### A Hands-on Approach to . . .

, 2009

"... These notes give a more detailed account of the material discussed in lectures. They are largely based on material found in books and research papers. In the ..."

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These notes give a more detailed account of the material discussed in lectures. They are largely based on material found in books and research papers. In the

### Nomura International plc.

, 812

"... A transform approach to compute prices and greeks of barrier options driven by a class of Lévy processes ..."

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A transform approach to compute prices and greeks of barrier options driven by a class of Lévy processes

### by

, 2009

"... These notes give a more detailed account of the material discussed in lectures. They are largely based on material found in books and research papers. In the ..."

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These notes give a more detailed account of the material discussed in lectures. They are largely based on material found in books and research papers. In the

### ON THE CONTINUOUS AND SMOOTH FIT PRINCIPLE FOR OPTIMAL STOPPING PROBLEMS IN SPECTRALLY NEGATIVE LÉVY MODELS

"... ABSTRACT. We consider a class of infinite-time horizon optimal stopping problems for spectrally negative Lévy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshol ..."

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ABSTRACT. We consider a class of infinite-time horizon optimal stopping problems for spectrally negative Lévy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki [26].

### OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT

"... Abstract. An asset manager invests the savings of some investors in a portfolio of defaultable bonds. The manager pays the investors coupons at a constant rate and receives management fee proportional to the value of portfolio. She also has the right to walk out of the contract at any time with the ..."

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Abstract. An asset manager invests the savings of some investors in a portfolio of defaultable bonds. The manager pays the investors coupons at a constant rate and receives management fee proportional to the value of portfolio. She also has the right to walk out of the contract at any time with the net terminal value of the portfolio after the payment of investors ’ initial funds, but is not responsible for any deficit. To control the principal losses, investors may buy from the manager a limited protection which terminates the agreement as soon as the value of the portfolio drops below a predetermined threshold. We assume that the value of the portfolio is a jump-diffusion process and find optimal termination rule of the manager with and without a protection. We also derive the indifference price of a limited protection. We describe numerical algorithms to calculate expected maximum reward and nearly optimal terminal rules for the asset manager and illustrate them on an example. The motivation comes from the collateralized debt obligations. 1.

### IN A JUMP DIFFUSION MODEL

, 2006

"... Abstract. We provide bounds for perpetual American option prices in a jump-diffusion model in terms of American option prices in the standard Black-Scholes model. We also investigate the dependence of the bounds on different parameters of the model. 1. ..."

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Abstract. We provide bounds for perpetual American option prices in a jump-diffusion model in terms of American option prices in the standard Black-Scholes model. We also investigate the dependence of the bounds on different parameters of the model. 1.

### Applying the Wiener-Hopf Monte Carlo simulation technique for Lévy processes to path functionals such as first passage times, undershoots and overshoots

, 2013

"... In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Lévy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many ap ..."

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In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Lévy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time allows sampling from. This includes classic examples such as stable processes, subclasses of spectrally one sided Lévy processes and large new families such as meromorphic Lévy processes. Finally we present some examples. A particular aspect that is illustrated is that the WHMC simulation technique performs much better at approximating first passage times than a ‘plain ’ Monte Carlo simulation technique based on sampling increments of the Lévy process.

### problems

, 2004

"... We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the so-called Canadization procedure suggested by P. Carr in ..."

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We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the so-called Canadization procedure suggested by P. Carr in [2] for the fast computation of American put option prices. In addition to the original application of this technique to optimal stopping problems, we provide an application to another problem in finance, namely the super-replication problem under stochastic volatility, and we show that the approximating value functions can be computed explicitly.