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8

The following paper gives you really all of the missing steps in a very detailed form: A Complete Solution to the Black-Scholes Option Pricing Formula by Ravi Shukla and Michael Tomas From the paper: "This presentation is purely for pedagogical purposes. In the course of doing work on option pricing, we found no complete solution for the Black-Scholes ...

8

It is important to note that he says: "In the risk-neutral world, $\frac{C(t,S_t)}{B_t}$ is a martingale." That is true by definition of what the risk-neutral measure is, also called martingale measure for exactly that reason. A risk-neutral measure is defined such that asset prices deflated by the numeraire (unit with which prices are measured) are ...

7

First, my notation. $K$ is the strike price, $S$ is the stock price, $r$ is the continuously compounded risk-free rate, $T$ is time at expiration, $t$ is time at issue, $\sigma$ is volatility, $\delta$ is continuously compounded dividend rate. The Black-Scholes formula for a European call is $C = Se^{-\delta (T-t)} N(d_1) - Ke^{-r(T-t)} N(d_2)$ $d_1 = \... 7$C= S_0 N(d_1) - K e^{-rT} N(d_2)C$,$S_0$and$K$have units of currency (e.g. USD).$N(d1)$and$N(d_2)$are unit-less (dimensionless), the formula is dimensionally correct. Considering,$d1 = \frac {ln{\frac {S_0} K} + r T + \frac {\sigma^2} {2} T} {\sigma \sqrt T }r$and$\sigma^2$have units of "per year", as they are stated on an annualized ... 6 What you need is to identify the distribution of the asset price$S_T$, conditional on the information set$\mathcal{F}_{t}$at time$t$, for$0\leq t < T. Note that \begin{align*} S_T &= S_t \exp\bigg(\int_{t}^T \Big(r_s-\frac{\sigma_s^2}{2}\Big)ds + \int_t^T\sigma_s dW_s \bigg). \end{align*} Let \begin{align*} P(t, T) = \exp\bigg(-\int_t^T r_s ds ... 5 TimeT$boundary condition is correct$u(T,x)=(x-K_1)^+-(x-K_2)^+$. Time$x\to 0$boundary condition is known and is equal to$0$. Time$x\to\infty$boundary condition is also known and is correct$\lim_{x\to\infty}u(t,x)=(K_2-K_1)e^{-r(T-t)}.$You need to be precise if you want your boundary be "absorbing" or "reflecting". 5 You miss the cross-derivative term in the Ito formula you use to express$d\left ( \frac {C_t}{S_t} \right)$. More specifically (see [Remark] below), $$d\left ( \frac {C_t}{S_t} \right) = \frac {1}{S_t} dC_t - \frac {C_t}{S_t^2} dS_t + \frac {C_t}{S_t^3} d\langle S_t, S_t \rangle {\color{green}{- \frac {1}{S_t^2} d\langle C_t, S_t \rangle}}$$ This last ... 5 What you need to do is to first make a variable change such as$u = \frac{x-\xi}{2\sqrt{t}}$. Then change the order of the limit and the integral. 5 (1) No, the stochastic differential equation for Heston model does not have an explicit solution. What does exist is an explicit formula for the Fourier transform of a call option price. See e.g. http://www.zeliade.com/whitepapers/zwp-0004.pdf for a decent survey. (2) Yes, implied vol always exists. You can check that the Black-Scholes price of an option ... 5 The PDE is defined for$x \in ]-\infty, +\infty[$but the finite difference scheme requires a truncated domain$[x_{\min}, x_{\max}]$, and the choice of$x_{\min}$and$x_{\max}$will affect the quality of the result, regardless of the scheme being explicit, implicit, or mixed. A good rule of thumb is to choose the truncation$[x_{\min}, x_{\max}]$such ... 4 According to wikipedia one chooses$\pi_t=-V_t+\frac{\partial V}{\partial S}S_t$. This means, you are shorten$V$and long$\frac{\partial V}{\partial S}$shares of$S$. The general theory of self-financing strategies assumes that your market consists of a$\mathbb{R}^{d+1}$process$S$, with$d$risky assets and one risk free (bank account). A trading ... 4 The following paper gives a simple derivation of the BSM (via a simple integration approach instead of the classical PDE approach) and the Greeks plus some intuition for each: Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Formulas by Garven, J. You find the derivation of the Greeks in chapter 4 (called "comparative ... 4 When a pay-off is piecewise linear plus jumps, it the same as the portfolio of calls and digital calls. Its price must agree with that of the portfolio by no arbitrage. Every time there is a jump we add in a digital call and every time there is a change in gradient we add in calls equal to the gradient change. Here we have a call struck at$K$. Just below$...

4

In short answer, Yes: the backward PDE solution with $v(t,L)=0$ and the expectation coincides under the Black-Scholes market. In the one dimensional case, this topic is mathematically treated in the theory of the scale function and the spead measure. See Revez-Yor 3rd.ed. Ch.VII.3 for details. I don't know whether there are some rigorous theories on the ...

4

The B/S PDE for a contingent claim $V(S, t)$ is $$\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial V^2}{\partial S^2} - r V = 0$$ subject to the terminal condition $V(S, T) = \ln(S / K)$. According to the hint, the solution to $V(S, t$) takes the form V(...

4

Ikonen and Toivanen don't say that the LCP is solved exactly, they simply say that the modified back-substitution is a valid algorithm to solve the LCP. A numerical error may arise around the location of optimal exercise, since it does not fall directly on the finite difference grid. I think that however, the error is of the same order as the discretization ...

4

Under the standard assumptions, generally speaking, any contract that depends on the current values of t and S, and which are paid for at the start satisfy this PDE. In the financial context, the boundary conditions would be different for the different contracts, so the solutions of the PDE would be different for the different contracts. Thus different ...

4

These are well known trivial solutions to the Black-Scholes PDE. The first one is just the price of the underlying stock and the second is interest bearing money in a bank. These are trivially true because there is no optionality involved (which is expressed in the boundary and terminal condition of the respective contract to price).

4

The option pricing formula must satisfy the PDE you have derived for all values of $K$. The only way this can be the case is if the two parts that you separate are both equal to zero. Suppose the joint PDE (before you separate it into two) is satisfied for some value of $K$. But suppose that the $Q$-part in parentheses on the second line is not zero. ...

3

Something is off in your plot. The value of a call should be very near zero with a strike price $10$ for the stock prices and times you have plotted. At first I thought you may have plotted "moneyness" defined as $S/K$ instead of $S$, but then your values are too low for that. May want to check your implementation. Besides that, the plot is telling you ...

3

Who gave you that idea? You absolutely can use Finite Differences for other PDEs. They are routinely used to solve hyperbolic PDEs (wave equation, both first and second order) and elliptic PDEs (steady state diffusion/heat equation). You can even mix and match the equation types and create PDEs that have characteristic of both hyperbolic and parabolic ...

3

Yes it can be done. However, bear in mind that a naive explicit FD scheme is not unconditionally stable (see CFL stability condition). As far as your initial/boundary conditions issue is concerned: [Time domain] Use terminal condition $V(S,T)=h(T)$ where $h(T)$ figures the payoff of the target derivative claim at maturity $T$ (e.g. $h(T)=(S-K)^+$ for a ...

3

In the Black-Scholes world, it is assumed that the option value $C(t, S_t)$ is replicable by an admissible self-financing trading strategy $\phi$, where $\phi_t=(\alpha_t, \beta_t)$. That is, \begin{align*} C(t, S_t) = \alpha_t B_t + \beta_t S_t, \end{align*} and \begin{align*} dC(t, S_t) = \alpha_t dB_t + \beta_t dS_t. \end{align*} Since $dB_t = rB_t dt$, ...

3

I believe the setup of the first part you presented is inaccurate. The whole point of the hedge argument is that you can setup a self-financing portfolio that only holds a certain amount of stock and invests/borrows at a specific financing rate. It can be shown that such portfolio almost surely has the same payoff as the option at maturity. The option ...

3

The above equation is the price of a call option. It has nothing stochastic inside it. It only depends on the current price and the time. So no Ito is needed. You should just compute the derivatives of your solution v (like you do for any deterministic multivariable function), plug them into the PDE and verify that it's satisfied.

3

You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think: A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff: $$V_t=\alpha_tS_t+\beta_t B_t$$ It further implies $$dV_t=\alpha_tdS_t+\beta_tdB_t$$ To replicate a derivative $C(S_t,t)$ by a self-...

3

It depends on the type of payoff you want to price. If it is a call option, you know that $V(0,t) = 0$ and $V(x,t) \approx x$ when $x \rightarrow +\infty$ so you can use a dirichlet condition $V(a,t) = 0$ and $V(b,t) = b$. Alternatively you can use a linear condition $\frac{\partial^2V}{\partial x^2} = 0$ which in practice works fine for a variety of payoffs....

3

The key point here is that the portfolio must be self-financing, namely the initial option premium $V_0$ should be enough to allow you to hedge it throughout its life. If not, the option price $V_0$ is either too low or too high. Because the option is written on the asset $S$, buying or selling $S$ is how you neutralize the changes in value of the option: ...

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