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The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S \geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,\infty)$, and rewrite as \begin{align} P_{t,T}(K) &=e^{-r(T-t)} \int_{-\infty}^{\infty}\left(K-S\right)^+ q_T^S(S)\,dS \\... 5 In swaptions, there is the expiration of the swaption into an underlying swap. When the dealers provide the vol surface, in the first column, they typically put the expiry of the swaption from earliest to farthest. Along the top row, they put maturity of the underlying swap from shortest to farthest. So when the dealers describe the upper left having high ... 3 Here are a few FX structured product examples: All of these can be notes or swaps, notes will pay back the notional at the end and carry no credit risk (and are normally set so that they are worth 100% at inception - i.e. they'll be worth 99% and the seller will take some profit/hedging costs). Swaps will either be set to be worth 0% (same deal as above, ... 3 For a very nice reference on this matter, I recommend Pykhtin and Zhu’s Guide to Modelling Counterparty Credit Exposure, a short paper that thoroughly defines these concepts. Expected Exposure EE(t) (also known as Expected Positive Exposure) for a trade with value V(t) is given by:EE(t)=\mathbb{E}[\max(0,V(t))]$$It is effectively “what you could ... 3 To lighten notation, we assume a constant accrual factor \tau, a swap rate S_n(T) which fixes at T and pays at T_p (e.g. T_p-T=\text{3 months}) and a simple CMS payoff of the form:$$\Phi(S_n(T))=(S_n(T)-K)$$fixed at time T=T_m. We are interested in pricing under a measure for which the underlying risk factor of interest (i.e. the swap rate) is ... 2 There are a few issues that need to be separated here. Issue "zero" is whether your MC is able to correctly represent the dynamics you've chosen for your assets. If you implement your MC properly, by construction it should converge in distribution to the postulated dynamics. No bias there. Variance yes potentially, because of discretisation, but no ... 2 You can use such an approximation but there are known analytical prices. You have a special case in which the stock price is normally distributed. See Bachelier Model. Set \mu=r-q (if you have dividends, or simply \mu=r if there are no dividends). So if you change from the real worl probability measure \mathbb{P} to the risk-neutral measure \mathbb{Q}... 2 Let P(t,T) denote the time t price of a zero-coupon bond maturing at time T and \mathbb{Q}_T be the associated equivalent martingale measure which uses P(t,T) as numeraire. Then, for any \mathcal{F}_T-measurable payoff \xi, the time t value of \xi is given by$$V_t=P(t,T)\cdot\mathbb{E}^{\mathbb{Q}_T} [\xi\mid\mathcal{F}_t]. The ...

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Insurers do use derivative pricing models such as Black-Scholes to price the sort of guarantees you describe. As far as I know, this used to be known as the "replication method" in the industry jargon, and it allows insurers to price guarantees in a market-consistent manner, hence enabling them to efficiently hedge them with traded instruments. In particular,...

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Yes it is equivalent of the dividend rate. The b in the function is cost of carry, so here it would be: $b=r_{USD}-r_{AUD}$ And r in the function is $r_{USD}$.

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Reposting comments as an answer: If you are doing Monte Carlo you'd have a new volatility function to use (rather than a constant vol like in black scholes) for each time and stock price from the local volatility model (It's usually written $\sigma (S, t)$). So you could use this local volatility function during a simulation regardless of the type of payoff....

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If the $(F^i_T)_i$ are lognormal, I'd choose their geometric average $\left(\prod_{i=1}^N F^i_T\right)^{\frac{1}{N}}$ because it's lognormal as well and hence the expectation is easy to compute. If they are normal, I'd choose the arithmetic average $\frac{1}{N}\sum_{i=1}^N F^i_T$, since it's gaussian as well.

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The SABR model itself is arbitrage-free even for high vol of vol. The question is whether the Hagan et al formula for implied volatility under the SABR model is arbitrage free - it isn't actually. For very low strikes arbitrage can occur using the Hagan et al formula for implied volatility, and perhaps also for very high vol of vol. Question: how do you ...

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A standard book in the volatility literature is Gatheral (2006). The book begins with stochastic volatility, llocal volatility and the Heston model. Then he adds jumps and default risks. He concludes with barrier options, exotic options and volatility derivatives. He includes many tables and graphs and writes rather well. The only downside is that he does ...

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