Questions tagged [martingale]
The martingale tag has no usage guidance.
1
vote
1answer
67 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
3
votes
1answer
44 views
How to prove martingality of forward rate under T-forward measure
Let $P(t,T)=\mathbb{E}_{Q_{R}}[e^{\int^{T}_{t}r(u)du}|\mathcal{F}_{t}]$ be the price of a 1-euro zero-coupon bond with maturity $T$ and $r(u)$ the interest rate process. Consider the the forward rate $...
1
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0answers
40 views
Unconditional Expectation vs. Conditional Expectation at time $0$
In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
2
votes
1answer
98 views
Fair price of a coupon paying bond
Consider a coupon paying bond with a maturity of $3$ years, that pays coupon annually. Let $c$ be the coupon rate (percentage) and let $F$ be the face value. This means that the holder of the bond ...
1
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0answers
45 views
What is the true “value” process of American derivatives?
Consider a continuous-time market where LOOP (law of one price) holds. The first fundamental theorem of asset pricing states explicitly that in the absence of arbitrage, the risk-neutral measure ...
2
votes
1answer
148 views
Risk neutral modelling of a stock
Suppose a stock $S$ follows
$$dS(t) = \alpha(t)S(t)dt + \sigma(t)S(t)dW(t),$$
where $W(t)$ is a Brownian motion under $P$. Also suppose there is a short rate process $r(t)$. My question would be is ...
2
votes
1answer
66 views
Finding the limit $\lim_{n \to \infty} P_0^n$ for a European Cash-or-Nothing put option with $P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$
Exercise :
Let $K>0$. A European Cash-or-Nothing put option $P$ has the following pay-out profile :
$$P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$$
Let $P_0^n$ be the no-arbitrage value at time $...
6
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0answers
106 views
Random variable minus Integral of Ito Generator is a Martingale under what conditions?
I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
3
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0answers
47 views
Martingale positive price process
I hope you can help me with this problem.
In my lecture notes, my professor stated that for a state price deflator $\phi\in L_{n+1}^2(P, F)$ (F being a filtration) and a strictly positive price ...
4
votes
1answer
84 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
0
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1answer
67 views
Risk neutrality coherence with risk aversion
I haven't been able to find an understandable explanation why the risk neutrality is coherent with the risk aversion implication of the expected utility hypothesis. I can see that when using the risk ...
3
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0answers
98 views
How to justify the martingale condition
By Radon-Nikodym theorem, the conditional expectation of $X$ with respect to a $\sigma$-algebra $\mathscr F$ is a nonnegative random variable denoted by $\def\E{\mathbf E}\E(X\mid \mathscr F)$, such ...
3
votes
1answer
83 views
Hedging Value-Financial Mathematics
EXERCISE
We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$
Let ...
3
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1answer
210 views
Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$
Exercise :
We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^...
2
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1answer
113 views
Showing that a market model has arbitrage and describing martingales
This is an exercise which I came upon while studying an introduction to financial mathematics.
Exercise :
Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let $\...
4
votes
1answer
258 views
Change of numeraire between T-forward and Bank Account
I follow a course, and get to the point that one bond price discounted by another one is a martingale:
$$
\frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale }
$$
I can not ...
3
votes
1answer
168 views
Libor Market Model (LMM) under risk neutral measure
I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get :
Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus:
$$ ...
3
votes
0answers
97 views
Equivalent martingale measure in time changed Levy models
I am investigating time changed Levy models. As far as I have seen, these models are usually directly described under the risk neutral measure $\mathbb{Q}$. However, I'm interested in first modelling ...
3
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0answers
32 views
Utility Maximization on a finite Probability Space. Possible mistakes in a paper?
I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the ...
9
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3answers
627 views
Measure theory in quantitative finance
When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to ...
5
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0answers
46 views
Martingale property of inhomogenous poisson process
I have found this martingale property for an inhomogenous poisson process with intensity $\lambda(s)$ which I don't know how to prove. The text itself advises: "proceed using Monotone class theorem".
...
2
votes
1answer
314 views
Calculating the stochastic integral of $\exp(-rt)S_t$
I am currently reading lecture notes which aim to show that if
$$
S_t = S_0 \exp (\mu t + \sigma W_t)
$$
then, under the probability measure $\tilde{\mathbb{P}}$ with density
$$
\gamma_T = \exp (c W_T ...
2
votes
1answer
86 views
Changing to martingale probability world
This question is really getting me annoyed and I'm struggling to do the final proof, I have no problem obtaining the adjustment to the drift rate necessary to collapse the drift term to make it a ...
1
vote
0answers
118 views
Martingale approach - Option pricing with Radom-Nikodym
I would like to get the price of an option which pays at time T the minimum between the logarithm of (S(1,T) / S(1,0)) and the logarithm of (S(2,T) / S(2,0), with the following processes:
(The two ...
1
vote
0answers
96 views
Hull Martingales and measures problem 27.16 7e?
Here's a question from Hull's Options Futures and Other derivatives which I'd appreciate if someone helped me to clarify. The question is from the chapter "Martingales and Measures"
Suppose that the ...
1
vote
0answers
66 views
Does pricing contingent claims under the EMM require us to define the distribution?
I am familiar with martingale pricing as primarily a notational abstraction which allows us to price contingent claims on $X_\tau$ by its conditional expectation. Usually, we interpret this to mean ...
0
votes
1answer
177 views
Equivalent Martingale Measure result Hull?
I've been reading Hull's chapter about Martingales and measures where he states that if you have the dynamics of two securities as follows:
\begin{align}
\frac{df}{f} = (r + \lambda \sigma_f) dt + \...
9
votes
2answers
1k views
Intuitive Explanation for Shannon's Demon?
I am reading Fortune's Formula by William Poundstone, and I am puzzled by a phenomenon called "Shannon's Demon", which Claude Shannon allegedly proposed in a series of lectures, and preserved only by ...
2
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0answers
138 views
Is the VIX a Martingale?
Say the S&P500 follows a Gaussian diffusion process, so that:
$$
VIX^2_{T,t}=\frac{1}{T}\mathrm{E}_t^\mathbb{Q}\left[\int_t^{t+T}\sigma_s^2ds\right]
$$
where $T$ is the tenor (assume fixed), $t$ ...
2
votes
1answer
256 views
Test if a process (with no drift) is a martingale
Consider the process
$$Z(t)=\int_{0}^{t} \frac{u^a}{t^a}dW_u$$ for some real constant $a$ and $W_t$ is a wiener process. I want to check whether this process is a $F_t^W$-martingale.
I noticed Lemma 4....
1
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0answers
123 views
Constant volatility and risk-free rate assumptions of Black Scholes
I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems ...
2
votes
2answers
116 views
The two fundamental theorems of Finance, as they relate to the martingale measure
I RECENTLY read this in an article by Battig and Jarrow, "the first fundamental theorem relates the notion of no arbitrage to the existence of an equivalent martingale measure, while the second ...
3
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0answers
54 views
Martingales with power-law tails and CLT
I'm writing a course paper on stable distributions and I couldn't find any source discussing limits of Martingales with power-law tails. Suppose we have a Martingale that produces IID observations at ...
3
votes
1answer
191 views
Equality under T-forward measure for convexity adjustment
I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,...
5
votes
1answer
315 views
Martingale measure result application for interest rates under T-forward measure?
I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality:
\begin{align*}
f_o = g_0 E^{g}\big(\frac{f_T}{...
1
vote
1answer
487 views
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{...
1
vote
1answer
303 views
Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
0
votes
0answers
573 views
Proof that integral of Brownian motion wrt time is not a martingale
Let $X_t=\int_0^t W_s ds$ where $W_s$ is Brownian motion, so $E[W_s]=0$.
Then $E[X_t]=\int_0^t E[W_s] ds=\int_0^t 0 ds=0$.
So $E[X_t|{\cal F}_s]=0\neq X_s$, almost everywhere. So by previous ...
4
votes
1answer
165 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$
with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$.
It is now often claimed that this ...
2
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0answers
35 views
Infinite Horizon Barrier Option Paradoxe [duplicate]
I've came across this question which is puzzling me. Imagine that interest rates are zero and that you observe a stock $S_t$ whose value today $S_0$ is equal to 1\$. We consider the derivative that ...
0
votes
1answer
113 views
Why calibration in $Q$ against option prices without showing that $Q$ is equivalent to $P$?
In practice, I have seen articles and financial textbooks on calibration of processes directly under the risk neutral world without showing that the measure is equivalent to a physical measure $P$. ...
2
votes
2answers
239 views
Pricing Equation for Best of Options
I am trying to derive a martingale pricing equation (closed form solution) for a best-of option. But I am getting stuck at a point.
There are 2 stocks $U(t)$ and $V(t)$ they both follow GBM with a ...
2
votes
2answers
158 views
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 ...
3
votes
2answers
226 views
Equivalent martingale measure price dynamics
Assume $S_0(t)=\exp(\int_0^t r(s) ds)$. Then $\mathbb{Q}\sim \mathbb
P$ is a martingale measure $\iff$ every asset price process $S_i$ has
price dynamics under $\mathbb Q$ of the form
$dS_i(t)...
5
votes
3answers
3k views
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 ...
2
votes
0answers
75 views
An arbitrage strategy involving forward contracts to show that LIBOR rates are martingales
I note $L_{t}^{[T_s, T_e]}$ the forward rate at time $t$ for the period $[T_s, T_e]$. Recall it is the strike making equal to $0$ the value at time $t$ of a forward contract for the period $[T_s, T_e]$...
3
votes
1answer
1k views
Girsanov Theorem application to Geometric Brownian Motion
I recently read this from a book on mathematical finance
The important example for finance the (unique) EMM for the geometric
Brownian. Let $S_{t}$ be the price of an asset, $${{d{S_t}} \over
{{...
3
votes
0answers
144 views
Martingale method for utility maximization - is the optimal strategy also a martingale?
The Martingale Method for utility maximization (seen in e.g. Björk's book) is based on separating the optimization problem $E^\mathbb{P}[U(X_T)]$ over a class of admissible strategies into the static ...
6
votes
1answer
3k views
Convexity Adjustment for Futures
Let $B_t$ be the cash account numeraire. The future and forward prices at time t are expressed as:
$$ 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{...
2
votes
1answer
78 views
What is the martingale measure requirement when $\mu(t,S(t)) = \mu(t)$?
It is known that if the numeraire is the bank account, then the martingale measure is determined by the fact that every asset has $r$ as its local rate of return.
However, the local rate of return is ...
