12 votes

How to derive the price of a square-or-nothing call option?

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 ...
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  • 20.4k
8 votes

How to derive the price of a square-or-nothing call option?

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{...
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  • 13.9k
6 votes

Contribution of an asset's variance to portfolio variance

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 ...
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  • 5,378
5 votes
Accepted

Boundary conditions Heston's stochastic volatility model

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 ...
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5 votes

Clarification on Paul Wilmott's derivation of Ito's Lemma

Given my earlier comment, the only open question is how $\frac{1}{2}\frac{d^2F(X(t))}{dX^2}\delta t$ becomes $\frac{1}{2}\int^{t+\delta t}_{t}\frac{d^2F(X(\tau))}{dX^2}d\tau\,.$ A more standard proof ...
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  • 1,419
5 votes
Accepted

Differential of integral of a stochastic process

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\...
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4 votes

Delta derivation from the expectation

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{\...
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  • 20.4k
4 votes
Accepted

How to derive Black-Scholes equation with dividend?

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*...
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4 votes
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Call option Delta

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-...
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3 votes
Accepted

Question on derivation step in portfolio replication under different borrowing and lending rates

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 ...
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  • 4,963
3 votes

Contribution of an asset's variance to portfolio variance

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 ...
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3 votes

Derivation of Call Delta from Black Scholes Model

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 ...
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  • 13.8k
2 votes

How to derive Black-Scholes equation with dividend?

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 ...
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2 votes
Accepted

Understanding the derivation of a ML-estimator

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, $\mathbf{1}'\mathbf{1} = T$ (...
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  • 2,310
2 votes

Derive an expression for the value of the asset as a function of time, V(t), t>=0

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 ...
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  • 2,108
2 votes

help with derivation of equation 8 in Derman and Kani's binomial tree for local vol

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 -...
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  • 4,963
1 vote
Accepted

How to get formulas for EWMA model with M-day records

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-\...
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1 vote

Delta-Gamma Neutral portfolio, derivation issue

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 ...
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