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2

risk-neutral. Really the forward measure. The price of the binary is struck at $K$ is $$ Z P( F_T > K) $$ with $Z$ the discount factor and $F_T$ the forward, and $P$ the probability in the forward measure. If rates are deterministic, the forward measure and the risk-neutral measure will agree.


4

Let M denote the number of underlying traded assets in the model excluding the risk free asset, and let R denote the number of random sources. Generically we then have the following relations: 1. The model is arbitrage free if and only if M ≤ R. 2. The model is complete if and only if M ≥ R. 3. The model is complete and arbitrage free if and only if M = R. ...


4

The uniqueness of the risk-neutral measure comes from the abundance of tradable assets. Let $B_t$ be the money-market account at time $t$. Let $Q_1$ and $Q_2$ be two risk-neutral measures. Then, for any tradable asset $X$ with maturity $T$, \begin{align*} E^{Q_1}\left(\frac{X_T}{B_T}\right) &= E^{Q_2}\left(\frac{X_T}{B_T}\right)\\ &=\frac{X_0}{B_0}. ...


2

essentially it comes down the fact that the dyadic quadratic variation of $W_t$ is $t$ with probability 1 and any measure change has to preserve this fact. Changing volatility would violate this invariance.


1

I think the terminology often leads to confusion. Risk neutral pricing is essentially based on the idea of state prices or Arrow securities. One imagines or attempts to replicate securities that represent a state of the market. The price that is the consensus price of the market participants is the price of the Arrow security and this defines the state ...


3

The quanto adjustment is required to achieve the martingale property for the discounted payoff after currency transformation. Since you do not require discounted asset values to be martingales for risk measurement you do not need a quanto adjustment. But of course you need to include the distribution of future FX-rates in your modelling (which might be what ...


3

There is only one real world! You would use the measure that best describes all the markets together. Bear in mind that for credit you are really interested in portfolio effects. What is the potential credit risk we could have to a particular name? This depends on all the contracts we have them regardless of currency and they need to be modelled ...


6

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 ...



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