From Diffusion to Jumps: Lévy Model in Finance

--

Financial markets do not always behave the way simple models suggest. Prices do not move smoothly all the time, and they do not follow the clean and continuous patterns assumed in the Black Scholes model. In reality, markets often experience sudden changes caused by things like earnings announcements, economic news, or liquidity issues. These events can lead to sharp and unexpected movements in prices.

This gap between theory and reality has been a major challenge in quantitative finance. While geometric Brownian motion is easy to work with mathematically, it does not capture important features seen in real market data, such as extreme movements, asymmetry, and sudden jumps. Because of this, models that rely only on continuous price changes tend to underestimate risk, especially during volatile periods.

To improve on this, jump diffusion models were developed to include sudden changes in price. However, these models can still be limited when the behavior of jumps becomes more complicated or happens more frequently. This leads to a more general and flexible framework known as Lévy processes.

Lévy processes extend classical models by allowing a wider range of jump behavior. They can capture both rare large movements and many small fluctuations. By combining continuous movement with jumps, they offer a more realistic way to describe how asset prices evolve.

In this article, we will look at how Lévy jump models help us better understand financial markets. We will introduce the basic ideas behind these models and discuss how they are used in areas such as option pricing and risk management.

Mathematical foundations :

Lévy Processes: Introduction

A stochastic process X = (Xₜ)ₜ≥₀ is said to have càdlàg paths if almost all its
sample paths are càdlàg — that is, continuous from the right with limits from the
left (from the French: continu à droite, limites à gauche).

Definition: Lévy Process

A real-valued process L = (Lₜ)ₜ≥₀ defined on a probability space (Ω, ℱ, ℙ) is
called a Lévy process if it satisfies the following conditions:

(i) Initialization

    L₀ = 0     ℙ-almost surely

(ii) Independence of Increments

For all 0 ≤ t₁ < t₂ < ··· < tₙ, the increments
    Lₜₙ − Lₜₙ₋₁ , … , Lₜ₂ − Lₜ₁

are independent random variables.
(iii) Stationarity of Increments

For all s < t, the increment Lₜ − Lₛ has the same distribution as Lₜ₋ₛ:

    Lₜ − Lₛ  ᵈ=  Lₜ₋ₛ

(iv) Continuity in Probability

For every ε > 0 and every s ≥ 0:

    lim   ℙ( |Lₜ − Lₛ| > ε ) = 0
   t → s

(v) Càdlàg Paths

L has càdlàg sample path : almost surely, the map t ↦ Lₜ(ω) is right-continuous
and possesses left-hand limits at every t > 0:

    Lₜ₋ := lim Lₛ     exists for all t > 0,  ℙ-a.s.
           s↑t
Note on the Conditions

Conditions (ii) and (iii) are the structural heart of the definition. Together they
imply that the law of L is completely determined by the one-dimensional marginal L₁,
since for any t ≥ 0:

         n
    Lₜ  ᵈ=  Σ  ( Lₖₜ/ₙ − L₍ₖ₋₁₎ₜ/ₙ )
        k=1

where each summand is i.i.d. with the same law as Lₜ/ₙ. This makes Lₜ infinitely
divisible for each fixed t.

Condition (iv) rules out fixed-time jumps — it does not, however, preclude
random-time jumps, which are a defining feature of Lévy processes. Condition (v) is
a regularity requirement on the paths: at any jump time τ, we have Lτ ≠ Lτ₋, and
the jump size is defined as:

    ΔLτ := Lτ − Lτ₋


Example 1: Brownian Motion with Drift

The process

    Xₜ = αBₜ + βt

is a Lévy process with continuous trajectories and characteristic triplet (β, α², 0).

Properties:
  • Drift component:       βt         (deterministic, linear)
  • Diffusion component:   αBₜ        (scaled Brownian motion)
  • Jump measure:          ν = 0      (no jumps — purely continuous paths)
  • Characteristic triplet: (β, α², 0)

Conversely, any Lévy process with continuous trajectories is necessarily a
Brownian motion with drift — making this the unique continuous Lévy process
up to scaling and shifting.


Example 2 :Poisson Process

Let (τᵢ)ᵢ≥₁ be a sequence of independent exponential random variables with
parameter λ > 0, and define the partial sums

    Tₙ = τ₁ + τ₂ + ··· + τₙ

The Poisson process N = (Nₜ)ₜ≥₀ is then defined by

    Nₜ = Σ  1{t ≥ Tₙ}
        n≥1

For any t > 0, the random variable Nₜ follows a Poisson distribution:

    ℙ(Nₜ = k) = e^(−λt) · (λt)ᵏ / k!     k = 0, 1, 2, …

Properties:
  • Intensity:              λ > 0
  • Characteristic triplet: (0, 0, ν)   where  ν = λδ₁
  • Jump measure:           ν = λδ₁     (Dirac mass at 1, scaled by λ)
  • Jump sizes:             always exactly 1
  • Paths:                  piecewise constant, jumping by +1 at each Tₙ
  • Mean:                   𝔼[Nₜ] = λt
  • Variance:               Var(Nₜ) = λt


Example 3: Compound Poisson Process

Let N be a Poisson process with intensity λ > 0, and let (Yᵢ)ᵢ≥₁ be a sequence
of i.i.d. random variables with common law μ, independent of N. The compound
Poisson process is defined by

         Nₜ
    Xₜ = Σ  Yᵢ ,     with the convention  Xₜ = 0  when  Nₜ = 0
        i=1

Interpretation: jumps arrive at the random times T₁, T₂, … of the Poisson
process, and each jump carries a random size Yᵢ drawn from μ.

Properties:
  • Jump arrival rate:      λ > 0       (Poisson intensity)
  • Jump size distribution: μ           (law of each Yᵢ)
  • Characteristic triplet: (β̃, 0, ν)  where  ν = λμ  and
                             β̃ = λ ∫ x 1{|x|≤1} μ(dx)
  • Jump measure:           ν = λμ      (intensity × jump law)
  • Paths:                  piecewise constant with jumps of random size
  • Mean:                   𝔼[Xₜ] = λt · 𝔼[Y₁]          (if 𝔼[Y₁] < ∞)
  • Variance:               Var(Xₜ) = λt · 𝔼[Y₁²]        (if 𝔼[Y₁²] < ∞)
  • Moment generating fn:   𝔼[e^(uXₜ)] = exp( λt ∫ (e^(ux) − 1) μ(dx) )

Note: The Poisson process in Example 9.1.3 is the special case where
μ = δ₁ (all jumps have size exactly 1), recovering Xₜ = Nₜ.

Real world Applications related to Risk:

Lévy Jump Processes in Risk Management



1. VALUE AT RISK (VaR) AND EXPECTED SHORTFALL (ES)


Under a Gaussian model, the α-level VaR is:

    VaR_α = μ + σ · Φ⁻¹(α)

This severely underestimates tail risk because it assigns near-zero probability
to large sudden losses. Under a Lévy model, the return distribution inherits
heavy tails directly from the jump measure ν(dx), so:

    ℙ( Xₜ < −VaR_α )  is materially larger than the Gaussian predicts

The jump measure ν(dx) contributes excess mass in the tails:

    Tail probability  =  diffusion tail  +  ∫ ν(dx)
                                           |x|>threshold

In practice, VaR and ES are computed as follows:

    Step 1:  Fit a Lévy model (VG, NIG, or CGMY) to historical log-returns
    Step 2:  Simulate N paths of Xₜ over the risk horizon T using the
             calibrated triplet (β, σ², ν)
    Step 3:  Compute VaR as the empirical α-quantile of simulated losses:

             VaR_α  =  −Q_α( X_T )

    Step 4:  Compute Expected Shortfall (CVaR) — the average loss beyond VaR:

             ES_α  =  −𝔼[ X_T | X_T < −VaR_α ]

Because ν can place significant mass on large negative moves, ES under a Lévy
model is substantially larger than its Gaussian counterpart — giving a more
honest and conservative picture of true tail exposure.


2. CREDIT RISK: DEFAULT AS A JUMP EVENT


In structural credit models, a firm defaults when its asset value Vₜ crosses
a fixed debt barrier D from above. Under pure Brownian motion:

    Vₜ = V₀ · exp(σBₜ + βt)

the crossing is always gradual — the process cannot gap below D — which forces
short-term credit spreads to collapse to zero. This flatly contradicts
observed market credit spreads on short-dated instruments.

Under a Lévy-driven asset value model:

    Vₜ = V₀ · exp(Xₜ)     where Xₜ has triplet (β, σ², ν)

the firm can jump below the barrier instantaneously. The default time is:

    τ  =  inf{ t ≥ 0 : Vₜ ≤ D }

and the default probability over horizon T is:

    ℙ( τ ≤ T )  =  ℙ(  min  Vₜ ≤ D )
                      0≤t≤T

The jump measure ν directly controls the likelihood of sudden default. A
heavier ν on the negative side raises ℙ(τ ≤ T) at short maturities, producing:

    • Positive and realistic short-term credit spreads        
    • Sudden default without prior financial distress         
    • Richer CDS term structure matching market quotes        

This makes Lévy models essential for pricing credit default swaps (CDS),
collateralized loan obligations (CLOs), and distressed corporate bonds.


3. OPERATIONAL RISK AND RUIN THEORY


The aggregate loss process in insurance or operational risk is modeled
naturally as a compound Poisson process — a canonical Lévy process:

         Nₜ
    Lₜ = Σ  Yᵢ
        i=1

where:
    Nₜ  =  number of loss events up to time t,  Poisson with intensity λ
    Yᵢ  =  size of the i-th loss,               i.i.d. with law μ
    ν   =  λμ                                   (the Lévy measure of Lₜ)

The firm's surplus (capital reserve) process is:

    Rₜ = u + ct − Lₜ

    u  =  initial capital reserve
    c  =  premium or revenue income rate
    Lₜ =  cumulative losses (compound Poisson)

Ruin occurs the moment the surplus turns negative. The ruin probability is:

    ψ(u)  =  ℙ( Rₜ < 0  for some  t ≥ 0 )

The Cramér-Lundberg theorem gives an exponential upper bound:

    ψ(u) ≤ e^(−Ru)

where R > 0 is the adjustment coefficient — the positive root of the
Lundberg equation:

    λ + cR  =  λ · 𝔼[ e^(RY) ]

Key risk management insight: the heavier the tail of μ (the jump size
distribution), the smaller R becomes, and the slower ψ(u) decays — meaning
the firm needs substantially more capital u to maintain ruin probability
below a regulatory threshold. This directly informs:

    • Basel III operational risk capital charges
    • Solvency II reserve requirements for insurers
    • Internal capital adequacy assessments (ICAAP)

Conclusion:

Lévy processes offer a more honest mathematical language for risk. By allowing jumps, they capture what Gaussian models routinely miss , sudden defaults, fat-tailed losses, and catastrophic operational events that arrive without warning. Across VaR and Expected Shortfall, credit risk, and ruin theory, the jump measure ν does the heavy lifting: it directly parameterizes how bad things can get and how fast they can get there. For any serious risk management framework, ignoring jumps is not a simplification it is a miscalculation.

!!!Happy Reading