# Questions tagged [martingale]

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### Martingale under risk neutral probability

I have a question to prove martingale under risk-neutral measure: Question Consider a discrete time process $S$, which at time $n \in \mathbb{N}$ has value $S_n = S_{0}\prod_{j=1}^{n}Z_j$, $S_0>0$ ...
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### multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
1 vote
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### Discounted price of an option

If the discounted price of any asset is a martingale under risk neutral measure, why is $E^Q[e^{-rT} (S_T-K)_+ | F_t]$, not merely $e^{-rt} (S_t-K)_+$? This is something I wanted to clarify, since ...
1 vote
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### Minimal entropy martingale measure and Bayes estimated under Kullback-Laibller divergence loss function

We know that no unique equivalent measure exists in an incomplete market. Therefore, we need to choose a pricing measure equivalent to the physical measure based on a criterion. One typical approach ...
23 views

### integral of adapted process with respect to semimartingale is a martingale

Fix $T > 0$ a finite time horizon. Let $H$ be an adapted (or progressively measurable, if needed) continuous process and S be a continuous semi martingale, both on $[0,T]$. Under what conditions is ...
180 views

### Is this an optimal stopping problem?

I am trying to work out how to approach a machine learning problem of 'learning' an optimal liquidation time/threshold, under some conditions, from historic data. The idea is a trader armed with this ...
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### Standard Brownian Motion and Exponential Martingale calculation [closed]

Let $W(t)$ be a standard brownian motion and let $Z(t) = \exp (\lambda W(t) - \frac{1}{2}\lambda^2 t).$ In Xinfeng Zhou's Green Book in the section on Brownian Motion (p.130) he writes as part of the ...
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### Expectation of Bt^4 given BS [closed]

What is the expectation of Bt^4 and Bt^3 given Bs? Given t>s. I understand that the expectation of Bt given Bs is Bs and that the expectation of Bt^2 given Bs is something like Bs - s + t.
1 vote
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### 4th Order Brownian Motion Martingale [closed]

I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
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### First known reference using martingale theory to derive BS formula

What is the first known paper which derives the Black-Scholes valuation formula for an option (1973) using martingale machinery - instead of PDEs?
1 vote
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### What happens trying to price derivatives starting from a non-geometric brownian motion?

To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question. The set-...
183 views

1 vote
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### Why the Esscher transform is the right transform for pricing formula?

A Wiener process has infinitely many states of the world at any time step. Does that not mean that there are infinitely many EMM's for any model that uses the Wiener process? But then if there is only ...
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### Why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?

why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire? For example, if I have the price of a forward price $f_t$ and a ...
102 views

### Is this term structure model valid? (Modeling the Zerobonds directly)

Let us define the dynamics of the discounted Zerobonds as $$\tilde{P}(t,T) = \int \sigma(t,T) dW_t + \tilde{P}(0,T)$$ Lets assume $\sigma(t,T)$ is s.t. $\tilde{P}(t,T)$ is a martingale and positive (...
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### Proving the discounted stock price is martingale

Let $\mathcal{K}_s$ be $$\mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$ where $\tilde{V}_t(\theta)$ is the discounted value process of the self financing ...
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### Solving an SDE using Ito's Lemma

Suppose that $Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$ with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process There is also ...
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### Martingale proof: Call-prices must be increasing in maturity

I have observed that IV is increasing with time to maturity by using market prices and plotting IV (from Black-Scholes) against log-moneyness, $\log(S_t/K)$. $S_t$ being the price of the stock at time ...
215 views

### Poisson process under equivalent martingale measure

I have a stochastic process $N(t)$ which is equal to $n$ with probability $P\{N(t) = n\}=\frac{\left(\lambda t \right)^{n}}{n!}e^{-\lambda t }$ where $t$ represents the time period. In other words, ...
1 vote
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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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### Efficient market hypothesis and martingales

One of the tasks in the book we´re using in introduction to finance is Stocks are expected to earn (much) more than the risk-free interest rate. This means that stock prices are expected to increase ...
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### Why do stock prices follow a martingale?

I have a quick question: why does the Efficient Market Hypothesis (EMH) assume that stock prices follow a martingale process? I understand that discounted prices under the risk-neutral probability ...
1 vote
228 views