Linked Questions

3
votes
3answers
406 views

Derivation of BS PDE problem using Delta hedging

I've always been confused with Delta hedging. It is well-known that for a (smooth enough) function of $(S,t)$ we have, due to Ito's lemma, that: \begin{eqnarray*} dC = \left(\frac{\partial C}{\partial ...
0
votes
2answers
809 views

Dynamic Delta Hedging And a Self Financing Portfolio

Let's assume the usual Black Scholes assumptions hold. My question is related to an answer on this question. There, the weights ($\Delta_t^1$,$\Delta_t^2$) are derived which form a locally risk free ...
9
votes
1answer
282 views

Replicating a portfolio with a certain payoff function

Assume there are two stocks $S_1$ with price $p_1(t)$ and $S_2$ with price $p_2(t)$ where $t$ indicates time. Assume, there is a hypothetical derivative $D$, which is such that, price of $D$ at a time ...
2
votes
3answers
272 views

Merton model riskless self-financing derivation

Suppose $dA_t = A_t[\mu dt+\sigma dW_t]$ (assets' value) under the physical measure, plus the other assumptions of the Merton model. Suppose further that debt and equity are tradeable assets that ...
3
votes
2answers
162 views

SDE for option value

Given an SDE for an underlying: $$dS(t) = \mu(S,t)dt+\sigma(S,t)dW(t)$$ the SDE for the value of the option $V=V(S,t)$ is given via Ito's lemma as: $$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\...
4
votes
1answer
144 views

Notion of risk-less portfolio in derivation of Black-Scholes

EDIT: As pointed out by Gordon in the comments, the portfolio I considered in my original post is neither self-financing nor (locally) risk-free. Though the central question is still open. Suppose ...
2
votes
2answers
191 views

What is a Short Option Hedging Portfolio?

In his book 'Stochastic Calculus for Finance II' Shreve uses the term: 'Short Option Hedging Portfolio' on page.156 (4.5.3). Can someone please explain this term with some kind of an example? It is ...
1
vote
1answer
88 views

Creating riskless portfolio in black scholes

$$\begin{align} d\pi &= \theta dV + dS \\[3pt] & = (\theta \partial V/\partial t + \theta \mu S \partial V/\partial S + \theta S^2 \sigma^2 \partial^2 V/2\partial S^2 +\mu S ) dt + (\theta \...
0
votes
1answer
80 views

Assumption in black scholes solution

Under the usual notations, In most textbooks on Quantative Finance, for deriving the Black-Scholes solution I find that authors, while setting up the riskless portfolio, assume that, $$\text{d} (\...
4
votes
0answers
58 views

Delta hedging and PF-value

Imagine buying a call option and shorting the delta. After some time $dt$, the stock price changes, and so does the delta and the call option value. We re-adjust our hedge using this new delta. ...