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22
votes
1answer
3k views

What is the role of stochastic calculus in day-to-day trading?

I work with practical, day-to-day trading: just making money. One of my small clients recently hired a smart, new MFE. We discussed potential trading strategies for a long time. Finally, he expressed ...
13
votes
1answer
4k views

Transformation from the Black-Scholes differential equation to the diffusion equation - and back

I know the derivation of the Black-Scholes differential equation and I understand (most of) the solution of the diffusion equation. What I am missing is the transformation from the Black-Scholes ...
10
votes
3answers
734 views

Deterministic interpretation of stochastic differential equation

In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by ...
9
votes
3answers
1k views

What tools are used to numerically solve differential equations in Quantitative Finance?

There are a lot of Quantitative Finance models (e.g. Black-Scholes) which are formulated in terms of partial differential equations. What is a standard approach in Quantitative Finance to solve these ...
9
votes
3answers
2k views

Is there an intuitive explanation for the Feynman-Kac-Theorem?

The Feynman-Kac theorem states that for an Ito-process of the form $$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$$ there is a measurable function $g$ such that $$g_t(t,x) + g_x(t, x) \mu(t,x) + ...
8
votes
1answer
420 views

Connections between random walk and heat equation (Material for ~)

I am preparing an undergraduate lecture in quantitative finance and I am looking for material that combines the topics: random walk and heat equation The material should be accessible ...
7
votes
2answers
290 views

Why does Black-Scholes equation hold on continuation region of American Option?

Explanation for Put Option: $ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0 $, where $\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S ...
6
votes
10answers
2k views

Using Black-Scholes equations to “buy” stocks

From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? ...
6
votes
2answers
211 views

How to get an analytic result for option price based on this model?

I defined such a model for stock price (1).... $$dS = \mu\ S\ dt + \sigma\ S\ dW + \rho\ S(dH - \mu) $$ , where $H$ is a so-called "resettable poisson process" defined as (2).... $$dH(t) = ...
4
votes
1answer
310 views

An equation for European options

So, any European type option we can characterize with a payoff function $P(S)$ where $S$ is a price of an underlying at the maturity. Let us consider some model $M$ such that within this model ...
4
votes
2answers
227 views

Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?

Both the Black-Scholes PDE and the Mass/Material Balance PDE have similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the ...
2
votes
1answer
129 views

General way to solve Partial differential equation using Feynman kac representation

Consider the following PDE on interval [0,T] $\left(\frac{\partial F}{\partial t}(t,x)+\mu (t,x)\frac{\partial F}{\partial x}+\frac{1}{2}\sigma^2(t,x)\frac{\partial^2F}{\partial ...
2
votes
1answer
87 views

How to get Black Scholes' Geometric Brownian Motion differential form form the closed form?

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that: $$S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu dt + ...
2
votes
2answers
243 views

SVCJ (SVJJ) Duffie et. al Model implementation in Matlab

I'm attempting to implement aforementioned SVCJ model by Duffie et al in MATLAB. so far without success. It's supposed to price vanilla (european) calls . parameters provided, the expected price is: ...
1
vote
0answers
73 views

PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
0
votes
0answers
42 views

Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation

Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation and show that the transformed payoff is g(x,τ) = be^[(1/2)((k+1)^2)τ+(1/2)(k−1)x]H(x),  ...