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1

Since variance is additive, your var swap at $t=t_1$ is the same as the realized cash pnl plus a new var swap traded on $t=t_1$ with strike being $K_1$ rather than $K_0$, with a variance amount being $\frac{T - t_1}{T}$ times the original variance amount, where $K_1$ is the fair strike on $t=t_1$ and $K_0$ is your old strike traded on $t=0$. If you would ...


1

The key variable is indeed the derivative's elasticity $\Omega$ (aka leverage, Lambda). It is defined by $$\Omega=\frac{\frac{\partial V}{V}}{\frac{\partial S}{S}}=\frac{\partial V}{\partial S}\frac{S}{V}=\Delta\frac{S}{V}.$$ Here, $V$ corresponds to the value of the derivative and $S$ to its underlying. This number measures how much riskier the derivative ...


2

Question 1 is answered in parts 1 through to 6: the idea is that each part slowly builds the tools required to derive the process equation for $S_t$ under the $S_t$ Numeraire. Question 2 & Question 3 are then answered in part 7. Part 1: Expectation of a function of a Random variable: Let $X(t)$ be some generic Random Variable with probability density ...


4

I provide a solution in two steps. The first steps carefully outlines how to split up the integral and what new measures are used. This first step does not require any special model assumption and holds in a very general framework. We derive a formula for your option that resembles the standard Black-Scholes formula. In a second step, I assume that the stock ...


2

Black scholes formula based on $S_t$ measure , theory, and formulas you mention are derived in detail in "Steven Shreve: Stochastic Calculus and Finance" draft pdf from 1997 , page 328 "stock price as numeraire".


3

It's just Girsanov's theorem. I suppose that under the risk neutral measure Q $$dS_{t}= r S_{t} dt + \sigma S_{t}dW_{t},$$ $$S_{t} = S_{0}\exp\left((r-\frac{\sigma^{2}}{2})T + \sigma W_{T}\right)$$ By multiplying by $e^{-rT}$ I have $e^{-rT}S_{T}$ which is a martingale so that I can change my measure under $Q$ to some equivalent probabilty $Q_{1}$ under ...


4

Give QuantLib a try: import QuantLib as ql today = ql.Settings.instance().evaluationDate averageType = ql.Average.Geometric option_type = ql.Option.Call strike = 120.0 exerciseDate = ql.TARGET().advance(today, 90, ql.Days) payoff = ql.PlainVanillaPayoff(option_type, strike) exercise = ql.EuropeanExercise(exerciseDate) option = ql....


2

Example 1. Buy-side buys notes whose coupon is $\min(\max(\rm{gearing}*(\rm{index}_1-\rm{index}_2),\rm{maximum}),\rm{minimum})$. Their motivation is often their view that the sell side is pricing these too cheap. Example 2. An insurance company wants to buy 20-year fixed-coupon GBP bonds. However they want higher yield than GUK, and they are willing to take ...


1

The Bermudan (American) callable/swaption is convex with respect to the strike. The payoff function of the Bermudan (American) callable/swaption is of the form, with implicit dependence on sample $\omega$, $$g(t,K)=\big(a(t)-b(t)K\big)_+$$ where $t$ is the time the swap (interest) rate is set and $K$ is the strike. $g(t,K)$ is obviously convex with respect ...


0

How about using the ATM option expiring closest to your futures contract delivery date? As in : Stock price : price today Futures price : price at a future date, usually the nearest expiring contract. Option price : price ATM at a comparable future date.


1

Ok as an example consider a 1yr-10yr 3pct Bermudan payer (the right to pay fixed at 3pct vs libor starting at any annual date from 1yr onwards with a maturity of 11yrs from today). For simplicity assume a flat yield curve. If rates are 1pct, the probability of exercise on the first date is low (a long way to cross the 3pct strike). If rates are 6pct, the ...


1

Given that liquidity of NIFTY options decreases rather quickly with moneyness, using the most liquid ATM option is your best bet (i.e. the least bid ask spread). Although, keep it consistent in that you use the same option (i.e. say the one that expiries 3M into the future) for different days. Also, better to use implied volatility as a performance measure ...


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