Levy process and random measure

Question

I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.

Answer 1

There is a whole literature on risk-neutral modeling with Levy processes.

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Consider an arbitrage-free market where asset prices are modeled by a stochastic process $(S_t)_{t \in [0,T]}, \mathcal{F}_t$ represents the history of the asset $S$ and $\hat{S}_t=e^{-rt}S_T$ the stochastic discounted value of the asset. The discounted expectation of the terminal payoffs under $Q$ is

$$\hat{S}_t = \operatorname{E}^{Q}[\hat{S}_T|\mathcal{F}_t]$$

There are two ways to define the risk neutral dynamics in the Black-Scholes model using a Brownian motion with drift:

1. Taking the exponential, i.e. $S_t=S_0 e^{B_t^0}$, where $B_t^0=(r-\sigma^2/2)t + \sigma W_t$, which is a brownian motion with drift.

2. Taking the stochastic exponential by applying Ito formula, i.e. $\frac{dS_t}{S_t}=rdt + \sigma dW_t=dB_t^1$, where $B_t^1=rt+\sigma W_t$.

Levy-processes are often used for modeling jump-processes (see Cox/Ross(1976) or Merton(1976)), especially in jump-diffusion models.

In the formulas above, we can generalize the Black-Scholes model to account for jumps, by replacing the brownian motion with drift by a Levy-process. Therefore, we get

1. $S_t = S_0 e^{rt + X_t}$, which is an exponential-Levy model as $(X_t)_{t \in [0,T]}$ describes a Levy-process.
2. Replace $B_t^1$ by a Levy-process $Z_t$ results in $dS_t=rS_tdt + S_tdZ_t$.

Furthermore, exponential-Levy models offer analytically tractable examples of positive jump processes. The availability of closed-form expressions for characteristic function of Levy processes also allows to use Fourier transform methods for option pricing.

I recommend the following literature on Levy-processes and their use in finance:

• Measure transformations for Levy-processes are discussed in Sato(1999)
• More general martingale measures for processes with independent increments are discussed in Grandits(1999).
• The absence of arbitrage and completeness for models with jumps is discussed in Bardhan(1999).
• Predictable representations for Levy processes in terms of a sequence of jump martingales were introduced by Nualart/Schoutens(2000) and Nualart/Schoutens(2001). A financial interpretation of their results in terms of hedging with vanilla options is given by Balland(2002).

Reference:

Cont/Tankov (2004), Financial Modelling With Jump Processes, Chapman & Hall/CRC Financial Mathematics Series