In complete, frictionless capital markets with no transaction costs and where the underlying securities follow geometric Brownian motions, the Black-Scholes framework [1] provides an elegant and tractable solution for pricing and hedging derivative securities, typically vanilla calls and puts. Unfortunately, actual financial markets are far more complex and empirical testing of the Black- Scholes model has highlighted its many shortcomings. Indeed, it is well docu- mented [2] [3] [4] that the observed properties of financial time series are not consistent with its underlying assumptions. Time-varying volatility, the presence of higher-order moments and serial correlation are now well established stylized facts of asset returns. Furthermore, [5] [6] [7] and [8] demonstrate that un- realistic assumptions about continuous-time hedging can lead to large hedging errors.

Over the past decade, several studies have proposed discrete time hedging models based on different objective functions (see for example [9] , [10] and [11] ). The idea of dynamic hedging, as detailed in [12] and [13] , is to find a self- financing optimal investment strategy that replicates a terminal payoff. In this paper, we build on the previous work of [14] [15] [16] [17] and [18] to derive a discrete time hedging strategy minimizing the expected risk of future contingent liabilities when asset returns follow a regime-switching random walk. Our hedging methodology is therefore robust to serially-correlated and non-Gaussian returns. Previous attempts to incorporate conditional returns in option pricing include GARCH models (see [19] for a complete review), stochastic volatility models [20] [21] [22] and jump models ( [23] and [24] to cite a few). These approaches have generally been successful at reproducing market prices, however none of them offer an effective, let alone optimal, hedging strategy.

Regime-switching models, popularized by [25] and [26] , have many charac- teristics that lend themselves nicely to financial time series modeling. They allow for time-varying conditional moments, a feature that is consistent with the observed non-Gaussian properties of financial returns while being easy to inter- pret. Regime-switching models have previously been used by [27] to capture interest rate dynamics and by [28] and [29] to model volatility. However, very few papers have attempted to apply regime-switching models to returns for option pricing and hedging. The aim of this paper is to demonstrate how to implement optimal hedging strategies and obtain derivatives prices under this class of models. We review the EM algorithm [30] , which provides an efficient estimation procedure, and propose a novel goodness-of-fit test for selecting the optimal number of regimes based on the work of [31] .

To empirically assess the relevance of our approach, we compare the performance of various hedging methodologies in the context of harvesting the negative market volatility risk premium. The implied volatility of equity index options is known to include a significant negative time-varying premium [32] . In economic terms, there is an insurance cost to holding assets that co-vary positively with volatility (such as options), because the latter tends to increase in unfavorable market states [33] [34] . In their seminal paper, [35] initiate short S & P 500 option positions and hedge the underlying risk until expiration. They empirically show gains from such strategy are proportional to the volatility risk premium. However, they derive daily optimal hedging ratios from the classical Black-Scholes approach adjusted for a GARCH conditional volatility. Even though their conclusion appears robust to such mis-specification, the impact on the strategy is unclear. In fact, investors should optimally account for time- varying volatility and discrete-time re-balancing when deriving hedging ratios. The proposed approach achieves this optimality under a quadratic (thus sym- metric) risk criterion for regime-switching models.

We first present empirical pricing results confirming the systematic down- ward bias in model prices relative to market prices. Secondly, we apply the proposed volatility-timing strategy. Intuitively, we expect the regime-switching methodology to under(over)-hedge relative to the Black-Scholes setting in low (high) volatility states. This is confirmed empirically as the proposed protocol decreases the annualized volatility of the strategy from 3.47% to 2.58% and increases the annualized expected return from 1.37% to 1.97% over the classical Black-Scholes delta-hedging approach.

The rest of the paper is structured as follows. Section 2 introduces the model by describing the filtering procedure to predict regimes, the estimation procedure and the goodness-of-fit test, together with an illustration for the S & P 500 index daily returns. The implementation of the estimation is presented in Appendix A and the implementation of the goodness-of-fit is presented in Appendix B. Then, in Section 3, we state the optimal dynamic discrete time hedging model for regime-switching processes and finally, in Section 4, we implement the proposed algorithm for European vanilla options written on the S & P 500 in the context of harvesting the volatility risk premium.

Regime-switching models are quite intuitive. At period t , given that the previous regime τ t − 1 has value i , the regime τ t = j is chosen with probability Q i j , and given τ t = j , the periodic return R t has cumulative distribution F j . Henceforth, F j is assumed Gaussian with mean μ j and standard deviation σ j . Following [25] , the non-observable regimes ( τ t ) t ≥ 1 , with values in { 1, ⋯ , l } , are modeled by a Markov chain with transition matrix Q , and stationary distribution ν . A (d-dimensional) discrete time process ( R t ) t ≥ 1 forms a regime-switching random walk if, given the regimes τ 1 = i 1 , ⋯ , τ n = i n , the random vectors R 1 , ⋯ , R n are independent, with respective densities f i 1 , ⋯ , f i n (associated to cumulative distributions F i 1 , ⋯ , F i n ). As a result, the stationary distribution of R t is a mixture, with density f ( x ) = ∑ i = 1 l ν i f i ( x ) .

The regime-switching geometric random walk model ( S t ) t ≥ 1 is then a process such that the associated (d-dimensional) log-returns process R t = log ( S t / S t − 1 ) forms a regime-switching random walk. In general, ( S t ) t ≥ 1 is not a Markov process, unless the regimes are serially independent. However, the process ( S t , τ t ) t ≥ 0 is Markovian. These models are particular cases of Hidden Markov Models (HMM).

The law of financial time series can be modeled adequately by a regime- switching model with Gaussian densities f 1 , ⋯ , f l , provided the number of regimes is large enough. Indeed, the serial dependence in regimes propagates to returns and captures the observed autocorrelation in financial time series. Also, the conditional distribution is time-varying, leading to conditional volatility, as well as conditional asymmetry and kurtosis. Finally, the model nests the Black- Scholes framework when the number of regimes is forced to one. Indeed, in such a case, the log-returns are independent and identically distributed with Gaussian density.

We now state results of practical interest pertaining to regime-switching models, including a regime prediction algorithm, an estimation algorithm and a goodness-of-fit test to determine the optimal number of regimes.

Denote the price process by S , that is, S t is the value of d underlying assets

at period t and let F = { F t , t = 0 , ⋯ , n } a filtration under which S is adapted. Further assume S is square integrable. Set Δ t = β t S t − β t − 1 S t − 1 , where the discounting factors β t are predictable (that is β t is F t − 1 -measurable for t = 1 , ⋯ , n ). We are interested in the optimal initial investment amount V 0 and the optimal predictable investment strategy ϕ → = ( ϕ t ) t = 1 n that minimizes the expected quadratic hedging error for a given payoff, C, at time n (for example a call option). Formally, the problem is stated as

m i n { V 0 , ϕ → } E [ { G ( V 0 , ϕ → ) } 2 ] , (8)

where G ( V 0 , ϕ → ) = β n ( C − V n ) , and V t is the current value of the replicating portfolio at time t . In other words, it is the current value of the optimal predictable investment strategy, ϕ → .

To solve (8), set P n + 1 = 1 , and define, for t = n , ⋯ , 1 ,

A t = E ( Δ t Δ t Τ P t + 1 | F t − 1 ) ,

b t = A t − 1 E ( Δ t P t + 1 | F t − 1 ) ,

α t = A t − 1 E ( β n C Δ t P t + 1 | F t − 1 ) ,

P t = ∏ j = t n ( 1 − b j Τ Δ j ) .

We can now state Theorem 1 of [18] , which is an extension of a result from [16] .

Theorem 1. Suppose that E ( P t | F t − 1 ) ≠ 0 P-a.s., for t = 1 , ⋯ , n . This condition is always respected for regime-switching models. Then, the solution ( V 0 , ϕ → ) of the minimization problem (8) is V 0 = E ( β n C P 1 ) / E ( P 1 ) , and

ϕ t = α t − β t − 1 V t − 1 b t ,   k = 1 , ⋯ , n . (9)

Remark 3.1. Note that V 0 is chosen such that the expected hedging error G is zero. [18] also showed that C t (defined by

β t C t = E ( β n C P t + 1 | F t ) E ( P t + 1 | F t ) ,   t = 0, ⋯ , n , (10)

is the optimal investment at period t so that the value of the portfolio at period n is as close as possible to C in terms of mean square hedging error G . As such, C t can be interpreted as the option price at period t . By increasing the number of hedging periods, C t tends to a price under a risk neutral measure [43] . For example, when there is only one regime and the density is Gaussian, C t tends to the usual Black-Scholes price, as the number of hedging periods tends to infinity.

The proposed optimal hedging implementation appears in Appendix C.

To exhibit the proposed hedging protocol, we systematically sell vanilla options on the S & P 500 and hedge them until expiration. We then assess the impact of model specification on the delta-hedging strategy by examining the statistical properties of the harvested realized variance risk premium [35] [32] . All hedging portfolios are re-balanced on a daily basis, as is often assumed in the volatility timing literature (see for example [44] ).

The market price of an option is defined as the last (i.e. at 4:15 PM EST) mid point between the bid and the ask. The price of the underlying is its listed close value. For simplicity, we neglect issues related to time-varying discount rates by assuming constant continuously compounded daily rates. Risk free rates, r, are linearly interpolated for a given maturity, n, from the zero-coupon U.S. yield curve.

In this paper, we propose a discretized version of the continuous time regime- switching model used by [25] , and demonstrate how to implement an optimal hedging strategy when the underlying asset returns are modeled as regime- switching random walks.

We first present estimation and filtering procedures for regime-switching models. Building on the work of [39] [40] , and [31] , we then propose a novel goodness-of-fit test for univariate and multivariate Markovian regime-switching models that builds from Rosenblatt’s transform. This test is particularly useful to determine the optimal number of regimes.

To illustrate the proposed methodology, we model the daily return series of the S & P 500. The resulting dynamics is consistent with commonly assumed characteristics of financial markets, such as the contemporaneous asymmetry between volatility and expected returns.

Furthermore, we state the regime-switching specialization of the optimal hedging algorithm [18] which generates optimal discrete-time (in our case, daily) exposures minimizing a symmetric quadratic criterion for the hedging error risk. It further performs pricing from which we confirmed a downward bias with respect to market prices. Such finding is consistent with the presence of a volatility risk premium and motivated the implementation of a systematic hedged short option strategy [35] . We compare our hedging results to the classical delta-hedging protocol and to optimal hedging protocols when the underlying asset returns are either modelled by a Gaussian or a Gaussian mixture distribution. Gaussian regime-switching models generate lower hedging errors than static models. Interestingly, they further yield greater profits and have favourable PNL distribution characteristics, such as less kurtosis and less negative skewness. Overall, the Sharpe ratio of the regime-switching hedging methodology doubles over the classical delta-hedging framework and the Omega ratio increases by roughly 0.2.

The hedging algorithm could easily be extended to American options, and further adapted to conditional volatility models, such as GARCH models. One limitation of our work is the weak dependence induced by the regimes. It would be preferable to have serial dependence involving also the past returns. This will be done in a future work.

This research was supported by the Montreal Institute of Structured Finance and Derivatives, the Natural Sciences and Engineering Research Council of Canada, the Institut de Finance Mathématique de Montréal, and Desjardins Global Asset Management.

Rémillard, B., Hocquard, A., Lamarre, H. and Papageorgiou, N. (2017) Option Pricing and Hedging for Discrete Time Regime-Switching Models. Modern Economy, 8, 1005-1032. https://doi.org/10.4236/me.2017.88070

The EM algorithm for estimating parameters consists of two steps, expectation and maximization:

(E-Step) Compute the conditional probabilities.

λ t ( i ) = P ( τ t = i | R 1 , ⋯ , R n )     and     Λ t ( i , j ) = P ( τ t = i , τ t + 1 = j | R 1 , ⋯ , R n ) ,

for all 1 ≤ t ≤ n and i , j ∈ { 1, ⋯ , l } .

(M-Step) Estimate the new parameters.

First, a rough estimate of the parameters must be provided. Then, the two- step procedure is repeated until a stopping criteria is met. The E-Step is described next for any densities, whilst the M-Step is stated only for Gaussian densities. For more details on the EM algorithm, see, for example, [36] .

First, define, for all i ∈ { 1, ⋯ , l } ,

η ¯ n ( i ) = 1 / l ,

η ¯ t ( i ) = ∑ β = 1 l     η ¯ t + 1 ( β ) Q i β f β ( R t + 1 ) ∑ α = 1 l ∑ β = 1 l     η ¯ t + 1 ( β ) Q α β f β ( R t + 1 ) ,   t = 1 , ⋯ , n − 1.

Then, for all i , j ∈ { 1, ⋯ , l } , one can check that

λ t ( i ) = η t ( i ) η ¯ t ( i ) ∑ α = 1 l     η t ( α ) η ¯ t ( α ) ,   t = 1 , ⋯ , n , (13)

Λ t ( i , j ) = Q i j η t ( i ) η ¯ t + 1 ( j ) f j ( R t + 1 ) ∑ α = 1 l ∑ β = 1 l     Q α β η t ( α ) η ¯ t + 1 ( β ) f β ( R t + 1 ) ,   t = 1 , ⋯ , n − 1 , (14)

Λ n ( i , j ) = λ n ( i ) Q i j .

We can now verify (13) and (14) are consistent. Indeed, for all 1 ≤ t ≤ n − 1 ,

∑ j = 1 l     Λ t ( i , j ) = η t ( i ) ( ∑ j = 1 l     Q i j η ¯ t + 1 ( j ) f j ( R t + 1 ) ) ∑ α = 1 l     η t ( α ) ( ∑ β = 1 l     Q α β η ¯ t + 1 ( β ) f β ( R t + 1 ) ) = λ t ( i ) ,

using the definition of η ¯ t . Also, ∑ j = 1 l     Λ n ( i , j ) = ∑ j = 1 l     λ n ( i ) Q i j = λ n ( i ) . Similarly, for all 1 ≤ t ≤ n − 1 ,

∑ i = 1 l     Λ t ( i , j ) = λ t + 1 ( j ) .

For the estimation procedure, we assume the densities f 1 , ⋯ , f l are Gaussian with means ( μ i ) i = 1 l , and covariance matrices ( A i ) i = 1 l .

The M-step consists of updating the parameters ( ν i ) i = 1 l , ( μ i ) i = 1 l , ( A i ) i = 1 l and Q according to

ν ′ i = ∑ t = 1 n     λ t ( i ) / n ,

μ ′ i = ∑ t = 1 n     x t w t ( i ) ,

A ′ i = ∑ t = 1 n ( x t − μ ′ i ) ( x t − μ ′ i ) Τ w t ( i ) ,

Q ′ i j = ∑ t = 1 n     Λ t ( i , j ) / ∑ t = 1 n     λ t ( i ) = 1 n ∑ t = 1 n     Λ t ( i , j ) / ν ′ i ,

for all i , j ∈ { 1, ⋯ , l } and where w t ( i ) = λ t ( i ) / ∑ l = 1 n     λ l ( i ) .

Note that ν ′ is not the stationary distribution for Q ′ since for any j ∈ { 1, ⋯ , l } ,

∑ i = 1 l     ν ′ i Q ′ i j = 1 n ∑ t = 1 n ∑ i = 1 l     Λ t ( i , j ) = 1 n ∑ t = 2 n + 1     λ t ( j ) = ν ′ j + λ n + 1 ( j ) − λ 1 ( j ) n ≠ ν ′ j .

However,

m a x 1 ≤ j ≤ l | ∑ i = 1 l     ν ′ i Q ′ i j − ν ′ j | ≤ 1 / n .

Hence, when n is large, ν ′ is close to the stationary distribution of Q ′ . In practice, we estimate the stationary distribution from Q ′ , rather than ν ′ , for consistency.

Interestingly, the first two sample moments are preserved in the Gaussian case. In other words, the two first fitted stationary moments are coherent with the sample mean and covariance matrix, defined as

x ¯ = 1 n ∑ t = 1 n     x t     and     S ¯ = 1 n ∑ t = 1 n ( x t − x ¯ ) ( x t − x ¯ ) Τ .

Indeed, from the M-step definitions, the stationary mean is

μ ′ ≡ ∑ i = 1 l     ν ′ i μ ′ i = ∑ i = 1 l ( 1 n ∑ l = 1 n     λ l ( i ) ) ( ∑ t = 1 n x t λ t ( i ) ∑ l = 1 n     λ l ( i ) ) = 1 n ∑ t = 1 n ∑ i = 1 l     x t λ t ( i ) = 1 n ∑ t = 1 n     x t = x ¯ ,

Similarly,

∑ i = 1 l     ν ′ i A ′ i = 1 n ∑ i = 1 l ∑ t = 1 n     λ t ( i ) ( x t − μ ′ i ) ( x t − μ ′ i ) Τ = 1 n ∑ i = 1 l ∑ t = 1 n     λ t ( i ) x t x t Τ + 1 n ∑ i = 1 l ∑ t = 1 n     λ t ( i ) μ ′ i ( μ ′ i ) Τ     − 1 n ∑ i = 1 l ∑ t = 1 n     λ t ( i ) x t ( μ ′ i ) Τ − 1 n ∑ i = 1 l ∑ t = 1 n     λ t ( i ) μ ′ i x t Τ = 1 n ∑ t = 1 n     x t x t Τ − ∑ i = 1 l     ν ′ i μ ′ i ( μ ′ i ) Τ .

Finally, from the previous result, the stationary covariance matrix is given by

A ′ ≡ ∑ i = 1 l     ν ′ i A ′ i + ∑ i = 1 l     ν ′ i μ ′ i ( μ i ) ′ Τ − μ ′ ( μ ′ ) Τ = 1 n ∑ t = 1 n     x t x t Τ − ∑ i = 1 l     ν ′ i μ ′ i ( μ ′ i ) Τ + ∑ i = 1 l     ν ′ i μ ′ i ( μ i ) ′ Τ − x ¯   x ¯ Τ = 1 n ∑ t = 1 n ( x t − x ¯ ) ( x t − x ¯ ) Τ = S ¯ .

In this Appendix, we state the goodness-of-fit test, which can be performed to assess the suitability of a regime-switching model as well as to select the optimal number of regimes, l * . The proposed test, based on the work of [39] , [31] and [37] , uses the Rosenblatt’s transform. For conciseness, we detail the implementation for two dimensional Gaussian regime-switching models, but the approach can be easily generalized.

Let i ∈ { 1, ⋯ , l } be fixed and R i be a random vector with density f i . For any q ∈ { 1, ⋯ , d } , denote by f i ,1 : q the density of ( R i ( 1 ) , ⋯ , R i ( q ) ) , and by f i , q the density of R i ( q ) given ( R i ( 1 ) , ⋯ , R i ( q − 1 ) ) . Further denote by F i , q the distribution function associated with density f i , q . By convention, f i ,1 denote the unconditional density of R i ( 1 ) . Then, the Rosenblatt’s transform

x ↦ T i ( x ) = ( F i ,1 ( x ( 1 ) ) , F i ,2 ( x ( 1 ) , x ( 2 ) ) , ⋯ , F i , d ( x ( 1 ) , ⋯ , x ( d ) ) ) Τ

is such that T i ( R i ) is uniformly distributed in [ 0,1 ] d .

For example, if f i is the density of a bivariate Gaussian distribution with mean μ i and covariance matrix

Σ i = ( v i ( 1 ) ρ i v i ( 1 ) v i ( 2 ) ρ i v i ( 1 ) v i ( 2 ) v i ( 2 ) ) ,

f i ,2 is the density of a Gaussian distribution with mean μ i ( 2 ) + β i ( y i ( 1 ) − μ i ( 1 ) ) and variance v i ( 2 ) ( 1 − ρ i 2 ) , with β i = ρ i v i ( 2 ) / v i ( 1 ) .

However, for regime-switching random walk models, past returns must also be included in the conditioning information set. For any x ( 1 ) , ⋯ , x ( d ) ∈ ℝ , the (d-dimensional) Rosenblatt’s transform Ψ t corresponding to the density (5) conditional on x 1 , ⋯ , x t − 1 ∈ ℝ d is given by

Ψ t ( 1 ) ( x t ( 1 ) ) = Ψ t ( 1 ) ( x 1 , ⋯ , x t − 1 , x t ( 1 ) ) = ∑ i = 1 l     W t − 1 ( i ) F i ,1 ( x t (1) )

and

Ψ t ( q ) ( x t ( 1 ) , ⋯ , x t ( q ) ) = Ψ t ( q ) ( x 1 , ⋯ , x t − 1 , x t ( 1 ) , ⋯ , x t ( q ) ) = ∑ i = 1 l     W t − 1 ( i ) f i , 1 : q − 1 ( x t ( 1 ) , ⋯ , x t ( q − 1 ) ) F i , q ( x t ( q ) ) ∑ i = 1 l     W t − 1 ( i ) f i , 1 : q − 1 ( x t ( 1 ) , ⋯ , x t ( q − 1 ) ) .

for q ∈ { 2, ⋯ , d } .

Suppose R 1 , ⋯ , R n is a size n sample of d-dimensional vectors drawn from a joint (continuous) distribution P . Also, let P be the parametric family of Gaussian regime-switching models with l regimes. Formally, the hypothesis to be tested is

H 0 : P ∈ P = { P θ ; θ ∈ Θ }   vs   H 1 : P ∉ P

Under the null, it follows that

U 1 = Ψ 1 ( R 1 , θ ) , U 2 = Ψ 2 ( R 1 , R 2 , θ ) , ⋯ , U n = Ψ n ( R 1 , ⋯ , R n , θ )

are independent and uniformly distributed over [ 0,1 ] d , where Ψ 1 ( ⋅ , θ ) , ⋯ , Ψ n ( ⋅ , θ ) are the Rosenblatt’s transforms conditional on the set of parameters θ ∈ Θ .

Since θ is unknown, it must be estimated by some θ n . Then, the pseudo- observations, U ^ 1 = Ψ 1 ( R 1 , θ n ) , ⋯ , U ^ n = Ψ n ( R 1 , ⋯ , R n , θ n ) are approximately uniformly distributed over [ 0,1 ] d and approximately independent. We next propose a test statistic based on these pseudo-observations.

The test statistic builds from the following empirical process:

D n ( u ) = 1 n ∑ t = 1 n ∏ q = 1 d I ( U ^ t ( q ) ≤ u ( q ) ) , u ≡ ( u ( 1 ) , ⋯ , u ( d ) ) ∈ [ 0 , 1 ] d .

To test H 0 against H 1 , we propose a Cramér-von Mises type statistic:

S n ≡ B n ( U ^ 1 , ⋯ , U ^ n ) = n ∫ [ 0 , 1 ] d { D n ( u ) − ∏ q = 1 d u ( q ) } 2 d u = 1 n ∑ t = 1 n ∑ k = 1 n ∏ q = 1 d { 1 − max ( U ^ t ( q ) , U ^ k ( q ) ) } − 1 2 d − 1 ∑ t = 1 n ∏ q = 1 d ( 1 − U ^ t ( q ) 2 ) + n 3 d .

Since U ^ i is almost uniformly distributed on [ 0,1 ] d under the null hypothesis, large values of S n should lead to rejection of the null hypothesis. Unfortunately, the limiting distribution of the test statistic will depend on the unknown parameter set, θ . Since it is impossible to construct tables, we use a different methodology, namely parametric bootstrap, to compute P-values. The validity of the parametric bootstrap approach has been shown for a wide range of assumptions in [31] . These results were recently extended to dynamic models [37] , including regime-switching random walks.

・ For a given number of regimes, estimate parameters with θ n computed from the EM algorithm applied to ( R 1 , ⋯ , R n ) .

・ Compute the test statistic,

S n = B n ( U ^ 1 , ⋯ , U ^ n ) ,

from the estimated pseudo observations, U ^ i = Ψ i ( R 1 , ⋯ , R i , θ n ) , for i ∈ { 1, ⋯ , n } .

・ For some large integer N (say 1000), repeat the following steps for every k ∈ { 1, ⋯ , N } :

- Generate a random sample { R 1 k , ⋯ , R n k } from distribution P θ n .

- Compute θ n k by applying the EM algorithm to the simulated sample, { R 1 k , ⋯ , R n k } .

- Let U ^ i k = Ψ i ( R 1 k , ⋯ , R i k , θ n k ) for i ∈ { 1, ⋯ , n } , and finally compute

S n k = B n ( U ^ 1 k , ⋯ , U ^ n k ) .

Then, an approximate P-value for the test based on the Cramér-von Mises statistic S n is given by

1 N ∑ k = 1 N     I ( S n k > S n ) .

For any d-dimensional vector x, let D ( x ) be the diagonal matrix with diagonal elements x ( 1 ) , … , x ( d ) , and further let e x p ( x ) denote the vector with components e x ( j ) , j = 1 , ⋯ , d . We assume a constant discounting factor, e − r n , with r n the daily continuously compounded discounting rate corresponding to the maturity, n, of C. Next, for every i ∈ { 1, ⋯ , l } , set

κ ( i ) = ∫ { e x p ( y − r 1 ) − 1 } f i ( y ) d y , B ( i ) = ∫ { e x p ( y − r 1 ) − 1 } { e x p ( y − r 1 ) − 1 } Τ f i ( y ) d y .

According to [18] , if ϕ t denotes the number of shares of the d risky assets in the portfolio at the beginning of period t − 1 , and V t is the value of the portfolio at period t , the choice of V 0 and ϕ 1 , ⋯ , ϕ n minimizing the mean square hedging error for a given payoff C at maturity n is V 0 = C 0 ( S 0 , τ 0 ) and

ϕ t = α t ( S t − 1 , τ t − 1 ) − V t − 1 D − 1 ( S t − 1 ) ρ t + 1 ( τ t − 1 ) , (15)

First, let ρ n + 1 ( i ) = { ∑ j = 1 l     Q i j B ( j ) } − 1 { ∑ j = 1 l     Q i j κ ( j ) } . Then, for all t = n , ⋯ , 1 and every i ∈ { 1, ⋯ , l } ,

γ t ( i ) = ∑ j = 1 l     Q i j γ t + 1 ( j ) { 1 − ρ t + 1 ( i ) Τ κ ( j ) } , (16)

ρ t ( i ) = { ∑ j = 1 l     Q i j γ t ( j ) B ( j ) } − 1 { ∑ j = 1 l     Q i j γ t ( j ) κ ( j ) } , (17)

C t − 1 ( s , i ) = e − r n γ t ( i ) ∑ j = 1 l     Q i j γ t + 1 ( j ) ∫   C t { D ( s ) e x p ( x ) , j }     × { 1 − ρ t + 1 ( i ) Τ ( e x p ( x − r n 1 ) − 1 ) } f j ( x ) d x , (18)

α t ( s , i ) = e − r n D − 1 ( s ) { ∑ j = 1 l     Q i j γ t + 1 ( j ) B ( j ) } − 1 ∑ j = 1 l     Q i j γ t + 1 ( j )     × ∫   C t { D ( s ) e x p ( x ) , j } { e x p ( x − r n 1 ) − 1 } f j ( x ) d x . (19)

(16) and (17) can be evaluated independently of the payoff, C, through offline computations. However, this is not the case for (18) and (19) for which integrals, or expectations, depending on C must be solved. This can be achieved with the Simulation/Interpolation method proposed in [17] , which we briefly describe next.

Expression (18) and (19) are of the form

g t ( s , i ) = ∑ j = 1 l     Q i j ∫ g t + 1 { π ( s , x ) , j } w t ( x , i , j ) f j ( x ) d x ,   t = n − 1 , ⋯ , 0 ,

where w 0 , ⋯ , w n and g n are known functions, and π ( k ) ( s , x ) = s ( k ) e x ( k ) − r , k = 1 , ⋯ , d . The methodology proposed in [47] for American options and in [17] for hedging is basically to use a Monte Carlo method to approximate g t ( s , i ) for all points s in some finite grid G. Since the values of g t + 1 are approximated at discrete points only, an interpolation method is necessary to evaluate g t .

To estimate g t ( s , i ) for every s ∈ G , one can:

・ Fix i ∈ { 1, ⋯ , l } .

・ For k = 1 , ⋯ , N , repeat the following steps:

- For every j ∈ { 1, ⋯ , l } , generate X k j ~ f j .

・ For every s ∈ G , set

g ^ t ( s , i ) = 1 N ∑ j = 1 l ∑ k = 1 N     Q i j g ^ t + 1 { π ( s , X k j ) , j } w t ( X k j , i , j ) .

where g ^ t + 1 { π ( s , X k j ) } is the linearly interpolated value from the discretized g t + 1 .