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One starts with the Black-Scholes equation $$\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}+ rS\frac{\partial C}{\partial S}-rC=0,\qquad\qquad\qquad\qquad\qquad(1)$$ supplemented with the terminal and boundary conditions (in the case of a European call) $$C(S,T)=\max(S-K,0),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\... 18 This is pure speculation: MFE's are really tailored toward valuation models (how can we develop a model to price x swap, etc.). You don't entirely have to worry about those details in order to trade them: you're just quoted a price based on these models. But if you go in-house at a bank and are working as a product quant (structured products, etc.), then ... 10 Martingales + Markovian Here is the motivation. Conditional expectations are martingales by the tower property of conditional expectations (an easy exercise to show). Suppose r=0, by the risk neutral pricing theorem E^\star\left[h(X_T)\bigg|\mathscr{F}_t,\,X_t=x\right] is the price of any derivative security with X as the underlying asset and payoff ... 9 Except in highly unusual cases, financial PDEs lack analytic solutions. The mathematical tools used are Monte Carlo, plus the usual ones for solving PDEs on grids, almost always one of the following: Trees, for very simple cases Explicit finite differencing, for throwaway projects or very specific cases Implicit or Crank-Nicolson finite differencing for ... 8 (1) You analytically solve a stochastic differential equation (SDE) using Ito's lemma. Your second equation (the discretized one) is how you could model one path over one step. To find the solution, you would model many of these paths over many steps and then take the expectation (i.e., Monte Carlo methods). The solution to the SDE models all of these paths ... 8 You can look at equity as a call option on the firm. In theory this illustrates the differences between holding equity or debt. The quick and dirty is that equity holders own the firm, but only after the debt holders are repaid. If you have a simple levered firm with one outstanding debt issue, it as though the equity holders have a call option on the firm ... 8 An equity represents ownership of a company and may be thought of as a derivative on the cash flow. For this reason, equities are valued through discounted cash-flow (DCF) analysis. An option is a right, though not an obligation, to buy or sell an asset at a fixed price at some point in the future. As per Black-Scholes, the value of an at-the-money option ... 7 You may want to look into these two open source projects: QuantLib which is aimed at providing a comprehensive software framework for quantitative finance. This is written in C++. JQuantLib the 100% Java implementation based on the first project. 7 The Feynman-Kac theorem primarily makes sense in a pricing context. If you know that some function solves the Feynman-Kac equation you can represent it's soluation as an Expectation with respect to the process. (confer this document) On the other hand a pricing function solves the FK-PDE. Thus often one would try solving the PDE to get a closed form ... 6 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 S_t from dS_t, you have to apply Ito's Lemma, see this question for details. This is the "classic" way you see it. If you want to do it the other way round,... 5 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 + \sigma \frac{\partial F}{\partial x}dW $$What does this equation actually represent? It suggests that a change in F (represented by \Delta F) equals a ... 4 The actual problem one solves for American options is an optimal stopping time problem, so the value of the option is$$ V_0 = \max_\tau E_{\tau}\left[e^{-r \tau} (S_\tau-K)^+ \right] where the maximum is taken over all stopping times (exercise strategies \tau>0 permissible in the contract). With a PDE operator such as you have, the instantaneous ... 4 I would start with explaining random walk (this should be fairly simple) and then making a connection to heat equation in discrete time. This paper is doing exactly this and by leaving out technicalities you should make this pretty intuitive for students. Basically the intuition is as follows: At each integer time unit, the heat at each point is spread ... 4 As far as I know, differential equations such as the Black-Scholes PDE are solved once analytically and then the result is used directly. If a given derivatives-pricing differential equation could not be solved analytically, it would probably be better to model it numerically using Monte Carlo methods than to derive a complicated PDE which must then be ... 4 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 \ge 0\} is a (\mathscr{F}_t, \mathbb{P})-martingale. On \mathscr{F}_T, we define the probability measure Q by \begin{align*} \frac{dQ}{dP}\big|_{\mathscr{F}... 3 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 Matlab's standard procedures don't perform well. In this case, a Levin-type method would perform much better. The reason that quadv produces NaN values is because ... 3 I think you provided the two that must be met. Pricing must be linear.V(\ldots, 2P) = 2 \cdot V(\ldots, P)$$And pricing must meet the law of iterated value. Where \tau \in (t, T)$$V_t(\ldots, \tau, V_{\tau}(\ldots, T, P)) = V_t(\ldots, T, P)$$These two laws must be met for any cash flow to prevent arbitrage. 3 (1) You can easily solve it in the case of constant coefficients. The answer will be \infty. In fact, this equation has no solution on any interval. The intuition is the following. For the SDE like$$ dS_t = \mu(t,S_t)dt+ \sigma(t,S_t)dw_t  you can mention that $dw_t = \xi_t\sqrt{t}$, where $\xi_t\sim\mathcal{N}(0,1)$ are standard gaussian i.i.d. random ...

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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 advection-diffusion PDEs. All we need is the Fourier transform: \begin{align*} \mathcal{F}[f] & = \int_{-\infty}^\infty e^{-i \omega y} f(y) dy, \end{align*} ...

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1) This last DE is implicit equation. This can't be solved analytically. I guess you can solve it by finite difference method. 2) The last term is indeed a differential term of order 1/2. However, it is the term for time difference and it can remain in the equation as it is. In the final formula as well it will come out to be as difference term, implying ...

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This question has been answered many times over already, though hopefully this will provide a bit more insight. If I understand your question correctly, you're basically asking if you can use BSM as a trading indicator. So let's think about what it really means to be trading an option. Every single variable (i.e. price of underlying asset, strike price, ...

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Physical equations tend to be forward equations, whereas in finance one deals with backward equations (e.g. Black-Scholes), so in my opinions analogies are a bit hard to make. The similarity is in the maths that you use, i.e. the PDE you need to solve.

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The way I think of it is that the PDE describes the flow of a time dependent probability distribution. The stochastic process describes individual realisations (random walks with a drift), but if you ran a large number of them you'd build up a distribution. The PDE says how that distribution changes in time (first term) due to deterministic drift (the ...

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As you have guessed correctly, these type of questions can be answered using Ito's Lemma.We have: $$d(M_t)= d(Z_t e^{\int_0^tF(Z_u)du})=d(Z_t) e^{\int_0^tF(Z_u)du}+Z_t d(e^{\int_0^tF(Z_u)du})+d(Z_t)d(e^{\int_0^tF(Z_u)du})$$ For the first two terms on R.H.S, we have: d(Z_t) e^{\int_0^tF(Z_u)du} = (f(W_t)dW_t + \...

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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 outdated. The backbone of affine models (such as SVJJ) is the characteristic function $\psi(u)$ of the log-price distribution, which is known in closed form. The ...

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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 http://en.wikipedia.org/wiki/Itō's_lemma In general every Markov process admits some kind of Ito's formula, known as Dynkin formula, which says that for a markov ...

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I am not sure any of the other answers mentioned this but the main reason you should not use an option model to buy/sell the underlying (BS or other) is that the option models are more about market-making in options and hedging using the underlying rather than forecasting the underlying. The layman way to understand this is that: using an option model, you ...

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This is basically how one would use BlackScholes to "purchase a stock"... double optionPrice = blackScholes(stock, strike, volatility, rate, time); if (optionPurchased) { if (stockPrice < thresholdPrice){ ExerciseOption(PurchaseStock()); } } Note that the purchase of the option would be on a futures exchange, not the regular stock ...

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I'm trying to understand what you are asking: you cannot use options directly, only buying stock thus, you want to use options as an indicator on whether or not you should buy a stock I don't think the BS model is a great indicator for stock direction because it's based of not knowing where the stock may go. That said, a lot of people use action in the ...

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As the BSM gives a call price as function of Stock price, volatility and other inputs, c(0) = BSM[Stock, Strike, volatility, riskfree rate, term], it seems to me you could use it in an analogous way to implied volatility (i.e., implied vol is the volatility input that produce a model output = traded market price). The analogous use, i think, would be to ...

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