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Here couple pointers that may make it clearer:
Drift can be replaced by the risk-free rate through a mathematical construct called risk-neutral probability pricing.
Why can we get away with that without introducing errors? The reason lies in the ability to setup a hedge portfolio, thus the market will not compensate us for the drift above and beyond the ...
25
The risk-neutral measure $\mathbb{Q}$ is a mathematical construct which stems from the law of one price, also known as the principle of no riskless arbitrage and which you may already have heard of in the following terms: "there is no free lunch in financial markets".
This law is at the heart of securities' relative valuation, see this very nice paper by ...
22
Being on the sell side and selling options you can intuitively think of it like this:
An option is like any other product that is being produced out of ingredients and because of the competitive situation of the producer this is done by the cheapest possible production process.
The ingredients are in a simple (Black Scholes) setting a stock and and a risk ...
10
I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality.
You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution).
I think there are no deeper truths to be found here.
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Q: What does the risk-neutral price represent if the option is not replicable?
In an incomplete market, there is no unique martingale measure but instead a set $Q$ of equivalent martingale measures. Consequently, there is an interval of arbitrage-free prices:
$ \Big( inf_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX], sup_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX] \Big)...
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In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example.
More precisely the assumption is that there is no $T\geq 0$ and self-financed portfolio $V$ such that $V_0 = 0$, $P(V_T < 0) = 0$ and $P(V_T > 0) > ...
10
You can price an asset paying $X_{t+1}$ in two ways:
$$P_t=\frac{1}{R_f}\sum_{\omega} Q(\omega)X_{t+1}(\omega)$$
$$P_t=\sum_{\omega} P(\omega)M_{t+1}(\omega)X_{t+1}(\omega)$$
As you can see, the price is making a joint statement (i.e. you can recover $Q(\omega)$) regarding both the probability of an event $P(\omega)$ and how much people dislike that event, i....
9
You may want to consider splitting two important, yet very different concepts:
Pricing a derivative security with contingent payoff and forecasting an asset.
Pricing a derivative can be achieved through setting up a hedge portfolio and track its evolution and "value" at any point in time before the derivative security pays off. Risk-neutral pricing is a ...
9
It depends on the purpose of your simulation.
If you want to model the asset price path for pricing some derivative then you need the risk-neutral measure (thus you take the risk-less rate as drift).
Why? Because the risk-neutral measure makes your pricing compatible with the pricing of other contracts in the market. It makes the prices consistent.
If ...
9
The short answer is:
As long as a derivative can be perfectly replicated via hedging in the underlying asset then the price of the derivative should be independent of investors' risk aversion and hence the application of risk-neutral probabilities and discounting of the future expected payoff under risk neutral probability leads to the same price of the ...
9
The risk neutral probability measure $Q$ is the true probability measure $P$ times the stochastic discount factor $M$ but rescaled so $Q$ sums to 1.
Simple derivation
For maximum simplicity, I'll work in a discrete probability space with $n$ possible outcomes. Everything goes through under measure theory in more general, infinite number of outcome ...
8
$$dS / S = \mu dt + \sigma dW \\
\\ dS / S -r dt= \mu dt - rdt + \sigma dW \\
\\ dS / S -r dt= [\frac{(\mu - r)}{\sigma}dt + dW]\sigma \\$$
Then, Girsanov tells us that, as long as the risk premium is bounded from below, we can write
$[\frac{(\mu - r)}{\sigma}dt + dW]\sigma$ as $\sigma d\tilde{W}$ where $\tilde{W}$ is simply another brownian motion with ...
8
A stochastic volatility model for a single risky asset can't be complete because you have two sources of randomness. But you can easily make it complete by adding a derivative whose value depends on the volatility. For example, if you add a variance swap in the Heston model then it becomes complete. This allows you to calibrate the model.
But your ...
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Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these have been relaxed.) In particular, there are two schools of criticism, one is that time homogeneity makes little sense in the real market. In fact, the Recovery ...
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This is a loaded question. Ross' recovery theorem has both flaws and insights. The single answer thus far did a great job of addressing the flaws from an economics perspective. No one questions that the math is wrong: it is correct. Here is a mathematical insight from Ross' work. Abstracting from the finance and economics, the purely probabilistic content of ...
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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 ...
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It is a very interesting question. There is a brief explanation in the book Martingale methods in financial modelling. Basically, it says that, the interest short rate $r_t$ can be modeled in any martingale measure $Q$, however, as long as the zero-coupon bond price $P(t, T)$ is defined by
\begin{align*}
P(t, T) = E^{Q}\Big(e^{-\int_t^T r_s ds}
\mid \...
8
The risk-neutral probability density function $q(.)$ is indeed given by
$$ q(S_T=s) = \frac{1}{P(0,T)} \frac{ \partial^2 C }{\partial K^2} (K=s,T) $$
where $P(0,T)$ figures the relevant discount factor. This is known as the Breeden-Litzenberger identity.
Because you do not observe a continuum of call prices in practice, you can use a finite difference ...
8
No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues.
This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention:
an auto-callable ...
8
In order to estimate a probability of an event with small probability, you might want to try to estimate the probability for a changed random variable that allocates a larger probability mass for the event happening. So, in your case, you might want to change the original $N(0, 1)$ to $N(100, 1)$ because for the second r.v. the probability of it being higher ...
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Recently I came across an interesting intuitive explanation:
Suppose driftless market. Market price is 105, strike price is 100. Call option costs 8, put option 3. (intrinsic value of call is 5, time value of both is 3)
Now the market starts drifting upwards massively. You say, that you would probably price call higher, e.g. at 10. Would you also price put ...
7
The classic argument using risk-neutral pricing is to assume that discounted stock prices are $\tilde{P}$-martingales where $\tilde{P}$ is the risk-neutral probability measure.
Then, you know that
$$\frac{S_t}{(1+r)^t}=\tilde{E}[\frac{S_T}{(1+r)^T} | \mathcal{F}_t]$$
by definition of a martingale process.
As the discounts are non-stochastic, you can ...
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Note first that this key equation is only assumed to hold true under some extra assumptions. Typically those assumptions are taken to be about absence of arbitrage, though it is possible to weaken them somewhat if you are willing to consider portfolio arguments or collectively agreeable objective function.
Anyway, the argument is this: if all the risk can ...
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In general these are the two basic approaches to QuantFinance:
Sell side (market maker, risk neutral): You use risk-neutral probabilities ("$\mathbb{Q}$") e.g. in option pricing (to e.g. calculate your greeks and hedge your portfolio), so that you live on the spread.
Buy side (market/risk taker): You use real-world probabilites ("$\mathbb{P}$") for e.g. ...
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Yes, you may as well take this as the definition of the risk-neutral probability $Q$.
I will now try to give you some intuition for that kind of construction.
Assume the risk-free interest rate $r$ is constant and that the world ends at time $T$. Suppose you have a security $B=B_t$ which is riskless, i.e. which follows the dynamics
$$ dB/B = r \, dt $$
so ...
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The risk-neutral probability is used as a convenient mathematical tool but, strictly speaking, it is not a necessary ingredient for the BS formula.
In fact, the formula can be derived by computing the expectation (under the physical probability) of the option payoff, properly discounted with the right stochastic discount factor:
$$ p_t = E_t[\;m_{t,T}\; f(...
7
If you imagine you have two risk-less assets that have a unit payoff at maturity $V_1(T) = V_2(T) = 1$ but their present value is not equal, e.g. $V_1(t) < V_2(t)$. You buy the cheaper, sell the more expensive, have a strictly positive cash-flow today and at maturity the cash-flows cancel out with certainty. This is a free lunch arbitrage. The same ...
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$\mathbb{P}$ is the true probability measure. Measure $\mathbb{Q}$ is a measure of convenience that allows risk neutral pricing. Stochastic discount factor $M$ takes you between the two.
If you care about prices you can either: (1) work under $\mathbb{Q}$ or (2) work under $\mathbb{P}$ with a stochastic discount factor $M$. There's an isomorphic ...
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First of all, you need to understand risk-neutral measures are not meant to make predictions of future prices, but they are meant to allow hedging (ie risk replication). Historical measures on their side, are meant to predict future prices.
I used the plural because they are many risk-neutral and historical measures. In both cases you choose one measure ...
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The result you're looking for is
$$ \left. \frac{d\Bbb{P}}{d\Bbb{Q}}\right\vert_{\mathcal{F}_t} = \left( \left. \frac{d\Bbb{Q}}{d\Bbb{P}}\right\vert_{\mathcal{F}_t} \right)^{-1} $$
This is a result from measure theory but since you mention it, let's see how we can show it based on Girsanov theorem.
Starting from the definitions you provide and introducing ...
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