Methodology

In this section we build on the ideas of [Chan and Kroese, 2011], as covered in section 2.5.3, in order to estimate the probabilities of extreme events for random variables which are Normal Inverse Gaussian (NIG) distributed. This is most directly applied to estimating the probability of a large loss of a financial stock.

By considering the functional form of the Normal Inverse Gaussian distri- bution (2.3.6), we note that when γ < 0, (sufficiently) large negative values will overwhelmingly be driven by extreme draws ofW, rather than Z. This is because for X to have an extreme value, we require either large W, or, alternatively, ex- treme values of both W and Z. However, these scenarios do not have the same chance of occurrence. The heavy tailed nature of the Inverse Gaussian distribution (W) implies it has a much higher likelihood of taking a high value than the Normal distribution (Z), during an extreme event. Furthermore, the heavy tail makes it significantly more likely that onlyW is large, as opposed to both W and Zbeing large, due to the rate of ’decay’ in the tail versus the growth in magnitude required. We use this intuition to calculate the probabilities for rare events of random variables which are NIG distributed (predominantly whenγ <0).

To allow us to do this we need to calculate the conditional probabilities associated with the NIG distribution. These are P(X < x|z), where (fixed)z ∼Z andP(X< x|w), where (fixed)w∼W.

6.2.1 Conditional Probabilities

Proposition 6.2.1. If X∼GH1(λ, χ, ψ, µ, σ2, γ) (a one dimensional GH distribu- tion), then P(X< x|w) = Φ x−µ−wγ √ wσ .

Proof. This result is a trivial application of 2.3.7.

Proposition 6.2.2. If X∼GH1(λ, χ, ψ, µ, σ2, γ) (a one dimensional GH distribu- tion), then whenγ <0,

P(X< x|z) =          1−FW(w−) if µ≥x FW(w+) + 1−FW (w−) if µ < x,(σz)2 >4γ(µ−x), z >0 1µ<x otherwise whenγ = 0, P(X< x|z) =          FW x−µ σz 2 if x_σz−µ ≥0, z >0 1−FW x−µ σz 2 if x_σz−µ ≥0, z <0 1µ<x otherwise and whenγ >0, P(X< x|z) =          FW(w+) if µ≤x FW(w+)−FW (w−) if µ > x,(σz)2 >4γ(µ−x), z <0 1µ<x otherwise

where FW is the cumulative distribution function ofW and

w+ = −σz+ p (σz)2_{−4γ(µ−x)} 2γ !2 and w−= −σz− p (σz)2_{−4γ(µ−x)} 2γ !2 .

Proof. We first consider the case whenγ <0.

quadratic formulae we can solve for√w, x=µ+γw+σ√wz =⇒(µ−x) +γw+σ√wz= 0 =⇒√w= −σz± p (σz)2_{−4γ(µ−x)} 2γ . (6.1)

For convenience, we define w+ and w− to be the two roots of 6.1, squared, assuming they exist, as defined above (in the proposition).

Since we only want real solutions of w, we need only consider positive roots of equation 6.1. For any positive roots to exist we require that

(σz)2 >4γ(µ−x) (6.2) and for an individual root to be positive we need (sinceγ <0)

±

q

(σz)2−4γ(µ−x)< σz. (6.3) We first consider the situations under which two positive roots of equation 6.1 will exist. For condition 6.3 to be satisfied for both roots, we require thatz≥0, as one root is positive. However for there to be two unique roots we requirez >0, as whenz= 0 the solution will degenerate to one unique root. Furthermore, we require 4γ(µ−x)>0, so that the positive root is less thanσz(aspy2_{− < y ⇐⇒ >0,}

∀y ∈ _R). Given that γ < 0, this simplifies to −(µ−x) > 0, which rearranges to x > µ. We also need to ensure condition 6.2 is true. Note that under these conditionsw+< w−.

If there are two roots,µ+γw+σ√wz will be less thanxwhenw < w+and when w > w−. This statement follows from the asymptotic behaviour of X with respect tow. As w→ ∞,µ+γw+σ√wz→ −∞< xfor any (fixed) choice of xor

z(whenγ <0).

This implies P(X < x|z) = FW(w+) + 1−FW(w−) when µ < x,(σz)2 > 4γ(µ−x), z >0.

We now consider the conditions under which only one positive root of equa- tion 6.1 will exist. If only one root exists it must be w−. We can see this by considering the opposing case and finding a contradiction; if only the root of equa- tion 6.1 linked to w+ _{was positive we would have +}q_(σz)2_{−4γ(µ−x) ≤ σz <}

−

q

(σz)2−4γ(µ−x), by condition 6.3, which is a contradiction for any choice of

For justw− to be positive we require 4γ(µ−x)≤0 (from condition 6.3). If this condition is true then condition 6.2 must also be satisfied. Given γ < 0 this requiresµ−x≥0, or alternativelyµ≥x.

By the same argument as for the two root case,µ+γw+σ√wz will be less thanx whenw > w−. This implies P(X< x|z) = 1−FW(w−) when µ≥x.

Finally we consider the case when there are zero roots (which must be every case not covered above). This means that the outcome of condition (µ+γw +

σ√wz < x) must be constant for all choices of w. When w = 0, the condition reduces toµ < x, implying P(X< X|z) =1µ<x.

For the case when γ > 0 a symmetric argument applies (which we do not provide here).

Whenγ = 0, we have x=µ+σ√wz, which rearranges to √w= x_σz−µ. Since we require √w > 0, we only have a solution when x_σz−µ > 0. Hence, the value of

P(X < x|z) depends on the sign of z (by the same arguments as for γ < 0). We have P(X< x|z) =          FW x−µ σz 2 if x_σz−µ ≥0, z >0 1−FW x−µ σz 2 if x_σz−µ ≥0, z <0 1µ<x otherwise.

6.2.2 Estimating Extreme Risk in the NIG Distribution

Using the conditional probabilities defined above, we can construct a scheme for estimatingP(X≤x), for choices of xwhich give a very low probability (conditional onZ). The scheme works as follows:

1. GenerateN samples from Z, denote them {zi}N_i=1. 2. EstimateP(X< x) using _N1Σ_iN₌₁P(X≤x|zi).

It should be clear that this estimator would converge to the correct value, given sufficient N. However it is not an unbiased estimator as it will underweight the probabilities that either both W and Zare extreme, or that Z<< 0. However as discussed in the introduction, whenγ < 0 we expect very little contribution from these cases. We will explore this hypothesis in the early results.

In the first example we also consider the opposite idea, where instead of using

P(X≤x|zi) we useP(X≤x|wi) and random samples fromW(conditional onW). We examine this in order to confirm our original intuition.

6.2.3 Extending to d Dimensions

We can also apply the above method to a linear combination of the individual dimensions of a ddimensional NIG distribution. This would allow us to apply the method to a portfolio of stocks, rather than just a single stock. We can do this by applying proposition 2.3.12 to compress this d dimensional problem down to one dimension and hence apply the method above.

Mathematically, we consider the estimation ofP(Y≤y) where Y:=Pd

i=1wiXi,X∼GHd(−1₂, χ, ψ, µ,Σ, γ) and wi ∈R>0 for 1≤i≤d. We note that by takingB ={w1, . . . , wd}we can apply proposition 2.3.12 to find that Y∼GH1(−1₂, χ, ψ, Bµ, BΣB0, Bγ). We can now apply the method in the previous section (6.2.2) to calculate the probabilities of a large loss.

This method is numerically tested in the third example by applying it to a simple basket of stocks.

In the following sections we apply this method (and hypothesis) to a simple test case and then to two financial examples, as outlined in the introduction (6.1).