Pricing Vasicek Bonds with State Price Deflators

# State price deflator in the Vasicek model


# State price deflator in the Vasicek model












I am trying to implement a simple bond pricing model using state price deflators in a Vasicek model. I am simulating paths of the processes

$$\mathrm{d}r^{P} =\kappa^{P}(\theta^P - r^P(t))\mathrm{d}t + \sigma\mathrm{d}W(t),$$ $$\mathrm{d}r^{Q} =\kappa^{Q}(\theta^Q - r^Q(t))\mathrm{d}t + \sigma\mathrm{d}W(t),$$

Where I use the relation between the risk neutral $\mathbb{Q}$-measure parameters and the real world $\mathbb{P}$-measure parameters in the Vasicek model given by

$$\kappa^{Q} =\kappa^{P} + \lambda_1 \cdot \sigma$$ $$\theta^Q = \frac{\kappa^P \cdot \theta^P - \lambda_0\cdot\sigma}{\kappa^Q},$$

where the parameters $\lambda_0$ and $\lambda_1$ define the market price of risk in a so called essentially affine model as proposed by Duffie and Kan, such that the market price of risk $\Lambda(t)=\lambda_0 + \lambda_1 \cdot r(t)$. For now, I considered only the case $\lambda_1 = 0$.

The state price deflator $\Pi(t)$ will then be the solution to the following SDE:

$$\frac{\mathrm{d}\Pi(t)}{\Pi(t)} = -r(t)\mathrm{d}t - \Lambda(t)\mathrm{d}W(t)$$

Now, having implemented a simulation loop for these processes, I would like to verify that the theoretical affine model zero coupon bond price

$$P(t,T) = e^{A(T)-B(T)r(t)}$$

coincides with my simulations, eg. I would like to verify that

$$P(0, 10)=E^Q\left[e^{-\int_0^{10}r(t)\mathrm{d}t}\right]=E^P\left[\frac{\Pi(10)}{\Pi(0)}\cdot 1\right]=E^P\left[\Pi(10)\right].$$ I do this by simulating $r$ under the risk neutgral $\mathbb{Q}$-measure for the exponential integral, and this coincides with the theoretical affine model zero coupon bond price $P(0,10)=e^{A(10)-B(10)r(0)}$.

Then, as to my understanding of state price deflators, I would need to use the $r$-process simulated under the real world $\mathbb{P}$-measure in order to calculate the zero coupon bond price using SPD, ie.

$$\frac{\mathrm{d}\Pi(t)}{\Pi(t)} = -r^P(t)\mathrm{d}t - \Lambda(t)\mathrm{d}W(t).$$

This does, unfortunately not yield the same result as discounting the risk-neutral $r$, which in turn yields the theoretical affine model bond price that is I get

$$P(0, 10)=E^Q\left[e^{-\int_0^{10}r(t)\mathrm{d}t}\right]\neq E^P\left[\Pi(10)\right].$$

Instead, I get the correct price using the $r$ process simulated under the $\mathbb{Q}$-measure, ie. using

$$\frac{\mathrm{d}\Pi(t)}{\Pi(t)} = -r^Q(t)\mathrm{d}t - \Lambda(t)\mathrm{d}W(t).$$

My question is: Is it correct that by using state price deflators for pricing, you use real world measure $\mathrm{P}$ for pricing, such that

$$E^Q\left[e^{-\int_0^{10}r(t)\mathrm{d}t}\right]=E^P\left[\Pi(10)\right]$$

At least, this is how I understood the link between risk neutral, real world and pricing kernels. What confuses me is that under my simulations, the equality

$$E^Q\left[e^{-\int_0^{10}r(t)\mathrm{d}t}\right]=E^Q\left[\Pi(10)\right]$$

seems to hold true. Am I doing something wrong? I have provided my code below:

```
for (int i = 1; i < steps; i++)
{
    // Simulating vasicek process
    double rQ = qRateProcess[i - 1];
    double rP = pRateProcess[i - 1];
    double dwP = randomGenerator.NextGaussian() * sqrtDt;
    double dwQ = randomGenerator.NextGaussian() * sqrtDt;
    double dwD = randomGenerator.NextGaussian() * sqrtDt;

    double drQ = KappaQ * (ThetaQ - rQ) * dt + Sigma * dwQ; // Risk neutral (Q-measure parameters)
    double drP = KappaP * (ThetaP - rP) * dt + Sigma * dwP; // Real world   (P-measure parameters)

    qRateProcess[i] = rQ + drQ;
    pRateProcess[i] = rP + drP;
    shortRateIntegral += qRateProcess[i] * dt;

    // Simulating deflator with (P-)vasicek interest rate process - should be the right method but is not?
    double deflator = deflatorProcess[i - 1];
    double dDeflator = -pRateProcess[i] * deflator * dt - deflator * (Lambda + LambdaY * pRateProcess[i]) * dwD;
    deflatorProcess[i] = deflator + dDeflator;
}
```