13 votes

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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  • 2,463
10 votes

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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  • 2,110
9 votes
Accepted

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 sample, than my previous one, though I am happy to have ...
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  • 2,463
8 votes
Accepted

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 ...
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  • 7,997
6 votes
Accepted

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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  • 9,347
6 votes

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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  • 20.4k
6 votes

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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  • 20.4k
5 votes
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Duration vs. Convexity Contradiction

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 ...
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  • 13.2k
5 votes
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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, ...
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  • 1,163
5 votes

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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  • 13.7k
5 votes
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What is gamma to do with realized volatility?

I like to think about this problem graphically. The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note ...
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  • 4,951
4 votes

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 ...
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  • 13.7k
4 votes

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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  • 20.4k
4 votes
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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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3 votes
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Derivation of convexity formula

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 ...
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  • 196
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-...
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  • 20.4k
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 ...
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  • 93
3 votes
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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 ...
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  • 13.7k
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.
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  • 790
3 votes
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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} $...
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3 votes

Girsanov theorem in CMS convexity derivation

Note that $$\frac{dQ_{T_p}}{dQ}|_{T_0} = \frac{P(T_0, T_p)}{P(0, T_p)}\frac{A(0, T_0, T_n)}{A(T_0, T_0, T_n)}$$. Then $$E^{Q_{T_p}}\big(S(T_0, T_n)\big) = E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{P(0, ...
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  • 20.4k
3 votes

What's the underlying idea of definition of constrained market in Skiadas' Asset Pricing Theory?

I have asked myself the very same question when I first read the book. As far as I can tell, the "scalability" condition is only imposed for technical reasons. It simplifies the subsequent proof of ...
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  • 2,310
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 ...
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3 votes
Accepted

SPX Convexity Spread

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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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 ...
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  • 5,603
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+...
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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. ...
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  • 1,662
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 ...
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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 \...
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3 votes
Accepted

Question in convex arbitrage

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. $\lambda=0.5$)...
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  • 5,358

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