10
votes
Three mathematical mistakes in Black-Scholes-Merton option pricing?
Don't take this as an answer per se, but as mentioned in my comment more a summary of imo Bjork's clear explanation that hopefully can convince you there is nothing wrong with the BS PDE and self-...
- 1,739
9
votes
Accepted
Book/reference to practice stochastic calculus and PDE for interviews
You may like:
Probability and Stochastic Calculus Quant Interview Questions by Ivan Matić, Radoš Radoičić, Dan Stefanica
150 Most Frequently Asked Questions on Quant Interviews, Second Edition by Dan ...
- 12k
6
votes
Question on Merton's self financing derivation
Having to use a typewriter in 1970 Merton tried to find a notation that is as intuitive as possible at the risk of looking unrigorous at first glance. Since the advent of LaTeX it is easy to ...
- 2,003
5
votes
Three mathematical mistakes in Black-Scholes-Merton option pricing?
The Black-Scholes formula is the proper limit of the binomial formula, and there seems little doubt that the derivation of the binomial model is correct as is uses no math beyond a bit of algebra. It ...
4
votes
Accepted
Stochastic integral involving Poisson Process
Let
\begin{align*}
X_t=\left(\int_0^t f(s,N_{s-}) d\tilde{N}_s\right)^2
\end{align*}
and
\begin{align*}
Y_t = \int_0^t f(s,N_{s-}) d\tilde{N}_s.
\end{align*}
By It$\hat{\text{o}}$'s product rule for ...
- 21k
4
votes
On first and last zeros before t in a Brownian Motion
Intuitively speaking, you generally have an event for which you do not know when it occurs (the time of the occurrence of the event is random), but you do know that it will occur at some point in the ...
- 1,736
4
votes
Accepted
Smile wings and varswap pricing
The main concern is usually for the far wing where strikes are low.
Variance swaps have a theoretical replication. The fair variance swap strike $K_{var}$ is computed as
$$ K^2_{var} = \frac{2*e^{rT}}...
- 8,163
4
votes
Why is the stochastic process of the volatility of a stock price square integrable?
You can define the Ito integral without square integrability but this makes working with applications like pricing more complicated, so the assumption is typically made in practice.
The question of ...
- 1,707
3
votes
Integrated Brownian motion
Your derivation is correct. Even if we fix your obvious typo the formula
$$
\textstyle\int_t^TW_s\,ds=\int_t^T(T-s)\,dW_s
$$
is wrong.
There is no doubt that
\begin{align}
TW_T&=\textstyle\int_0^...
- 2,003
3
votes
Three mathematical mistakes in Black-Scholes-Merton option pricing?
There is no doubt that the the Black & Scholes formula for the European call
$$\tag{1}
C(S_t,t)=S_t\Phi(d_1)-e^{-r(T-t)}K\Phi(d_2)
$$
where
$$\tag{2}
d_{1,2}=\frac{\log(S_t/K)+r(T-t)\pm\sigma^2(T-...
- 2,003
3
votes
Vega hedge of a barrier option
Too long for a comment. I find Bergomi's sentence vague, so here follows an equally imprecise attempt at an answer.
A claim that can be statically replicated in a model-free manner is in fact immune ...
- 1,739
3
votes
Accepted
Volatility swaps hedging
Although this question seems Taylor-made for me, I shall resist promoting my own work and refer you instead to Carr and Lee's seminal paper Robust replication of volatility derivatives.
Basically what ...
- 1,739
2
votes
Accepted
Is this the correct discretisation of the Hull-White SDE for building a python model?
I am led to believe that your discretization is fine but uses too large of a step-size. This answer will tackle discretizing SDEs in general and then apply it to the Hull-White model.
The simplest and ...
- 225
2
votes
How is variance derived in BS?
Assume a flat (both in strike and time) volatility input, $\sigma$. Then, the variance a GBM accumulates from $t_0$ up to time $t_1$ is
$$
\text{Var}(t_0, t_1) = \sigma_{t_0}^2 (t_1 - t_0).
$$
Now ...
- 588
2
votes
How to hedge a dual digital option
Dispersion trading is a way to mitigate correlation risk. The book "Foreign Exchange Option Pricing A Practitioners Guide" (Chapter 10 Multicurrency Options) introduces an analysis framework....
2
votes
Accepted
Did I derive the Kelly criterion correctly?
Might be a typo but you dropped the $\alpha$ on the noise term after solving the SDE: in $\exp(...)$ you should have $\alpha \sigma W_t$ instead of $\sigma W_t$. For deriving the Kelly criterion, it ...
- 225
2
votes
Accepted
Deriving an Analytical Expression for Standard Deviation of Log Returns
You solution formula in the question is not correct, it should be
$$Y(t) = Y(0)\cdot \exp\left(\int_0^t\left(r(s)-\frac{1}{2}\eta^T(s)\cdot \mathbf{W}(s) \right)dt +\int_0^t\ \eta^T(s)d\mathbf{W}(s) ...
- 1,009
2
votes
Accepted
If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns?
If the returns follow Geometric Brownian Motion model then by definition the returns are independent of past returns. You are right in that returns are affected by past returns but there are models ...
- 444
2
votes
Can the PDE of Black and Scholes really be derived from the CAPM?
I was looking at this just this morning. It can be derived from CAPM, depending on what is meant by 'the market', in addition to some other (simplifying) assumptions which I write below:
So let's ...
- 1,739
1
vote
Accepted
Solving Equation for estimation risk averse parameter
You can think about it like this: given $\mu,\sigma,r$, a risk aversion parameter $\gamma$ will induce an optimal weight $w(\gamma)$, which in turn will induce some value at risk $VaR_{\alpha}$.
Hence ...
- 6,470
1
vote
Time-shifted power law in path dependent volatility
The time-shifted power-law kernels $K_1 (t)$ and $K_2 (t)$ assign a weight to past returns and squared returns, respectively. Each kernel is a function of the following parameters: lag parameter $\tau&...
- 1,406
1
vote
Accepted
Bloomberg FXFM: what is the point of knowing risk neutral probabilities?
Risk neutral probabilities are immensely useful. As you might know, they are the building blocks used to calculate derivatives prices across most asset classes. An entire industry is based on that.
...
- 16.6k
1
vote
Accepted
Mean level of the state variables under the risk-neutral measure in Arbitrage-free Nelson Siegel
The reason for setting $\theta^{\mathbb{Q}}=0$ is that, along with other restrictions, it identifies the parameters of the model uniquely, which means that the model is then well-defined, and there is ...
- 1,736
1
vote
What is the PDE for this interest rate derivative?
You can work out the specific PDE by applying the multi-dimensional Itô formula to the (sufficiently smooth) value function $\Pi_t = f(t, r, \sigma)$.
Follow the steps from the regular Heston PDE ...
- 41
1
vote
Construction of Itos integral
I would like add another answer for completeness, should someone stumble upon this in the future.
If $(X_t:t\geq 0)$ is simple adapted process, square-integrable defined by:
$$X_t = \sum_{j=0}^{n-1}Y_{...
- 208
1
vote
Milstein Scheme for Jump-Diffusion models
Those terms represent iterated integrals of the type
$$ \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} dW(z) dJ(s) $$
and
$$ \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} dJ(z) dW(s) $$
Which seem to be the third and ...
- 51
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