PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 79(93) (2006), 65–72 DOI: 10.2298/PIM0693065O

MARKOVIAN BLACK AND SCHOLES

E. Omey and S. Van Gulck

Communicated by Slobodanka Janković

Abstract. We generalize the classical binomial approach of the model of Black and Scholes to a Markov binomial approach. This leads to a new formula for the cost of an option.

1. Introduction

Consider a call option with strike price X and exercise time t. We divide the time t into the time points t/n, 2t/n, …, nt/n. During each time unit the price goes up by a factor u or down by a factor d. The value after n time units is given by

S(t) = S(0)u^{S_{n}}d^{n−S_{n}}

where S(0) is the price at time t = 0 and where S_{n} denotes the number of ups during n time periods. The cost of the option that does not give rise to an arbitrage is given by

(1)  K = r_{0}^{−n}E( max(S(0)u^{S_{n}}d^{n−S_{n}} − X, 0))

where r_{0} = 1 + rt/n is the nominal interest rate. In the usual Black–Scholes approach, cf. Ross [1999, Chapter 7], Cox et al. [1979], one assumes that S_{n} has a binomial distribution given by S_{n} ∼ BIN(n, p) and one takes

(2)  u = exp (a\sqrt{t/n}), d = exp ( − b\sqrt{t/n})

and

p = \frac{1 + rt/n − d}{u − d}

or

p = \frac{r^{∗} − d}{u − d}

where r^{∗} = exp(rt/n). The basic assumption in this binomial model is that the ups and the downs appear independently from each other, i.e., the sum S_{n} is made up of independent 0 − 1 variables. In the present paper we assume that the ups and the downs are governed by a Markov Chain and provide a Black and Scholes formule for this case.

2000 Mathematics Subject Classification: Primary 60J20; Secondary 62P05, 91B28.

2. Markovian approach

We shall consider the case where the ups and downs are governed by a Markov chain as follows. Let Y_{i} = 1 if the price goes up at the i − th time unit and let Y_{i} = 0 otherwise. Assume that P (Y_{1} = 1) = p and that the transition probabilities are given by

P = (\begin{matrix}p_{0,0} & p_{0,1} \\ p_{1,0} & p_{1,1} \end{matrix})

In the paper we assume that the transition probabilities are strictly between 0 and 1. The number of ups is given by S_{n} = ∑_{i=1}^{n} Y_{i} and formula (1) holds.

It is useful to note that the Markov chain has a unique stationary vector given by (x, y) where

(3)  y = \frac{p_{0,1}}{p_{0,1} + p_{1,0}} and x = 1 − y

The eigenvalues of P are given by 1 and λ = 1 − p_{0,1} − p_{1,0} = p_{1,1} − p_{0,1}. Note that |λ| < 1. Properties of S_{n} can be found in e.g. Omey, Santos and Van Gulck [2006]. As a special case we also consider correlated Bernoulli trials studied by Dimitrov and Kolev [1999], see also Edwards [1960] or Wang [1981]. In this case the transition matrix is given by

P (p, ρ) = (\begin{matrix}q + ρp & p(1 − ρ) \\ q(1 − ρ) & p + ρq \end{matrix})

and now we have P (Y_{i} = 1) = p = y, for all i and λ = ρ = ρ(Y_{i}, Y_{i+1}) ≠ 0. In the Markov chain setting we have the following result concerning moments of S_{n}.

Proposition 1 (Omey et al. [2006]). (i) We have

E(S_{n}) = ny − (y − p)\frac{1 − λ^{n}}{1 − λ}

and

Var(S_{n}) = n\frac{1 + λ}{1 − λ}xy + ∑_{k=0}^{n−1}(C(1)λ^{k} + C(2)λ^{2k} + C(3)kλ^{k})

where C(1), C(2), C(3) are given in the remark below. As n → ∞ we have

E(S_{n}) ∼ ny and Var(S_{n}) ∼ nxy \frac{1 + λ}{1 − λ}

(ii) If P = P (p, ρ) we have E(S_{n}) = np and

Var(S_{n}) = \frac{pq}{1 − ρ}(n(1 + ρ) − 2ρ\frac{1 − ρ^{n}}{1 − ρ} )

Remark. Using a(1) = (y − p)(y − x) and a(2) = (y − p)^{2} the constants are given by

C(1) = (a(1)(1 − λ) − 2xyλ − 2a(2))/(1 − λ)
C(2) = a(2)(1 + λ)/(1 − λ)
C(3) = 2a(1)

For large values of n we can approximate the distribution of S_{n} by a normal distribution. We have the following central limit theorem.

Theorem 2 (Omey et al. [2006]). As n → ∞ we have

\frac{S_{n} − ny}{\sqrt{nθ}} ⇒^{d} Z ∼ N(0, 1),

where θ = xy(1 + λ)/(1 − λ)

3. Markovian Black and Scholes

In view of (1) we define W by the following relation:

u^{S_{n}}d^{n−S_{n}} = exp(W)

Assuming that (2) holds, we have

(4)  W = (a + b)\sqrt{t/n}S_{n} − b\sqrt{tn}

Using (4) and Proposition 1(i) we find that

(5)  E(W) = \sqrt{nt}((a + b)y − b) − (a + b)(y − p)\sqrt{\frac{t}{n}} \frac{1 − λ^{n}}{1 − λ}

and

(6)  Var(W) ∼ (a + b)^{2}txy \frac{1 + λ}{1 − λ}

In order to obtain useful estimates in (5) and (6), we make the following assumptions about the transition probabilities. First we introduce some extra notations. Let α and β denote real parameters and let

(7)  r_{u} = exp(αt/n) and r_{d} = exp(βt/n)

For the transition probabilities we assume that there are constants A, B, C, D such that

(8)  p_{0,1} = A + B \frac{r_{u} − d}{u − d} and p_{1,1} = C + D \frac{r_{d} − d}{u − d}

Later we shall reduce the number of parameters in (8). With model (8) we want to take into account the difference between going from a ‘down’ to an ‘up’ and from an ‘up’ to another ‘up’.

Proposition 3. We have

(9)  p_{0,1} = A + B(\frac{b}{a + b} + \sqrt{\frac{t}{n}}(\frac{α − ab/2}{a + b} + o(1)))

and

(10)  p_{1,1} = C + D(\frac{b}{a + b} + \sqrt{\frac{t}{n}}(\frac{β − ab/2}{a + b} + o(1)))

Proof. Let us consider p_{0,1}. Using (2) and (7), a Taylor expansion shows that

r_{u} − d = α\frac{t}{n} + b\sqrt{\frac{t}{n}} − b^{2} \frac{t}{2n} + O(1)n^{−3/2}

and

u − d = (a + b)(\sqrt{\frac{t}{n}} + (a − b)\frac{t}{2n}) + O(1)n^{−3/2}

Now observe that

(a + b)\frac{r_{u} − d}{u − d} − b = \frac{(a + b)(r_{u} − d) − b(u − d)}{u − d}

Using the Taylor expansions, we readily obtain that

(a + b)\frac{r_{u} − d}{u − d} − b ∼ (α − \frac{ab}{2})\sqrt{\frac{t}{n}}

From this and (8) we obtain (9). In a similar way also (10) follows. □

Note that as n → ∞ we have

p_{0,1} → A + B \frac{b}{a + b}

p_{1,1} → C + D \frac{b}{a + b}

Since 0 < p_{i,j} < 1, these expressions show that the parameters A, B, C, D should satisfy some restrictions. Using (3) we also obtain that y → y^{∗} and that λ → γ where

y^{∗} = \frac{A(a + b) + Bb}{(A + 1 − C)(a + b) + b(B − D)}

and

γ = C − A + \frac{b(D − B)}{a + b}

Note that

y^{∗} = \frac{A + B \frac{b}{a+b}}{1 − γ}

In view of (5) we choose A, B, C, D in such a way that

y^{∗} = \frac{b}{a + b}

If, for example A = 0 and B = D = 1 − C, then we obtain that y^{∗} = b/(a + b) and in this case we have γ = C.

Now we can proceed in studying W, cf. (5), (6).

Theorem 4. (i) As n → ∞, we have W ⇒^{d} W^{∗}, where W^{∗} = μ + σZ, with Z ∼ N(0, 1) as in Theorem 2 and with

μ = t\frac{1}{1 − γ} (B(α − \frac{ab}{2}) + bD(\frac{β − ab/2}{a + b}) − bB(\frac{α − ab/2}{a + b}))

and

σ^{2} = tab\frac{1 + γ}{1 − γ}

(ii) As n → ∞, we have

P (S(0) exp(W) > X) → P (Z > \frac{log(X/S(0)) − μ}{σ})

Proof. (i) Since by Theorem 2, S_{n} is asymptotically normal, also W is. We have to determine E(W) and Var(W) as n → ∞. First consider E(W) and observe that

(a + b)y − b = \frac{I}{II}

where I = (a + b)p_{0,1} − b(p_{0,1} + p_{1,0}) and II = p_{1,0} + p_{0,1}. Using II = 1 − λ we have

II → 1 − γ

As to I we have I = ap_{0,1} − b + bp_{1,1}. Using (9) and (10) we readily obtain that

I = K(1)\sqrt{\frac{t}{n}}(1 + o(1))

where

K(1) = B(α − \frac{ab}{2}) + bD(\frac{β − ab/2}{a + b}) − bB(\frac{α − ab/2}{a + b})

We conclude that

(a + b)y − b = \sqrt{\frac{t}{n}}(\frac{K(1)}{1 − γ} + o(1))

Using this result, we find that

E(W) → t\frac{K(1)}{1 − γ}

For the variance we find that y → y^{∗} and x → 1 − y^{∗}. It follows that

Var(W) → tab\frac{1 + γ}{1 − γ}

This proves the result.

(ii) This follows from (i) □

Remark. With the choice A = 0 and B = D = 1 − C, we find the following simpler expressions: we have γ = C and

μ = t ((α − \frac{ab}{2}) + b(\frac{β − α}{a + b})) and σ^{2} = tab\frac{1 + γ}{1 − γ}

Taking also α = β = r, we can simplify more and find that

μ = t(r − \frac{ab}{2}) and σ^{2} = tab\frac{1 + γ}{1 − γ}

If we take a = b = σ_{p} where σ_{p} represents the volatility of the underlying security, then we find

μ = t(r − \frac{σ_{p}^{2}}{2} ) and σ^{2} = tσ_{p}^{2} \frac{1 + γ}{1 − γ}

The case where γ = 0 corresponds to the usual Black and Scholes model. Here we have the extra parameter γ. Using p_{0,1} → (1 − γ)/2 and p_{1,1} → (1 + γ)/2 we see that γ is closely connected with the probability of arriving at an ‘up’ starting from a ‘down’ or an ‘up’. The parameter γ heavily influences σ^{2} (and hence also K, see below). Taking γ = −0.5, γ = 0 and γ = 0.5 we see that σ^{2} varies from σ^{2} = \frac{1}{3}tσ_{p}^{2} to σ^{2} = tσ_{p}^{2} and σ^{2} = 3tσ_{p}^{2} respectively.

Returning to the option cost the following result follows from (1) and Theorem 4.

Theorem 5. As n → ∞, we have

K = exp(−rt)E( max(S(0) exp W^{∗} − X, 0))

where W^{∗} ∼ N(μ, σ^{2})

Using standard formulas for the normal distribution, we find that

K = S(0) exp(−rt + μ + σ^{2}/2)Φ(w) − exp(−rt)XΦ(w − σ)

where

w = \frac{σ^{2} + μ − log(X/S(0))}{σ}

and where Φ(w) is the standard normal distribution function.

Remarks. 1) If the parameters are chosen in such a way that rt = μ + σ^{2}/2, we find that

K = S(0)Φ(w) − exp(−rt)XΦ(w − σ)

which is similar to the classical Black and Scholes formula.

2) As a special case we consider the case where P = P (p, ρ). Now we assume that

p = A + B \frac{r_{∗} − d}{u − d}

where r_{∗} = exp(αt/n). Using p_{0,1} = p(1 − ρ) and p_{1,1} = ρ + (1 − ρ)p and the previous analysis can be used. In this case we have α = β and

y^{∗} = A + B \frac{b}{a + b}

We have to assume that y^{∗} = b/(a + b). Using the notations as in the proof of Theorem 4, we find that K(1) = B(1 − ρ)(α − ab/2). Now we find that μ = tB(α − ab/2) and that σ^{2} = tab(1 + ρ)/(1 − ρ). A convenient choice seems to be A = 0 and B = 1.

Corollary 6. If P = P (p, ρ) and p = (r_{∗} − u)/(u − d), then Theorem 4 holds with μ = t(α − ab/2) and σ^{2} as before.

4. Final remarks

1) A correlated binomial distribution has been introduced and studied by Madsen [1993], Altham [1978], Kupper and Haseman [1978], Mingoti [2003]. Examples and applications can be found e.g., in quality control, Lai et al. [1998]. See also Edwards [1960], Wang [1981].

2) Many stochastic processes are based on a counting process {N(t), t ⩾ 0}, where N(t) denotes the number of times a certain event occurs in the time interval (0, t]. In many processes one models N(t) with a Poisson, binomial or negative binomial distributions. In Minkova [1999, 2001], Dimitrov and Kolev [1999], the authors study inflated processes by introducing an additional parameter ρ. We introduce this process by using another approach as follows. For fixed n let S_{n} ∼ BIN(n, p) and for fixed ρ let W (ρ) denote a geometric distribution. The generating function of S_{n} is given by (1 − p + pz)^{n} and the generating function of W (ρ) is given by K(z) = (1 − ρ)z/(1 − ρz). We define a new random variable N by defining its generating functions: E(z^{N}) = (1 − p + pK(z))^{n}. The r.v. N is said to have an inflated-binomial distribution with parameters p, n and ρ; notation N ∼ IBIN(n, p, ρ). In the context of stochastic processes, Minkova [2001] studied N(t) where N(t) ∼ IBIN(n, t/α, ρ). It could be of interest to use this type of inflatedbinomial in the context of the formula of Black and Scholes.

Acknowledgement

The authors take pleasure in thanking an anonymous referee whose comments made it possible to simplify the proofs of our results.

References

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[11] E. Omey, J. Santos, and S. Van Gulck (2006), A Markov Binomial Distribution, J. Math. Sci., to appear

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EHSAL, Stormstraat 2 (Received 23 12 2005) 1000 Brussels Belgium

        edward.omey@ehsal.be
    stefan.vangulck@ehsal.be