State-Price Deflators
Bridging the gap between market price consistency and real-world valuations


Stochastic modeling of non-linear financial products has become increasingly important over the last few years. The complex nature of the guarantees that exist in many financial products has generally required the use of computationally intensive Monte Carlo approaches. As this modeling has evolved, it has divided down two paths—real-world and risk-neutral. Still both paths can be re-united through the use of deflators.

Portfolios and balance sheets in general include products with embedded options that can be properly valued with standard risk-neutral valuation techniques. Still, determining the future value of such products (as required for regulatory or economic capital calculations) remains difficult as future outcomes are not simulated correctly. When using risk-neutral simulations, asset prices are assumed to grow with the risk-free interest rate, which is not realistic. When utilizing real-world simulations, future outcomes are correctly simulated since asset prices grow at the actual expected return (the risk-free rate combined with a risk premium), however valuations using a standard (risk-neutral) discount factor are inconsistent, since the returns are not risk-neutral but the discount factor is. Therefore real-world simulations are needed to correctly simulate future values of the variables, and a proper stochastic discount factor (the Deflator or SDF) is needed to calculate the present value of financial contracts. We explore the combination of these two methods in order to correctly simulate future outcomes, value the products and determine the uncertainty in their future value.

The Real World
In many stochastic applications, the requirement is to test the robustness of clients' portfolios, of products' designs (or even of businesses' strategies) and to quantify the range of possible financial outcomes. For this type of application, the scenarios must represent the real world. the outcomes for each scenario produced by the stochastic economic generator used must represent a path that could occur in the future. The range of the scenarios represents the population of possible future outcomes.

The Risk-Neutral World
The valuation or pricing of financial products, of options and guarantees, and increasingly of benefits, lines of businesses or companies requires a stochastic process for the full financial intricacies to be captured. For many applications, the fundamental requirement is that there is consistency with the techniques used to value or price assets—so that both sides of the balance sheet are consistent. As a consequence, there is a requirement that the economic scenarios are "risk neutral". When such scenarios are used, discounting the projected cash flows at the risk-free rate for each scenario and taking the mean gives a value or price that is consistent with a market valuation of the assets. There are differences between the two worlds, and one needs to make sure that the right type of generator is used in each application.

State-Price Deflators
Discounting the projection results from real-world scenarios does not yield market-consistent valuation (i.e. a valuation that measures riskiness in the same way as capital markets do). Fortunately, a solution exists: State-Price Deflators. Deflators bridge the gap between real-world and risk-neutral scenarios, so that they may be used to calculate market consistent valuations of any cash flow stream using real-world scenarios.

Technically speaking, deflators are path-dependent stochastic risk discount factors, and separate factors are associated with each real-world scenario. To summarize, their effect is to put a greater emphasis on those scenarios in which risky assets perform badly, therefore compensating for the positive risk premium associated with risky assets in real-world expectations. The riskiness and downside aversion that is experienced in the market valuation of assets is absorbed within the deflator values. This contrasts with risk-neutral valuations, where it is absorbed within the economic scenarios themselves.

Obtaining appropriate deflators is a complicated exercise. A stochastic economic generator is first developed to generate the deflator values alongside other simulated economic outcomes (interest rates, equity returns, inflation indexes, etc.). The key property of deflators is that their values are dependent only on scenario and time and are independent of the assets and liabilities to which they are applied. This means that they can be used to calculate a value on any stream of cash flows that varies according to the economic assumptions used. The market-consistent valuation of these cash flows is always the mean value of the deflated cash-flows.

The Model
The following model shows how real-world simulations and the deflator together are able to correctly value and simulate an asset price (a stock price in our example). The interest rate is considered to be constant, just as in the BSM model, and the process for the stock price (St) is simulated according to real-world expectations. Under these assumptions the stock prices grow on average with the actual expected return (with the risk-free rate (rf) plus a risk premium (π)).

The process for the stock price under a real world probability measure (P) is shown in equation (1a) and equation (1b) states this process under the risk-neutral probability measure (Q).

dSt = ( rf + π ) Stdt + σSt
dWt(P)     (1a)
dSt = ( rf ) Stdt + σStdWt(Q)           (1b)

The expected value of the stock at time (T), is obtained by applying Ito's lemma to equations (1a) and (1b).

(P) [St] = S0 exp(( rf + π - 1/2 σ2 ) T + σdWT(P) )     (2a)
E(Q) [St] = S0 exp(( rf - 1/2 σ2 ) T + σdWT(Q) )          (2b)

Discounting the value of the stock at time T with the proper discount factor should yield the initial value of the stock price (S0) under both probability measures. Under the risk neutral measure, this discount factor (DF) is well known, whereas the discount factor under the real-world probability measure can be found by applying Girsanov's theorem.

Girsanov's theorem states that changing a probability measure can be obtained by using the Radon-Nikodym derivative. Multiplying this derivative (L) with the discount factor under the risk-neutral measure yields a proper stochastic discount factor, or deflator. Using the correct (stochastic) discount factors, equation (3) should hold.

(Q)[St . DF] = E(P) [St . DF . L ] =  S0     (3)

The first part of this equation is under a risk-neutral measure and the second part under a real-world measure. Under the risk-neutral probability measure, equation (3) is known to hold and standard techniques can be applied to prove this equality. Though algebraically complicated, intuitively it is straightforward. The stock price grows with the risk-free rate under this measure and is discounted with the same risk-free rate, so that there is no change, see equation (4).

(Q)[St . DF] = E(Q) [(S0 exp((rf
- 1/2 σ2) T + σdWT(Q) )) exp( -∫0T( rSds )) |F0 ]     (4)

Under the real-world probability measure, equation (3) can also be proved to hold.
By taking the expectation of the discount factor times the Radon-Nikodym derivative (which equals the deflator or SDF) and by then substituting the result in equation (3) and once again applying Ito's lemma we obtain:

E(P)[St . SDF] = E(P) [(S0 exp(( rf-1/2σ2 )T +σdWT(P) ) exp( -rT -π/σ WT(P) -1/2(π/σ)2T)] = S0

The above shows that the valuation of a stock results in the correct value (S0) using both methods, even when using real-world simulations. Using the real-world probability measure, the stock grows at the actual expected rate of return. Therefore the discount factor should also be corrected for the resulting value to equal the value under the risk-neutral probability measure. This valuation method can be extended to any sort of product with or without embedded options. Furthermore the volatility of the stock can also be modeled in a stochastic manner, just as other economic random variables can be incorporated in the model.

The main advantage of the valuation under the real-world probability measure is that the correct value of a product can be obtained through the stochastic discount factor and the simulations can also be used for predicting realistic stock prices. Combining real-world simulations with a stochastic discount factor can be very useful for banks, insurers as well as portfolio managers. They can use this method to estimate the value of their products along their term structure and, more importantly, estimate the uncertainty in this value. 
In terms of risk management, normal practice dictates that capital calculations are based on a 99% one year Value at Risk (VaR). When using real-world simulations and a standard discount factor, estimated average values can be incorrect, which means that the VaR calculations may also be incorrect. When using risk-neutral valuation to estimate the VaR, only current market conditions are taken into account. Current market conditions are not necessarily a good measure for future outcomes, which could lead to incorrect VaR estimates. Combining the deflator with real-world simulations resolves these inconsistencies and results in more accurate VaR calculations.
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