Pricing Bivariate Options In Complete Markets

Let us consider a derivative contract that is written on two underlying assets, which we denote as 51 and S2. The information structure is represented in the usual way by a filtered probability space St, P generated by the stochastic processes S1(t) and S2(t), t e 0, T . Throughout the discussion, we will assume that S1(t), S2(t) are continuous random variables with non-negative support. If, for the sake of simplicity, we take the bivariate derivative to be European, its pay-off may be written...

Bibliography

Anderson, R. & Sundaresan, S. (1996) Design and valuation of debt contracts, Rev. Finan. Stud., 9, 37-68. Arvanitis, A. & Gregory, J. (2001) Credit The Complete Guide to Pricing, Hedging and Risk Management . Risk Books, London. Avellaneda, M. & Paras, A. (1996) Managing the volatility risk of portfolios of derivative securities the Lagrangian uncertain volatility model, Appl. Math. Finance, 3, 21-52. Avellaneda, M., Levy, A. & Paras, A. (1995) Pricing and hedging derivative...

N 012

Notice that in particular such a function is non-negative h gt 0 and non-increasing h' lt 0 . Given the notion of complete monotonocity, we have the following Theorem 3.13 If the inverse of a strict generator is completely monotone, then the corresponding copula entails PQD Among Archimedean copulas, we are going to consider in particular the one-parameter ones, which are constructed using a generator ya t , indexed by the real parameter a. By choosing the generator, one obtains a subclass or...

F230 exp12530226

While their joint survival probability, according to the Marshall-Olkin model, is CM F x , F2 y min exp -1.25 gt 0 exp -0.5x , exp -0.75x exp -y exp -1.25y - 0.5x , x lt y exp -0.75x y , x gt y since m 1 3, n 1 5. The joint survival probability beyond x y t 3 years for instance is CM Fi 3 , F2 3 exp 3 -1.25 - 0.5 0.5248 11 For an exponential r.v. the expected value is the reciprocal of the intensity. 2.6 DENSITY AND CANONICAL REPRESENTATION This section introduces the notion of density and...

Copula Methods In Finance A Primer

Up to this point, we have seen that the three main frontier problems in derivative pricing are the departure from normality, emerging from the smile effect, market incompleteness, corresponding to hedging error, and credit risk, linked to the bivariate relationship in OTC transactions. Copula functions may be of great help to address these problems. As we will see, the main advantage of copula functions is that they enable us to tackle the problem of specification of marginal univariate...

N i Tf ji

Copula Family

Where t-1 v , q2 t-l z , and the copula itself is absolutely continuous. Since, as recalled by Roncalli 2002 , given a couple of r.v.s X, Y , jointly distributed as a Student's t, the conditional distribution of given X x, is a Student's t with u 1 degrees of freedom, the conditional distribution via copula C ,1u v, z is It follows that an equivalent expression for the bivariate Student's copula is If u gt 2, each margin admits a finite variance, u u - 2 , and pXY can be interpreted as a linear...

Contents

List of Common Symbols and Notations xv 1 Derivatives Pricing, Hedging and Risk Management The State of the Art 1 1.2 Derivative pricing basics the binomial model 2 1.2.1 Replicating portfolios 3 1.2.2 No-arbitrage and the risk-neutral probability measure 3 1.2.3 No-arbitrage and the objective probability measure 4 1.2.4 Discounting under different probability measures 5 1.2.5 Multiple states of the world 6 1.3 The Black-Scholes model 7 1.3.3 The martingale property 11 1.4 Interest rate...

Incomplete Markets

The most recent challenge to the standard derivative pricing model, and to its basic structure, is represented by the incomplete market problem. A brief look over the strategy used to recover the fair price of a derivative contract shows that a crucial role is played by the assumption that the future value of each financial product can be exactly replicated by some trading strategy. Technically, we say that each product is attainable and the market is complete. In other words, every contingent...