44
votes
Accepted
How to estimate real-world probabilities
The risk-neutral measure $\mathbb{Q}$ is a mathematical construct which stems from the law of one price, also known as the principle of no riskless arbitrage and which you may already have heard of in ...
- 14.4k
14
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 ...
- 15k
13
votes
Theoretical limits for contango and backwardation
This is a basic fact about futures trading and the storage of commodities.
The phrase that was used by futures traders in the old days (and probably still today) was "the contango is limited by the ...
- 9,222
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 ...
- 1,497
11
votes
How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?
Let $P$ be the risk-neutral measure. We define the measure $P_S$ such that
\begin{align*}
\frac{dP_S}{dP}\big|_t &=\frac{S_t}{e^{rt}S_0}\\
&=e^{-\frac{1}{2}\sigma^2 t+\sigma W_t}.
\end{align*}
...
- 20.8k
11
votes
Present and future role of pricing quants
FO is shrinking across the large investment banks. The market is not developing new products that will need new pricing formulas, if anything it is reverting to more vanilla structures.
Nowdays FO ...
- 4,287
10
votes
Innovative ways of visualizing financial data
Although quite simple connected scatterplots can give interesting new insights on how time series perform together:
http://steveharoz.com/research/connected_scatterplot/
As an example: Gold vs. S&...
- 27.2k
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 ...
- 9,222
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 ...
- 11.2k
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)
$$
\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 ...
- 5,612
8
votes
Accepted
What is a Constant Maturity Swap (CMS) rate?
A constant maturity swap (CMS) rate for a given tenor is referenced as a point on the Swap curve. A swap curve itself is a term structure wherein every point on the curve is the effective par swap ...
8
votes
Accepted
Option on an Option
The proof is relatively long, so I focus on displaying the reasoning and major steps.
We work on a Black-Scholes model. Without loss of generality, we focus on an option with strike $P$ to buy at $t_e$...
- 7,705
7
votes
Accepted
Why Must Dividends Be Reinvested to Use Risk-Neutral Pricing?
What do you mean by "so we can price as usual"? What you showed is that for every $c \in \mathbb R$ we can find a probability measure such that the drift of $S$ is $c$. But that does not really say ...
- 3,896
7
votes
Heston Model Integration Oscillations
There has been a huge amount of work on this. Generally a Fourier transform approach is used.
First, be careful to use the form of the characteristic function that does not wind about zero in order ...
- 6,843
7
votes
Using a Constant as a Numeraire
Either $r=0$ in which $B_t$ is constant and is a valid numeraire (as is any multiple of it.)
or $ r \neq 0$ in which case an asset of constant value would give an arbitrage since we could take
$$
B_t ...
- 6,843
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,
\...
- 20.8k
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 ...
- 7,705
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 ...
- 6,824
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
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 ...
- 2,906
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 ...
- 10.3k
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 $\...
- 15k
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 ...
- 6,869
7
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, ...
- 15k
6
votes
Accepted
the cash flows behind closing out futures positions
Futures contracts are marked to market every day. This mean that each evening cash is added to or subtracted from your account as a result of the price movement that day. When you close out your ...
- 9,222
6
votes
What is a Constant Maturity Swap (CMS) rate?
In simple terms: An ordinary swap might be a 10 year swap of Libor vs a fixed rate; this fixed rate is determined in the marketplace every day and is published by Reuters, Bloomberg etc. as the '10 ...
- 9,222
6
votes
Accepted
Derive vega for Black-Scholes call from this formula?
Note that,
\begin{align*}
\frac{\partial{C}}{\partial{\sigma}} &=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\frac{-1}{\sigma})(d_-)-\frac{Ke^{-rt}}{\sqrt{2\pi}}e^{\frac{-d_-^2}{2}}(\frac{-1}{\...
- 20.8k
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.
- 1,389
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