Portfolio hedging: offsetting risk

Quantitative Risk Management in Python

Jamsheed Shorish

Computational Economist

Portfolio stability

VaR/CVaR: potential portfolio loss for given confidence level
Portfolio optimization: 'best' portfolio weights
- But volatility is still present!
Institutional investors: stability of portfolio against volatile changes
- Pension funds: c. USD 20 trillion

Image of sunglasses on a yellow flotation raft

Investment portfolio: sunglasses company
- Risk factor: weather (rain)
- More rain => lower company value
- Lower company value => lower stock price
- Lower stock price => lower portfolio value

Image of sunglasses on a yellow flotation raft Image of raindrops on a glass pane

Investment portfolio: sunglasses company
- Risk factor: weather (rain)
- More rain => lower company value
- Lower company value => lower stock price
- Lower stock price => lower portfolio value
Second opportunity: umbrella company
- More rain => more value!

Image of multicolored folded umbrellas in a stand

Investment portfolio: sunglasses company
- Risk factor: weather (rain)
- More rain => lower company value
- Lower company value => lower stock price
- Lower stock price => lower portfolio value
Second opportunity: umbrella company
- More rain => more value!
Portfolio: sunglasses & umbrellas, more stable
- Volatility of rain is offset

Image of sunglasses on a yellow flotation raft Image of multicolored folded umbrellas in a stand

Hedging: offset volatility with another asset
Crucial for institutional investor risk management
Additional return stream moving opposite to portfolio
Used in pension funds, ForEx, futures, derivatives...
- 2019: hedge fund market c. USD 3.6 trillion

Derivative: hedge instrument
European option: very popular derivative
- European call option: right (not obligation) to purchase stock at fixed price $X$ on date $M$
- European put option: right (not obligation) to sell stock at fixed price $X$ on date $M$
- Stock = "underlying" of the option
  - Current market price $S$ = spot price
- $X$ = strike price
- $M$ = maturity date

Option value changes when price of underlying changes => can be used to hedge risk
Need to value option: requires assumptions about market, underlying, interest rate, etc.
Black-Scholes option pricing formula: Fisher Black & Nobel Laureate Myron Scholes (1973)
- Requires for each time $t$:
  - spot price $S$
  - strike price $X$
  - time to maturity $T := M - t$
  - risk-free interest rate $r$
  - volatility of underlying returns $\sigma$ (standard deviation)

¹ Black, F. and M. Scholes (1973). "The Pricing of Options and Corporate Liabilities", Journal of Political Economy vol 81 no. 3, pp. 637–654.{{3}}

Market structure
- Efficient markets
- No transactions costs
- Risk-free interest rate
Underlying stock
- No dividends
- Normally distributed returns
Online calculator: https://www.math.drexel.edu/~pg/fin/VanillaCalculator.html
Python function black_scholes(): source code link available in the exercises

S = 70; X = 80; T = 0.5; r = 0.02; sigma = 0.2

option_value = black_scholes(S, X, T, r, sigma, option_type = "put")

print(option_value)

10.31222171237868

Hedge stock with European put option: underlying is same as stock in portfolio
Spot price $S$ falls ($\Delta S < 0$) => option value $V$ rises ($\Delta V > 0$)
Delta of an option: $\Delta := \frac{\partial V}{\partial S}$
Hedge one share with $\frac{1}{\Delta}$ options
Delta neutral: $\Delta S + \frac{\Delta V}{\Delta} = 0$; stock is hedged!
Python function bs_delta(): computes the option delta
- Link to source available in the exercises

Quantitative Risk Management in Python