The maths of how the general valuation works.

Further Reading

The mathematics of the general valuation and further references can be found in this document or here. A summary follows below.

Products

A product is an encapsulation of a financial contract. It only knows:

  • Which market obserables determine its cashflows.
  • How the observables are used to calculate the cashflows.
  • On what dates the cashflows will take place.

A product does not in general have any implementation of a valuation or knowledge of the models that could be used to value it.

Market Observables

These are quantities that can be observed unambigiously in the market and are referenced in contracts betweem parties to determine (either directly or indirectly) what the amount of money one party needs to the other.

Examples include:

  • A share price
  • A floating rate index
  • A value for the consumer price index
  • The value of one currency in terms of units of another currency, i.e. an exchange rate

Valuation Usage

A typical valuation model will follow this process to find the value of a product.

  1. The model will tell the contract after which date cashflows will be needed.
  2. The model will then ask the contract which market obserables it depends on.
  3. For each of these observables the model will:
    1. Ask the contract at which dates it needs to know the indices.
    2. By some means produce realizations of these indices and set them in the product.
  4. At this point the product knows the values of the market observables it cares about and can calculate its cashflows.

This same method can be used to get realized cashflows for settlement purposes.

Valuation Formula

In general any cashflow \( CF \) at time \(T\) valued at time \(t_0\) is a function of market obserbles \( S_i \) observed at times \( t_i \leq T \) will have the canonical valuation formula: $$V(t_0) = \mathbb{E} ^ \mathbb{Q} \left[ \frac{1}{B(t_0,T)} CF(S_1(t_1), S_2(t_2), ...) \right]$$ When there is some optionality or exercise decision in general the formula can become slightly less well defined. After the last possible exercise time the cashflows can be valued the same as above. Meaning we can use the above to find \( V(T_{En}) \) the value at the last exercise date \( T_{En} \)

Forward Valuation

When an expected positive exposure (EPE) or potential future exposure (PFE) is required then the expected forward value of cashflows in a contract is required. If the forward date is \( T_F \) then for a cashflow at \( T \geq T_F \) we have: $$V(T_F) = \mathbb{E} ^ \mathbb{Q} \left[ \frac{1}{B(T_F,T)} CF(S_1(t_1), S_2(t_2), ...)\middle| \mathcal{F}(T_F) \right]$$ To evaluate this one would need to perform another Monte Carlo simulation. But we know that this conditional expectation is \( \mathcal{F}(T_F) \) measurable and each of the market observables are functions of some underlying stochastic factors \( X_1(t), X_2(t), ...,X_m(t) \) so it must be possible to write: $$\mathbb{E} ^ \mathbb{Q} \left[ \frac{1}{B(T_F,T)} CF(S_1(t_1), S_2(t_2), ...)\middle| \mathcal{F}(T_F) \right] = f\left( X_1(T_F), X_2(T_F), ...,X_m(T_F) \right)$$