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# Questions tagged [martingale]

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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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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 $... • 11 15 votes 2 answers 20k views ### Why is this stochastic integral a martingale? Suppose that:$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$is a ... • 840 9 votes 1 answer 601 views ### Prove$E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$given$Y_t$is a martingale Edit years later: No idea why I'm upvoted. I actually am not sure how I'm correct. But maybe I haven't forgotten conditional expectation as much as I thought I have. We are given a filtered ... • 923 5 votes 2 answers 2k views ### Uniqueness of equivalent martingale measure in Black Scholes-Model Let's consider standard Black-Scholes model with price process$S_t$satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where$B_t$is standard Brownian Motion for probability$\mathbb{P}$. I ... • 53 37 votes 5 answers 9k views ### 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 ... • 4,337 27 votes 6 answers 21k views ### What is a martingale? What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis? • 2,707 27 votes 5 answers 17k views ### 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 ... • 763 17 votes 2 answers 9k 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 ... • 3,015 12 votes 1 answer 9k views ### Convexity Adjustment for Futures Let$B_tbe 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{... • 326 10 votes 2 answers 4k views ### 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}, ... • 151 8 votes 1 answer 281 views ### 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 ... • 2,633 6 votes 1 answer 931 views ### 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 ... • 61 4 votes 3 answers 2k 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{... • 282 3 votes 3 answers 4k 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 ... 2 votes 1 answer 939 views ### 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} $$I ... • 181 2 votes 1 answer 538 views ### Discounted price process - martingale I have a process S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}, where X_t is a Levy process and I want to check for which m the process e^{-(r-q)t}S_t is a martingale. The third condition of a ... • 443 2 votes 2 answers 192 views ### Calculate E^{\mathbb{Q}}\left[e^{-\int_{0}^{T_2}r_t\,dt} \frac{S\left(T_2\right)}{S\left(T_1\right)}\right] Let S\left(t\right) be a tradable financial security that doesn't generate cash flow (eg no dividend). S\left(t\right) follows an unknown stochastic process. We now have a financial derivative ... • 483 2 votes 2 answers 2k views ### Conditional expectation of a geometric brownian motion I'm reviewing stuff from the past and I'm very confused all of a sudden. Some verification would help about the following.$$ \mathbb{E}[e^{\sigma W(t)}|{\cal F}_s] = \mathbb{E}[e^{\sigma (W(t) - W(... • 255 1 vote 1 answer 303 views ### Prove uniqueness, and proveY_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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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 ...