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4

This drift comes from making the discounted stock a martingale in the risk-neutral measure $\mathbb Q$ You start with a stock in $\mathbb P$ having this form: $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ You also have a discount factor $e^{rt}$. The idea is to remove the drift of the discounted process in $\mathbb Q$ so you get (after applying Girsanov's ...


3

You only have one asset in your portfolio which means that you can only statically hedge. By the definition of self financing, $V_0=\phi_0 S_0$, $V_1=V_0+\phi_1 (S_1-S_0)$, and $V_1= \phi_1 S_1$. Putting these last two together, $V_0=\phi_1 S_0 $. Hence $\phi_1=\phi_0$ and you have a static position. Intuitively, this is because you cannot trade in ...


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?.


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

I don't know the BS formula you are trying to use. The price is the expected value of the discounted payoff under the risk neutral probability measure (I.e. Under which S is a martingale) So the you need to compute the risk neutral probabilities for S to go up or down. The probabilities given in the problem have no impact. They are just there to trick 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

The first consideration is to set prices which do not generate arbitrage opportunities. The existence of a risk-neutral probability measure ensures that the model is arbitrage-free. In the Black-Scholes setting, as you mentioned, the market is complete and there as a unique martingale measure, hence only one possible price for each derivative. In a ...


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I would consider the financial applications of the Greeks: hedging. The "main" greeks, viz. Delta, Gamma, Theta, Vega and Rho, all have intuitive financial meanings. Gamma is the rate of change of your Delta (how many shares of stock to own) with respect to the stock price, so a high Gamma implies you will be rebalancing in large quantities (often ...


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


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In the BS model, with friction-free markets in continuous time, the cost of the hedging portfolio is the initial cost of setting up the portfolio. There are no costs over time as the hedging portfolio is self-financing: any purchase of the underlying is paid for by borrowing money and any selling of the underlying is invested at the risk-free rate. At ...


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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.



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