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A risk-neutral measure is a probability measure that yields an expected present value (discounted at the risk-free rate) which is equal to the current market price. The risk-neutral measure is also called an equivalent martingale measure.

1 vote

Market Price of Risk for Consumption Asset - Hull's Example 28.1

Note that just before Equation (28.9), Hull writes $-$ my emphasis: The market price of risk of [asset] $\theta$ measures the trade-offs between risk and return that are made for securities depend …
0 votes

Extending an incomplete market to generate a complete one

I believe I have found an exact answer to my question in Thomas Bjork's book, Arbitrage Theory in Continuous Time, on page 122 (third edition): Meta-theorem 8.3.1 Let $M$ denote the number of underly …
7 votes

Why discounted derivative price is a martingale?

Under a Black-Scholes framework, the dynamics of the stock price under the risk-neutral measure $\mathbb{Q}$ are given by ... $$ S_t = r S_tdt +\sigma S_tdW^{\mathbb{Q}}_t $$ ... and those of the ri …
3 votes

General Dynamics of a Tradable Asset under the Risk Neutral Measure

Our market has a tradeable asset $S$ and a risk-less money market account $B$, that is, the numéraire of the risk-neutral measure. We assume the following standard conditions, which are widely applica …
3 votes

Formal proof for risk-neutral pricing formula

You have two main papers that show this result: In a finite framework and in a somewhat simplified continuous framework, see Harrison & Pliska (1981), Corollary on page 228, Proposition 2.9 and Prop …
4 votes

Calculate $E^{\mathbb{Q}}\left[e^{-\int_{0}^{T_2}r_t\,dt} \frac{S\left(T_2\right)}{S\left(T_...

We assume a Black-Scholes world except the dynamics of the stock price, namely: No arbitrage opportunities. No dividend payments from the stock. Existence of a riskless asset yielding the risk free …
1 vote

How can I use the Radon-Nikodym theorem to show that forward measure is indeed measure?

Introduction Technically, I don't think you need the Radon-Nikodym theorem here. That theorem assumes the existence of two equivalent probability measures $Q_1$ and $Q_2$ and states that there must e …
2 votes

Proving a process is martingale under the Risk Neutral Measure

We define the process $Y_t=Y(t,S_t)$ as follows: $$Y_t=\left(\frac{S_t}{S_0}\right)^\lambda \exp\left\{-\left(r\lambda-\lambda(1-\lambda)\frac{\sigma^2}{2}\right)t\right\}$$ Method 1 Let: $$\alpha=\ …
1 vote

Why can only non-dividend paying assets serve as numeraire?

I am not familiar with the deep mathematical intricacies of advanced no-arbitrage theory, an extremely technical subject. However, from reading literature reviews, I suspect this is an historical lega …
2 votes

Log-normal risk-neutral price derivation from binomial trees, not clear about step in deriva...

Let $X\sim N(0,1)$ be a standard normal variable and $\alpha:=\sigma\sqrt{T}$, then by definition of the expectation and the distribution of normal variables: $$\begin{align} \mathbb{E}\left(e^{\alpha …
5 votes

Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$

Assume that: $$ S_0^1(1+r)\leq a,b $$ Arbitrage for a portfolio $V_t$ is defined as: $$V_0\leq0, \quad P(V_1\geq0)=1, \quad P(V_1>0)>0$$ Consider borrowing at rate $r$ to buy the risky asset such …
4 votes

Pricing a contract

The proof strategy consists on showing the quantity of interest is normally-distributed, then using the moment-generating function of a normal variable to obtain its third moment. Under measure $\math …
12 votes

How do we determine the "correct measure"?

Recall that any traded asset divided by a numéraire is a martingale under the measure associated to that numéraire. For the 3 interest rates you mention, the natural measure (namely the one that makes …