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Quant Risks - Hedge Simple Options and Bond

Introduction of Risk. Hedging strategy aginst market risk for (Vanilla) Options and Bond

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By KOO Tin Lok, Story
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Quant Risks - Hedge Simple Options and Bond

1 Source of Risk

Typical sources of Risks

  1. Market Risk - Risk of change in portfolio value due to underlying assets (e.g. stock, bond, fx, commod)
  2. Credit Risk - Risk of not receving promised repayment on outstanding investment (e.g. bonds)
  3. Operational Risk - Risk of losses from internal process

In this chapter we will focus on Market Risk.

1.1 Market Risk

Market risk is the change in portfolio value due to the change of market value of the underlying assets (e.g. asset price, exchange rate, commodity price, etc). This chapter will cover two examples of assets and how to hedge their Risk

PortfolioSources of riskHedge method
Options $V_t$Asset Price $S_t$Delta, Gemma, Theta, Vega, Rho
BondInterest Rate $r_t$Bond Immunization

2 Options

2.1 Option pricing

Option price with [[Black-Scholes Merton Model]]:

CALL Option

\[c(t, S_t) = S_t e^{-q(T-t)} N(d_1) - K e^{-r(T-t)} N(d_2)\]

PUT Option

\[p(t, S_t) = K e^{-r(T-t)} N(-d_2) - S_t e^{-q(T-t)} N(-d_1)\]

where

\[\begin{align} d_1 &= \frac{\ln \frac{S_t}{K}+(r-q+\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t} } \\ d_2 &= \frac{\ln \frac{S_t}{K}+(r-q-\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t} }= d_1-\sigma \sqrt{T-t} \\ \end{align}\]

Put-call Parity: The price of CALL options and PUT options can be related by

\[p_t + S_t e^{-q(T-t)} = c_t + K e^{-r(T-t)}\]

2.2 Option Greeks

Major Greeks - The most important greeks

Option GreekDefinitionOrderPropertyMath formula for Put/Call
Delta $\Delta$$\Delta = \frac{\partial V}{\partial S_t}$1Call: +ve (higher price, more payoff)
Put: -ve (higher price, less payoff)		$\Delta_c = e^{-q\tau}N(d_1)$
$\Delta_p = e^{-q\tau}(-N(-d_1))$
Gamma $\Gamma$$\Gamma = \frac{\partial^2 V}{\partial S_t^2}$2> 0
(convex curve,
and higher price, the delta goes more positive)		$\Gamma_c=\Gamma_p=e^{-q\tau}\phi(d_1)\frac{1}{S_t \sigma \sqrt{\tau}}$
Theta $\Theta$$\Theta = \frac{\partial V}{\partial t}$1time is measure in year.
need to $\frac{1}{252}$ when use		$\Theta_c=q S_t e^{-q\tau} N(d_1) -r K e^{-r\tau} N(d_2) - S_t e^{-q\tau} \phi(d_1) \frac{\sigma}{2\sqrt{\tau}}$
$\Theta_p=q S_t e^{-q\tau} (-N(-d_1)) -r K e^{-r\tau} (-N(-d_2)) - S_t e^{-q\tau} \phi(d_1) \frac{\sigma}{2\sqrt{\tau}}$
Vega $\Lambda$$\Lambda = \frac{\partial V}{\partial \sigma}$1> 0
(with higher vol, will be more in-the-money and more out-of-money
but the negative penalty of out-of-money is 0)		$\Lambda_c=\Lambda_p=S_t e^{-q\tau} \phi(d_1) \sqrt{\tau}$
Rho $\rho$$\rho = \frac{\partial V}{\partial r}$1Call: >0 (r increase, reduce the cost (PV) of exercise LONG Call (-cash + Stock)
Put: <0 (r increase, reduce the payment (FV) of exercise LONG Put (-Stock + cash)		$\rho_c=\tau K e^{-r \tau} N(d_2)$
$\rho_p=\tau K e^{-r \tau} (-N(-d_2))$

Minor Greeks - Some names of the minor greeks, not typically used.

GreekDefinitionOrder
Vanna$\frac{\partial^2 V}{\partial S_t \partial \sigma}$2
Vomma$\frac{\partial^2 V}{\partial \sigma^2}$2
Charm$\frac{\partial^2 V}{\partial S_t \partial t}$2
Speed$\frac{\partial^3 V}{\partial S_t^3}$3
Color$\frac{\partial^3 V}{\partial S_t^2 \partial t}$3

2.2.1 Delta Hedging

2.2.1.1 Construction

  • Assume you short sell European Call option, to hedge the Delta risk, you also LONG $\Delta_t$ unit of asset
  • Your portfolio is $f_t=-1c_t + \Delta_tS_t$

  • Where we use [[Black-Scholes Merton Model]] to model $S_t$

    $S_t=S_0 exp((\mu-\sigma^2/2)t+\sigma W_t)$

    and the differential form

    $d S_t = \mu S_t d_t + \sigma S_t d W_t$

  • Your change of portfolio is $d f_t = - d c + \Delta_t d S_t + \Delta_t q S_t dt$ (assume continuous dividend q for asset)

  • using [[Ito’s Lemma]]

    \[\begin{align} dc(t, S_t) &= c_t dt + c_S \cdot d S_t + \frac{1}{2} c_{SS} \cdot d{\langle S_t \rangle}_t \\ \text{where } dS_t = \mu S_t d_t + \sigma S_t dW_t \text{ and } d{\langle S_t \rangle}_t = (\sigma S_t)^2d_t \\ &= c_t dt+\mu S_t c_S + \sigma S_t c_S \cdot dW_t + \frac{1}{2} c_{SS} \cdot \sigma^2 S_t ^2 \cdot dt \\ &= (c_t + \mu S_t c_S + \frac{1}{2} c_{SS} \cdot \sigma^2 S_t ^2)d_t + \sigma S_t c_S \cdot dW_t \end{align}\]

  • There $df_t$

    \[\begin{align} df_t &= -[(c_t + \mu S_t c_S + \frac{1}{2} c_{SS} \cdot \sigma^2 S_t ^2)d_t + \sigma S_t c_S \cdot dW_t] + \Delta_t d S_t + \Delta_t q S_t dt \\ &= -[(c_t + \mu S_t c_S + \frac{1}{2} c_{SS} \cdot \sigma^2 S_t ^2)d_t + \sigma S_t c_S \cdot dW_t] + \Delta_t (\mu S_t d_t + \sigma S_t d W_t) + \Delta_t q S_t dt \\ &= (-c_t - \mu S_t c_S - \frac{1}{2} c_{SS} \cdot \sigma^2 S_t ^2 + \Delta_t \mu S_t + \Delta_t q S_t)d_t + (-\sigma S_t c_S + \Delta_t \sigma S_t) dW_t \end{align}\]

  • The random part is only

    \[(-\sigma S_t c_S + \Delta_t \sigma S_t) dW_t\]
  • By choosing $\Delta_t = c_S$ this random terms goes to 0. (i.e. no Risk)
  • When $\Delta = 0 \implies \text{No Change in portfolio value even when asset price changes}$
  • If we have a portfolio, with total $\Delta=0$ then this portfolio is Delta Neutral (and should have return of Risk-free rate $r$)

2.2.1.2 How to Delta Hedge

Find the $\Delta$ of the portfolio, then Buy the asset with $\Delta$ unit

2.2.2 Gamma Hedging

2.2.2.1 Construction

  • When $\Gamma$ is small, the change of asset price gives us small $\Delta$ (i.e. the portfolio value varies slightly)
  • When $\Gamma$ is large, the change of asset price gives large $\Delta$ (i.e. the portfolio value varies a lot)
  • more frequent balancing is needed in the case of large $\Gamma$
  • Want $\Gamma = 0 \implies \text{No Change in the delta even when asset price changes}$
  • this risk is due to we ignore the higher order terms in Ito’s formula
  • Not complete-risk free as we still ignore more higher-order terms, but they are usually very small

2.2.2.2 How to Gamma Hedge

Find the $\Delta$ and $\Gamma$ of the different portfolio Create a new weighted portlio $w$ s.t. $\vec{w}^T\cdot\vec{\Delta} =0$ and $\vec{w}^T\cdot\vec{\Gamma} =0$

2.3 Talyor Series Expansion and second-order-approximation:

Formula

Single variant: $f(x) = \sum_{0}^{\infty}{\frac{1}{n!}f_x^{(n)}(a)(x-a)^n}$

Multi variant: $f(x, y) = \sum_{0}^{\infty}{\frac{1}{n!}((x-a)\frac{\partial}{\partial x} + (y-a)\frac{\partial}{\partial y})^n(f(a,b))}$

What talyor series means?

  • Example 1a:
    • Image you have a line $y=f(x)=2x +C$
    • Given $f(10) = 8$
    • What is $f(12)$ (without finding $f(x)$, because in general$f(x)$ can be complicated, and the derivatives are actually our Greeks)
    • First find the derivatives $f’(x) = \frac{dy}{dx}=2$ , and hence $f’‘(x)=0$
    • intutitively, $f(12) = f(10) + slope * \text{difference in x}$
    • i.e. $f(12) = f(10) + f’(10) (12-10) = 8+2*2=12$
    • This is the basic of talyor series. we can find any value of $f(x)$ without knowing the explicit form of $f(x)$
    • we call this (1st order) talyor series expansion of $f(x)=f(10)+f’(10)(x-10)$ at 10
    • Noted: choosing the origin point 10 to expand could be critical, we hope this point to be
      • continuous
      • differentiable
  • Example 1b:
    • What happens if the curve is not a line? (e.g. 2nd order polynomials or even higher order)
    • Let $f(x)=2x^2-7x+C$, given $f(10)=8$, what is $f(12)$
    • First, for you to verify $C=-122$

    • We can still do it be talyor series up to 2nd order derivatives

      \[f'(x)=4x-7, f''(x)=4, f'''(x)=0\]

    • Hence

      \[f'(10)=33, f''(10)=4\]
    • Then lets expand $f(x)$ at $x=10$
    \[f(x) = f(10) + f'(10)(x-10) + f''(10)(x-10)^2\]
    • \[\begin{align} \text{L.H.S.} &= 2\cdot12^2-7\cdot12-122 = 82 \\ \text{R.H.S.} &= f(12) \\ &= f(10) + f'(10)(12-10) + \frac{1}{2} f''(10)(12-10) \\ &= 8+33\cdot2+\frac{4\cdot2^2}{2} \\ &=82 = \text{L.H.S.} \\ \end{align}\]

3 Bond

3.1 Risk of bonds

Typical sources of Risks

  • Market risk: Interest risk
  • Credit risk: Default risk In this chapter we will focus on Market Risk.

3.2 Bond Investment model

  • Investment Horizon: $H$ years
  • At $t=0$ invests capital into a bond portfolio
  • At $t=H$, sells the bond
  • During $t= (0, H)$, receive coupons from bond and reinvest into risk-free asset
  • Risks:
    • Reinvetment risk: interest earned from reinvestment in risk-free asset
    • price risk: the price of selling the bond at $t=H$
  • Assume interest rate is annualized, compound annually

3.3 Sensitivity of bond

  • Duration: $\frac{\partial P}{\partial i}$
  • Convexity: $\frac{\partial^2 P}{\partial i^2}$
  • Modified Duration

    3.3.1 Bond price

  • discounted value of all future cash flows
  • price $P(i)$
\[P(i) = \sum_{k=1}^n{\frac{C_k}{(1+i)^{t_k}}}\]

3.3.2 Duration:

  • Duration: $P’(i)$, where $i$ is the annual effective yield rate

    \[\begin{align} D(i) = P'(i) &= \sum_{k=1}^n{C_k (-t_k) (1+i)^{-(t_k+1)}} \\ &= -\frac{1}{1+i}\sum_{k=1}^n{C_k t_k (1+i)^{-t_k}} \quad \square\\ \end{align}\]
  • commonly we used the modified duration $D_{mod}(i)$ which is the Duration normalized by $P(i)$ as we are more concern on the percentage change instead of the price changes. So that we can compare the bonds.
  • We also noticed that this is always -ve, we want to make it positive for easier comparison, so this is defined as -ve

  • Modified Duration: $-\frac{P’(i)}{P(i)}$, where $i$ is the annual effective yield rate

    \[\begin{align} D_{mod}(i) &= - \frac{P'(i)}{P(i)} \\ &= - \sum_{k=1}^n{\frac{C_k (-t_k) (1+i)^{-(t_k+1)}}{P(i)}} \\ &= \frac{1}{1+i}\sum_{k=1}^n{\frac{t_k \cdot C_k }{P(i) \cdot (1+i)^{t_k}}} \quad \square\\ \end{align}\]

  • We defined Macaulay duration as the weighted average of time required to receive the cashflows

    \[\begin{align} D_{mac}(i) &= \sum_{k=1}^n{\frac{t_k \cdot C_k }{P(i) \cdot (1+i)^{t_k}}} &\quad \square \\ D_{mac}(i) &= (1+i)D_{mod}(i) &\quad \square \\ \end{align}\]

3.3.3 Convexity

  • Convexity:

    \[\begin{align} C(i) = P''(i) &= \sum_{k=1}^n{C_k (-t_k)(-(t_k+1))(1+i)^{-(t_k+2)}} \\ &= \sum_{k=1}^n{C_k \cdot t_k \cdot (t_{k} + 1) \cdot (1+i)^{-(t_k+2)}} \\ &= \frac{1}{(1+i)^2} \sum_{k=1}^n{C_k \cdot t_k \cdot (t_k + 1) \cdot (1+i)^{-t_k}} &\quad \square\\ \end{align}\]

  • Modified Convexity:

    \[\begin{align} C_{mod}(i) &= \frac{P''(i)}{P(i)} \\ &= \frac{1}{(1+i)^2} \sum_{k=1}^n{\frac{C_k \cdot t_k \cdot (t_k + 1) }{P(i) (1+i)^{t_k}}} &\quad \square\\ \end{align}\]

3.3.4 Use of Duration and Convexity

  • Let $i_0$ be the initial yield rate

  • Second-order approximation of the change in bond price

    \[\begin{align} P(i) &\xrightarrow[i_0]{Talyor Expansion} P(i_o) + P'(i_0)(i-i_0) + \frac{P''(i_0)(i-i_0)^2}{2!} + \frac{P^{(3)}(i_0)(i-i_0)^3}{3!} + \frac{P^{(4)}(i_0)(i-i_0)^4}{4!} + ... \\ &\approx P(i_o) + P'(i_0)(i-i_0) + \frac{P''(i_0)(i-i_0)^2}{2!} \\ \\ \frac{P(i) - P(i_0)}{P(i_0)} &\approx \frac{P'(i_0)}{P(i_0)}(i-i_0) + \frac{1}{2} \frac{P''(i_0)}{P(i_0)} (i-i_0)^2 \\ &= -D_{mod}(i_0)(i-i_0) + \frac{1}{2}C_{mod}(i_0)(i-i_0)^2 \quad\square\\ \end{align}\]
  • Portfolio of bonds
    • if all bonds have the same yield rate
    • Then $D_{mod}$ and $C_{mod}$ is just the weighted average of the duration
    • $D_{mod}(i) = \sum_{k=1}^n{\frac{P_k(i)}{P(i)} \cdot D_{mod}^k(i)}$
    • $C_{mod}(i) = \sum_{k=1}^n{\frac{P_k(i)}{P(i)} \cdot C_{mod}^k(i)}$

3.3.5 Internal Rate of return (IRR)

  • In bond, we use IRR to check the annual effecitve interest rate compouded interest

  • $FV_H = P_0 (1+i_{IRR})^H$

    \[IRR = (\frac{FV_H}{P_0})^\frac{1}{H} - 1 \quad \square\]

3.4 Bond Price Immunization - Hedge risk against i

3.4.1 Construction

To study Bond price moment, we assumes

  • at $t=0$, interest rate = $i_0$
  • at $t > 0$, interest rate change to $i$
  • This $i$ will not change for all the time horizon

    3.4.2 IRR

  • Further expands $IRR$
  • let $m$ be the last coupon received before $H$, i.e. $t_m <= H < t_{m+1}$
  • $P_H$ is the bond price at $H$

  • $S_H$ is the cashflow at H received by coupons reinvested with $i$

    \[\begin{align} FV_H(i) &= P_H(i) + S_H(i) \\ P_H(i) &= \sum_{k=m+1}^{n} {\frac{C_k}{(1+i)^{t_k-H}}} \\ S_H(i) &= \sum_{k=1}^{n} {C_k(1+i)^{H-t_k}} \\ FV_H(i) &= \sum_{k=m+1}^{n} {\frac{C_k}{(1+i)^{t_k-H}}} + \sum_{k=1}^{n} {C_k(1+i)^{H-t_k}} \\ & = (1+i)^H \sum_{k=1}^{n} {C_k (1+i)^{-t_k}} \\ & = (1+i)^H P_0(i) \\ IRR &= [\frac{(1+i)^H P_0(i)}{P_0(i_0)}]^{1/H} - 1 \\ &= \frac{1}{P_0(i_0)^{1/H}}(1+i)P_0(i)^{1/H} - 1 \quad \square \end{align}\]

3.4.3 Bond Immunization $\frac{\partial IRR}{\partial i}=0$

  • Set $\frac{\partial IRR}{\partial i}=0$ (as this is unsolvable, we only let this to 0 for $i=i0$)

    \[\begin{align} \frac{\partial IRR}{\partial i} &= [\frac{P_0(i)}{P_0(i_0)}]^{1/H} + (1+i) [\frac{1}{P_0(i_0)}]^{1/H}\frac{1}{H} P_0(i)^{1/H-1}P'_0(i) \\ \frac{\partial IRR}{\partial i}|_{i=i_0} &= [\frac{P_0(i_0)}{P_0(i_0)}]^{1/H} + (1+i_0) [\frac{1}{P_0(i_0)}]^{1/H}\frac{1}{H} P_0(i_0)^{1/H-1}P'_0(i_0) \\ 0 &= 1 + (1+i_0)\frac{1}{H^*}\frac{P'_0(i_0)}{P_0(i_0)} \\ H^* &=(1+i_0)(-\frac{P'_0(i_0)}{P_0(i_0)}) \\ &= (1+i_0)D_{mod}(i_0) \\ &= D_{mac}(i_0) \quad \quad \square \end{align}\]
  • The Solution is that we can find a $H^*=D_{mac}(i_0)$ such that the $IRR$ have no movement verus $i$
  • How can we choose make $H^*=D_{mac}(i_0)$?
    1. Choose a bond, find the $D_{mac}(i_0)$ and change our investment time horizon to $D_{mac}(i_0)$
    2. Find a bond in the market that its $D_{mac}(i_0)$ matches our investment time horizon $H$
    3. Construct a bond portfolio with different $D_{mac}(i_0)$ such that it matches our investment time horizon $H$.
      • Recall the portfoilo Duration is just weighted sum $D_{mod}(i) = \sum_{k=1}^n{\frac{P_k(i)}{P(i)} \cdot D_{mod}^k(i)}$

3.4.4 Optimal Portfolio

  • There could have more than onr Portfolio that have the same $D_{mac}(i_0)$, is there anyone that is the most optimal?
  • Choose the portfolio with the highest Convexity $C_{mod}$

  • Proof by second order approximation

    \[\begin{align} P_0(i) &= P_0(i_0) + (-D_{mod}(i_0)P_0(i_0))\cdot(i-i_0)+\frac{1}{2}(C_{mod}P_0(i_0))\cdot(i-i_0)^2 + ... \end{align}\]
  • If there are two portfilio $A$, $B$, which have the same $D_{mac}(i_0)$ then they have the same $P_0(i_0)$, $D_{mod}(i_0)$ therefore, the $C_{mod}$ can give higher $IRR$

    3.4.5 Extend to multiple changes of ir

  • In the above model, we assumed only 1 change of $i$ at $t>0$
  • if this changes multiple times, we need to reblance it
  • For each payment day (after coupon is received) and before i changes.
  • we need to rebalance to match the future Duration
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