Asset Pricing Theory — Homework 1

Send individually to juan.imbet@dauphine.psl.eu before October 5, 23:59.

1 Problem 1 — SDF geometry in \(\mathbb{R}^2\)

Two equally likely states (\(\mathbb{P}=(\tfrac12,\tfrac12)\)). One asset with payoff \(x=(2,1)\) and price \(p=1.5\). No arbitrage holds. Use the pricing inner product

1.1 1) Find one SDF \(m=(m_1,m_2)\) that prices the asset

State the SDF pricing condition and identify one valid \(m\).

1.2 2) Plot the payoff line and all SDFs in the same space.

1.3 3) Decompose an SDF into parallel/orthogonal components to \(x\)

Define \(m_{\parallel}\) and \(m_{\perp}\) relative to \(x\) using the pricing inner product, and explain which component determines price.

1.4 4) Only the parallel component prices the asset

Show algebraically that only \(m_{\parallel}\) affects \(p=\langle m,x\rangle\) and interpret geometrically.

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2 Problem 2 — Risk-neutral probabilities in a one-step trinomial model

Consider an extension of the binomial model seen in class. At date 1 the stock takes \(\{uS,\,S,\,dS\}\) with \(u>1>d>0\). Risk-free asset: price \(1\), payoff \(1+r_f\). There is an ATM European call (strike \(K=S\)) with price \(C\). All three states are equally likely.

2.1 Tasks

1. Find the risk neutral probabilities in this economy.
2. Determine the bounds on \(C\) so that there are no arbitrage opportunities.