Numerical Example

Consider the ergodic Markov transition matrix \(\Pi\) with nearest neighbor priority persistence \[ \begin{align*} \Pi^{'} = \left[ \begin{array}{cccc} 0.5 & 0.2 & 0.1 & 0.1 \\ 0.3 & 0.5 & 0.25 & 0.175\\ 0.15 & 0.2 & 0.5 & 0.225 \\ 0.05 & 0.1 & 0.15 & 0.5 \end{array} \right] \hspace{16mm} \mu_{0} = \left[ \begin{array}{c} 0.22183 \\ 0.33100 \\ 0.27598 \\ 0.17117 \\ \end{array} \right] \end{align*} \] The columns of \(\Pi^{'}\) are conditional transition probabilities to each of \(N=4\) states as a function of the current state, which is indexed by the column number. The vector \(\mu_{0}\) is the invariant measure for this process: \(\mu_{0}^{'} \Pi = \mu_{0}^{'}\).

State Prices are Digital Options Prices

State prices are obtained from the list of levels of the numeraire good \[ X \ni x = \left[ \begin{array}{c} 1.05 \\ 2.05 \\ 3.05 \\ 4.05 \end{array} \right] \hspace{8mm} v = x^{-\gamma} = \left[ \begin{array}{c} 0.86384 \\ 0.11608 \\ 0.03525 \\ 0.01505 \end{array} \right] \] by calculating the marginal valuations. Valuations are quantified using CRRA preferences with risk aversion parameter \(\gamma =\) 3, shown in the vector \(v\) above. We also report logarithmic preferences as a benchmark for CRRA with \(\gamma \rightarrow 1\).

The pricing kernel contains marginal utility scaled probabilities that reflect the market prices of risk for these states \[ \begin{align*} \boldsymbol{M}_{t, 1}(x_{n}) &:= \left[\begin{array}{c} v(x_{1}, x_{n}) \pi(x_{1}, x_{n}) \\ v(x_{2}, x_{n}) \pi(x_{2}, x_{n}) \\ \vdots \\ v(x_{N}, x_{n}) \pi(x_{N}, x_{n}) \end{array} \right] \end{align*} \] The kernel is normalized to obtain the martingale measure \[ \widehat{\boldsymbol{M}}_{t, 1}(x_{n}) := \frac{1}{v_{n}}\boldsymbol{M}_{1, t}(x_{n}) \] with \(v_{n} := \sum_{x_{m} \in X} v(x_{m}, x_{n}) \pi(x_{m}, x_{n})\). The CRRA conditional martingale measures are \[ Q^{'} := \left[ \begin{array}{cccc} \widehat{\boldsymbol{M}}_{t, 1}(x_{1}) & \widehat{\boldsymbol{M}}_{t, 1}(x_{2}) & \widehat{\boldsymbol{M}}_{t, 1}(x_{3}) & \widehat{\boldsymbol{M}}_{t, 1}(x_{4})\end{array} \right] \] In our numerical example, we obtain \[ \begin{align*} Q^{'} = \hspace{4mm} \left[ \begin{array}{cccc} 0.91357 & 0.72179 & 0.63854 & 0.70717 \\ 0.07365 & 0.2425 & 0.21450 & 0.16629 \\ 0.01118 & 0.02945 & 0.13026 & 0.06492 \\ 0.00159 & 0.0063 & 0.0166 & 0.06162 \end{array} \right] \hspace{16mm} q_{0} = \left[ \begin{array}{c} 0.89150 \\ 0.09138 \\ 0.01473 \\ 0.00239 \\ \end{array} \right] \end{align*} \]

The valuation mechanism is strikingly evident from the invariant \(Q-\) measure \(q_{0}\). \(q_{0}\) contains the long-run risk-adjusted probabilities relative to the long-run physical probabilities \(\mu_{0}\).

Option Prices

If markets are complete, any option \(a \in \mathbb{R}^{N}\) can be priced by replication \[ \begin{align*} P_{a}(x_{n}) = a \cdot \boldsymbol{M}_{t, 1}(x_{n}) \\ = a \cdot Q^{'} 1_{n} \end{align*} \]

where \(1_{n} = 1_{x_{t}=x_{n}} \in \mathbb{R}^N\) indicates the current state by placing a one in its position in the lattice \(X\) and setting all other entries to zero.

Using the risk neutral probabilities \(q(x_{m}, x_{n}) = \tfrac{1}{v_{n}} v(x_{m}, x_{n}) \pi(x_{m}, x_{n})\), a put option paying \(x_{n-1}-x_{j}\) for \(j \leq n-1\) and a call option paying \(x_{j} - x_{n+1}\) for \(j \geq n+1\) are priced according to their expected payoffs \[ \begin{align*} \text{put}_{1}(n,n-1) = \sum_{X \ni x_{m} \leq n-1} q(x_{m}, x_{n})\left[x_{n-1} - x_{m}\right] \\ \text{call}_{1}(n, n+1) = \sum_{X \ni x_{m} \geq n+1} q(x_{m}, x_{n})\left[x_{m} - x_{n+1}\right] \end{align*} \]

Put and Call Option Conditional Dependence

Call options on procyclical levels are more responsive than puts near the money and more responsive than otehrwise identical longer maturity calls.