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9

$\theta$ is the "mean" for this process. If $X_t > \theta \implies (\theta - X_t) < 0 $, which means that the drift for the process is negative and tends towards $\theta$. The opposite case can be made for $X_t < \theta$ ; the process will have positive drift when $X_t$ is below $\theta$. Therefore we can consider $\kappa$ to be the "speed" of mean ...


5

No. Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated. Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have ...


3

Actuarial science traditionally focuses on estimation of joint probabilities using real data where math finance is on valuation of contracts under an arbitrary distribution. It means the first one deals with methods of estimation of future distributions (the number of accidents of a given kind, the probability of someone with a given profile to have a ...


3

There is a good quick well-known approximation for at-the-money options: $$\textrm{Call,Put} = 0.4 S \sigma \sqrt{T}.$$ See further discussion at What are some useful approximations to the Black-Scholes formula?.


2

I think to gain intution you have to understand that the same agents that value the stocks will value the options. And agents compensate for volatility by demanding higher expected returns. Therefore you should ask: Why are stocks priced as they are in the first place? In your example, the stock with higher volatility has much lower expected return. This ...


2

Yes and No. In the absence of arbitragers, the price of the option will be different for each speculator based on their drift expectations (and each speculator has a risk in his position and will limit his ability to trade large sizes to avoid bankruptcy) and the option price will converge to priced off a supply-and-demand driven drift expectation. ...


2

Because you can hedge. Once you have delta hedged, the pay-off is symmetric about up and down moves so drift doesn't matter. Also the delta-hedged call and the delta hedged put have to have the same value since they have the same pay-off. (Put-call parity) Yet any argument that the call should be worth more because of drift says that the put should be ...


2

I don't know where you would have read that, but no, time value cannot be negative. Time value is option value minus intrinsic value. Intrinsic value is a model-imdependent no-arbitrage bound on option value. For an out-of-the-money payoff, intrinsic value is zero, and since the call or put payoff is non-negative this is a clear lower bound. For an ...


2

The claim payoff you describe, $g(M)$, looks to me like a tight butterfly spread that pays off only in one state of the world. Can't you just replicate that by short two calls with strike $K_0$ and long two calls, with strikes one either side at $K_0\pm 1$? Then the price of your option would be $C(K_0+1)+C(K_0-1)-2\cdot C(K_0)$. This is effectively the ...


2

this is probably the most asked question in quantitative finance... There are many answers. One nice example to consider is what if the calls were struck at zero. The call then pays the stock price at time $T$ and so it's value today must the stock price today since we can replicate by holding one unit of stock. This will be true regardless of the drift of ...


2

if only one person can make a choice, it strikes me as unlikely that it can reduce value. Ultimately, a choice means that the holder can choose between one of a number of portfolios on a given date. They will choose the one of maximal value. As long as the without choice portfolio was one of the ones they could have chosen, value can only go up.


2

Try this paper by Rolf Poulsen : http://colloquium.mathfinance.de/abstracts/poulsen.pdf. He derives barrier option prices in the Black-Scholes model using only reflection and Girsanov's Theorem, and then discusses extensions.


2

I'll provide an extreme counterexample. Suppose your non self-financing replicating portfolio is to do the following. At time t through T you own no stock and no bonds. Then you observe X at time T and place X dollars in the portfolio. It would be incorrect to say that the options value at time t is zero. The idea of a replicating portfolio is that at ...


1

The binomial model certainly is self-financing. First, get the value at every node by working backwards using risk-neutral evaluation. Then at each step and node, you get the value in the up node and the down node from where you are. You can fit a straight line as a function of stock through the two. You hold stocks and bonds to fit this straight line with ...


1

I think that you can find the answer to this question here: http://people.stern.nyu.edu/wsilber/chuang-silber%20approx%20option%20value.pdf


1

Option pricing is all about intrinsic value and time value. The intrinsic value is the difference between the strike price and the underlying market price. A call is a right to buy the underlying. Therefore intrinsic value of a call is positive when the strike price is below the underlying market price. You can buy for less than the market offer. A put is ...


1

A pithy way to put it is "implied volatility is the wrong number to put in the wrong formula to get the right price." That is, implied volatility is by definition the parameter $\sigma$ to plug into the Black-Scholes option pricing formula to get the market price of a vanilla option. This is called "volatility," but in reality it isn't the same as the ...


1

I don't know of any libraries for this. There is a pretty good literature on the problem you mention though. I suggest https://cs.uwaterloo.ca/~paforsyt/numuncert.pdf as a good paper to follow; they study numerical techniques, document pitfalls, and even prove something about convergence of their preferred approach.


1

Practically, it is very difficult to get a measurement of a stock's true drift while there are very well-documented processes to estimate volatility. It is therefore very convenient mathematically to select the risk neutral pricing measure that eliminates idiosyncratic drift. At its heart, Black Scholes constructs a dynamic, replicating portfolio for an ...


1

As always I recommend reading Rennie and Baxter for an introduction to option pricing that's not too technical and gives intuition about how it all works.


1

If you know the stock will finish above the strike, then the call option becomes a forward contract since it will always be exercised. We therefore price it as a forward. Its value at maturity is $$S_T - 60.$$ We can synthesize $S_T$ with one unit of stock costing $70.$ We can synthesize $60$ with $60$ ZC bonds which costs $60/1.055$ since the yield is ...


1

Important assumptions: - we have zero interest rate, - option is perpetual, EDIT: with probability 1, share price will hit the barrier $H$ (in fact this is a hidden assumption that price changes continuously or we can at least trade at the very moment when $S_t = H$). No, we can't assume that, because , as @q.t.f noted, it would imply arbitrage. ...


1

Unfortunately I cannot upvote user2142's answer because I lack the reputation, but his reasoning makes sense to me: the price is $\$1/H$ because as the seller of the option you buy $1/H$ shares for the premium. You sell them when the $S_t$ hits $H$ to obtain the $\$1$ you have to pay to the option buyer. I think the price is model free for any model with ...


1

Let $T= \inf\{t>0: S_t = H\}$. Then the option payoff is given by $\mathbb{1}_{\{T < \infty\}}$, and the value of the option is given by $\mathbb{P}(T< \infty)$. We assume that the stock price process is a geometric Brownian motion, that is, for $t>0$ $$ S_t = \exp\big(-\frac{1}{2}\sigma^2 t + \sigma W_t\big),$$ where $\{W_t, t \geq 0\}$ is a ...


1

The classical connection is the http://en.m.wikipedia.org/wiki/Esscher_transform developed for actuaries in 1932 which essentially transforms the objective probability measure into the risk neutral one used in quant finance.



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