Questions tagged [derivation]
The derivation tag has no summary.
34 questions
0 votes
3 answers
280 views
How to derive a valuation formula for an "Asymptotic" Call option?
Suppose you have an "Asymptotic" Call option with a payoff equal to $ K * max[1 - K/S_{T}, 0] $, where $K$ is the strike price and $S_{T}$ is the underlying asset's price at maturity. How ...
- 21
0 votes
0 answers
85 views
How do I derive the effective rate per period for backward looking overnight interbank borrowing?
The formula for the effective rate per period (of say $n$ days) for backward looking overnight interbank *borrowing has been quoted as follows in my textbook: $$ \left( \prod_{i=1}^{n} \left(1 + \frac{...
1 vote
0 answers
192 views
How to adjust an assets position to target volatility in a long-short portfolio?
I have a portfolio of weights $\mathbf{x}$ where some positions in $\mathbf{x}$ are short s.t. $\Sigma_i x_i=0$ (dollar neutral). The standard way to estimate the volatility contribution per asset is ...
- 435
4 votes
0 answers
342 views
Balland - SABR goes normal
To summarise this very long post : please help me understand the undetailed proof of the quoted paper. I am not comfortable using a result I do not fully understand. I am reading Balland & Tran ...
- 2,750
6 votes
0 answers
421 views
In-depth derivation of implied volatility in the SABR model
I'm working through the derivation of Hagan's formula (Hagan et al, 2002) for the implied volatility of an option in the SABR model. I'm finding it pretty confusing. Most of my hang-ups are coming ...
3 votes
1 answer
163 views
Proving lognormality of security in Black-Scholes market
Can someone prove that for some security $S_t$ with drift $\mu$ and volatility $\sigma^2$ in a Black-Scholes market we have that $Y_t = (S(t))^{1/3} \sim \text{Lognormal}$, w.r.t. the risk-neutral ...
- 61
1 vote
0 answers
84 views
Hedge return of foreign asset
Given that $S(T)$ is the value of an asset in a foreign currency, $X(T)$ is the spot domestic/foreign rate, $P_t$ is the value of a Portfolio in the the domestic currency (invested in $S$) and $z^{f}$ ...
0 votes
1 answer
419 views
Clarification on Paul Wilmott's derivation of Ito's Lemma
I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
- 101
2 votes
1 answer
103 views
Question on derivation step in portfolio replication under different borrowing and lending rates
I'm currently trying to understand the derivation of a pricing PDE on a european claim that considers stock lending fees: https://cs.uwaterloo.ca/~paforsyt/hjb.pdf In Appendix A.2, the author talks ...
- 133
3 votes
1 answer
1k views
Boundary conditions Heston's stochastic volatility model
I'm trying to derive the following boundary conditions for heston's stochastic volatility model. This is p. 289 of Shreve's Stochastic calculus for finance \begin{align} c(T, s, v) &=(s-K)^{+} \...
- 33
0 votes
0 answers
207 views
Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution
\begin{align} \arg \min_w \enspace & -w^\top \mu \\ \mathrm{s.t.} \enspace & 1_N^\top w = 1 \\ & w_i \geq 0 \enspace \forall i=1,\dots, N \end{align} is the optimization problem for ...
- 3,140
5 votes
2 answers
1k views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$? I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
- 3,140
6 votes
0 answers
165 views
$\frac{\partial C_{BS}}{\partial T}$ in local volatility derivation in terms of implied volatility
In Gatheral's book, in the derivation of local volatility in terms of implied volatility, we use the regular Dupire formula $$ \frac{\partial C}{\partial T} = \frac{1}{2} \sigma^{2}K^{2}\frac{\partial^...
- 129
2 votes
1 answer
189 views
help with derivation of equation 8 in Derman and Kani's binomial tree for local vol
in this paper "The Volatility Smile and Its Implied Tree" - Derman and Kani 1994 i understand the derivation of all equations up to 7. But eq 8 i cannot figure out how to derive! i have ...
- 796
0 votes
0 answers
164 views
What is the differential Value-at-Risk?
I am currently working on a Machine Learning Project, implementing portfolio optimization algorithms according to different risk measures. I have found sufficient information on Sharpe Ratio ...
- 1