# Tag Info

### Convexity of an American put option

It is indeed. The price of an American option is the Bermuda option in the limit that the exercising interval approaches zero. The Bermuda option at any exercising time can be evaluated inductively ...
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### Why are FRA/futures convexity adjustments necessary?

This has been posted a few times now, so I will invest the time on a full response. FRA / Futures convexity has nothing to do with profits/losses being immediately recognised on the future through ...

### Why is there a convexity adjustment if the payment date differs from Libor end date?

Let us denote $\delta$, the Libor's tenor (e.g. 3M), $P(t, T)$ the price of a zero coupon bond price paying 1 unit of currency at $T$, and $L_t(T, T + \delta)$ the forward 3M Libor starting at $T$ and ...
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### Convexity of an American put option

Here is a much more straightforward proof of the convexity of the American option with respect to a parameter, if it is independent of time and deterministic, than my previous one, though I am happy ...

### What is the correct convexity adjustment for an Interest Rate Swap with unnatural reset lag?

Consider a date sequence \begin{align*} 0 \leq t_0 \leq T_s < T_p < T_e, \end{align*} where $t_0$ is the valuation date, $T_s$ is the Libor start date, $T_p$ is the payment date, and $T_e$ is ...
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### A very simple question about convexity of a bond

The chart you posted does not give a correct visual representaion of convexity . Convexity is not $\frac{\partial^2 P}{\partial y^2}$ but $\frac{1}{P}\frac{\partial^2 P}{\partial y^2}$. So you have to ...
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### B-splines: convexity in IV/Price

No, and this is wrong. The implied vols (from market prices) are actually not necessarily convex but yet may be still arbitrage-free, there are many examples of this for various equities. Furthermore, ...

### Convexity in a DV01 neutral trade

Let’s say you do a 2s-10s steepener, dv01 neutral. What does this mean ? It means you are using the current dv01s of the 2s and 10s, which are approximately 1.99 and 9.12, to weight the relative ...
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The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is $$\frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2,$$ where $D$ is the duration and $C$ is ...

### Can two bonds have same yield and price but different convexity?

To directly answer the question: bond A= one day to maturity , price 100, yield 2%. Bond B: 10 years to maturity, price 100 yield 2%. This is perfectly possible. Bond B has greAter convexity but ...

### Convexity of an American put option

Let $\mathscr{T}$ be the set of stopping times with values in $[0, T]$. Note that, for any $\tau \in \mathscr{T}$, $\lambda_1\ge 0$, $\lambda_2 \ge 0$, and $\lambda_1+\lambda_2 =1$, \begin{align*} &...
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### Interest Rate Convexity - Fundamental Question

This is indeed just a convention, as you point out. It comes from the fact that zero coupon bonds, by convention, do not have any volatility exposure. Rather, it is assumed the prices of ZCBs are ...

### Why is there a convexity adjustment if the payment date differs from Libor end date?

Consider a date sequence \begin{align*} 0 \leq t_0 \leq T_s < T_e < T_p, \end{align*} where $t_0$ is the valuation date, $T_s$ is the Libor start date, $T_e$ is the Libor end date, and $T_p$ is ...
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### How to Take Advantage of Arbitrage Opportunity of Two Options

I think it is far easier to understand by just drawing the payoffs. You have two put options: A European put option on a non-dividend paying stock with strike price 80 is priced at 8 dollars, and a ...
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You can think of both ( difference and ratio ) indicators as some aggregated measure of difference between flat vol (ATM vol) and "total vol" than includes skew and kurtosis effects.
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You have left out the chain rule term in the first derivative and second derivative. First derivative should be: $$\frac{\partial P}{\partial YTM} = \frac{1}{2(1+YTM/2)} \sum_{i=1}^N \frac{-2 t_i ... 3 votes ### High convexity vs low convexity bond definition Do not forget the effect of passing time (the theta) on your portfolio. If two portfolios have the same value and duration, then the portfolio made up of the difference has locally zero sensitivity ... 3 votes ### Change of numeraire from bank account to Zcb Let B_t= e^{\int_0^t r_sds} be the money-market account value at time t, and P(t, T) be the value of the zero-coupon bond with maturity T and unit face amount. Moreover, let Q be the risk-... 3 votes ### Interest Rate Convexity - Fundamental Question OK, so I think I have this figured out in my head now in terms of martingale measure theory. Thanks dm63 for pointing me in the right direction! Just for my own peace of mind and perhaps to help ... 3 votes ### What is the correct convexity adjustment for an Interest Rate Swap with unnatural reset lag? This seems to be a (short term, only 3 months) CMS swap. I wrote a paper about the different approaches to price them, available here. You can pick the one best fitted for your needs. 3 votes Accepted ### Proof of the convexity adjustment formula Well, you need to know what is the stochashtic model you are using for y_T, if you assume it's a geometric brownian motion you have this process : y_T = y_0 e^{\sigma W_T - \frac{1}{2} \sigma^2T} ... 3 votes ### Why is there a convexity adjustment if the payment date differs from Libor end date? The other two answers do a good job of explaining, within the context of mathematical financial models, why a convexity adjustment is necessary, but I think a more tangible perspective can also be ... 3 votes Accepted ### Jensen’s inequality in Convexity adjustment premium P_2-P_1 where: P_1=\frac{1000}{\left(1+\frac{0.06+0.04}{2} \right)\left(1+0.05 \right)} P_2=0.5\frac{1000}{\left(1+0.06 \right)\left(1+0.05 \right)}+0.5 \frac{1000}{\left(1+0.04 \right)\left(1+... 3 votes ### Why is portfolio optimization a convex problem if variance is concave? I'm in no way a portfolio theory expert, but the negative of a convex function is concave and vice versa. You can look at minimizing a concave function as maximizing a convex function and vice versa. ... 3 votes ### Leveraged ETF pair trade, where's the gamma/convexity? Both products actually have positive convexity, they will buy more underlying (SP500) when the price goes up and sell it when it goes down. However, if you hedge every day, you will just cancel out ... 3 votes ### Leveraged ETF pair trade, where's the gamma/convexity? I disagree that these products are convex*. At any point in time, the ETF exposure to the underlying is linear, it's just that it changes through time. A 2x ETF will just have 2x exposure to the ... 3 votes ### Leveraged ETF pair trade, where's the gamma/convexity? As @Lliane explains, you are actually describing a position in which the underlying is rebalanced everyday, hence the compounding effect of the leveraged ETF vanishes. Maybe a bit of modelling can be ... 3 votes ### Bond Convexity & Interest Rates Assume we are using continuously compounding rates, and that discount factors are given by the ZCBs P(0, t_i) = e^{-y_i \cdot t_i}. The price of a fixed bond is given by$$B = \sum_1^n N \cdot \...
See the graph below. Let's define the PNL as the position's payoff at expiry plus accrued initial investment, i.e. collected / paid option premia. Assuming $K_1=95,K_2=100,K_3=105$ (i.e. $\lambda=0.5$)...
We define a convex risk measure as $$\rho( \lambda X_1 + (1-\lambda) X_2) \le \lambda \rho( X_1 ) + (1-\lambda) \rho(X_2),$$ for $\lambda \in(0,1)$. A coherent risk measure is subadditive and ...