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# 数学代写|金融数学代写Financial Mathematics代考|Volatility trading using options

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## 数学代写|金融数学代写Financial Mathematics代考|Taking volatility position using straddles and strangles

To benefit from an increasing volatility of the underlying asset price process in the future, one can long one at-the-money call and one at-the-money put with the same strike to form a straddle (see Fig. 1.2a). However, to establish the straddle portfolio, one has to pay the upfront premium of two expensive at-the-money options. To establish a less costly portfolio with similar volatility exposure, one may switch to a strangle which consists of one out-of-the-money put and one out-of-the-money call (see Fig. 1.2b). In order to achieve a positive terminal payoff in a strangle, the asset price has to go beyond an interval with bounds specified by the strikes of the out-ofthe-money call and put. The cost required to establish the strangle is lower when the interval becomes wider, but there is a higher chance of getting zero terminal payoff under more widened interval.

## 数学代写|金融数学代写Financial Mathematics代考|Volatility exposure generated by delta hedging options

Equity options provide exposure to both direction risk of the asset price and volatility risk. According to the Black-Scholes hedging principle, one can remove the exposure to the asset price risk using delta hedging. However, delta hedging can never be perfect since the real world financial markets violate many of the Black-Scholes model assumptions, like volatility cannot be accurately estimated, asset price may move discontinuously, liquidity may not be available, frequent trading would incur high transaction costs, etc.

We would like to show that delta hedging an option based on some chosen timedependent hedge volatility generates a profit and loss $(\mathrm{P} \& \mathrm{~L})$ that is related to the realized variance and the cash gamma position (defined as product of option gamma and square of asset price). Consider the underlying asset price which follows a continuous semimartingale process under a risk neutral measure $Q$ as specified by
$$\frac{\mathrm{d} S_{t}}{S_{t}}=(r-q) \mathrm{d} t+\sigma_{t} \mathrm{~d} W_{t},$$
where $r$ and $q$ are constant risk-free rate and dividend yield, respectively, $\sigma_{t}$ is the instantaneous volatility process, and $W_{t}$ is a standard Brownian motion under $Q$. The assumed dynamics of $S_{t}$ allows no jump. Let $\sigma_{t}^{i}$ be the time-dependent implied volatility derived from traded option prices at varying times. We write the time- $t$ option price as $V_{t}^{i}=V\left(S_{t}, t ; \sigma_{t}^{i}\right)$ with reference to implied volatility $\sigma_{t}^{i}$. Suppose an option trader sells an option at time zero priced at the current market implied volatility $\sigma_{0}^{i}$, the option price is given by $V_{0}^{i}=V\left(S_{0}, 0 ; \sigma_{0}^{i}\right)$. The seller’s short position in the option is delta hedged at some chosen time-dependent hedge volatility $\sigma_{t}^{h}$ for the remaining life of the option. In summary, for $t \in[0, T]$, there are 3 volatilities: (i) $\sigma_{t}$ is the actual instantaneous volatility of $S_{t}$, (ii) $\sigma_{t}^{i}$ is the implied volatility derived from market option price $V_{t}^{i}$ at varying time $t$, (iii) $\sigma_{t}^{h}$ is the time varying hedging volatility adopted by the hedger.

## 数学代写|金融数学代写FINANCIAL MATHEMATICS代考|VOLATILITY EXPOSURE GENERATED BY DELTA HEDGING OPTIONS

$$\frac{\mathrm{d} S_{t}}{S_{t}}=(r-q) \mathrm{d} t+\sigma_{t} \mathrm{~d} W_{t},$$

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