6

Consider a coin independently tossed 10 times. Assume under the measure $P$, $Pr(H)$ > 0.5 but not equal to 1. Let a risk neutral person be iteratively given the gamble between getting atleast $n$ heads versus atleast $n$ tails. Clearly the person always chooses getting atleast $n$ heads. Let $X$ represent number of heads and $Y$ the number of tails. Thus ...


4

Let $Y_t= e^{B_t}$ and $Z_t = B_{t}-t / 2$. Then, \begin{align*} dX_t &= Z_t dY_t + Y_t dZ_t + d\langle Y, Z\rangle_t\\ &=(B_{t}-t / 2)e^{B_t}\big( dB_t + 1/2\,dt \big) + e^{B_t}\big(dB_t -1/2\, dt\big) + e^{B_t} dt\\ &=e^{B_t}(B_t-t / 2+1)dB_t + e^{B_t}(B_t/2-t / 4 -1/2+1)dt\\ &=e^{B_t}(B_t-t / 2+1)d\big(B_t+1/2t\big). \end{align*} We ...


2

(Now that I saw Gordon's solution, I can finish my attempt; I had noticed $dB_t +1/2dt$ immediately from product rule for $V_t$, zero quadratic covariation between $t/2$ and $e^{B_t}$, but hours later :) I was still perplexed by $U_t$.) $$ X_t = U_t - V_t $$ $$ V_t = e^{B_t}t/2$$ $$ U_t = e^{B_t}B_t $$ $$ dV_t = \boxed{1/2e^{B_t} dt} + 1/2V_t(dB_t + 1/2dt) $...


1

The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions. I'm transcribing the abstract below: We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually ...


1

$N_t$ process comes with its own Poisson law (probability measure) $P$ defined via intensity $\lambda$. Under it, $N_t-\lambda t$ is a martingale wrt ${\cal F}_t =\sigma(N_u | u\in [0,t])$ (as $E^P[N_t]=\lambda t$ and $N_t-\lambda t$ has independent increments). Any other equivalent Poisson law, $Q$, defined via a given intensity $\gamma$, can be built using ...


1

Consider a radon nikodym derivative, the Random variable: $Z(w)= 1_{N(T,w)=1}+1-Pr(N(T)=1)$. It is admissible since it is always positive and has an expectation 1. This will lead us to the formation of an equivalent measure, which I will denote by $'$. We start with the easy theorem that $E'(X)=E(XZ)$ for any random variable X, and RND $Z$ $E'(1_{N(T)=1})=E(...


1

It sounds to me like you understand everything apart from how Girsanov's theorem defines the EMM. Girsanov's theorem tells us that if $B_t$ is standard Brownian motion under $P$, then for any adapted process $\gamma_t$ (satisfying certain conditions) the process $\hat{B}_t$ defined by: \begin{equation} d\hat{B}_t = \gamma_t dt +dB_t \end{equation} is ...


1

I try to clarify. First, let us be under the real world probability measure $\mathbb{P}$. Say that the process $Y_t$ is a $\mathbb{P}$ standard Brownian motion. Then the following is true by definition (assume wlg. that s < t): $(Y_t - Y_s) \sim N(0, t-s)$ Now, using your definition of $W_t$: $\mathbb{E} \left[(W_t - W_s)\right] = \mathbb{E} \left[(B_t ...


1

Let $X_{T_1}$ be a random quantity known (fixed) at $T_1$ (measurable wrt $T_1$-information), $B$ be the standard bank account and $P$ standard zero-coupon bond price. From standard pricing, for the second contract: $$E_t\left[B_t B_{T_2}^{-1} \left(X_{T_1} - F_2\right) \right] =0 $$ implies $$ F_2= E_t\left[B_t B_{T_2}^{-1} X_{T_1}\right] P(t,T_2)^{-1},$$ ...


1

Note that just before Equation (28.9), Hull writes $-$ my emphasis: The market price of risk of [asset] $\theta$ measures the trade-offs between risk and return that are made for securities dependent on $\theta$. Additionally, some lines below $-$ my emphasis: Chapter 5 distinguished between investment assets and consumption assets. An investment ...


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