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 ...
- 21.1k
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{...
- 14.6k
7
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 ...
- 6,554
6
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 ...
- 15.9k
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 ...
- 2,023
5
votes
Derivation of Call Delta from Black Scholes Model
Here's a mathematical derivation of the Black-Scholes delta.
The call option price under the BS model is
$$ C = S_0 N(d_1) - e^{-rT} K N(d_2) \quad\text{with}\quad
d_{1,2} = \frac{\log(S_0\,e^{rT}/K)}...
- 91
5
votes
Accepted
Differential of integral of a stochastic process
Under some probability space $(\Omega,\mathcal{F},\Bbb{P})$ equipped with the (augmentation of the) natural filtration ${\bf{F}}=(\mathcal{F}_t)_{t \geq 0}$ of a $\mathbb{P}$-Wiener process $(W_t)_{t\...
- 14.6k
4
votes
Proving lognormality of security in Black-Scholes market
To prove this and similar transformations of securities we resort to Ito's Lemma. Let us define $f(t, S) = (S)^{1/3}$ with derivatives $\frac{\delta f(t, S)}{\delta t} = 0, \frac{\delta f(t, S)}{\...
4
votes
Accepted
Call option Delta
First note that delta is the derivative w.r.t. to the spot and not the strike. The latter is often called "dual delta". Also, you don't need any knowledge of Black-Scholes as this is a model-...
- 6,034
4
votes
Accepted
How to derive Black-Scholes equation with dividend?
We assume that the stock price process $\{S_t,\,t>0\}$ satisfies, under the real-world probability measure $P$, an SDE of the form
\begin{align*}
dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big),
\end{align*...
- 21.1k
3
votes
Accepted
Question on derivation step in portfolio replication under different borrowing and lending rates
Noting that
$$ B= V -\alpha S = V - (\alpha S)^+ + (\alpha S)^- $$
$$ = (V - (\alpha S)^+)^+ - (V - (\alpha S)^+)^- + (\alpha S)^-,$$
a clearer way to write the dynamics of the funding costs (funding ...
- 5,043
3
votes
Contribution of an asset's variance to portfolio variance
The Lagrangian 'solution' can yield negative contributions to portfolio risk, which is a bad look. An alternative definition is via the symmetric square root of the covariance, $\Sigma^{1/2}$. For ...
- 547
3
votes
Derivation of Call Delta from Black Scholes Model
Look here for a detailed derivation of the formula for $\Delta$ (be aware that this particular website uses $r_d$ to denote the risk-free rate and $r_f$ to denote the dividend yield). You can always ...
- 15.9k
2
votes
How to derive Black-Scholes equation with dividend?
The only difference in the derivation when you have a dividend-yield paying stock lies in the value of the Riskless Portfolio $\Pi_t$.
The financial meaning here is the key: to delta-hedge your ...
2
votes
Derive an expression for the value of the asset as a function of time, V(t), t>=0
Let's suppose $P$ is total annual deposits made continuously, then the change in value of total deposits $dV_t$ is (assuming no condition on additional deposits)
$$dV_t= V_t r dt + P dt $$
where we ...
- 2,238
2
votes
help with derivation of equation 8 in Derman and Kani's binomial tree for local vol
Let's start from (EQ 5) (introduce $w$ notation for wealth factor and $C_i$ for call price).
$$ 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 -...
- 5,043
1
vote
How to derive the CAPM from maximizing the Sharpe ratio?
I imagine that after 3 years you don't need to maximize the shape ratio anymore, but I actually stumbled across the same question when studying for the exam and thus ended up at your post here, so ...
- 11
1
vote
How to derive Dupire's local volatility?
First a comment, as discussed here the (correct) expression you have stated here is not the one stated and derived in Rouah.
As for calculating the derivatives, the best case would be if your ...
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-\...
- 6,204
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