On the Binomial Approximation of the American Put
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On the Binomial Approximation of the American Put Damien Lamberton1
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract We consider the binomial approximation of the American put price in the Black– Scholes model (with continuous dividend yield). Our main result is that the error of approximation is O((ln n)α /n), where n is the number of time periods and the exponent α is a positive number, the value of which may differ according to the respective levels of the interest rate and the dividend yield.
1 The Binomial Approximation Consider the Black–Scholes model, in which the stock price at time t is given by St = S0 e(r −d−
σ2 2 )t+σ Bt
,
where, under the risk-neutral probability measure, (Bt )t≥0 is a standard Brownian motion. Here, r is the instantaneous interest rate, and d is the dividend rate (or the foreign interest rate in the case of forex options). We assume r > 0 and d ≥ 0. Denote by P the price function of the American put with maturity T and strike price K , so that P(t, x) =
sup Ex e−r τ f (Sτ ) , 0 ≤ t ≤ T , x ∈ [0, +∞),
τ ∈T0,T −t
with f (x) = (K − x)+ , and Ex = E (· | S0 = x). Here, T0,t denotes the set of all stopping times with respect to the Brownian filtration, with values in the interval [0, t]. For technical reasons (especially for the derivation of regularity estimates for the second time derivative of the price function), it is more convenient to use the log-stock price. So, we introduce
B 1
Damien Lamberton [email protected] Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, Projet Mathrisk INRIA, 77454 Marne-la-Vallée, France
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Applied Mathematics & Optimization
X tx = x + μt + σ Bt , with μ = r − d −
σ2 , 2
and U (T , x) = sup E e−r τ ϕ(X τx ) , τ ∈T0,T
with ϕ(x) = (K − e x )+ . We then have P(t, x) = U (T − t, ln(x)), t > 0, x > 0. Note that U (t, x) satisfies the following parabolic variational inequality ∂U + (A − r )U , ϕ − U = 0, max − ∂t with the initial condition U (0, .) = ϕ. Here, A is the infinitesimal generator of X , namely A=
σ 2 ∂2 ∂ +μ . 2 2 ∂x ∂x
˜ ) ≤ ln(K ) such that Recall that, for each T > 0, there is a real number b(T ˜ ). U (T , x) > ϕ(x) ⇔ x > b(T In fact, if (b(t), 0 ≤ t ≤ T ) is the exercise boundary of the American put with ˜ maturity T , we have b(t) = ln(b(T − t)). We will also need the European value function, defined by U¯ (T , x) = E e−r T ϕ(X Tx ) . Note that U¯ (0, .) = ϕ and −
∂ U¯ + (A − r )U¯ = 0. ∂t
Note that, in Sect. 3, the function U¯ will be denoted by u ϕ . We now introduce the random walk approximation of Brownian motion. To be more precise, assume (X n )n≥1 is a sequence of i.i.d. real random variables satisfying EX n2 = 1 and EX n = 0, and define, for any positive integer n, the process B (n) by (n)
Bt
=
T /n
[nt/T ]
Xk , 0 ≤ t ≤ T ,
k=1
where [nt/T ] denotes the greatest integer in nt/T .
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Applied Mathematics & Optimization
We will assume the following about the common distribution of the X n ’s (cf.
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