Questions tagged [martingale]
The martingale tag has no usage guidance.
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Explaining the Risk Neutral Measure
What is the Risk Neutral Measure?
I don't believe this has been answered on the internet well and with all the parts connecting.
So:
What is the risk neutral measure/pricing?
Why do we need it?
How ...
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Why discounted derivative price is a martingale?
Usually after showing that discounted stock price process is martingale under the risk-neutral measure, most authors say that this implies that the discounted derivative price process is a martingale ...
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Intuition for Stock Price Numeraire Drift
I would like to ask whether there is an intuition for the drift of price processes under the Stock numeraire.
I find it intuitive that the martingale measure under the Money Market numeraire induces ...
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PDE for Pricing Interest Rate Derivatives
Suppose that interest rate $r(t)$ follows some short-rate models, say Vasicek, so that$dr = a(b-r) dt + \sigma dZ$, with constants $a,b,\sigma$.
It is well known that the price of zero-coupon bond $...
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Why is this stochastic integral a martingale?
Suppose that:
$W^*_t$ is a Wiener process under probability measure
$\mathbb{P}^*$ and;
$\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$.
In my lecture notes, it says that $\tilde{S}_t$ is a ...
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Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Edit years later: No idea why I'm upvoted. I actually am not sure how I'm correct. But maybe I haven't forgotten conditional expectation as much as I thought I have.
We are given a filtered ...
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Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
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Strictly local martingales: what is the intuition behind them?
A process $X_t$ is a local martingale if there exists an increasing sequence of stopping times $\{\tau_k,k=1,2,...\}$, with $\tau_k \to \infty$ almost surely, such that each stopped process is a ...
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What is a martingale?
What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?
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Is the stock price process a martingale or a Markov process?
Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property".
Are they unrelated? Is it that the Markov property implies some sort of ...
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Intuitive Explanation for Shannon's Demon?
I am reading Fortune's Formula by William Poundstone, and I am puzzled by a phenomenon called "Shannon's Demon", which Claude Shannon allegedly proposed in a series of lectures, and preserved only by ...
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Convexity Adjustment for Futures
Let $B_t$ be the cash account numeraire. The future and forward prices at time t are expressed as:
$$ Fut = E_t^Q\left[S_T\right],$$
$$ Fwd = \frac{E_t^Q[S_T/B_T]}{E_t^Q[1/B_T]}.$$
Where
$$ \frac{...
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Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
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FTAP a-la Harrison, Kreps and Pliska
I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some ...
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Why aren't american put options martingales?
I don't understand what's wrong in the following argument.
Assume that we have a no-arbitrage market where the following products are traded:
a risky asset $S$,
a risk-free bond $B$,
an American put ...
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Change of measure between T-forward and T*-forward contract?
I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation:
\begin{align*}
P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big).
\end{...
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How to prove martingality of forward rate under T-forward measure
Let $P(t,T)=\mathbb{E}_{Q_{R}}[e^{\int^{T}_{t}r(u)du}|\mathcal{F}_{t}]$ be the price of a 1-euro zero-coupon bond with maturity $T$ and $r(u)$ the interest rate process. Consider the the forward rate $...
user40884
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Martingale pricing with time-dependent risk-free rate
I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e.
$$
d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du}
$$
I ...
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Discounted price process - martingale
I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
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Calculate $E^{\mathbb{Q}}\left[e^{-\int_{0}^{T_2}r_t\,dt} \frac{S\left(T_2\right)}{S\left(T_1\right)}\right]$
Let $S\left(t\right)$ be a tradable financial security that doesn't generate cash flow (eg no dividend). $S\left(t\right)$ follows an unknown stochastic process.
We now have a financial derivative ...
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Conditional expectation of a geometric brownian motion
I'm reviewing stuff from the past and I'm very confused all of a sudden. Some verification would help about the following.
$$
\mathbb{E}[e^{\sigma W(t)}|{\cal F}_s] =
\mathbb{E}[e^{\sigma (W(t) - W(...
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Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$
Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $...
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Is the stock price process a martingale or a random walk in efficient markets?
What is the difference between RWH and EMH?
In efficient market, the price will be fully reflected by available information.
If there is no news, the price would be unchanged.
If there is a news, ...
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1
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Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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