From the differential formulas (9.3.14) and (9.3.15) for the...

From the differential formulas (9.3.14) and (9.3.15) for the stock and discounted exchange rate in terms of the Brownian motions under the domestic risk-neutral measure, derive the differential formulas (9.3.22) and (9.3.23) for the redenominated money market account and stock discounted at the forcign interest rate and written in terms of the Brownian motions under the foreign risk-neutral measure.

Key Concepts

Risk-Neutral Measure

The risk-neutral measure is a probability measure under which the discounted prices of tradable assets become martingales. This framework is critical in derivative pricing and financial modeling, as it ensures that the expected growth rate of asset prices, when adjusted for the cost of money via discounting, is zero. The concept allows practitioners to price derivatives by simply taking expectations under the appropriate risk-neutral measure, whether domestic or foreign, rather than having to directly deal with the actual physical probabilities of asset returns.

Change of Numeraire

The change of numeraire is a technique used in financial mathematics to switch the unit of measurement or basis of pricing from one asset (or currency) to another. In practice, this change transforms the underlying stochastic processes and requires an associated change in the probability measure to maintain the martingale property of discounted asset prices. This is especially useful when switching from a domestic to a foreign perspective, as is necessary when pricing assets in different currencies.

Girsanov's Theorem

Girsanov's theorem is a key result in stochastic calculus that facilitates the change of measure by adjusting the drift terms in stochastic differential equations driven by Brownian motion. When moving from the domestic risk-neutral measure to the foreign risk-neutral measure, Girsanov's theorem provides the mathematical foundation for transforming the dynamics of the processes – such as the stock price and the money market account – ensuring that the modified processes under the new measure remain valid martingales.

Stochastic Differential Equations (SDEs) and Itô's Lemma

Stochastic differential equations (SDEs) describe the evolution of random processes over time, incorporating both deterministic trends and random fluctuations via Brownian motion. Itô's lemma is a fundamental tool in handling SDEs, as it allows one to compute the differential of a function of a stochastic process. In the context of the question, Itô's lemma is used to derive the dynamics of the redenominated assets after changing currencies and measures.

Exchange Rate Dynamics

Exchange rate dynamics involve modeling the random behavior of currency conversion factors over time. These dynamics play a crucial role when assets are priced in different currencies, as converting from domestic to foreign currency requires adhering to the underlying stochastic process governing the exchange rate. The derived differential formulas for redenominated assets reflect these dynamics and ensure the coherent temporal evolution of asset prices under the new risk-neutral framework.

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