Quant Finance

Some Quant Finance

Posted by TX on February 12, 2025

Some Quant Finance (inexhaustive list of knowledge)

Greeks

Delta

  • ATM call has delta 0.5
  • OTM call has delta 0
  • ITM call has delta 1

Buy Call-Put Parity: $ C - P = S - K e^{-rT} $, buying a call and a shorting a put is equivalent to buying the stock with borrowed money of $K e^{-rT} $

Obviously

\[- C + (P + S - K e^{-rT})= 0\]

So to hedge a short position in call, one would need to long the stock, buy a put, and borrow $K e^{-rT} $ of cash

Gamma

High means convex trade (delta moves faster than underlying)

Gamma is Highest for ATM, and Close to 0 for OTM and ITM options

  • Gamma scalping
    Use straddle (call + put) to bet on positive gamma

Stock up -> delta up -> short stock at higher price to flatten delta, profit in mean time

Stock down -> delta down -> long stock at lower price, Profit

Each hedging brings profit. But can’t hedge at infinitely high frequencies

During volatile times, option would be even more volatile, need to hedge more frequently

Vega

W.R.T. Change in implied vol (IV)
Gamma and vega usually always positive

While B.S. assumes constant vol, to fit the option price under B.S. framework, one computes that the IV.

IV typically varies across both strike ($K$) and expiry ($T$), forming a volatility surface ($\sigma(T, K)$). This is opposed to realized vol (RV), which is the fixed vol of the underlying, computed by sample SD of some assets over some window, or as a fraction of the absolute return of the asset.

Models for RV include GARCH, HAR, etc. primarily for risk purposes.
Models for IV include Stoch Vol models like Heston, Kalman filter (refer here ), for trading the IV surface.

Note the reason we have an IV surface is to fit the BS framework under constant vol and lognormal assumption. The true asset need not be lognormal, and one can obtain a constant vol also matching all option prices, by noting that

\[p(x) \propto \frac{\partial^2}{\partial K^2} Call\]

Where we obtain the market-calibrated pdf of the asset $p(x)$. Then the vol is given by $Var(X)$ under this pdf.

Note vol $\sigma_t$ is not observable, and the VIX is not tradable (but VIX future is)

IV is usually higher for ITM and OTM options than ATM

Shorting Vega means selling options and profiting from the risk premium (but faces an unlimited risk)

Brownian motion

1.$B_0 = 0$, for any $0,t1…,tn$, increment $B_{t2} - B_{t1}$, are independent
2.stationary & normal: $B_{t+h} - B_t \sim N(0, h)$ depends on h not t
3.Continuous path over t

Martingale

1.Xt is Ft measurable
2.E[Xt] is finite for all t
3.E[Xt|Fs] = Xs (expecation of future given past information is current value)

This suggests there is no systematic way to predict future changes and make a guaranteed profit.

Black-Scholes

Assumptions

  1. Efficient market (all past information is priced in)
  2. Stock price follows GBM, and Log(St) is normal
  3. Volatility is assumed to be constant
  4. Risk free rate constant and known
  5. Arbitrage free

Volatility smile

implied volatility (IV) is higher for options that are deep in-the-money or out-of-the-money compared to at-the-money options.

Cannot compute IV analytically, have to numerical (bisection or newton)

  • Why does it exist? ITM and OTM options are more in demand than ATM options Long equity funds buy OTM put (K < S) as insurance for stock crash, and speculators buy OTM calls (K > S) in hope of a large rise

Note by call-put parity, C and P shares same vol, rise in price for either C or P pushes up the vol.

volatility trading

During periods of high volatility, price fluctuations are more frequent, creating more opportunities for market makers to capture the bid-ask spread.
At same time, higher risk too. To hedge, may need wider spread.
Less volatile times (more liquidity), can have tight spread, less risk of unfavorable move

Generally, buying options is long vol, and selling options is short vol. Gamma scalping with a straddle can be seen as long vol with flat delta

Risk Management

Metrics

VaR = Upper bound of Confidence interval for the loss
ES = E[ loss | loss > VaR]

VaR ignores tail risk. One can say 99% confident Loss won’t exceed 1M, but within the 1% chance, don’t know if it’s 1M or 1B. ES addresses this

Also VAR is not subadditive: risk(A+B) < risk(A) + risk(B), which encourage diversification

PCA / ICA

Can use PCA to find which risks contribute to portfolio’s risk

Look for dimension of max variance, opposed to ICA which look for least entropy (least normal)

Also find which factors drive asset returns (macro factors)

Questions on Rates

  • What are the key risk factors in a fixed income portfolio?
    Interest rate risk, credit risk, liquidity risk, inflation risk, and reinvestment risk.

  • If 10-year Treasury yields rise by 20 bps, how would you expect corporate bond spreads to behave and why?
    Long-term yield may be due to two reasons:
    1. Investors expect long-term inflation, or central banks being hawkish (raise interest rate), hence demand higher yield
    2. Investors have confidence in economy, and shift into riskier assets (stocks), causing long term treasury bond price drop
      In first case, corporate bonds spread will widen – investors demand compensation for credit risk. Second case is the reverse – spreads will tighten

  • A zero-coupon bond matures in 5 years. How does its price sensitivity to interest rates compare to a 5-year coupon bond?
    Zero-coupon bonds are more sensitive to interest rate changes, because all cash flows come at maturity.

  • How to price exotic options with no closed-form solutions
    Price with Monte Carlo: Simulate payoff paths (usually geometric BM), and compute average payoffs.
    One might also use numerical integration, if the payoff density is known (say $p(x)$), then the avg payoff is just

    \[\int x p(x) dx\]

Which can be computed with trapizoidal rule, simpson rule or quadrature.

With MC integration this is evaluated as

\[E_q[x p(x)/q(x)]\]

By sampling from some proposal $q(x)$. More can be found here

MC converges at $O(1/\sqrt{N})$, very slow and expensive

It is better than numerical integration when dim of data > 6

Inefficiency vs Arbitrage-free

A market can be inefficient but not arbitrage-free.

A stock may exhibit price momentum because markets fail to quickly incorporate information.

Traders could exploit this pattern (inefficiency) by predicting future price movements, but this involves risk.

At the same time, the market could be arbitrage-free because there are no riskless opportunities to exploit inconsistencies in pricing.

CATALOG