13

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 via the dynamic programming principle as the maximum of the payoff and the risk-neutral expected value of the Bermuda option price at the next exercise time. The ...


10

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 ending at $T+\delta$, seen from $t$: $$L_0(T, T + \delta) = \frac{1}{\delta} \left(\frac{P(0, T)}{P(0, T + \delta)} - 1 \right)$$ The vanilla case: payment at ...


9

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 made the connection amongst the dynamics programming principle, the discrete time process and the continuous time process there. Let $g(t,\omega,x)$ be the ...


8

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 margin settlement, and potential reinvestment, whilst deferred on the FRA. Although the opposite seems to be a very common belief amongst many practitioners (...


6

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 normalize for P. The 4 curves you plot have very different P. When the curves are redrawn normalized so they go through the same point $(y_0,P_0)$ you will ...


6

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 the Libor end date. Let $\Delta_s^e = T_e-T_s$. For $0\le t \le T_s$, define \begin{align*} L(t, T_s, T_e) = \frac{1}{\Delta_s^e}\bigg(\frac{P(t, T_s)}{P(t, T_e)}-...


6

I dont think you can see convexity in such a plot, since each of these prices are not observed from a single bond deliverable, but from different coupon bond deliveries. If the delivery was always based on same coupon type bond and quite similar maturity (http://www.cmegroup.com/trading/interest-rates/us-treasury/10-year-us-treasury-...


6

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*} &\ \max(\lambda_1 K_1+\lambda_2 K_2 -S_{\tau}, 0) \\ =&\ \max\big(\lambda_1 (K_1-S_{\tau})+\lambda_2 (K_2 -S_{\tau}), 0\big)\\ =&\ \lambda_1\max(K_1-...


5

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 convexity. For small $\Delta y$ the square is much smaller. Thus the duration component dominates.


5

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, preserving convexity is not necessarily enough either. In terms of implied variance $w(y)=\sigma^2 T$ as a function of log-moneyness $y=\ln\frac{K}{F}$, the no ...


5

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 principal amounts of the bonds. Now, the key point is, when the market moves, these dv01s move and you no longer have a balanced trade. That is the convexity ...


4

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 it also has substantially more risk.


4

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 the payment date. Let $\Delta_s^e = T_e-T_s$. For $0\le t \le T_s$, define \begin{align*} L^e(t, T_s, T_e) = \frac{1}{\Delta_s^e}\bigg(\frac{P(t, T_s)}{P(t, T_e)...


4

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 put option on the same stock with strike price 90 dollars is priced at 9 dollar The difference between the two payoffs is equal to 10 dollars (90 strike puts ...


3

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-neutral measure and $Q_T$ be the $T$-forward measure. If the interest rate $r_t$ is deterministic, then \begin{align*} P(t, T) &= E\left(e^{-\int_t^T r_s ds} \...


3

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 others in the future, my understanding is as follows: Vanilla Swap: We observe the LIBOR $L(T_i, T_{i+1})$ at time $T_i$ and payment occurs at $T_{i+1}$. Therefore ...


3

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 given. Now, you can replicate a regular fra with strike K exactly using ZCBs: Long one ZCB with maturity T(i) and short (1+alpha K) ZCBs with maturity T(i+1)...


3

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

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} $ If you compute the expectation and variance you get $ \mathbb{E}(y_T) = y_0$ and $Var(y_T) = {y_0}^2( e^{\sigma^2 T }-1)$ As $y_0 $ is constant you ...


3

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 $$E^{Q_{T_p}}\big(S(T_0, T_n)\big) = E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{P(0, T_p)}\frac{A(0, T_0, T_n)}{A(T_0, T_0, T_n)}\bigg) \\ = \frac{A(0, T_0, T_n)}{P(0, T_p)} E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{A(T_0, T_0, T_n)}\bigg).$$ That ...


3

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 the Fundemental Theorem of Asset Pricing in constrained markets. There are several papers that have shown that the theorem is valid for conic constraints. ...


3

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 to yields and is delta hedged. Since the sum of (modified) theta (the derivative to time minus the position funding, in essence the carry) and local yield ...


3

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.


3

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 useful. Consider two forward rate agreements (FRA) to receive fixed and pay floating, with the same fixing date $T_s$ and end date $T_e$. The first pays on the ...


3

$P_2-P_1$ where: $P_1=\frac{1000}{\left(1+\frac{0.06+0.04}{2} \right)\left(1+0.05 \right)}$ $P_2=0.5\frac{1000}{\left(1+0.06 \right)\left(1+0.05 \right)}+0.5 \frac{1000}{\left(1+0.04 \right)\left(1+0.05 \right)}$


3

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. Also, the optimization problem is over the weights, and not over densities (which variance is concave in as your link shows). Portfolio variance is convex in ...


3

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 $$B = \sum_1^n N \cdot \delta \cdot K \cdot e^{-y_i t_i} + N \cdot e^{-y_nt_n},$$ where $\delta = t_i-t_{i-1}$, and $K$ is the coupon. Now bump all discount rates (yields) by $\epsilon$ ...


3

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. $\lambda=0.5$), the orange payoff diagram below belongs to a setup where $C_2<\lambda C_1 + (1-\lambda) C_3$: You paid some net fee initially, and you obtain a position ...


2

The upper bound for the 80 call is C(90) + 10, or 30. At least assuming no arbitrage. Let's start by assuming the risk-free rate is 0 (this isn't a problem, but the math is clearer without it), so we don't have to discount the price. Then, the call price is given by $C(K) = E_t[(S_T - K)^+]$, which gives: \begin{array} $C(K - 10) &= E_t[max(S_T - (K - ...


2

There is no generic solution. However, the KKT conditions are of the forms \begin{align*} \begin{cases} Qy + \lambda_1 \mu +\lambda_2 Py = 0,\\ \mu^T y = 1,\\ y^TPy \leq k^2 \sigma^2,\\ \lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0. \end{cases} \end{align*} Here, the condition $$\lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0 $$ means that two cases need to ...


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