13
votes
Accepted
General way to solve Partial differential equation using Feynman kac representation
Let's skip to the stochastic differential equation (SDE):
$$ dF=\left[\frac{\partial F}{\partial t}+\mu \frac{\partial F}{\partial x}+\frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} \right]dt +...
- 1,829
7
votes
Accepted
How to get Black Scholes' Geometric Brownian Motion differential form form the closed form?
You derivation here is flawed because you are deriving with respect to two processes and you do not take into account that the variable $W_t$ is stochastic and hence $S_t$ is as well.
So, to derive $...
- 10.9k
7
votes
Transformation from the Black-Scholes differential equation to the diffusion equation - and back
I'd like to give an alternative derivation not involving the clever (mystifying?) transformation to the heat equation and thus present a more general technique for solving constant coefficeint ...
- 2,748
7
votes
The solution to arithmetic brownian motion
To solve the SDE you should use the so called variation of constant method.
Define a process $Y_t=e^{-\mu t}X_t$, so that using Itô we obtain:
$$dY_t=-\mu Y_t dt+ e^{-\mu t}X_t=e^{-\mu t} \sigma dW_t $...
- 493
6
votes
Accepted
Idea of using logarithm for solving SDE in Black-Scholes model
Black and Scholes (1973) were not the first ones to use the geometric Brownian motion as a model for stock prices. For example, Samuelson did it before them.
It all started with a Brownian motion as ...
- 13.8k
6
votes
Transformation of local volatility model
Consider a function $f(X_t)$. Ito's lemma gives:
$$df(X_t)=\text{time terms}+f'(X_t)\sigma(X_t)dW_t$$
Now any $f$ satisfying:
$$f'(X_t)\sigma(X_t)=\text{constant}$$
gives a constant volatility for $f(...
- 1,337
6
votes
Accepted
Transformation of local volatility model
Yes it is called the Lamperti transform. This document, in particular Theorem 2, page 7, describes what the Lamperti transform is.
- 4,871
5
votes
Accepted
Riccati Equation in spot rate model
As you noted, this is a Riccati type ODE and it can thus be simplified using the standard transformations for this class - see e.g. Wikipedia. We start by defining
\begin{equation}
C(t, T) = \frac{1}{...
- 5,800
4
votes
Accepted
Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
By Bayes' rule for conditional expectation (or here),
$$E_{\mathbb Q}[X_t | \mathscr F_u] E[L_T| \mathscr F_u] = E[X_tL_T| \mathscr F_u]$$
$$ \to E_{\mathbb Q}[X_t | \mathscr F_u] L_u = E[X_tL_T| \...
- 841
4
votes
Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$
We consider the case where the Novikov condition is satisfied, that is,
\begin{align*}
E\left[\exp\left(\frac{1}{2}\int_0^T \theta^2_s ds \right)\right] < \infty.
\end{align*}
Then $\{L_t \mid t \...
- 20.4k
4
votes
How to solve this PDE using Feynman-Kac?
Martingale Approach
As you noted, you need to solve
\begin{eqnarray}
F(0) & = & e^{-r T} \mathbb{E} \left[ \left( X_T - K \right)^2 \right]\\
& = & e^{-r T} \left( \mathbb{E} \left[ ...
- 5,800
4
votes
PDE for Pricing Interest Rate Derivatives
Note that
\begin{align*}
M(r_t, t) &\equiv Q(r_t, t) e^{-\int_0^t r_u du} \\
&=E\left(e^{-\int_0^T r_u du} h(r_T, T) \mid \mathscr{F}_t \right)
\end{align*}
is a martingale. Moreover,
\begin{...
- 20.4k
4
votes
Accepted
Pricing the Passport option
You maximize the terms in $q$ in the PDE because this is a consequence of the Bellman principle of optimality in dynamic programming. The intuition is that the global optimal strategy $\{q_t\}_{0 \leq ...
- 5,542
4
votes
Evaluating the SDE $dX_t = t\,dS_t$
Using Itô's Lemma, notice that:
$$d(tS_t)=tdS_t+S_tdt=dX_t+S_tdt$$
Hence:
$$X_t=tS_t-\int S_udu$$
Using independence of Brownian increments, $E(S_udW_u)=E(S_u)E(dW_u)=0$, and the chain rule for the ...
- 7,101
3
votes
Accepted
Market Making Strategies Found by Hamilton-Jacobi-Bellman Equation
At the terminal time $T$, the terminal condition is $g(T, q) = -\alpha q^2$, this implies,
$$
\begin{aligned}
g(T, q) &= \frac{1}{\kappa} \log{\omega(T, q)} = -\alpha q^2\\
\Rightarrow \omega(T,q)...
- 496
3
votes
Accepted
The PDE of caplet and floors
It must be a typo for the equation in the book. That is, the equation for a caplet is of the form
\begin{align*}
\frac{\partial V}{\partial t} + LV - r_t V +\max(r_t-r^*, 0) = 0,
\end{align*}
which ...
- 20.4k
3
votes
Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$
This is a corollary of Feynman-Kac theorem. For self-containedness,
I re-produce the proof as follows.
Assume that there exists a $C^{1,2}$-function $F=F(t,x)$ defined
on $[0,T]\times\mathbb{R}$ that ...
3
votes
How to understand the market price of risk
I think you misunderstood the underlying idea of the risk-neutrality and the market price of risk.
The basic idea is to price the option with a portfolio consisting of the underlying asset $S$ and ...
- 247
3
votes
Accepted
SVCJ (SVJJ) Duffie et. al Model implementation in Matlab
About the integration problem: Your integrand is highly oscillatory, and the adaptive quadrature of Matlab doesn't handle such integrands very well. In general, I would recommend Mathematica when ...
- 2,320
3
votes
Accepted
Differential of time over Browninan motion
Edits have been made, but the original question asked about $\frac{\mathrm{d}t}{W_t}$.
The random variable does exist
In your [original] question you ask about $\frac{\mathrm{d}t}{W_t}$, although I ...
- 1,359
3
votes
Accepted
What are the advantages and limitations of predicting future stock prices using stochastic differential equations?
The SDE you are describing is called the Geometric Brownian Motion. In the end its just a model, which underlies certain assumptions, which are usually not met in the real world scenarios. There are ...
- 342
3
votes
What are the advantages and limitations of predicting future stock prices using stochastic differential equations?
Take the analogy of equations modelling something in physics.
Just because you write down an equation, it does not mean it has to be connected to anything in reality. It only do so to the extent you ...
- 571
3
votes
Black Scholes to Heat Equation
Let
$n=\sigma^2(T-t)$
$$dn=-\sigma^2dt$$
$\frac{\partial {V}}{\partial {t}} = \frac{\partial {V}}{\partial {n}}\frac{d {n}}{d {t}}$
$\frac{\partial {V}}{\partial {t}} = -\sigma^2\frac{\partial {V}}{\...
- 356
3
votes
Accepted
Proof verification : risk free rate
You have:
$$r(t):=\theta+(r_0-\theta)e^{-kt}\tag{1}$$
Then:
$$r^\prime(t)=-k(r_0-\theta)e^{-kt}\tag{2}$$
which is clearly equal to:
$$k(\theta-r(t))\tag{3}$$
based on $(1)$.
- 7,101
2
votes
SVCJ (SVJJ) Duffie et. al Model implementation in Matlab
I would suggest that you use a more 'modern' method to recover option prices from characteristic functions.
The approach of this papers (for practical calculations of option prices) is somewhat ...
- 4,217
2
votes
How to get an analytic result for option price based on this model?
I didn't work out the explicit details, but you can reproduce Black&Scholes methodology using the Ito's formula for Jump Diffusions. See for example, the sectio about Poisson jump processes in ...
- 331
2
votes
Accepted
How to price the American style Asian option with recent N day average
In convertible bond pricing there is something similar called a "soft call" with similar properties so you might want to search for literature on them. The main difference is that soft calls are an ...
- 14.4k
2
votes
How to apply the chain rule for partial derivatives to transformations?
As you state in your comment, you only have trouble with the second partial derivative w.r.t. the spot. So you understand how the first partial derivative is obtained
\begin{equation}
\frac{\partial ...
- 5,800
2
votes
The PDE of the probability hitting the barrier before T
May be I have overlooked something, but I believe that
\begin{align*}
Q(t, S) = \mathbb{P}\left(\tau_{B} \le T \mid \mathcal{F}_t\right).
\end{align*}
Then $\{Q(t, S), \, 0<t < T\}$ is a ...
- 20.4k
2
votes
why futures contract has no value
The mathematical analysis above is correct, but to understand WHY we say that "a futures has no value" it is helpful to understand how a Futures Exchange works.
When you enter into a position (for ...
- 9,462
Only top scored, non community-wiki answers of a minimum length are eligible