36
votes
Explaining 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 $M$...
- 15.2k
21
votes
Accepted
Explaining the Risk Neutral Measure
Intro:
Great answer given by Kevin. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage&...
- 5,998
13
votes
Accepted
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 $d\langle W_s(...
- 21k
13
votes
Intuitive Explanation for Shannon's Demon?
Whether it's called volatility pumping, rebalancing premium, or Shannon's Demon it would just be a form of replicating a short gamma option strategy (eg. selling straddles). Intuitively, you are ...
- 3,595
12
votes
Accepted
How do we determine the "correct measure"?
Recall that any traded asset divided by a numéraire is a martingale under the measure associated to that numéraire. For the 3 interest rates you mention, the natural measure (namely the one that makes ...
- 7,989
12
votes
Heston stochastic volatility, Girsanov theorem
Consider the Heston (1993) model under the real world measure ($\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&...
- 15.2k
10
votes
Accepted
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 ...
- 7,989
10
votes
Accepted
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 ...
- 7,989
10
votes
Accepted
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^{...
- 216
9
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 ...
- 15.2k
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*}
\...
- 21k
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 ...
- 7,989
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 ...
- 27.3k
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 ...
- 1,396
7
votes
Accepted
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 ...
- 15.2k
7
votes
Intuition for Stock Price Numeraire Drift
I have a take on the intuition part of the question. Isn't it a simple consequence of Jensen's inequality? Thus, assuming $r=0$ for simplicity, we have in the money market measure: $E(S_T)=S_t$, ...
- 16.5k
7
votes
Explaining 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)
...
- 1,875
6
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*}...
- 21k
6
votes
Accepted
Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Bayes' rule for conditional expectation (or here) gives us
$$E_{\mathbb Q}[X_t | \mathscr F_u] E[L_T| \mathscr F_u] = E[X_tL_T| \mathscr F_u]$$
Use martingale property and iterated expectation:
$$E_{\...
- 921
6
votes
Accepted
Steven Shreve: Stochastic Calculus and Finance
Note merely that $B_t=B_s+(B_t-B_s)$ which is the sum of independent normally distributed random variables. In particular, $B_s$ is $\mathbb{F}_s$-measurable and $B_{t-s}$ is independent of $\mathbb{F}...
- 15.2k
6
votes
Accepted
Pricing call option using risk-neutral martingale approach with squared stock price boundary?
You do not really need the dynamics of $S_t^2$. You can simply apply your standard technique from risk-neutral pricing. The time zero price of a European-style contract with payoff $X$ is given by $$...
- 15.2k
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 ...
- 8,438
6
votes
Accepted
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 ...
- 2,570
6
votes
Accepted
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 ...
- 2,570
6
votes
Discounted price of an option
The process $Y_t:=(S_t-K)^+$ cannot be the price of a traded asset because of Jensen's inequality. Instead, it is the price of the option which is a martingale.
In the Black-Scholes model, the ...
- 7,989
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.
...
- 6,853
5
votes
Accepted
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 ...
- 7,989
5
votes
Accepted
How to prove martingality of forward rate under T-forward measure
For the instantaneous forward, please see the last page of this note: T-Forward Measure by Fabrice Douglas Rouah (http://www.frouah.com/finance%20notes/The%20T-Forward%20Measure.pdf).
For the simple ...
- 6,204
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 $s<T$:
$$E^T[Fo(t,T)|F_s] = E^T[E^T[S_T|F_t]|F_s] = E^T[S_T|F_s] = Fo(s,T)$$
(...
- 146
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