12

I provided an answer, based on an elementary approach, to an exactly same question yesterday. However, that question has disappeared, even though I like to keep a record for what I wrote. I would suggest that people do not delete their questions as they may be helpful for others. Here, I re-post that answer. We assume that, under the risk-neutral measure, ...


8

See this excellent paper by @MarkJoshi which defines/discusses the use of power numeraires. Starting from a dynamics specified under the risk-neutral measure $\mathbb{Q}$ \begin{align} &\frac{dS_t}{S_t} = (r-q) dt + \sigma dW_t^{\mathbb{Q}}\\ \iff& S_T\ \vert\ \mathcal{F}_t = S_t e^{(r-q-\frac{\sigma^2}{2})(T-t) + \sigma(W_T-W_t)} \tag{EQ.0} \end{...


6

In this answer, I am assuming that you want to keep correlations constant. To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written as $$ \Sigma = \mathbf{SRS} $$ where $\mathbf{S}$ is a diagonal matrix of the simple volatilties $\sigma_i$, and $\mathbf{R}$ is the correlation matrix. Thus in ...


5

You can't really derive or prove boundary conditions. You impose them and try to economically motivate them. Let's consider a European-style call option and go through the boundary conditions step by step. $S=0$ When the underlying asset's value is zero, then the option to buy this asset is worthless. Thus, $$C(t,S=0,v)=0.$$ $S\to\infty$ As the underlying ...


5

Under some probability space $(\Omega,\mathcal{F},\Bbb{P})$ equipped with the (augmentation of the) natural filtration ${\bf{F}}=(\mathcal{F}_t)_{t \geq 0}$ of a $\mathbb{P}$-Wiener process $(W_t)_{t\geq 0}$, consider the Itô process $$ X_t = X_0 + \int_0^t \mu(s) ds + \int_0^t \sigma(s) dW_s \tag{1} $$ for some sufficiently well-behaved functions $\mu$ ...


4

Since $Y=e^{(r-\frac{\sigma^2}{2})\tau + \sigma \sqrt{\tau}Z}$, then \begin{align*} xY > K \Leftrightarrow Z > -d_2, \end{align*} where \begin{align*} d_2 = \frac{\ln \frac{x}{K} + (r-\frac{\sigma^2}{2})\tau}{\sigma\sqrt{\tau}}. \end{align*} Consequently, \begin{align*} e^{-r\tau}\mathbb{E}\big(Y \mathbb{1}_{\{xY >K\}} \big) &= e^{-\frac{\sigma^...


4

We assume that the stock price process $\{S_t,\,t>0\}$ satisfies, under the real-world probability measure $P$, an SDE of the form \begin{align*} dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big), \end{align*} where $\{W_t, \, t >0\}$ is a standard Brownian motion. Here, we need to consider the total return asset $e^{qt}S_t$, that is, the asset with the dividend ...


4

First note that delta is the derivative w.r.t. to the spot and not the strike. The latter is often called "dual delta". Also, you don't need any knowledge of Black-Scholes as this is a model-independent result. The result follows from the general expression of the call price \begin{equation} C_0 = e^{-r T} \mathbb{E}_\mathbb{Q} \left[ \left( S_T - K \right)...


3

Noting that $$ B= V -\alpha S = V - (\alpha S)^+ + (\alpha S)^- $$ $$ = (V - (\alpha S)^+)^+ - (V - (\alpha S)^+)^- + (\alpha S)^-,$$ a clearer way to write the dynamics of the funding costs (funding account by funding account, the borrowing one, the lending one, and the one that funds at lending rate corrected by stock borrowing fees) is: $$dB = r^l (V - (\...


3

The Lagrangian 'solution' can yield negative contributions to portfolio risk, which is a bad look. An alternative definition is via the symmetric square root of the covariance, $\Sigma^{1/2}$. For portfolio $\vec{w}$ define $$ \vec{r} = \Sigma^{1/2}\vec{w}. $$ The norm of $\vec{r}$ is the volatility of the portfolio. Moreover, this definition is equivariant ...


3

Look here for a detailed derivation of the formula for $\Delta$ (be aware that this particular website uses $r_d$ to denote the risk-free rate and $r_f$ to denote the dividend yield). You can always ask for more specific help regarding a particular step in the derivation. It is easy to see that $\mathbb{Q}[\{S_T\geq K\}]= \Phi(d_2)$. Just replace $S_T=S_0\...


2

The only difference in the derivation when you have a dividend-yield paying stock lies in the value of the Riskless Portfolio $\Pi_t$. The financial meaning here is the key: to delta-hedge your option you buy a quantity $\Delta$ of the stock $S$, and only the stock is paying you the dividend, so you have to add this contribution in time to your hedge. The ...


2

I agree with vanguard2k's comment: A few more details on the notation would be helpful. But, as far as I can tell, the second equality is a simple expansion. First, $\mathbf{1}'\mathbf{1} = T$ (assuming the vectors are elements of $\mathbb{R}^T$). The expression $\mathbf{1} (\mathbf{1}'\mathbf{1})^{-1} \mathbf{1}'$ is therefore nothing else than a $T\times ...


2

Let's suppose $P$ is total annual deposits made continuously, then the change in value of total deposits $dV_t$ is (assuming no condition on additional deposits) $$dV_t= V_t r dt + P dt $$ where we assumed $r$ is constant. Solving above differential equation, we have: $$V_T = V_0 e^{rT} + \frac{P}{r} (e^{rT} -1)$$ Assuming $t_1$ is the time period at ...


2

Let's start from (EQ 5) (introduce $w$ notation for wealth factor and $C_i$ for call price). $$ wC_i = \lambda_i (F_i -S_i)(S_{i+1}-S_i)^{-1} (S_{i+1}-s_i) +\Sigma $$ I have used (EQ 3) $p_i = (F_i -S_i)(S_{i+1}-S_i)^{-1}$. This is equivalent to: $$ wC_i (S_{i+1}-S_i)= \lambda_i (F_i -S_i) (S_{i+1}-s_i) +\Sigma (S_{i+1}-S_i)$$ $$ wC_iS_{i+1} -wC_iS_i=\...


1

Let's try the mean formula, and you can then apply the same logic to variance and covariance. We have: $\mu_t=\left(1-\lambda\right)r_{t-1}+\lambda \mu_{t-1}$ Which means: $\mu_{t-1}=\left(1-\lambda\right)r_{t-2}+\lambda \mu_{t-2}$ $\mu_{t-2}=\left(1-\lambda\right)r_{t-3}+\lambda \mu_{t-3}$ Now we can take the first formula: $\mu_t=\left(1-\lambda\...


1

let $\frac{\partial C}{\partial S}=\delta_c$ let $\frac{\partial^2 C}{\partial S^2}=\Gamma_c$ let $\frac{\partial C_0}{\partial S}=\delta_0$ let $\frac{\partial^2 C_0}{\partial S^2}=\Gamma_0$ we want $\frac{\partial V}{\partial S}=\frac{\partial C}{\partial S}=\delta_c$ and $\frac{\partial^2 V}{\partial S^2}=\frac{\partial^2 C}{\partial S^2}=\Gamma_c$ ...


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