The main objective of this note is to review Risk-Neutral and Real-World valuation set-ups and related concepts. Additionally, we state basic mathematical background that is the basis of the valuation of the financial instruments.
Market consistent valuation of a financial instrument (specifically asset/liability in insurance world) is the value in which it could be traded in the market by two knowledgeable, willing partners. Although the insurance products are not traded in the market, we assume that insurance liabilities are traded by the shareholders of the insurance companies in a deep liquid market.
Risk-neutral valuation is a mathematical tool based on the principle of no-arbitrage opportunities and reflects the market view of the risk; it is designed for market consistent valuation. In this method the cash flow of the asset/liability is replicated with the assets traded in a deep liquid market and the value will be the cost of the replicating portfolio. Especially, the main objective of the risk-neutral valuation is to infer a quasi-market-price for the insurance product for the insurance companies. On the other hand, a real world valuation is designed to capture a future observable state of the world and the dynamic of market prices in order to understand the possible risks. The choice of these frameworks depends on the modelling purpose.
The absence of arbitrage opportunity is equivalent to the fact that there exists a martingale measure (or risk-neutral measure) from a mathematical point of view. It is a measure of the event space that makes all the discounting processes martingales. More precisely, let \( B_t \) be the discounting process, i.e., the value of 1 dollar cash flow seen at time 0, then
$$E_t [B_{t+1} S_{t+1}]=B_t S_t, $$
$$E_0[B_1 S_1]=S_0,$$
where \( S_t \) denotes the price process of a specific numeraire security.
Consider a deep liquid market in which non-dividend securities are traded continuously, s.t., the prices of securities are defined in probability space \( (\Omega, \mathcal{F},\mathbb{P}) \), where
$$\mathcal{F}= \{ \mathcal{F}_t=\sigma \{ W(s) : \,\, 0 \leqslant s \leqslant t \} \,\, : \,\,\, 0 \leqslant t \leqslant T \}, $$
is the filtration and \( W \) is multi-dimensional Brownian motion.
We say \( V \) is \( T \)-maturity derivative if \( V(T) \) is \( \mathcal{F}_T \)-measurable and moreover we consider that the derivative pays \( V(T) \) at time \( T \).
Then a zero coupon bond (ZCB) is a \( T \)-maturity derivative that pays 1 (or \( P(T,T) \)) unit of the currency at \( T \) and has the price \( P(t,T) \) at any time \( t \) between 0 and \( T \).
On the other hand \( \beta(t) \), the value of bank account at time \( t \), has the following dynamic
$$d\beta (t) =r(t)\beta (t) \, dt,$$
$$\beta(0)=1,$$
where \( r \) is short rate and therefore
$$\beta (t) = \exp(\int_0 ^t r(u)\, du ).$$
We know that for every numeraire \( N \), there exist a probability measure \( \mathbb{P} ^N \) s.t., the price of \( V \) normalized by \( N \) is martingale under \( \mathbb{P} ^N\).
Risk-Neutral measure \( \mathbb{Q} \) is defined as the corresponding measure for numeraire \( \beta(t) \) s.t., \( \frac{V(t)}{\beta(t)} \) is martingale under the measure \( \mathbb{Q} \). Thus for \( t \leqslant T \)
$$V(t)= \beta (t) \mathbb{E }_t ^\mathbb{Q} [\frac{V(T)}{\beta(T)}].$$
The ZCB price is then
$$P(t,T)=\mathbb{E}_t ^{\mathbb{Q}}[e^{- \int _t ^T r(u)\, du}].$$
Moreover no arbitrage argument implies
$$P(t, T+\tau)= P(t,T) \, \mathbb{E}_t ^{\mathbb{Q}} [P(T, T+\tau)], \,\,\, \,\,\, \tau>0.$$
For \( \tau>0 \), the time \( t \) forward price of ZCB spanning \( [T,T+\tau] \), is
$$P(t, T, T+\tau):= \mathbb{E}_t ^{\mathbb{Q}} [P(T, T+\tau)] = \frac{P(t,T+\tau)}{P(t, T)}. $$
Also, we define forward LIBOR rate \( L \) as
$$1+\tau L(t, T, T+\tau)= \frac{1}{P(t, T, T+\tau)}. $$
