Volterra equation for pricing and hedging in a regime switching market

Abstract It is known that the risk minimizing price of European options in Markov-modulated market satisfies a system of coupled PDE, known as generalized B–S–M PDE. In this paper, another system of equations, which can be categorized as a Volterra integral equations of second kind, are considered. It is shown that this system of integral equations has smooth solution and the solution solves the generalized B–S–M PDE. Apart from showing existence and uniqueness of the PDE, this IE representation helps to develop a new computational method. It enables to compute the European option price and corresponding optimal hedging strategy by using quadrature method.


Introduction
In recent years, a large amount of research is being done in the field of derivative pricing in Markovmodulated market. In such a market, floating rate of interest of a money market account, growth rates, and volatility coefficients of stock prices are taken as functions of an observable finite state continuous time Markov chain. The stock price processes are modeled as Markov-modulated geometric Brownian motions. Due to the presence of additional randomness, such regime switching ABOUT THE AUTHORS Many published articles on the topic of "pricing in a regime switching market" lack mathematical rigor, leading to an ambiguity. This work is accomplished with an aim to re-examine the existing literature on this topic and present a mathematically rigorous treatment of the problem. The model, discussed, has further possible extensions. The results reported in this paper also have the potential to be extended to those generalizations.
The first author of this paper works on various other topics in Applied Probability. Those include equilibrium of non-cooperative semi-Markov games under ergodic cost, portfolio optimizations and optimal control under risk sensitive cost, fluid limit in queuing networks, PDE techniques in stochastic control and differential games, stability analysis of SDE, etc.

PUBLIC INTEREST STATEMENT
In a financial market, several financial instruments are available to the investors for purchase. Evaluation of the fare price of those is, therefore, extremely important to the investors. On the other hand, since the market is subject to various uncertainties, the computation of the price and the seller's replication strategy of the instrument, solely using the received price, are often tricky.
In this paper, we consider a theoretical market model which generalizes the famous Black-Scholes-Merton model to incorporate random variability of the parameters namely interest rate, mean growth rate, and the volatility coefficient. The state-of-the-art approach suggests solving a partial differential equation to find the price and the hedging strategy. In this paper, we have shown that those can also be obtained by solving an integral equation. This finding enables to propose a new more efficient and robust numerical procedure to compute European option price and the hedging. model leads to an incomplete market. Therefore, the option pricing is rather involved. Indeed, there are contingent claims which are not attainable by self-financing strategies. Furthermore, existence of multiple equivalent martingale measures leads to multiple no-arbitrage prices of the same contingent claim. To address this difficulty, option pricing in an incomplete market is studied by several approaches Basak, Ghosh, and Goswami (2011), Buffington and Elliott (2002), Deshpande andGhosh (2008), DiMasi, Kabanov, andRunggaldier (1994), Guo (2002), Guo andZhang (2004), Heath, Platen, andSchweizer (2001), Jobert and Rogers (2006), Mamon and Rodrigo (2005), Schweizer (2001), Tsoi, Yang, and Yeung (2000), etc.
To price and hedge a claim of European type in the above incomplete market, we would consider the locally risk minimizing pricing approach by Föllmer and Schweizer (1991). It is shown in Deshpande and Ghosh (2008) that the locally risk minimizing price of an option of European type can be derived from the unique solution of a Cauchy problem, where the PDE is a generalization of Black-Scholes-Merton PDE (see Deshpande & Ghosh, 2008 for details). In a recent paper by (Basak et al., 2011), an implicit stable Crank-Nicholson (C-N) scheme is developed to solve that Cauchy problem numerically. The present paper also deals with numerical computation of locally risk minimizing price but it adopts a completely different approach. In this paper, we study a system of equations which can be categorized as Volterra integral equations of second kind. It is shown that this system of integral equations has unique smooth solution and the solution solves the generalized B-S-M PDE given in Basak et al. (2011). Or in other words, the risk minimizing option price is characterized as unique solution of a system of Volterra equations. Finally, we develop a stable scheme to solve this system numerically. This finding resolves various computational challenges. First of all, it enables development of an alternative numerical approach to find the option price by using quadrature method. In principle, C-N scheme (to solve B-S-M type PDE) involves inversion and N times multiplication of a matrix of order M, where M is proportional to the space discretization (Basak et al. 2011). Therefore, T PDE (N, M), the corresponding computational complexity to solve the PDE is O(NM 3 ). Here, N is the number of equi-spaced points on time horizon [0, T]. On the other hand, we have the following result. Let T IE (N, M) denote the computational complexity to solve the IE with above grid, using step-by-step quadrature method. Then, we have Secondly, we are also able to find a Volterra equation for optimal hedging strategy. Needless to mention, this equation can also be solved by a similar numerical method. Therefore, calculating hedging strategy becomes as easy as calculating option price. Needless to mention, solving the PDE for hedging strategy is generally much harder than solving the PDE for option price. We also study one typical example of a regime switching market and carry out computation for solving the PDE as well as the IE. The computational elapsed times are recorded for both the cases with varying M for the purpose of comparison. The elapsed time data collated in a single plot clearly shows how the proposed scheme outperforms the C-N scheme for large values of M.
This paper is organized in the following way. The Markov-modulated market model is presented in Section 2 along with the main results of the paper. We present the proofs of Theorems 2.1 and 2.2 in Section 3. In Section 4, a step-by-step quadrature method is developed to solve the IE for option price. This section also contains the proof of stability of the scheme and a detailed calculation of computational complexity. Section 5 includes performance comparison of the scheme with that in Basak et al. (2011) by considering a typical numerical example. Finally, some remarks about immediate generalization of the present work are given in Section 6.

Model and main result
Let (Ω,  , P) be the underlying complete probability space. Let = {1, 2, … , k} be the state space of an irreducible Markov chain {X t , t ≥ 0} with transition rule denotes the generating Q-matrix of the chain and are the transition probabilities from state i to state j. We consider a market where the financial parameters, namely interest rate, drift coefficient, volatility coefficient are functions of the observed Markov chain X t . Let {B t , t ≥ 0} be the price of money market account at time t where, spot interest rate is r(X t ) and B 0 = 1. We have We consider a market consisting only one stock as tradable risky asset. The stock price process S t solves where {W t , t ≥ 0} is a standard Wiener process independent of {X t , t ≥ 0}. Let  t be a filtration of  satisfying usual hypothesis and right continuous version of the filtration generated by X t and S t .
Clearly, the solution of above SDE is an  t semimartingale with almost sure continuous paths. To price a claim H of European type in the above incomplete market, we would consider the locally risk minimizing pricing approach by Föllmer and Schweizer (see Föllmer & Schweizer, 1991;Heath et al., 2001). A hedging strategy is defined as a predictable process The components ξ t and ε t denote the amounts invested in S t and B t , respectively, at time t. An optimal strategy is the one for which the quadratic residual risk (see Föllmer & Schweizer, 1991 for details) is minimized subject to a certain constraint. It is shown in Föllmer and Schweizer (1991) that the existence of an optimal strategy for hedging an  t measurable claim H is equivalent to the existence of Föllmer-Schweizer decomposition of discounted claim H * : (1). Further, H * appeared in the decomposition constitutes the optimal strategy. Indeed, the optimal strategy = ( t , t ) is given by and B t V * t represents the locally risk minimizing price at t of the claim H. Hence, the Föllmer-Schweizer decomposition is the key thing to verify. Now onward we consider a particular claim i.e. a European call option on {S t } with strike price K and maturity time T. In this case, the  t measurable contingent claim H is given by Before stating the main results we recall that in the Black-Scholes-Merton model (Black & Scholes, 1973) the  t measurable claim H is attainable and the price η(t, S t ) at time t ∈ [0, T] is given by Moreover, the solution φ(t, s, i) of (4) and (5) is the locally risk minimizing price of H (as in (2)) at time t with S t = s, X t = i. (5) with above grid using step by step quadrature method. Then we have  (4) and (6) give (t, s, i) = i (t, s) and (t, s, i) = i (t, s) s respectively. Hence the B-S-M price and hedging can be recovered from Equations 4-5 and 6-7, respectively.

Equations of pricing and hedging
Consider the following system of partial differential equations for t < T, s > 0 and i = 1, 2, … , k with the boundary condition where φ is of polynomial growth. Note that if Λ is a null matrix i.e. the case when the Markov chain X t does not transit almost surely, the Equation 9 coincides with that of standard B-S-M model. In view of this, the above system can be considered as a generalization of Black-Scholes equation for a Markov-modulated market where the extra coupling term represents the correction term arising due to the regime switching. Nevertheless, the fact, the solution of above problem gives locally risk minimizing price, needs a proof. To this end, we quote the following theorem from Deshpande and Ghosh (2008).  (ii) An optimal strategy = ( t , t ), is given by where Proof of Theorem 2.1. We prove the first part of Theorem 2.1 primarily by constructing a smooth solution of (4)-(5). In order to do that let (Ω, ,P) be a complete probability space which holds a standard Brownian motion W and a Markov process X independent of W such that the rate matrix of X is the same as that of X. Let S t be given by and  t be the underlying filtration satisfying usual hypothesis. Thus, P is risk-neutral measure for the risky asset S given by (11). Let Y t represent holding time i.e. the amount of time the process X t is at the current state after the last jump. Let the consecutive jump times be 0 = T 0 < T 1 < T 2 < ⋯ and n(t) : = max{n ≥ 0|T n ≤ t}. Hence, T n(t) = t − Y t . Clearly, f (y|i) : = i e − i y is the conditional probability density function of holding time and F(y|i) = 1 − e − i y is the corresponding CDF where i = − ii . Here, we recall the following obvious relation Because of Markovity of (S t ,X t ), we know that there is a measurable function : [0, T]×[0, ∞) × → ℝ such that (t, 0, i) = 0 and holds for all t ∈ [0, T] where Ẽ is expectation under P . Due to irreducibility of X t , for any fixed X 0 ,S 0 , the map φ (as in (12)) is defined uniquely almost everywhere on [0, T]×[0, ∞) × . Now by conditioning at transition times and using the conditional lognormal distribution of stock price process, we have where i (t, s) is the standard Black-Scholes price of European call option with fixed interest rate r(i) and volatility σ(i). Again using irreducibility of Markov chain, we can replace (S t ,X t ) by generic variable (s, x) in the above relation and thus conclude that φ is a solution of (4)-(5). The first term on the righthand side is clearly in C 1,2 ((0, T)×ℝ + × ). The continuous differentiability in t of the second term follows from the fact that the term (t + v, x, j) is multiplied by C 1 ((0, ∞)) function in v and then integrated over v ∈ (0, T − t). Now twice continuous differentiability in s of the second term follows from direct calculation. Thus (t, s, i) is in C 1,2 ((0, T)×ℝ + × ). Finally, the continuity of φ on [0, T]×ℝ + follows trivially. We note that the right side of (4) can be considered as the image of φ under a contraction on a suitable Banach space. Hence, uniqueness follows from Banach fixed point theorem.
The proof follows by differentiating both sides of (4) with respect to s.

Numerical method
To solve (4)-(5), we use the step-by-step quadrature method. Let Δt and Δs be the time step and stock price step sizes, respectively. For m, m′, l positive integers and i ∈ , set Now we use the following quadrature rule over successive intervals [0, nΔt] for a function on this interval, we use where ω n (l) are weights to be chosen appropriately. Applying the above procedure in (4), we obtain the following set of equations with We choose a repeated trapezium rule by which the weights ω n are given by Convergence of the above scheme is obvious, the issue of stability is addressed below.
Theorem 4.5. Let a : = max i e −( i +r(i)) . For the scheme (13) is strictly stable with respect to an isolated perturbation. Moreover, the scheme displays uniformly bounded error propagation.
Δt ≤ e −aT a Proof. We first note that (m, m � , l, i) corresponds to a lognormal density and the holding time densities f(·|·) are bounded. Let δ n be an additive error in n m (i) ∀m and i. Now it is easy to show that the effect of the isolated perturbation δ n in N m (i)(N : = ⌊ T Δt ⌋) is additive and given by If Δt is sufficiently small and satisfies (15), we get n < δ n , i.e. the scheme is strictly stable with respect to an isolated perturbation. Let δ n be bounded by a fixed constant δ. Now the total effect of the perturbation in the value N m (i) is given by Hence the result follows.
□ Now we are ready to prove Theorem 2.3.
Proof of Theorem 2.3. To organize better, before computation of (13), we evaluate and store the values of known functions on the entire grid, so that those values can directly be used at later stages. Let C be the number of operations, required to accomplish that. We first estimate C. Let the constant c η be the number of elementary operations required to evaluate η at a single entry. Similarly, let c  and c exp be the constants corresponding to the functions  and exponential respectively. Hence in view of (13), we obtain directly Let C (i) m (n) denote the number of additional computational operations which are required to obtain n m (i) from (13) for fixed n(≥1), m and i assuming that values of n−1 m (i) are known for all m and i. We allow C (i) m (0) to represent the computational complexity of initial data at each entry. Hence, C(n, M) : = ∑ i∈ , m≤M C (i) m (n) represents the total complexity at nth stage for each n ≤ N.
It is evident from (14)  In this section, we have developed a numerical scheme to compute option price using a quadrature method. It is natural to ask if this has any advantage over the one based on solving the PDE (9)-(10) using Crank-Nicholson implicit scheme. In order to compare the computational complexities, we present a brief description of the corresponding Crank-Nicholson scheme below. where M is a large positive integer. We carry out computation for solving (9)-(10) as well as (4)

Conclusion
This work comprises theoretical derivations as well as numerical experiments. It also presents a selfcontained proof of existence and uniqueness of generalized B-S-M PDE while proving the Theorem 2.1. It seems that the Volterra equation of optimal hedging has been studied for the first time in this paper. This paper makes it clear that such equation for hedging can also be obtained for more general semi-Markov-modulated market in the exactly similar manner. Needless to mention that this observation opens up an opportunity of practical application. Δt = 1∕8, Δs = 10∕M