16
votes
Accepted
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
I provide a solution in three steps.
The first step carefully outlines how to split up the expectation and what new measures are used. This first step does not require any special model assumption ...
- 16.2k
13
votes
Accepted
Derivation of VIX Formula
The piece you are missing is an approximation via the Taylor formula of the logarithm:
$$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$
Apply this to the first term in the final formula of the technical ...
- 1,527
10
votes
Which models do Bloomberg/Reuters use to derive implied volatility for interest rate derivatives with negative forward rates?
Short Version
Market standard (since the low interest rate environment after 2008) is to use Normal Vol (used in the Normal / Bachelier model)
Market data comes from contributors like Tullett, ICAP ...
- 9,394
10
votes
Accepted
When would open interest equal trading volume?
Futures are in "zero net supply", or "for every long there is a short", which means that at any time there are investors who are long a certain number of contracts and other investors who are short an ...
- 9,402
10
votes
Accepted
Impact on DV01 of cbot bond futures by changing coupon from 6% to 4%
It's complicated.
Assuming there is no CTD switches, then yes, the theoretical modified duration should be unchanged and the DV01 will be lower.
For simplicity, imagine that there is only one bond ...
- 11.8k
10
votes
Accepted
Relationship between Vega and Gamma in Black-Scholes model
Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends)
$$
\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 ...
- 5,702
10
votes
Accepted
Risk Neutral Valuation, Drifts and Calibration
There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing.
As an example, ...
- 16.2k
9
votes
Accepted
Option on an Option
The proof is relatively long, so I focus on displaying the reasoning and major steps.
We work on a Black-Scholes model. Without loss of generality, we focus on an option with strike $P$ to buy at $t_e$...
- 8,169
8
votes
What is meant by the funding cost of a derivative?
The theory for pricing derivatives is based on self-financing trading strategies that replicate all the payoffs of the derivative. Hence, derivatives pricing requires funding at the risk free rate, ...
- 9,394
8
votes
Accepted
Calculating DV01 for Treasury Futures with CTD switch risk
You are trying to calculate the so-called "option-adjusted DV01" (OA DV01). The nice thing about OA DV01 is that it's a smooth function of yield shifts. I'm going to be lazy here and simply ...
- 11.8k
7
votes
Heston Model Integration Oscillations
There has been a huge amount of work on this. Generally a Fourier transform approach is used.
First, be careful to use the form of the characteristic function that does not wind about zero in order ...
- 6,993
7
votes
Using a Constant as a Numeraire
Either $r=0$ in which $B_t$ is constant and is a valid numeraire (as is any multiple of it.)
or $ r \neq 0$ in which case an asset of constant value would give an arbitrage since we could take
$$
B_t ...
- 6,993
7
votes
Accepted
Black-Scholes formula for Poisson jumps
We assume that the process $\{J_t, \, t\ge 0\}$ is defined at the jump times of the Poisson process $\{N_t, \, t \ge 0\}$, and all the jump sizes are independent and identically distributed. That is,
\...
- 21.2k
7
votes
Why discounted derivative price is a martingale?
Under a Black-Scholes framework, the dynamics of the stock price under the risk-neutral measure $\mathbb{Q}$ are given by ...
$$ S_t = r S_tdt +\sigma S_tdW^{\mathbb{Q}}_t $$
... and those of the ...
- 8,169
7
votes
Accepted
The dice game and derivatives trading
The interviewer meant that he's smart. Quoting Senior VP of People operations at Google,
On the hiring side, we found that brainteasers are a complete waste of
time. How many golf balls can you ...
- 7,004
7
votes
CMS Pricing - Convexity Adjustment by Replication
The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate.
Let us start with the fair ...
- 1,012
7
votes
Radon Nikodym derivative when changing numeraires
Any nonnegative random variable $Z$ with expectation 1 is a Radon-Nikodym derivative:
$$
\mathbb{E}^{\mathbb{P}} \left(Z\right) = \mathbb{E}^{\mathbb{P}} \left(\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\...
- 2,680
7
votes
Accepted
Static vs Dynamic Hedging: when is each one used?
It depends a little bit what you're trying to do.
If you can statically replicate the payoff of a position at $t=0$,
then putting on that hedge will insulate you from all risk coming
from the ...
- 3,056
7
votes
How do market makers calculate the IV for options?
They do not calculate it, they set it at a market clearing level based on supply and demand. It is similar to the way equity market makers set the price of a stock: a lot of buyers => raise the ...
- 11.6k
7
votes
Accepted
What is the Radon-Nikodym derivative in the Heston model?
Let
\begin{align*}
\mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\
\mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t},
\end{align*}
where $\...
- 16.2k
7
votes
Accepted
Differences between main classes of interest pricing derivatives models
I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not.
1 ) STIR: as the term ...
- 9,394
6
votes
Using a Constant as a Numeraire
A Numeraire must be a tradeable asset. If you can find a constant tradeable asset, then yes a constant can be used as a numeraire.
- 1,439
6
votes
What is a Constant Maturity Swap (CMS) rate?
In a vanilla swap, the IR on the floating leg usually depends on the reset period/swap frequency. If frequency is 6m, 6m LIBOR is used for reset, 3m LIBOR for quarterly resets etc. In a floating CMS ...
- 61
6
votes
Accepted
What are the books in which to study the basics of the derivative financial instruments?
As mentioned by @Adam, Stochastic Calculus for Finance by Shreve is a good start if you have a reasonably strong mathematical background. Volume I is simpler, as it presents derivative pricing methods ...
- 8,169
6
votes
Accepted
Replicating a portfolio with a certain payoff function
A general hedging strategy
Let assume that $S_1(t)$ and $S_2(t)$ are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM):
$$\forall \, i \in \{1,2\}, dS_i(t) =...
- 8,169
6
votes
Accepted
Replicating a square derivative with calls and puts
Note that
\begin{align*}
S_T^2 = 2\int_0^{S_T} k dk.
\end{align*}
Then
\begin{align*}
S_T^2 &= 2S_T^2-2\int_0^{S_T} k dk\\
&=2S_T\int_0^{S_T}dk-2\int_0^{S_T} k dk\\
&=2\int_0^{S_T} (S_T-k)...
- 21.2k
6
votes
Cash-settled swaptions
The advantage of cash-settled swaptions is that the payoff only depends on one variable: the corresponding swap rate which is directly observable in the market:
$$
\mathrm{Payoff}(T) = f(S_T) = A^{\...
- 3,946
6
votes
Refer some most recent books of derivatives pricing by C++
If by "modern C++" you mean C++11 and beyond, I'm afraid you won't find such a book at this time.
If you are content with idiomatic C++ 03, as in "using the STL and smart pointers instead of managing ...
- 7,853
6
votes
Accepted
The positivity of the market price of risk
No, it can be negative. The price of risk is what you agree to receive on average in exchange for positive returns when the risk measure is high, and determined by the covariance of the risk measure ...
- 932
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