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12

We created the SABR model because we realized that (a) option values were nonlinear in the volatility, and (b) volatilities are stochastic. This means that if one had an option (or portfolio of options) which have positive gamma in the volatility dimension, on average we'd make money from fluctuations in the volatility, and we'd lose money with negative vol-...


7

If they were a bank, or insurer, utility etc, then some regulator would likely encourage them do everything that others do, whether they like it or not, or whether it makes any sense. But if no regulator tells them to... and if if they don't have exposure to interest rates beyond 1 year... well, let's look at USD 1 year swap rate, for example, https://fred....


6

This is more of a math question than a quant question. Under Black Scholes dynamics (assuming $r=0$ for simplicity), as everyone knows we have $$C=SN(d_1)-KN(d_2)$$. In this case, we are interested in large negative $d$, since $lnS$ is large and negative. There is an asymptotic series for $N(x)$ whose first term for large negative x is $$N(x)=-\phi(x)/x$$,...


6

Let $P(t, T)$ be the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value. Consider the pricing of the caplet with payoff $(L(t_1; t_1, t_2)-K)^+$ at time $t_1$, where $0<t_1 < t_2$ and, for $0\le s \le t_1$, \begin{align*} L(s; t_1, t_2) = \frac{1}{t_2-t_1}\left(\frac{P(s, t_1)}{P(s, t_2)}-1\right) \end{align*} is the forward ...


5

In a past life, I was an equity strategist at a sell-side bulge bracket firm. In 2008 (obviously) the bank decided to take a long hard look at the funding costs of its derivatives books. So they appointed an MD in IBD/corporate finance who would obviously lack “skin in that game”, and I was his chosen “research guy” (deliberately not then from a fixed income ...


4

I assume you want to price a derivative product that pays $\int_0^T\ln S_tdt$ at maturity time $T$, from time $t=0$. I'll ignore generalization to time $t$ because it is trivial (split the integral in two, before and after $t$ as you did). The first trick it to do an integration by part on $\ln S_t dt$: $d(t\ln S_t) = \ln S_t dt + \frac{t}{S_t}dS_t + \frac{t}...


3

The problem seems to be that you forgot the mean of the process. If $ds_t = rs_tdt + \sigma dW_t^\mathbb Q$, then the solution of the SDE is given by $$s_T = s_te^{r(T-t)} + \sigma\int_t^Te^{r(T-u)}dW^\mathbb Q_u.$$ Since the last integral is Gaussian, the distribution of the terminal price is given by $$s_T \sim\mathrm N\left(s_te^{r(T-t)}, \frac{\sigma^2}{...


3

If you sell a call option, in the language of vol trading you are short gamma and vega, and long theta. So yes, as the underlying wiggles around, if you keep delta-hedged by trading it you will lose some money. This can alternatively be seen as gamma losses (coming from the second-order move in the underlying, which you have not hedged) or as coming from the ...


2

It would be an Asian strike call option, with the Asianing being computed over some period $[0;\tau]$. Not a big deal that the average is not computed from 0 to T. It even seems more natural to be done that way in practice. Before $\tau$, pricing would be similar to your option Asianing until T. For $t>=\tau$, you already know the value of the Asian ...


2

I would add to the answers here, the concept of margin of error. If your swaptions and options are short dated the effect of discounting them is going to be minimal, for a one year option at 1% OIS you have a 0.99 discount factor, i.e a 1% error in valuation if you substitute a discount factor at 1.00 and ignore discounting. If your models are similarly ...


2

Since dividends and interest rates mature annually and the time of expiry is two years, we can model the option as a two-step binomial tree, structured as the one in figure: Data from OP question: $S_0 = 45$ current price of the underlying $S$; $\hspace{2.5cm}$ $\sigma = 0.2\;\text{per annum}$ volatility of $S$; $r = 0.02\;\text{per annum}$ interest rate; ...


2

Achieving gamma neutrality refers to your whole portfolio situation. If you have a portfolio $P$ made up of $n_S$ shares of stock $S$, and of $n_1$, $n_2$ option calls $C_1$, $C_2$ on $S$ (options 1 and 2 differ in strike price or expiration), pursuing a gamma hedging strategy would imply to achieve neutrality on the Delta $\Delta_P$ and Gamma $\Gamma_P$ ...


2

Case I Let us consider a derivative with a payoff $H(L(T_{f},T_{S},T_{E}))$ which is paid at time $T_{p}$. Note that: $T_{f}$ - LIBOR fixing date; $T_{S}$ - LIBOR start date; $T_{E}$ - LIBOR maturity date; $T_{p}$ - derivative payment date. Also, $T_{f}=T_{S}=t_{1}$ and $T_{E}=T_{p}=t_{2}$ in the question. In your first case $H(L(T_{f},T_{S},T_{E}))=(L(T_{...


2

Most, but not all, columns on the typical risk report will double if you double the notional ; and can be also netted for two different trades. Moneyness (how far the is the option from being at the money) is usually expressed as how much the underlying needs to move, so it does not depend on the notional. Market data (implied vol, interest rates, yields, ...


2

To extend @d_797's answer, then this stems from Jensens inequality (see this): For a function $V: I \rightarrow \mathbb{R}$ for $I$ being an interval in $\mathbb{R}$, then V is convex if it satisfies: $$ V(S_1 \lambda + (1-\lambda)S_2) \leq \lambda V(S_1)+(1-\lambda)V(S_2)$$ for any two points $S_1,S_2 \in \mathbb{R}$ and $\lambda\in [0,1]$. Now, if $V(\cdot)...


1

I'm not 100% sure, please double-check. I think that both in the ITM and the OTM case (requested), a model-free answer cannot exist. In particular, the rate at which: ITM: $C(S_0) \rightarrow S_0$ as $S_0 \rightarrow \infty$ and OTM: $C(S_0) \rightarrow 0$ as $S_0 \rightarrow 0$ depends on the model-specific risk-neutral transition density $p^Q(S_T, T | ...


1

Edit: the original question didn't specify "model independence" and so the below focuses on the BS framework. Also, i focused on speed of convergence rather than order of convergence: I will try to update my answer with some thoughts on order of convergence later on. Not sure this answers your question, but the speed at which the option prices ...


1

The definition of linear is just the usual definition, it would imply that $V(S) = aS$ for some constant $a$ (on an interval like $[S_1,S_2]$). The definitions of convexity, concavity are more the "first-principles" definition and are equivalent to conditions related to positivity or negativity of $V''(S)$ that one sees in introductory calculus. (...


1

I will try to provide an answer to your questions: In retrospect, I do not believe that there is any industry standard for calculating historical volatility (I could be wrong on that part). As long as you have a consistent (and unbiased) estimator of variance (quadratic variation) you're set. As described in the original article of the Yang-Zhang estimator, ...


1

On an expiration basis, your put protected long underlying makes money above $80 and you have a locked in loss of \$5 below \$75. Note that long underlying plus long put is synthetically equal to a long call. Pretending no carry cost or dividend, your position is the same as buying the \$75 call for \$5 and the P&L is the same as stated above. There's ...


1

I am actually more interested in it from the other perspective. If we have a price shock what is the likely IV change that will affect the options pricing? If you're using Python, I would recommend the Mibian library (http://code.mibian.net/). You can simulate a price shock by increasing the volatility parameter (which is HISTORICAL volatility in this case) ...


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