Computation of Hedging Coefficients for Mortgage Default and Prepayment Options: Malliavin Calculus Approach

Abstract

This study explores the hedging coefficients of the financial options to default and to prepay embedded into mortgage contracts based on the change in spot rate, underlying house price and its volatility. In the computations, the finite-dimensional Malliavin calculus is applied since the distribution of both options is unknown and their payoffs are non-differentiable. Naturally, the hedging coefficients are obtained as a product of option’s payoff and an independent weight, which permits the user to derive estimations for the hedging coefficients by running a crude Monte Carlo (MC) algorithm. The simulations reveal that the financial options to default and to prepay are both more sensitive to a change in spot rate than a change in underlying house price and its volatility. There are two potential usages of the hedging coefficients: first, they allow the user to determine the effects of spot rate and underlying house price change and its volatility on the default and prepayment options, and second, borrowers and lenders can replicate and hedge their main portfolio by using the balance between these coefficients and the default and prepayment options.

Appendices

Appendix A: Monthly Payment and Principal Balance

In the case of FRM, the mortgage loan is repaid by a series of equal monthly payments (MP) on pre-determined payment dates. The MP of an FRM for a householder and the outstanding balance (OB)following each payment (OB) are calculated using standard annuity formulas,

$$\begin{array}{@{}rcl@{}} MP &=& OB(0)\frac{(\frac{c}{12})(1+\frac{c}{12})^{m}} {(1+\frac{c}{12})^{m}-1},\\ OB(t) &=& OB(0)\frac{(1+\frac{c}{12})^{m}-(1+\frac{c}{12})^{t}} {(1+\frac{c}{12})^{m}-1}, \end{array} $$

where c is the fixed yearly mortgage rate, OB(0) is the initial loan amount and m is the life of the mortgage loan in months.

Appendix B: A Brief Review on Malliavin Calculus

The Malliavin calculus is a powerful tool to deal with anticipating processes. This theory becomes a natural tool to analyze the Greeks starting with the pioneering studies (Fournié et al. 1999; Fournié et al. 2001) since it makes possible to differentiate with respect to the chance parameter.

Consider a real separable Hilbert space, H, with a scalar product denoted by $\left \langle \cdot ,\cdot \right \rangle _{H}$. Then, the class of smooth random variables $\left (\mathcal {S}\right )$ contains random variables F of the form,

$$ F=f\left( W(h_{1}),\ldots,W(h_{n})\right), $$

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where $f:\mathbb {R}^{n}\longrightarrow \mathbb {R}$ represents the set of all infinitely continuously differentiable functions with all of its derivatives having polynomial growth $(f \in C_{P}^{\infty }\left (\mathbb {R}^{n}\right ))$, and h_i ∈ H, i = 1, 2,…, n for n ≥ 1 and $W=\left \{W(h), h\in H\right \}$ is an isonormal Gaussian process defined on a complete probability space $\left ({\Omega }, \mathcal {F},P\right )$. Here, W is a centered Gaussian family of random variables and satisfies for all h, g ∈ H (Nualart 2006). Then, the Malliavin derivative is given with the following definition.

Definition 2

Let $F \in \mathcal {S}$ and $H=L^{2}([0,T], \mathcal {B}, \mu )$. Then, the Malliavin derivative $D:\mathcal {S}\mapsto L^{2}\left (\left [0, T\right ],\beta ,\mu \right )$ of F is given by

$$ DF=\sum\limits_{i = 1}^{n}{\frac{\partial f}{\partial x_{i}}\left( W(h_{1}),\ldots, W(h_{n})\right)h_{i}}, $$

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where $\frac {\partial f} {\partial x_{i}}$ is the derivative of $f \in C_{P}^{\infty }\left (\mathbb {R}^{n}\right )$ with respect to its i th variable.

Note that the derivative operator, D, is closable from L²(Ω) to $L^{2}\left ({\Omega }\times \left [0, T\right ]\right )$ (Nualart 2006). Therefore, it can be extended to the stochastic Sobolev space $\mathbb {D}^{1,2}$, which is closure of $\mathcal {S}$ with respect to the norm

Moreover, the domain of the Malliavin derivative, $\mathbb {D}^{1,2}$, is a Hilbert space with a scalar product,

where $F, G \in \mathbb {D}^{1,2}$.

Now, the chain rule for the Malliavin derivative can immediately be introduced.

Proposition 4 (Chain Rule)

LetF = (F₁,…, F_n) bea random vector whose components are in thespace$\mathbb {D}^{1,2}$and$\varphi :\mathbb {R}^{n}\mapsto \mathbb {R}$isa function in space$f \in C_{P}^{\infty }\left (\mathbb {R}^{n}\right )$.Then,$\varphi (F)\in \mathbb {D}^{1,2}$and

$$ D_{t} \varphi(F)=\sum\limits_{i = 1}^{n}{\frac{\partial \varphi}{\partial x_{i}}(F)D_{t}F_{i}}, a.s.\quad t \in [0,T]. $$

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As an immediate result of Proposition 4, the following lemma can be introduced.

Lemma 1

Suppose that the sequence$F_{n}\in \mathbb {D}^{1,2}$convergingto F in the space$L^{2}\left ({\Omega },\mathcal {F},\mathbb {P}\right )$satisfying . Then, $F\in \mathbb {D}^{1,2}$ and DF_n weakly converges to DF in $L^{2}\left ({\Omega }\times \left [0, T\right ]\right )$.

Now Proposition 4 can be generalized to the functions even not necessarily differentiable.

Proposition 5

Given a functionφthatsatisfies, for a positive constant$K\in \mathbb {R}$,

$$\left|\varphi(x)-\varphi(y)\right|\leq K\left|x-y\right|, \quad x,y\in\mathbb{R}^{n} $$

and $F\in \mathbb {D}^{1,2}$. Then $\varphi (F)\in \mathbb {D}^{1,2}$and there exists an n-dimensional random vector$G\in \mathbb {R}^{n}$, $\left |G\right |<K$such that

$$D(\varphi(F))=\sum\limits_{i = 1}^{n}{G_{i}DF_{i}}. $$

The Malliavin derivative is a closed linear, unbounded operator with a dense domain in space L²(Ω). Now, an adjoint operator δ of this operator may be defined such that $\delta :L^{2}\left ({\Omega }\times \left [0, T\right ]\right )\mapsto L^{2}\left ({\Omega }\right )$ and denote the domain of adjoint operator δ by Dom(δ).

Definition 3

Let $u\in L^{2}\left (\left [0,T\right ],\beta ,\mu \right )$ and u ∈ Dom(δ). Then, $\forall F\in \mathbb {D}^{1,2}$, one has

where c is a constant depending on u,

$$\delta(u)={{\int}_{0}^{T}}{u_{t}\delta W_{t}} $$

is in L²(Ω) and the duality formula holds,

Now, based on the previous definitions the integration by parts formula can be introduced.

Proposition 6 (Integration by Parts Formula)

Suppose that$F\in \mathbb {D}^{1,2}$andFh ∈ Dom(δ) forh ∈ H.Then, the following equation is satisfied

$$\delta\left( Fh\right)=FW(h)-\left\langle DF, h\right\rangle_{H}. $$

Moreover, if F = 1 a.s.,

$$\delta(h)=W(h). $$

Remark 1

In particular if $H=L^{2}([0,T], \mathcal {B}, \mu )$, where μ is a σ-finite atomless measure on measurable Borel space $([0,T], \mathcal {B})$, Proposition 6 becomes

$${{\int}_{0}^{T}} F h_{t} \delta W_{t} =F {{\int}_{0}^{T}} h_{t} \delta W_{t} -{{\int}_{0}^{T}}{D_{t}F h_{t} dt}. $$

Remark 2

The domain of the Skorohod integral also contains the adapted stochastic processes in the space $L^{2}\left ({\Omega }\times \left [0, T\right ]\right )$. If the integrand is adapted, the Skorohod integral coincides with the Itô integral (see Nualart 2006), i.e.

$$\delta(h)={{\int}_{0}^{T}}{h_{t} dW_{t}}, $$

and

$${{\int}_{0}^{T}} F h_{t} dW_{t} = F{{\int}_{0}^{T}}{h_{t} dW_{t}}-{{\int}_{0}^{T}}{D_{t}F h_{t} dt}, $$

where $F\in \mathbb {D}^{1, 2}$ and .

In this part it is necessary to give the following Remark.

Remark 3

In the case when F is a d-dimensional random column-vector and h_t is a matrix process with a dimension d × d, Remark 2 translates to

$$\delta(hF)={{\int}_{0}^{T}} F h_{t} dW_{t} = F{{\int}_{0}^{T}}{h_{t}DW_{t}}-{{\int}_{0}^{T}}{Tr((D_{t}F)u_{t})dt}, $$

with the convention that the Itô integral for a matrix process is a column-vector. Here Tr represents the trace of the matrix.

The first variation process of multi-dimensional process X_t is the derivative of X_t with respect to its initial value x for $t\in \left [0, T\right ]$, and it is introduced by following definition.

Definition 4

Let (X_t)_{t∈[0, T]} be an $\mathbb {R}^{2}$ valued Itô process whose dynamics are driven by the SDE given with Eq. 3. Then,

is known as the first variation process. Here, the primes denote the derivatives and σ_i is the i^th column vector of σ. Moreover, corresponds to a 2 × 2 identity matrix and Y_t = D^xX_t. Note that β and σ are at least twice continuously differentiable functions with bounded derivatives, and X_t has continuous trajectories.

It is true that one can represent Y_t as Y_t = ∇_xX_t, and moreover, since β and σ are continuously differentiable functions with bounded derivatives and the process X_t is continuous, the Malliavin derivative of X_t can immediately be written as follows (See Fournié et al. 1999 for further details)

Moreover, the following remark also can be given.

Remark 4

Note that $Y_{t}^{11}$ can be expressed in terms of the house price, i.e.

$$Y_{t}^{11}= \frac{1} {h_{0}} h_{t},\quad a.s. $$

Now, a definition is given for a square integrable option payoff ${\Phi } = {\Phi }(X_{t_{1}},\ldots , X_{t_{n}})$, which is evaluated at time t₁, t₂,…, t_n. Then, the price of a claim is the discounted value and it is expressed as

The objective of the computation of Greeks is to differentiate the contingent claim value v with respect to the model parameters. In order to make the proofs, it is necessary to assume the diffusion matrix σ to be uniformly elliptic, that is to say

$$ \exists\eta>0, \text{such that}\quad \xi^{\top} \sigma(x)^{\top} \sigma(x)\xi \geq \eta\left|\xi\right|^{2},\quad \text{for any}\quad \xi, x\in\mathbb{R}^{n}. $$

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B.1 Variations in the Initial Condition

In order to obtain a valid computation result, it is necessary to guarantee that the weights do not degenerate with the probability one. Therefore, Fournié et al. (1999) demonstrates the following set of square integrable functions

$${\Gamma}=\left\{\alpha\in L^{2}\left( \left[0, T\right]\right);{\int}_{0}^{t_{i}}{\alpha(t)dt}= 1, \forall i = 1,\cdots,n\right\}, $$

in $\mathbb {R}$ to avoid degeneracy. Here t_i’s are a partition of [0, T] as introduced above.

The Bismut-Elworthy-Li formula introduced in Bismut (1984) and Elworthy and Li (1994) is required.

Proposition 7 (Bismut-Elworthy formula)

Assume that the diffusion matrixσisuniformly elliptic and Φ isa continuously differentiable function with bounded gradient. Then, for anyα(t) ∈Γ

B.2 Variations in the Drift Coefficient

In order to assess the sensitivity of the price of the contingent claim v to changes in the drift coefficient the perturbed process below is introduced,

$$dX_{t}^{\epsilon} =\left[\beta(X_{t}^{\epsilon})+\epsilon\gamma(X_{t}^{\epsilon})\right]dt + \sigma(X_{t}^{\epsilon})d\mathbb{W}_{t}, \quad X_{0}^{\epsilon}=x, $$

where 𝜖 is a scalar and γ is a bounded function. The following Proposition tells the user how sensitive the price of a claim on the perturbed process is to 𝜖 in the point 𝜖 = 0.

Next consider a continuously differentiable and square integrable continuous function Φ satisfying,

Proposition 8

Suppose that the diffusion matrixσsatisfiesthe uniformly ellipticity condition and the option payoff Φ isa square integrable continuous function. For the perturbed price process

the following is obtained

B.3 Variations in the Diffusion Coefficient

In the computation of the contingent claim, v, sensitivity with respect to it’s volatility parameter, the perturbed process below is used

$$dX_{t}^{\epsilon} =\beta\left( X_{t}^{\epsilon}\right) + \left[\sigma\left( X_{t}^{\epsilon}\right)+\epsilon\gamma(X_{t}^{\epsilon})\right]d\mathbb{W},\quad X_{0}^{\epsilon}=x, $$

where the parameter 𝜖 is a scalar and γ is a continuously differentiable function with bounded derivatives. Furthermore, σ + 𝜖γ satisfies the uniform ellipticity condition given with Eq. 17. To proceed the computations, it is also needed to introduce the variation process with respect to the scalar 𝜖

$$dZ_{t}^{\epsilon} =\beta^{\prime}\left( X_{t}^{\epsilon}\right)Z_{t}^{\epsilon} dt +\sum\limits_{i = 1}^{2}{\left( \sigma_{i}^{\prime}\left( X_{t}^{\epsilon}\right)+\epsilon\gamma(X_{t}^{\epsilon})\right)Z_{t}^{\epsilon} d{W_{t}^{i}}}+\gamma(X_{t}^{\epsilon})d\mathbb{W}_{t} $$

with an initial value $Z_{0}^{\epsilon }= 0_{n}$. Here, one may notice that $\frac {\partial X_{t}^{\epsilon }}{\partial \epsilon }=Z_{t}^{\epsilon }$. Also, $Z_{t}^{\epsilon }|_{\epsilon = 0}$ is defined as Z_t. Furthermore, the set of square integrable functions on $\mathbb {R}$ are defined as

$$ {\Gamma}_{n}=\left\{\alpha\in L^{2}\left( [0,T)\right):{\int}_{t_{i-1}}^{t_{i}}{\alpha(t)dt}= 1,\quad \forall i = 1,\ldots,n, t_{0}= 0\right\}. $$

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Now, based on the perturbed process (10) the following proposition may be given.

Proposition 9

Assume that the diffusion matrixσsatisfiesuniformly ellipticity condition (17) and for$B_{t_{i}}=Y_{t_{i}}^{-1}Z_{t_{i}}$,i = 1,…, nthevalueσ^− 1(X_t)Y_tB_t ∈ Dom(δ) exists.Then, for the price process

and for any α ∈Γ_n the following is obtained

where

Remark 5

If $B_{t}\in \mathbb {D}$, the Skorohod integral $\delta \left (\sigma ^{-1}(X_.)Y_.\tilde {B}_.\right )$ may be calculated as

$$\begin{array}{@{}rcl@{}} \delta\left( \sigma^{-1}(X_.)Y_.\tilde{B}_.\right)&=&\sum\limits_{i = 1}^{n}\left\{B_{t_{i}}^{\top}{\int}_{t_{i-1}}^{t_{i}}{\alpha(t)\left( \sigma^{-1}(X_{t})Y_{t}\right)^{\top}}d\mathbb{W}_{t}\right.\\ &-&{\int}_{t_{i-1}}^{t_{i}}{\alpha(t)Tr\left( \left( D_{t}B_{t_{i}}\right)\sigma^{-1}(X_{t})Y_{t}\right)dt}\\ &-&\left.{\int}_{t_{i-1}}^{t_{i}}{\alpha(t)\left( \sigma^{-1}(X_{t})Y_{t}B_{t_{i-1}}\right)^{\top} d\mathbb{W}_{t}}\right\}. \end{array} $$