SOME NOTES ABOUT THE MARTINGALE REPRESENTATION THEOREM AND THEIR APPLICATIONS

An important theorem in stochastic finance field is the martingale representation theorem. It is useful in the stage of making hedging strategies (such as cross hedging and replicating hedge) in the presence of different assets with different stochastic dynamics models. In the current paper, some new theoretical results about this theorem including derivation of serial correlation function of a martingale process and its conditional expectations approximation are proposed. Applications in optimal hedge ratio and financial derivative pricing are presented and sensitivity analyses are studied. Throughout theoretical results, simulation-based results are also proposed. Two real data sets are analyzed and concluding remarks are given. Finally, a conclusion section is given.


Introduction
The martingale representation theorem states that any martingale adapted with respect to a Brownian motion can be expressed as a stochastic integral with respect to the same Brownian motion. It has many applications in construction hedging strategies for various types of assets with different stochastic dynamics, see [1]. In the current note, the time series features of this important theorem are proposed. Before going future, an important lemma is proposed. Let B t be the standard Brownian motion on (0,∞) and F t is the sigma-field constructed by history of B s , s ≤ t, i.e. F t = σ{B s |s ≤ t}. Hence, if s ≤ t; then F s ⊆ F t . Indeed, F t is the augmented filtration generated by standard Brownian motion B t . Also, assume that X, Y are the future values of two stochastic processes at some known future time T . According to the martingale representation theorem, it is necessary to assume that both of X, Y are squared integrable random variables with respect to F ∞ to use this theorem for X and Y (see, [1]). These assumptions are kept fixed for all further discussions of the paper. (a) The correlation ρ xy between X, Y , is given as follows here u s and v s are two predictable processes used in martingale representation theorem applied to X, Y , respectively.
(b) Suppose that E(X|Y = y) = ay + b, E(Y |X = x) = cx + d. Then, where µ A and σ 2 A are the mean and variance of A = X, Y , respectively.
The martingale representation theorem implies that (see [2]) there exist two predictable processes u t , v t such that For a review in stochastic calculus, see [7]. It is easy to see that By multiplying above two equations and taking expectation, it is seen that Notice that [7] Hence, it is seen that covariance between X and Y, i.e., σ xy = cov (X, Y ) is given by Also, using the Ito isometric lemma (see [7]), it is seen that Therefore, Similarly, Thus, the proof is complete. (b) The proof is straightforward by using the iterated expectation law (see [2]). Therefore it is omitted. Example 1. Here, to give an example for assumption of part (b), a special case is considered. For a specific t, h > 0, let X be a martingale with respect to F t , then according to the martingale representation theorem, we have Here, E(X) is constant and independent of t. Let Y = X t+h . For special case, suppose that u s is a deterministic real-valued function. Then, Clearly, X t is an independent increment process and Γ has a normal distribution with zero mean and variance t+h t u 2 s ds. Notice that X Y is a linear combination of X Γ as follows and since X Γ has a joint normal distribution with mean vector E (X) 0 and covariance matrix has a joint distribution with mean vector E (X) E (X) and covariance matrix Here, The correlation between X, Y is Thus (see [6]) Also, notice that E (X t+h | X t ) = X t . Hence, the parameters of a, b, c, and d of the theorem are The rest of the paper is organized as follows. In the next section the application of above Lemma 1 in deriving optimal hedge ratio is discussed. Section 3 uses the (b) of Lemma 1 to approximate the conditional mean and it is applied to financial derivative pricing. Simulation results is given throughout theoretical sections. Real data sets analysis are given in Section 4. Finally, a Conclusion section is given.

Optimal hedge ratio
Here, the application of above discussion in portfolio management is discussed. The cross hedging procedure is the construction of an almost riskless portfolio by using one unit of the first asset X in long position and h units of Y in short position (at the maturity) (see [4]).
Let Z = X − hY be the value of portfolio at maturity T . The variance of Z is given by By minimizing σ 2 z with respect to h, it is seen that the optimum hedge ratio h opt is given by Hence, the optimum value of portfolio at maturity is The variance of Z at maturity is σ 2 x (1 − ρ 2 xy ). Next, suppose that the risk free interest rate is zero, then the value of X, Y at maturity T is the following (see [2]) Hence, The following proposition summarizes the above discussion. (a) Under the martingale representation, the optimum hedge ratio for cross hedging X by Y , is given by The replicating ratio for rebalancing portfolio the dynamic hedging portfolio is P r o o f. See the above discussions.
Next, consider the martingale representation theorem as follows Let G(t) = E(u 2 t ) and g(t) = log (G (t)) and g ′ be its first derivative. According to Lemma 1, (a) and Ito isometric lemma, the correlation coefficient ρ t (h) between X t+h , X t is given by The second term g ′′ (t) could be added to mentioned approximation, which is not necessary in practice.

Conditional mean approximation
Here, using the second part of Lemma 1, the conditional mean of E(X t |X t+h ) is approximated and then its financial application is seen.

Approximation
Notice that one can see that X t is a martingale with respect to filtrationḞ t , the σ-field generated by X s , s ≤ t. Next, assume that the conditional expectation of E(X t |X t+h ) is well-approximated by linear combination aX t+h + b. Then, using the Lemma 1, (b), it is seen that ). The following proposition summarizes the above discussion. Proposition 2. Assuming E(X t |X t+h ) is well-approximated by a linear function of X t+h , then Here, G (t) = E(u 2 t ) and g (t) = log(G (t)) and g ′ be its first derivative.
P r o o f. See the above discussions.

Remark 1.
Here some sensitivity analysis are discussed. Indeed, we have the following properties.
(a) As h → 0, then ρ t (h) → 1 which is clear (since the correlation of each variable with its-self is one). As h → ∞, then ρ t (h) → 0 which is clear since X t+h and X t are enough far from each others. Also, ∂ρ ∂h = −0.5g ′ (t) (1 + hg ′ (t)) −3/2 which converges to the −0.5g ′ (t), as h → 0. When h → ∞, then ∂ρ/∂h goes to zero which is clear since the variation of ρ t (h) is too small at infinity.
(b) It is easy to see that ∂ρ ∂t For example, when t = 0.1, 0.5, the following Fig. 1 shows the behavior of , h ∈ (0, 1), respectively. As more t → 0, then the curvature of ρ t (h) is more close to the horizontal axis. Notice that It is clear because as t → ∞ or t → 0, then ρ t (h) → 1 and its variation is too small. This is an interesting phenomena that as t gets large, then correlation B 2 t with its future values is large for each h. For special case, when h = t, then ρ t (h) = √ 2/2. Also, let h = q(t), for some real valued function q, and suppose that q(t)/t converges to α (β) as t → 0 (t → ∞), then ρ t (h) tends to 1/ √ 1 + α (1/ √ 1 + β). Fig. 1 shows the behavior of ρ 0.1 (h) and ρ 0.5 (h) which verifies the above discussion. For another example, as extension of Brownian motion, consider the Ornstein-Uhlenbeck process U t defined by dU = −αU dt + σdB.
The Ito lemma implies that X = X t = e αt U t satisfies the stochastic differential equation dX = σe αt dB which is martingale with respect to F t . Using the Example 1, it is seen that and It is seen that  Equivalently, From Dambis, Dubins-Schwarz (DDS) theorem (see [5, p. 204]), it is seen that whereB is another Brownian motion. Again, using the results of the previous example, the same results are obtained. Using the results of Remark 1, part (b), it is seen that g (t) = log σ 2 2α + log(e 2αt − 1). Then, Hence, ∂ρ ∂t → 0 as t → ∞.

Pricing
In this section, the application of above approximation in pricing of financial derivative is studied. Consider the price of financial derivative f at time t which expires at maturity T (t ≤ T ) written on a given underlying financial asset. Then, where r, Q, T, and F t are risk free rate, risk neutral probability measure, maturity of financial derivative and the σ-field of price time series s u , u ≤ t, (see [7]). Here, under the risk neutral probability measure, the dynamic of price of underlying asset is given by ds = rsdt + σdB at which σ is volatility of price. According to the Black-Scholes formula, the price of financial derivative satisfies the partial differential equation Let X t = e −rt f t . Then, using the Ito lemma, it is seen that dX = e −rt ∂f ∂s σsdB. Then, Thus, where ∆ is the Greek letter delta representing the sensitivity parameter of financial derivative with respect to variation of s. Notice that and where l (t) = log (E Q ∆ 2 s 2 ) and g ′ (t) = −2r + l ′ (t) and ρ t (h) = 1 1 + h(−2r + l ′ (t)) .
In practice, the quantity E Q ∆ 2 s 2 is approximated using a Monte Carlo simulation. The following proposition summarizes the above discussion.
Proposition 3. For the financial derivative with price f t then the correlation coefficient ρ t (h) between f t+h , f t is given by where l (t) = log (E Q ∆ 2 s 2 ) and g ′ (t) = −2r + l ′ (t).
P r o o f. The result is a direct consequence of previous discussions.

Remark 2.
Hereafter, the sensitivity analysis of ρ t (h) to its parameters σ, h is verified. For trivial derivative we have f = s, then ∆ = 1, and under the risk neutral measure Q, we have The solution is s t = s 0 e (r−σ 2 /2)t+σBt .

Real data sets
In this section, throughout real data sets the computational aspects of above theoretical results are studied.
Example 3. In this example, the application of the formula for backward forecasting of daily stock price of Apple co. for period of 3 December 2019 to 2 December 2020 (including 254 observations) is studied. Backward forecasting is useful for checking the correctness of guess of traders about future price of a specified share (see [3]). According to the Proposition 2, the backward forecasting in a martingale process is given as follows: As follows, error analyses is given to verify the accuracy of the above formula. Using the first 80 percent of data set (i.e., 202 observations, dated from 3 December 2019 to 21 September 2020), the following Ito process if fitted to the Apple co. stock price, which has solution s = 64.31e 0.00254t+0.0305B . Here, is martingale, and s = 64.31X t e 0.003t .
Next, assuming observations 21 September 2020 to 2 December 2020 are known this is the assumption of the trader about the future, the available data (data for 3 December 2019 to 21 September 2020) are forecasted, backwardly. Here, we used the remaining 20 percent of data, as trader conjecture about future. However, in practice, he may used own dada obtained by his techniques for fundamental analysis. The following Fig. 2 gives the error obtained by different actual and backward forecast for period of 3 December 2019 to 21 September 2020. It is seen that trader guess about future is true.
Example 4. In this example, the daily stock prices of Amazon co. for period of 2 October 2017 to 30 September 2019 (including 502 observations) are studied. It is seen that σ = 0.0199, r = 0.05 per year and s 0 = 959.19. Consider a call option with strike price k = 970, with maturity T = 1 (12 months) and European type. The delta parameter is Here, Φ is the normal standard distribution function. The following Fig. 3 shows the ρ t (0.2), ρ t (2) for various values of t. To simulate l(t), a Monte Carlo simulation with 1000 repetitions is performed. Also, the variance reduction method is applied. It is seen that as h becomes large then, naturally, ρ t (h) becomes small. As follows, the Black-Scholes (BS) price of a call option is compared with the approximate price. Also, ∆ is an important sensitivity Greek letter to obtain a riskless portfolio. Then, actual ∆ is compared with its approximation. Based on these comparisons, the following table is derived. Here, min, q i , i = 1, 2, 3, and max are the minimum, the first, second, third quartiles and the maximum of errors (differences between BS price and ∆, with their approximations), respectively. It is seen that the approximation works well.

Conclusion
In this paper, first, the correlation between two stochastic processes, satisfying the martingale representation theorem format, are derived. This correlation is used to obtain the optimal hedge ratio in a portfolio where two assets have the above mentioned stochastic process behaviors. Then, the results are developed to the serial correlation between a stochastic process and its lags. Then, this serial correlation is approximated. Sensitivity analyses of serial correlation to the time and lags and the parameters of underlying stochastic processes are studied and some interesting results about the relationship of process to its lags in long term (when t tends to ∞) are proposed. Using the serial correlation, the backward forecast of price of financial assets such as share, equity, stocks or financial derivatives are presented. Forecasts are done using the backward conditional which is well approximated. Throughout, simulated examples and real data sets applicability of proposed methods are seen.