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4

Fubini's theorem is only used to reverse the order of integration. We have: $\int_{-\infty}^{\infty}{e^{i\nu k} \left( C \int_k^{\infty} \left( e^x - e^k \right) q(x) dx \right) dk} = \int_{-\infty}^{\infty}{\int_k^{\infty}{C e^{i\nu k} \left( e^x - e^k \right) q(x) dx} dk} $ Now, let $f(x, k) = C e^{i\nu k} \left( e^x - e^k \right) q(x)$, ...


4

Since the volatility is not changing, we can assume that the only change is the underlying asset price $S$. Then \begin{align*} C(S+\Delta) &\approx C(S) + Delta \times\Delta +\frac{1}{2} Gamma \times \Delta^2 \\ &=11.50 + 0.58 \times 0.5 + \frac{1}{2}\times 2 \times (0.5)^2\\ &=12.04. \end{align*}


3

A possible reason may be your computation of maturity period. Exchange compute the maturity in minute till expiry and then divide it by total trading minute in a year to arrive at maturity. An another possible reason may be your choice of risk free interest rate. There are various proxy for risk free interest rate like Treasury rate and LIBOR of different ...


3

you just add in any auxiliary variables accumulated along the path that determine the pay-off to the regression variables. So path-dependence is not a problem. If you have previous decisions, you may need to do different regressions based on their possible values or make them into a continuous variables that can be used for regression.


2

As your code works for the short maturity case, I assume that it is correct. The volatility of $80 \%$ is simply huge. Thus the area covered by the paths is huge too. As you can read e.g. here the sampling error is proportional to the variance of the process, which is huge in your case. As a brute force solution you can just enlarge the number of samples. ...


2

The easiest way to think of this is as follows: Settlement Price - Price at which the exchange margins all accounts for those options. Closing Price - Mid/Bid/Ask of Active Market at the exchanges last trade time. E.g. for TY Contracts this is at 5pm EST vs. a Settle Time of 3pm EST. Last Trade Price - Not all options trade every day. This is the price ...


2

As barrycarter stated in the comment - the value of a set of [European!] options is the sum of the values of the individual options. This is simply follows from integral of a sum being a sum of integrals. $$butterfly\,option\,price = \\ \int_0^\infty butterfly\,payoff(S) dS = \\ \int_0^\infty (call\,payoff(S,K)+call\,payoff(S,K')+call\,payoff(S,K'')) dS ...


2

A butterfly in general has a payoff of the form \begin{align*} (X_T-K_c)^+ + (K_p-X_T)^+-(X_T-K_{atm})^+-(K_{atm}-X_T)^+, \end{align*} where $X_T$ is the asset value at maturity $T$, while $K_c$, $K_p$, and $K_{atm}$ are strike levels.


1

Probably because volib assumes that the Black-Scholes holds which as we not is not true. A better way to compute implied volatility is to use a Moment-Free-Implied-Measure. One possibility is to closely following the model-free estimate proposed by Demeter et al. (1999) and Carr and Madan (1998) who show that if one owns a portfolio of options across all ...


1

We consider the forward value, which can be employed to estimate the equity value. Let $T_1=0.5$ be the dividend payment time, and $T=1$. Moreover, let $r_1=5\,\%$ be the annualized interest rate to $T_1$, $r=6\,\%$ be the interest rate to $T$, and $d=5$ be the dividend payment. Then, the forward value, under the risk-neutral measure with the deterministic ...


1

The relationship between interest rates and equity prices being at best unstable and weak, I'll assume that the level of interest rate is irrelevant here. So the answer to your question (price of the equity in a year) is 95, everything else being equal. Of course it's unlikely that the equity will actually price at 95 in a year due to market movements, but ...


1

There are a couple of options that you can use to account for dollar amount dividends. Firstly, if dividends are expected to increase or decrease in proportion to the stock price, you can convert the dividends into a percentage by dividing the latest dividend by the last stock price on the day the dividend was declared and multiply by the number of dividends ...


1

you need a positive dividend rate or a negative interest rate. Without these, it is a model-free result that early exercise is never optimal for a call option.


1

What if you write $$ P[R_{n+1} = d|F_n] = 1 - P[R_{n+1} = u|F_n] ? $$ Let us write $P(u) = P[R_{n+1} = u|F_n]$ Then the part to show is $$ u \bar{S}_n P(u) + d \bar{S}_n (1-P(u)) $$ and this $$ \bar{S}_n \left(d +(u-d)P(u) \right), $$ where we just expanded terms and then extracted the coefficients.


1

Your adjusted scheme is correct. Basically, taking a maturity $T$, you can consider the forward price process $F_t^T = S_t e^{r(T-t)}$. You apply the Andersen scheme to $F_t^T$ and then note that \begin{align*} S_{t+\Delta} &= F_{t+\Delta}^T e^{-r(T-(t+\Delta))}\\ &=F_t^T \exp(\ \Box \ ) e^{-r(T-(t+\Delta))}\\ &=S_t e^{r(T-t)}\exp(\ \Box \ ) ...



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