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6

By definition, the payoff of a log-contract of maturity $T$ writes $$ \phi(S_T) = \ln\left(\frac{S_T}{S_0}\right) $$ Let $\Pi_t$ denote the $t$-value of such a contingent claim. We are interested in the price at $t=0$, best known as the option premium. Theory tells us that the latter premium can be computed as $$ \Pi_0 = e^{-rT} E^{\mathbb{Q}} \left[ ...


5

Time $T$ 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".


3

This formula is used for replication of certain payoffs, for example, the log-payoff in Variance replication using options. The value of $\kappa$ can be set to any number, for example, $\kappa=E(S_T)$. This is a decomposition of the payoff, which is not a valuation of the payoff itself, and then further valuation is still needed. For example, based on the ...


3

well solve for the value of $q$ that makes the value of the stock divided by the bond be a martingale. You will find that only one value does so. It is the one you posted. If you then define the discounted value of an option to be its expectation of the discounted pay-off, its discounted value is a martingale. So the discounted value of everything is a ...


3

well there are lots of things to get right... first you need to the non-callable version right, to get that right requires getting the smile right since a callable range accrual is really just a bunch of digitals with timing effects. these days discounting and forwarding are done with different curves so you'll need to get that right too. then you'll ...


3

it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$ \frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A, $$ with $A$ the annuity is remarkably good. One way of deducing these formulas is to do an asymptotic/Taylor expansion about $\sigma=0.$


2

Yes, your broker could have used one or combination of many factors: estimated volatility surface from historical returns of your target index, historical returns of similar indexes, implied volatility of similar indexes, existing inventory,etc. Check out these two approaches to deriving surfaces from returns starting slide 14


2

I assume this is a plot of option value versus price of the underlying. The only case where it ought to be symmetric is if the pdf of the underlying is symmetric eg normally distributed. I'm guessing your chart assumes a lognormal underlying, which is a non symmetric pdf, so the graph is non symmetric.


1

Your valuation is NOT for the knock-out option that you have specified. Let \begin{align*} \tau = \inf\{t \mid 0 \le t \le T, S_t \ge L\}. \end{align*} Here, we set the infimum of an empty set to $\infty$. Then, the payoff of the knock-out option is of the form \begin{align*} (S_T-K)^+ 1_{\tau = \infty}. \end{align*} Under the Black-Scholes setting, this ...


1

This Quandl Page provides you the informations you need: a lot of programming languages and other tools are linked to Quandl.


1

Local vol model gives a "too shallow" forward skew. Derivatives of which the price are depending on the forward skew will be mispriced. If i remember correctly, Hagan's paper


1

When you say 'overprice' I assume you mean model price > market price. In my experience this is true for all reasonable models. It's due to excessive supply of the Bermudan structure in the market.


1

In order to define option price we should follow Black Scholes construction to construct riskless portfolio at t then to state that instantaneous rate of return of this portfolio equal risk free rate r ( t ) where r is a random on [ t , t + dt ] interval. We actually then arrive at the problem which could not be embedded in BS pricing world.


1

The most rigorous approach I have seen so far eliminating the risk premium is this one: Emanuel Derman: The Perception of Time, Risk and Return During Periods of Speculation (2002) Equation 2.23 on page 11 derives $\mu$ ~ $r$ but it only holds in the limit when you hypothesize countless uncorrelated stocks in a diversifiable market. Still an interesting ...



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