasserman, and Kou (1997). The option parameters are S(0) = 110, K =100, =0:30 per year, r =0:1, andt =0:2 year, which represents roughly 50 trading days. Remark. The above result was proposed in Broadie, Glasserman, and Kou (1997), where it is proved for four cases: down-and-in call, down-and-out call, up-and-in put, and up-and-out put, while in this paper all eight cases are covered in a simpler proof. To get a feeling of the accuracy of the approximation, Table 2.1 is taken from Broadie, Glasserman, and Kou (1997). The numerical result suggests that even for daily monitored discrete barrier options, there can still be big di erences between the discrete prices and the continuous prices. The improvement by using the approximation, which shifts the barrier from H to He pt=m in the continuous time formulae, is signi cant. 3. The Proof of Theorem 2.1 We shall only prove the case for the discrete up-and-out call option case (of course with H K and H > S(0)), as the other ones follow easily by symmetry and by the fact that the sum of two otherwise identical in- and out- put (call) options is a regular put (call) option. Note that the case of the discrete up-and-out call option is not covered by the theorem in Broadie, Glasserman, and Kou (1997). The following three results are needed in the proof. Proposition 3.1. (Discrete Girsanov Theorem). For any given probability measure P,we 5

6 can introduce a new probability ^P, de ned by the following formula ( d ^P m dp =exp X a i Z i 1 ) mx a 2 i ; 2 where a i, i =1; :::; n, are arbitrary constants, and Z i are standard normal random variables, N(0; 1), undertheprobabilitymeasure P. Then under the probability measure ^P, for every 1 i m, ^Z i := Z i a i is a standard normal random variable. The proof of this follows easily by checking the likelihood ratio identity, hence being omitted. Proposition 3.2. (Rescaling Property). Under any probability space P, for Brownian motions with drifts ¹ and, and standard deviation 1, P (W ¹ (1) x; (c; W ¹ ) > 1) = P ³W¹( 2 ) x ; ( c; W¹) > 2 : Proof. This holds because the process W(t) = ¹t + B(t) has the same joint distribution as ¹t + B( 2 t) = 2 ¹t + B( 2 t) : 2 Theorem 3.1 (Siegmund-Yuh, 1982, Siegmund, 1985, pp ). Let the stopping times 0 (b; U) for discrete random walk and (b; U) for continuous time Brownian motion be de ned as 0 (b; U) =inffn 1:U n b p mg; (b; U) =infft 0:U(T ) bg; where U(t) :=vt + B(t) and U n is a random walk with a small drift (as m!1), nx U n := Z i + p v ; m 6

7 where Z i s are independent standard normal random variables. Then, for any constants b y and b>0, asm!1, P U m <y p m; 0 (b; U) m = P (U(1) y; (b + = p m; U) 1) + o(1= p m); where = ³(1=2) p 2¼. Here the constant was calculated in Cherno (1965). Corollary 3.1. Under the same setting as in Theorem 3.1, we have for any constants b y and b>0, P U m y p m; 0 (b; U) >m = P (U(1) y; (b + = p m; U) > 1) + o(1= p m): Proof. A simple algebra yields P U m y p m; 0 (b; U) >m = P 0 (b; U) >m P U m <y p m; 0 (b; U) >m = P U m b p m; 0 (b; U) >m P U m <y p m; 0 (b; U) >m = P U m b p m P U m b p m; 0 (b; U) m P U m y p m + P U m <y p m; 0 (b; U) m : Since by Theorem 3.1, P U m b p m; 0 (b; U) m = P U(1) b; (b + = p m; U) 1 + o(1= p m); P U m <y p m; 0 (b; U) m = P U(1) y; (b + = p m; U) 1 + o(1= p m); we have P U m y p m; 0 (b; U) >m = P (U(1) b) P U(1) b; (b + = p m; U) 1 P (U(1) y)+p U(1) y; (b + = p m; U) 1 + o(1= p m) = P (b + = p m; U) > 1 P U(1) y; (b + = p m; U) > 1 + o(1= p m) = P (U(1) y; (b + = p m; U) > 1) + o(1= p m); from which the corollary is proved. 2 7

8 Proof of Theorem 2.1. First of all, E (e rt (S m K) + I( 0 (H; S) >m)) = E (e rt (S m K)I(S m K; 0 (H; S) >m)) = E (e rt S m I(S m K; 0 (H; S) >m)) Ke rt P (S m K; 0 (H; S) >m) = I Ke rt II (say). Using the discrete Girsanov theorem in Proposition 3.1, with a i = p t, we have that the rst term is given by ( I = E (e rt S(0) exp ¹m t + p ) mx t Z i I(S m K; 0 (H; S) >m)) = S(0)E "exp ( T + p t ) # mx Z i I(S m K; 0 (H; S) >m) = S(0) ^E(I(S m K; 0 (H; S) >m)) = S(0) ^P (S m K; 0 (H; S) >m): Under ^P, log S m has a mean ¹m t + p t m p t =(¹ + 2 )T instead of ¹T under the measure P. Therefore, the price of a discrete up-an-out-call option is given by V m (H) = S(0) ^P W m log(k=s(0)) p ; 0 (a=( p T );W) >m t Ke rt P W m log(k=s(0)) p ; 0 (a=( p T );W) >m ; t where under ^P, W m = P m ( ^Z i + f(¹+2 )=g p T m ), and under P, W m = P m (Z i + (¹=)p T m ); and ^Z i and Z i are standard normal random variables under ^P and P, respectively. Now using Corollary 3.1 with yields, as m!1, V m (H) = S(0)P Ke rt P y = log(k=s(0)) p ;b= T p W (¹+ 2 ) T W ¹ p T a p T = log(h=s(0)) p y; T (1) log(k=s(0)) p ; (b + = p p m; W (¹+ 2 ) T T (1) log(k=s(0)) p ; (b + = p p m; W ¹ T ) > 1 T 8 ) > 1 + o(1= p m);

9 where the notation W c (t) means a Brownian motion with drift c andstandarddeviation1. By Proposition 3.2 (the rescaling property), we get V m (H) = S(0)P W ¹+ 2 (T) log(k=s(0)) ; (b p q T + T=m;W ¹+ 2 ) >T Ke rt log(k=s(0)) P W ¹ (T) ; (b p q T + T=m;W¹ ) >T + o(1= p m) Since (b p T + qt=m;w)= ( a q + T=m;W) = (He p T=m ;S); we have V m (H) = S(0)P S(0)e (¹+2 )T +B(T ) K; (He p T=m ;S) >T (3.1) Ke rt P S(0)e ¹T +B(T ) K; (He p T=m ;S) >T + o(1= p m): Similarly, by using the continuous time Girsanov theorem, the continuous time price V (H) can be written as ³ V (H) = S(0)P S(0)e (¹+2 )T +B(T ) K; (H; S) >T (3.2) ³ Ke rt P S(0)e ¹T +B(T ) K; (H; S) >T : Comparing (3.1) and (3.2) yields from which the result is proved Discussion V m (H) =V (He p T=m )+o(1= p m); This paper does two things. First, it simpli es the proof in Broadie, Glasserman, and Kou (1997) paper, and secondly it also generalizes the result to include more cases of discrete barrier options. We conjecture that, however, the technique used in Broadie, Glasserman, and Kou (1997), along with the discrete Girsanov theorem, could be used to prove the similar barrier correction formulae for two dimensional barrier options, which is still an open problem. We are investigating this. 9

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