Georgian Mathematical Journal Volume 10 (2003), Number 2, 325–334

TIME-DEPENDENT BARRIER OPTIONS AND BOUNDARY CROSSING PROBABILITIES

A. NOVIKOV, V. FRISHLING, AND N. KORDZAKHIA

Abstract. The problem of pricing of time-dependent barrier options is considered in the case when interest rate and volatility are given functions in Black–Scholes framework. The calculation of the fair price reduces to the calculation of non-linear boundary crossing probabilities for a standard Brownian motion. The proposed method is based on a piecewise-linear approximation for the boundary and repeated integration. The numerical example provided draws attention to the performance of suggested method in comparison to some alternatives.

2000 Mathematics Subject Classification: 91B28, 60J65.

Key words and phrases: First passage times, Wiener process, Girsanov transformation, numerical integration.

1. Introduction

In the diffusion equation for an underlying asset S_{t} let us assume the coefficients μ(t) and σ(t) to be time-dependent,

dS_{t} = μ(t)S_{t}dt + σ(t)S_{t}dW_{t}, 0 ≤ t ≤ T < ∞,  (1)

W_{t} is a standard Wiener process given on a probability space (Ω, F, P). We assume a bank account process B_{t} is driven by the equation dB_{t} = r(t)B_{t}dt and hence

B_{t} = exp ( ∫_{0}^{t} r(s)ds),

where r(t) is a positive function of time, the so-called spot interest rate. The solution of equation (1) is

S_{t} = S_{0} exp \{ ∫_{0}^{t} μ(s)ds − \frac{1}{2} ∫_{0}^{t} σ^{2}(s)ds + ∫_{0}^{t} σ(s)dW_{s}\}.  (2)

We assume here that μ(s) and σ(s) are square-integrable and nonrandom functions. Further, we also assume that μ(s) = r(s), 0 ≤ s ≤ T . This assumption means that we use the free-arbitrage approach to pricing of options (see details, e.g., in [1] or [2]). Then the process {S_{t}/B_{t}, t ≥ 0} is a martingale with respect to the information flow F_{t} = σ{S_{s}, 0 ≤ s ≤ t} and probability measure P defined above.

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It is well known that under the free-arbitrage assumption the fair price of an option with a payoff function f_{T} is given by the formula

C_{T} = E[f_{T} /B_{T}],

where E(·) is a symbol of expectation with respect to measure P (see details, e.g., in [1] or [2]).

A down-and-out call option is a call option that expires if the stock price falls below the prespecified “out” barrier H. “Down” here refers to an initial price of stock S_{0} being above of the barrier H. A down-and-in call is a call that comes into existence if the stock price falls below the “in” barrier at any time during the option’s life. Note, if we buy a down-and-out call and a down-and-in call with the same barrier price, H, strike price K, and time to expiration, T, the payoff of the portfolio is the same as for a standard call option. In the case of up-and-out option, the barrier lies above the initial stock price, and if the stock price ever rises above the barrier, then the option becomes worthless. Similarly, there exist up-and-in options. Below we consider the case of up-and-out barrier option with time-dependent upper barrier H(t). In this case the payoff function is

f_{T} = (S_{T} − K)^{+}I{τ > T } = (S_{T} − K) I{S_{T} > K, τ > T }

where we use the notation I{A} for the indicator function of a set A and

τ = inf {t ≥ 0 : S_{t} ≥ H(t)} .

2. Pricing of Time-Dependent Barrier Options

The problem is to find a fast and accurate algorithm for the calculation of the fair price of up-and-out barrier option

C_{T} = E[(S_{T} − K)I{S_{T} > K, τ > T }/B_{T} ].

This problem has been addressed, e.g., by Roberts and Shortland in [3]. For simplicity of the notation and further exposition, we assume the volatility function is a constant: σ(s) = σ > 0. The following proposition reduces the pricing problem to the calculation of boundary crossing probabilities by the standard Wiener process with respect to measure P.

Proposition 1. The fair price of a up-and-out European call option with a single upper barrier H(t) is given by

C_{T} = S_{0}p_{1} − K exp ( − ∫_{0}^{T} r(s)ds)p_{0},  (3)

where

p_{1} = P{σW_{T} + σ^{2}T > G; σW_{t} + σ^{2}t < g(t), t ≤ T },

p_{0} = P{σW_{T} > G; σW_{t} < g(t), t ≤ T },

G = ln (\frac{K}{S_{0}} ) + \frac{1}{2} σ^{2}T − ∫_{0}^{T} r(s)ds,

g(t) = ln (\frac{H(t)}{S_{0}}) + \frac{1}{2} σ^{2}t − ∫_{0}^{t} r(s)ds.

Proof. Using (2) with σ(s) = σ we have

C_{T} = E[I\{S_{T} > K, τ > T \}\frac{S_{T}}{B_{T}} ] − E[I\{S_{T} > K, τ > T \}\frac{K}{B_{T}} ]

= S_{0}E[I\{S_{T} > K, τ > T \} exp \{σW_{T} − \frac{1}{2}σ^{2}T \}]

− K exp { − ∫_{0}^{T} r(s)ds}P{S_{T} > K, τ > T }.

To see that P{S_{T} > K, τ > T } = p_{0} one needs just express S_{t} and τ in terms of W_{t}.

Denote the Girsanov exponent

Z_{T} (f) = exp \{ ∫_{0}^{T} f(s)dW_{s} − \frac{1}{2} ∫_{0}^{T} f^{2}(s)ds\}.

By the Girsanov theorem (see, e.g., [2]) for any square-integrable nonrandom function f(s) and an event A ∈ F_{T}

E[I{A}Z_{T} (f)] = P̃{A}  (4)

where probability measure P̃ is such that the process

{W̃_{t} := W_{t} − ∫_{0}^{t} f(s)ds, t ≥ 0}

is a standard Wiener process with respect to (F_{t}, P̃). Applying this fact with f(s) = σ we have

p_{1} = P̃{σW_{T} > G; σW_{t} < g(t), t ≤ T }

= P̃{σW̃_{T} + σ^{2}T > G; σW̃_{t} + σ^{2}t < g(t), t ≤ T }

= P{σW_{T} + σ^{2}T > G; σW_{t} + σ^{2}t < g(t), t ≤ T } . □

Remark 1. For other types of barrier options, such as double barrier options or partial barrier options, the equation (3) still holds with modified values of p_{1} and p_{0}.

We now need tools for the calculation of probabilities p_{1} and p_{0} in Proposition 1. In fact, the calculation of boundary crossing probabilities has other important applications besides the pricing of barrier options. This problem arises in various fields such as psychology (see [4]), clinical trials (see [5]) and many other areas as physics, insurance, and nonparametric statistics. While the time of calculation for the purpose of evaluating the fair price of barrier options is very important, in other applications like clinical trials or physics a high degree of accuracy becomes more important than the time of calculation. For calculation other methods could be used, such as partial differential equations (PDE), see [6], integral equations [7] and Monte Carlo simulation approaches. We now introduce a method based on numerical integration, proposed by Wang and Pötzelberger [8] and then developed by Novikov et al. [9] which led to an another work by Pötzelberger and Wang [10]. This method may in fact have certain advantages over the other methods. One of the advantages of this approach is that it can be used in the case of boundaries which may even be discontinuous. Another important advantage is that we can control the accuracy of the approximation as it will be shown below.

Let ĝ(s) be the boundary on the interval [0, T] which is considered as an approximation for function g(s) defined in Proposition 1. For example, one may consider ĝ(s) as piecewise-linear continuous functions with nodes t_{i}, t_{0} = 0 < t_{1} < ⋯ < t_{n} = T (in general, this function might be discontinuous or nonlinear). Denote

p(i, ĝ | x_{i}, x_{i} + 1) = P {W_{s} ≤ ĝ(s), s ∈ (t_{i}, t_{i+1}) | W_{t_{i}} = x_{i}, W_{t_{i+1}} = x_{i+1}}

When ĝ(t) is a linear function on the interval (t_{i}, t_{i+1}) the last probability is given by (see, e.g., [8], [9])

p(i, ĝ | x_{i}, x_{i} + 1)

= I\{ĝ(t_{i}) > x_{i}, ĝ(t_{i+1}) > x_{i+1}\}[1 − exp \{ − \frac{2(ĝ(t_{i}) − x_{i})(ĝ(t_{i+1}) − x_{i+1})}{t_{i+1} − t_{i}}\}].

The next formula gives the representation for a boundary crossing probability of the form

P (ĝ, K, T) := P {W_{t} ≤ ĝ(t), t ≤ T; W_{T} > K}

as an n-fold repeated integral of p(i, ĝ|x_{i}, x_{i} + 1) and the transition probability of the Wiener process:

P (ĝ, K, T) = E[I{W_{T} > K}] ∏_{i=0}^{n−1} p(i, ĝ | W_{t_{i}}, W_{t_{i+1}}).  (5)

This formula seems to be firstly noted by Wang and Pötzelberger [8] for the case of piecewise one-sided continuous linear boundaries. Its generalization to the case of nonlinear two-sided boundaries was presented by Novikov et al. [9], see also [10].

Denote the density of the standard Wiener process W_{t} by φ(x, t) = \frac{1}{\sqrt{2πt}} × exp(−\frac{x^{2}}{2t}). The following recurrent algorithm indicated in [9] for the general case of two-sided boundaries can be used to evaluate the boundary crossing probability P (ĝ, K, T) in (5).

Let z_{0}(x) = p(0, ĝ_{n} | 0, x)φ(x, Δt_{1}),

z_{k+1}(x) = ∫_{−∞}^{ĝ_{n}(t_{k})}z_{k}(u)p(k + 1, ĝ_{n} | u, x)φ(x − u, Δt_{k})du, k = 0, …, n − 1.

Then

P (ĝ, K, T) = ∫_{K}^{ĝ_{n}(T)}z_{n−1}(u)φ(u, T)du.  (6)

We should note that formula (3′) in [9] contains a misprint which is corrected in (6). An example which illustrates calculation using this algorithm will be presented in Section 3. Here we discuss an accuracy of the approximation.

In [9] it is demonstrated how the measure transformation method based on the Girsanov theorem can be used for estimating difference between probabilities P (g, K, T) and P (ĝ, K, T) in terms of the distance

Δ_{T} (ĝ, g) := ∫_{0}^{T} (\frac{d}{ds} (ĝ(s) − g(s)))^{2}ds.

Here the same technique is used to get an estimate for an accuracy of approximation for option prices.

We will always assume that the boundaries g(t) and ĝ(t) which define the price C_{T} and its approximation Ĉ_{T}, respectively (see the notation in Proposition 1) have the following properties:

g(s) − ĝ(s) is a continuous function, g(0) − ĝ(0) = g(T) − ĝ(T) = 0, Δ_{T} (ĝ, g) < ∞.

Theorem 1.

|C_{T} − Ĉ_{T}| ≤ C\sqrt{Δ_{T} (ĝ, g)},

where

C = \sqrt{\frac{2}{πσ^{2}}} E[(S_{T} − K)^{+}/B_{T} ].  (7)

Proof. We will use the following representations for C_{T} and Ĉ_{T} in terms of the original Wiener process:

C_{T} = E[(S_{0}B_{T} exp \{σW_{T} − \frac{σ^{2}T}{2} \} − K)^{+}I\{σW_{t} < g(t), t ≤ T \}/B_{T} ],

Ĉ_{T} = E[(S_{0}B_{T} exp \{σW_{T} − \frac{σ^{2}T}{2} \} − K)^{+}I\{σW_{t} < ĝ(t), t ≤ T \}/B_{T} ].

Let probability measure P̃ be defined by formula (4) with

f(s) = \frac{d}{ds} (g(s) − ĝ(s))/σ.

Then by the Girsanov theorem the process

{W̃_{t} = W_{t} + (ĝ(t) − g(t))/σ, t ≥ 0}  (8)

is a standard Wiener process with respect to (F_{t}, P̃). Note that due to the assumption ĝ(T)−g(T) = 0 we have the equality W̃_{T} = W_{T}. Besides, expressing W_{t} via W̃_{t} from (8) and substituting it into the representation for Z_{T} (f) we also have

(Z_{T} (f))^{−1} = exp \{ − ∫_{0}^{T} \frac{d}{ds} (ĝ(s) − g(s))/σdW̃_{s} − \frac{Δ(ĝ(s), g(s))}{2σ^{2}}\}.

As E( . ) = Ẽ[(Z_{T} (f))^{−1}( . )] , we have

C_{T} = Ẽ[(Z_{T} (f))^{−1}(S_{0}B_{T} exp \{σW_{T} − \frac{σ^{2}T}{2} \} − K)^{+}
× I{σW_{t} < g(t), t ≤ T }/B_{T} ]

= Ẽ[(Z_{T} (f))^{−1}(S_{0}B_{T} exp \{σW̃_{T} − \frac{σ^{2}T}{2} \} − K)^{+}

× I{σW̃_{t} < ĝ(t), t ≤ T }/B_{T} ]

= E[(Z_{T} (−f))(S_{0}B_{T} exp \{σW_{T} − \frac{σ^{2}T}{2} \} − K)^{+}

× I{σW_{t} < ĝ(t), t ≤ T }/B_{T} ].

Using this representation we get

|C_{T} − Ĉ_{T}| = |E[(Z_{T} (−f) − 1)(S_{T} − K)^{+}I{σW_{t} < ĝ(t), t ≤ T }/B_{T} ]|
≤ E[|Z_{T} (−f) − 1|(S_{T} − K)^{+}/B_{T} ].

Here the random variables Z_{T} (−f) and S_{T} are independent as they are functions of Gaussian random variables ∫_{0}^{T} f(s)dW (s) and W_{T} which are independent.

Indeed, due to the properties of stochastic integrals and the choice of function f(s) the covariance of those random variables is

E[W_{T} ∫_{0}^{T} f(s)dW (s)] = ∫_{0}^{T} f(s)ds = (g(T) − ĝ(T) + ĝ(0) − g(0))/σ = 0

Hence

E[|Z_{T} (f) − 1| (S_{T} − K)^{+}/B_{T}] = E[|Z_{T} (f) − 1|] E[(S_{T} − K)^{+}/B_{T} ]

To complete the proof we note that a random variable log(Z_{T} (f) is normally distributed with mean −Δ_{T} (ĝ, g)/(2σ^{2}) and variance Δ_{T} (ĝ, g)/(σ^{2}). By direct calculation we have the equality

E|Z_{T} (f) − 1| = 2(Φ(\sqrt{Δ_{T} (ĝ, g)/σ^{2}} ) − \frac{1}{2}),

where Φ(x) is a standard normal distribution. As Φ(x) − 1/2 ≤ x/\sqrt{2π}, x > 0 it follows that

E|Z_{T} (f) − 1| ≤ \sqrt{\frac{2Δ_{T} (ĝ, g)}{πσ^{2}}} . □

Remark 2. The price of the ordinary call option E[(S_{T} − K)^{+}/B_{T}] in (7) is easy to evaluate by the famous Black–Scholes formula. If we assume that the boundary g(t) is a twice continuously differentiable function and the lengths of intervals (t_{i}, t_{i+1}), i = 1, …, n − 1, for a piecewise-linear approximating function ĝ_{t} are equal (i.e., a uniform partition is considered), then, obviously, Δ_{T} (ĝ, g) = O(\frac{1}{n^{2}} ) as n → ∞. Hence by Theorem 1 we have

|C_{T} − Ĉ_{T}| = O(\frac{1}{n}).

We can essentially improve this estimate by using Theorem 3 from [9] along with Proposition 1.

Proposition 2. Let g(t) be a twice continuously differentiable function and ĝ(t) be a piecewise-linear continuous function such

ĝ(t_{i}) = g(t_{i}), t_{i} = \frac{iT}{n} , i = 0, …, n.

Then as n → ∞

|C_{T} − Ĉ_{T}| = O(\frac{log n}{n^{3/2}}).

Theoretically, we can improve this estimate for the rate of convergence if we allow the use of a non-uniform partition. In the context of boundary crossing problems it has recently been shown by Pötzelberger and Wang [10] that under some conditions on boundaries with the use of a specifically designed non-uniform partition

|P \{W_{t} < g(t), t ≤ T \} − P \{W_{t} < ĝ(t), t ≤ T \}| = O(\frac{1}{n^{2}} ).

Applying this fact along with Proposition 1 we get

|C_{T} − Ĉ_{T}| = O(\frac{1}{n^{2}} ).

Note that a search for an optimal non-uniform partition could be a rather time-consuming procedure especially for large n.

3. Numerical Example

This section contains a numerical example of the calculation of the fair price of a barrier option which was considered by Roberts and Shortland in [3]. In this paper the Vasicek model is used for the risk-free interest rate I_{t}:

I_{t} − r = a + ∫_{0}^{t} (r − I_{s})ds + σ Ŵ_{t},

where Ŵ_{t} is a standard Wiener process independent of W_{t}. Then r(t) = EI_{t} = r + ae^{−t} and ∫_{0}^{t} r(s)ds = rt + a(1 − e^{−t}). Note that the interest rate is now considered to be stochastic rather than deterministic as in Section 2.

Roberts and Shortland considered in [5] the example with S_{0} = 10, σ = 0.1, r = 0.1, and a = 0.05. The style of option was the up-and-in European call option with boundary H = 12, strike price K = 11, and maturity at T = 1. To price this option we use that the sum of prices of “up-and-down” and “upand-in” options equals to the price of “standard call” and hence the assertion of Theorem 1 is true for “up-and-in” options also.

The boundary function g(t) for this example is

g(t) = ln(H/S_{0}) + σ^{2}t/2 − ∫_{0}^{t} r(s)ds = 0.18232 − 0.095t − 0.05(1 − e^{−t}).

By using an analytic approximation Roberts and Shortland obtained the following bounds for the fair price:

0.51675 ≤ C_{T} ≤ 0.51796.

They also used the Monte-Carlo method to evaluate the fair price of the option. By simulating 1 million sample paths of the stock price with step size 0.001 they obtained

C_{T} = 0.513903

with standard error 0.016. This value of the fair price is less than the lower bound, although a 95% confidence interval for C_{T} does include these bounds. In order for a 95% confidence interval to have comparable width to the analytic bounds, we would require about 700 million sample paths with step size 0.001. The computational time required to do this would clearly prevent the direct Monte Carlo method from being useful. However, the use of the variance reduction technique might dramatically reduce the required sample size.

Using the suggested numerical integration method with piecewise-linear approximation for 50 and 400 uniformly spaced nodes, we obtained for both cases the following value for the approximation of the fair price:

Ĉ_{T} = 0.51683 .  (9)

This value is within the analytic bounds obtained by Roberts and Shortland. Note that by Theorem 1 the upper bound for errors of these estimates are 9·10^{−4} and 1.1 · 10^{−4}, respectively for n = 50 and n = 400. The stability of numerical integration is verified by using the Gaussian quadrature method with 32 and 64 nodes, the reported numbers are the same as in (9).

For the calculation of boundary probabilities in Proposition 1 we also used the integral equation method from [7]. Solving the integral equation iteratively, for three iterations only we obtained the fair price as C_{T} = 0.51695. This is also within the bounds given by Roberts and Shortland.

By using the PDE approach we obtained C_{T} = 0.51671 as the fair price. It is noteworthy that this is slightly less than the lower bound obtained by Roberts and Shortland, although the difference is only in the fifth digit. However this is an acceptable accuracy for the bank practice.

Acknowledgements

The first author was supported by ARC Large Grant A0010474.

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(Received 17.02.2003)

Author’s addresses:

A. Novikov Steklov Mathematical Institute 8, Gubkin St., Moscow GSP 117966 Russia Department of Mathematical Sciences University of Technology Sydney, Broadway, PO Box 123 NSW 2007 Australia E-mail: Alex.Novikov@uts.edu.au.

V. Frishling and N. Kordzakhia Commonwealth Bank of Australia 175, Pitt St., Sydney, NSW 2000 Australia E-mail: frishlv@cba.com.au

kordzani@cba.com.au