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You don’t just need self-financing in a risk-neutral world but it’s a much more fundamental principle. If you look at a portfolio that is not self-financing, i.e. you can inject or withdrawal funds at any time, you can hedge any derivative easily. If you can always add the amount of money you need, then hedging becomes trivial. Thus, one requires the self-...


4

I think it is far easier to understand by just drawing the payoffs. You have two put options: A European put option on a non-dividend paying stock with strike price 80 is priced at 8 dollars, and a put option on the same stock with strike price 90 dollars is priced at 9 dollar The difference between the two payoffs is equal to 10 dollars (90 strike puts ...


4

In practice, the self-financing condition can be regarded as an economic consequence of market competition. Take the perspective of an investment bank trading in hedgeable derivatives. If the hedging strategy is not self-financing, then it must be either: Generating cash outflows for the bank. It is therefore uneconomical for the bank to trade this product; ...


2

This was proved by Vasiceck in his 1977 paper. If you suppose that the price of a pure discount bond depends only on a markovian short rate $r(t)$ with SDE \begin{equation} dr(t)=\mu(t,r(t))dt + \sigma(t,r(t))dW(t) \end{equation} then you can assume that $P(t,T)=F(t,r(t);T)$. Now, with similar arguments used in the derivation of the Black-Scholes formula, ...


2

My question now is what further adjustments do these stressed curves need? One subject that comes to mind is the "no arbitrage-ness" of the curve. Do I have to make sure that the curve does not present arbitrage opportunity? If so, how? No there is no such thing as arbitrage arising from a risk free zero curve. Any zero rate you get is by definition the ...


2

There is the following paper by Roger Lee and Dan Wang on Displaced Lognormal Diffusion implied volatility. They give some no arbitrage bounds. It could be more useful for interest rates than normal volatilities as you may actually want to bound your rates from below. Lee & Wang, Displaced Lognormal Volatility Skews, 2009


2

No arbitrage means that you can't have a portfolio with a positive expectation without risk. let's suppose that the value of option with jumps is lower than $C_{BS}(0,S_0)$ Please consider the following portfolio at time 0: Sell BS hedge on option with jumps with price $C_{BS}(0,S_0)$. Buy option with jumps with price $P_J(0,S_0)$. Keep the rest of money $...


2

It really simplifies your life when dealing with valuation. As stated already a non-self-financing portfolio either generate or absorb cashflow. Such cash flows would need to be taken into account when valuing a certain derivative based on replication. So, in general, is way easier to just deal with a self-financing portfolio. On the other hand, when you ...


1

If there is no interest rate, the european and american put prices are the same for every strike. More details can be found in my answer for the question below: Longstaff Schwartz Algrorithm in R


1

Basically, if a contingent claim is replicable its value today is the value of the replicating strategy. If you suppose that the market is complete and that there are two equivalent pricing measures $\mathbb{Q}^1$ and $\mathbb{Q}^2$, the price of a claim $A$ is given either by $\mathbb{E}^{\mathbb{Q}^1}\left( A \right)$ or by $\mathbb{E}^{\mathbb{Q}^2}\left(...


1

You can find the answers to most of your questions in the Taylor's series and/or approximation theory articles, but I will add a bit more detail below (in order): A simplistic example would be $y=a+bx$ vs $z=bx$, so greeks being equal does not necessarily mean that the prices will be equal. But you can use hedging/replicating argument, though it needs more ...


1

If $X$ is a lognormally distributed variable, $X = e^{\mu + \nu Z}$, so $\ln X$ has mean $\mu$ and variance $\nu^2$, and $Z$ is normally distributed, then $$ E\left[ X^n \right] = e^{n\mu + \frac{1}{2} n^2\nu^2} $$ This solves your question with $X = S_T/S_t$, $n = \beta$, $\mu = (r -\frac{1}{2} \sigma^2) (T-t)$ and $\nu = \sigma \sqrt{T-t}$ in the Black ...


1

It essentially boils down to: same random variable, different probability measures. So when you set u and d, you fix the values that the random variable can take. Probability Measure does not change that- it only re-weights the probability in a way. The probability $p_1$ and $p_2$ are the probabilities of the two states under the P(physical) measure, and ...


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The simple explanation is that in the absence of calendar spread arbitrage, we should observe monotonic option prices with respect to maturity. And option prices are monotonic with respect to increase in volatility. Let $(X_t)_{t \geq 0}$ be a martingale, $L>0$ and $0\leq t_1, t_2$, then we have $$E[(X_{t_{2}} - L)^{+}] \geq E[(X_{t_{1}}-L)^{+}]$$ for ...


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