By Hans Bühlmann (auth.), G. Ottaviani (eds.)
This e-book, released with the contribution of the Italian coverage corporation INA, comprises the invited contributions offered on the third foreign AFIR Colloquium, held in Rome in 1993. The colloquium used to be aimed toward encouraging examine at the theoretical bases of actuarial sciences, its interplay with the speculation of finance and of company finance, including mathematical tools, resembling likelihood and the speculation of stochastic procedures. within the spirit of actuarial culture, recognition used to be given to the hyperlink among the theoretical method and the operative difficulties of economic markets and associations, and insurance firms specifically. The e-book is a vital reference paintings for college kids and researchers of actuarial sciences and finance. it may possibly even be advised to practitioners with theoretical interests.
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We label this probability measure as the Q measure. It is sometimes called the equivalent martingale measure because under this measure the price process for the risky asset suitably normalized is a martingale 6. The equivalence between the existence of the equivalent martingale measure and the absence of arbitrage represents one of the most fundamental results in modern financial economics. The valuation of any payoff in this model can be accomplished in terms of the Arrow Debreu securities. Suppose we have a European call option with strike price K.
Is the conditional expectation taken with respect to the risk-adjusted measure of the real economy. 40 Franco Moriconi Real unit ZCBs. If Xs is equal to a unit of consumption good we are considering a real unit ZCB with maturity s. Obviously the time t price b(t; s) of this bond is obtained by solving (9) under the terminal condition Xs = 1 , but in this case the valuation equation can be simplified into a form similar to the valuation equation (1) of the one-factor CIR model; its solution is: (12) w here the functions Al (,) and B 1 ( ,) are independent on x and have exactly the same form obtained in the single-factor model (equations (5) and (6».
The quantity: P(r(t), t) := -q(r(t), t) g(r(t), t) is usually referred to as the risk premium, and 1* (r( t), t) := f(r(t), t) - P(r(t), t) is the so-called risk-adjusted drift. Once the parameters of the spot rate process and the market price of risk are specified, the price of the IRS security is obtained by solving the fundamental valuation equation (1) subject to the appropriate boundary conditions. Analyzing Default-Free Bond Markets by Diffusion Models 27 Price of zero-coupon unit bonds. Let us denote by v(r(t), t; s) the price at time t of a zero-coupon unit bond (ZCB) with maturity s ~ t.
Financial Risk in Insurance by Hans Bühlmann (auth.), G. Ottaviani (eds.)