28 votes

What is the Risk Neutral Measure?

Life Without a Risk-Neutral Measure How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$...
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  • 13.8k
17 votes

Strictly local martingales: what is the intuition behind them?

I think to understand the martingale/local martingale distinction, it helps to bring in a third class of processes, the uniformly integrable martingale. I would argue that the local martingale and ...
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16 votes
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Why is this stochastic integral a martingale?

In the integral $$\int_0^t S_u dW^{*}_u \, ,$$ $dW^{*}_u \equiv W^{*}_{u+du} - W^{*}_u$ is independent from the integrand $S_u$. So, $\mathbb{E}\left[ \int_0^t S_u dW^{*}_u\middle\vert \mathcal{F}...
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  • 1,118
13 votes
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Convexity Adjustment for Futures

We assume that, under the probability measure $Q$, \begin{align*} dS_t &= S_t\big(r_t dt + \sigma dW_s(t)\big),\\ dr_t &= -k\, r_t dt + \alpha dW_r(t),\tag{1} \end{align*} where $d\langle W_s(...
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  • 20.4k
13 votes

Intuitive Explanation for Shannon's Demon?

Whether it's called volatility pumping, rebalancing premium, or Shannon's Demon it would just be a form of replicating a short gamma option strategy (eg. selling straddles). Intuitively, you are ...
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  • 3,250
12 votes
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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 ...
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12 votes
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What is the Risk Neutral Measure?

Intro: Great answer given by KeSchn above. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no ...
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11 votes
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Intuition for Stock Price Numeraire Drift

As a general principle, I would be wary of economic or financial interpretations of change of measure techniques. Changing numéraires is merely a mathematical tool to ease pricing, see for example the ...
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11 votes
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Numeraire correlated to the traded asset

As @ilovevolatility explains, the seminal reference for this matter is Geman, El Karoui & Rochet (1995). We assume none of the assets are dividend paying, and they are strictly positive. There are ...
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10 votes

Why is this stochastic integral a martingale?

I aim to give a careful mathematical treatment to this answer, whilst following the fantastic book "Basic Stochastic Processes" by Brzezniak and Zastawniak. The reason I am putting this answer on is ...
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10 votes

Heston stochastic volatility, Girsanov theorem

Consider the Heston (1993) model under the real world measure ($\mathbb{P}$) \begin{align*} \mathrm{d}S_t&=\mu^\mathbb{P} S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}^\mathbb{P}, \\ \mathrm{d}v_t&...
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10 votes
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

First, let's check if these models are abritrage free. The first fundamental theorem of asset pricing says that if there exists an equivalent probability measure under which $\frac{S_t}{\beta_t} = e^{...
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8 votes

Intuition for Stock Price Numeraire Drift

The drift is the expectation of the return over an infinitesimal interval. Let $Q$ be the risk-neutral measure and $Q^S$ be measure associated with the stock price numeraire defined by \begin{align*} \...
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8 votes

Why aren't american put options martingales?

European Contracts It's a really important question and as @noob2 commented, the FTAP is normally applied to European-style derivatives, even if they are (strongly) path-dependent, including barrier ...
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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 ...
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7 votes

Intuitive Explanation for Shannon's Demon?

You may find the following paper worthwhile. It addresses most of the above points (and many more) in a systematic way: Dubikovsky, Vladislav and Susinno, Gabriele, Demystifying Rebalancing Premium ...
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7 votes

Measure theory in quantitative finance

Measure theory helps us overcome some of the drawbacks of constructing measures (measure of probability when ranged at $[0,1]$). Classic probability theory is effective for probability models whose ...
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  • 1,346
7 votes
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Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale

Let $(W_t)$ be a standard Brownian motion and $a>0$. We define $X_t=e^{aW_t-\frac{1}{2}a^2t}$. Then, the process $(X_t)$ is adapted and integrable which are the first two conditions of being a ...
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6 votes
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Steven Shreve: Stochastic Calculus and Finance

Note merely that $B_t=B_s+(B_t-B_s)$ which is the sum of independent normally distributed random variables. In particular, $B_s$ is $\mathbb{F}_s$-measurable and $B_{t-s}$ is independent of $\mathbb{F}...
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6 votes
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Pricing call option using risk-neutral martingale approach with squared stock price boundary?

You do not really need the dynamics of $S_t^2$. You can simply apply your standard technique from risk-neutral pricing. The time zero price of a European-style contract with payoff $X$ is given by $$...
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  • 13.8k
6 votes

What is the Risk Neutral Measure?

I believe the other answers are nearly exhaustive; but here's a bit of intuition I'd like to add: Think of the decision (= equilibrium price) of a market as: Decision = f(probabilities, risk aversion) ...
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  • 1,662
6 votes

Fama: Efficient Capital Markets: A Review of Theory and Empirical Work - are martingales incorrect?

The way I understand it is: In equation 2 $x_{j, t + 1}$ is defined as the change in of $p_j$ over the period $t$ to $t + 1$. The formula says that the expectation of the change is zero which is the ...
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  • 7,597
6 votes
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If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?

The market is complete iff there is a unique risk-neutral measure: when every contingent claim is attainable, its unique no arbitrage price is the cost of the replicating portfolio. In the case of an ...
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  • 2,260
6 votes
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Proving that a stochastic process is a martingale using Ito's Lemma

$$ d Y \left(t\right) := d \left[\int_0^t{a \left(s\right)\mathrm{d}W_s}\right] = a \left(t\right) dW_t $$ Note that since $Y$ is a driftless process, it is a local martingale, and because $a$ is ...
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  • 2,260
5 votes
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Difference betweem martingale property and adapted filteration

Let's consider a random process $X$. If $X$ is an adapted process, then we know, without any uncertainty, what its value is at the present time. This idea is formalized with measure theory. For $X$ ...
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  • 1,780
5 votes

Martingale representation theorem

The martingale representation theorem says that for any martingale $M$, there exists a unique stochastic process $\nu_t$ such that \begin{align*} M_t = \mathbb{E}(M_0) + \int_0^t\nu_sdW_s. \end{align*}...
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  • 20.4k
5 votes

Why discounted derivative price is a martingale?

We decree that $D_t$ has a certain process which makes it a martingale. In particular, we let $$ D_t = \mathbb{E} ( D_T \, | \, \mathcal{F}_t) $$ This is trivially a martingale by the tower law. ...
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5 votes
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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 ...
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5 votes

How to prove martingality of forward rate under T-forward measure

By definition, $$Fo(t,T)=E^T[S_T|F_t]$$ Note that expectation is taken under $T$-forward measure. Now, evaluating at $s<T$: $$E^T[Fo(t,T)|F_s] = E^T[E^T[S_T|F_t]|F_s] = E^T[S_T|F_s] = Fo(s,T)$$ (...
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