We consider factor models for interest rates and asset prices where the riskneutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward-backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5: [511][512][513][514][515][516][517][518][519][520][521][522][523][524][525] 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4): 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511-525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.Keywords Affine models · Forward-backward stochastic differential equations · Stochastic flows · Bond price · Futures price · Forward price JEL Classification E43 · G12 · G13 Mathematics Subject Classification (2000) 60G35 · 60H20 · 60H30 · 91B28 · 91B70

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We completely characterise the futures price and forward price of a risky asset (commodity) paying a stochastic dividend yield (convenience yield). The asset (commodity) price is modelled as an exponential affine function of a Gaussian factors process while the interest rate and dividend yield are affine functions of the factors process. The characterisation we provide is based on the method of stochastic flows. We believe this method leads to simpler and more clear-cut derivations of the futures price and forward price formulae than alternative methods. Hedging a long term forward contract with shorter term futures contracts and bonds is also examined.

The guaranteed minimum withdrawal benefit (GMWB) rider, as an add on to a variable annuity (VA), guarantees the return of premiums in the form of periodic withdrawals while allowing policyholders to participate fully in any market gains. GMWB riders represent an embedded option on the account value with a fee structure that is different from typical financial derivatives. We consider fair pricing of the GMWB rider from a financial economic perspective. Particular focus is placed on the distinct perspectives of the insurer and policyholder and the unifying relationship. We extend a decomposition of the VA contract into components that reflect term-certain payments and embedded derivatives to the case where the policyholder has the option to surrender, or lapse, the contract early.