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6

Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic function of the realised interest rate $R_t=\int_0^t r_s\text{d}s$ is \begin{align*} \varphi_{R_t}(u)=\mathbb{E}\left[e^{iuR_t}\right] = A_t(u)e^{B_t(u)r_0}, \end{...


4

A partial but general answer: Let $\mathcal{F}_t^W$ be the filtration generated by $W$. Since $X_T = \int_t^T v_u du$ is $\mathcal{F}_T^W$ measurable, the Clark-Ocone-Haussman formula states $$ X_T = E_t[X_T] + \int_t^T E_u \left[ D_u^W X_T \right] dW_u $$ with $D_u^W X_T$ denoting the Malliavin derivative of $X_T$ with respect to $W_u$. Hence, $$ Var(X_T) = ...


3

Ito's lemma is for twice differentiable functions of the form $f(t,W(t))$. You speak of $W(t)dW(t)$ - this is informal notation and doesn't have a mathematical meaning. Although once you put the integral sign, it becomes mathematically precise. So there's nothing known as $W(t)dW(t)$, but $I(t)=∫_0^{t}W(u)dW(u)$ is well defined via the definition of an Ito ...


0

The answer is that $S_t$ is a random variable which has realizations that can be solved for using a monte carlo or numerical methods. By solving for this value many times (which is what a monte carlo does for instance) you can find a distribution of prices of the underlying at a given time. This is because the process is random so each solve should be ...


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 ...


4

Duffie et al. (2000) show how to obtain the characteristic function of the log asset price in a fairly general affine jump diffusion model. Among others this includes the Black-Scholes (1973) model, the Heston (1993) model, the Bates (1996) model, the Merton (1976) model and the Kou (2002) model. This case also allows you to add stochastic interest rates. ...


2

$$ \lim_{x \to \infty} \frac{\sigma - \sigma \Phi(x)}{(\mu + \sigma x)\phi(x)} $$ $$\stackrel{0/0}{=} \lim_{x \to \infty} \frac{-\sigma \phi(x)}{\sigma\phi(x) + (\mu + \sigma x)\phi'(x) } $$ $$ =\lim_{x \to \infty}\frac{-\sigma \phi(x)}{\sigma\phi(x) - (\mu + \sigma x)x\phi(x) } = 0$$ (Used $\phi'(x) = -x\phi(x) $ and $\Phi'(x) =\phi(x)$, just like in ...


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Regarding the second question, if we assume that $A$ is lognormal (shifted lognormal) and imply its two (three) moments from respective spot rate components under the appropriate probability measure, then we have its probability density function $f_A$ which allows us to compute the expectation of the payoff the usual way: $$ E\left[\left(\frac{1}{K}-\frac{1}{...


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