MFIN 354: Risk Management
in Financial Institutions
Ibrahim Jamali, PhD
How Traders Manage Risk
1. A Brief Taxonomy of Risk
• Market risk: Defined as the risk to a financial portfolio from
movements in market prices such as equity prices, foreign exchange
rates, interest rates and commodity prices
• Liquidity risk: Risk from conducting transactions in markets with
low liquidity (i.e., low trading volume or large bid-ask spreads)
• Operational risk: Risk of loss due to physical catastrophe, technical
failure and human error in the operation of a firm, including fraud,
failure of management and process errors. Exposure to operational risk
offers very little returns so, ideally, firms should eliminate it
• Credit risk: Risk that a counterparty may become less likely to fulfill
its obligations in full or in part on the agreed upon date
2. How Traders Manage (and View) Risk
Financial Institution
Front Office
(Trading
Function)
Middle Office
(capital
adequacy,
overall risk,
regulatory
compliance)
Back office
(record keeping
function)
2. How Traders Manage (and View) Risk
• Front office: Hedges risk by ensuring that exposure to individual
market variables is not too great
• Middle office: aggregates the exposures of all traders
Risk Decomposition
Identify risks on a one-by-one
basis and handle each separately
vs.
Risk Aggregation
Reduce risk by
being well diversified
• Traders have a directional view of risk (more in line with risk
decomposition)
3. Delta
• The delta of a portfolio with respect to a market
∆𝑃
variable
∆𝑆
• Using calculus, delta is the partial derivative of the portfolio value with
𝜕𝑃
respect to the value of the variables
𝜕𝑆
• Example: Trader works on a gold portfolio
Value of the portfolio (currently) = $317,000
Price of gold = $1,300 per ounce
Suppose that there is a small increase in the price of gold from $1,300
per ounce to $1,300.10. Also suppose that this $0.10 increase in the price of
gold decreases the value of your portfolio by $100.
Sensitivity of the portfolio = -100/0.1 = -1,000
3. Delta
Example (continued): This is the delta of the portfolio. The portfolio
loses value at rate of $1,000 per 1$ increase in the price of gold.
Similarly, it gains value at a rate of about $1,000 per $1decrease in the
price of gold.
The trader can eliminate the delta exposure by buying 1,000 ounces of
gold. This is because the delta of a long position in 1,000 ounces of gold
is 1,000.
This is know as delta hedging. When the hedge is combined with the
existing portfolio, the resulting portfolio has a delta of zero. Such a
portfolio is referred to as delta neutral
3. Delta: Linear Products
• Linear product: one whose value at any given time is linearly
dependent on the value of an underlying market variable (Example:
Forward or futures contracts; Counterexample: Options)
• A linear product can be hedged relatively easily (hedge and forget)
• The value of a forward contract where the holder has to sell (short)
the asset for a delivery price of K is:
𝐾 − 𝐹0 𝑒 −𝑟𝑇
where F0 is the forward price (today), K: delivery/settlement price, r:
Interest rate, T: Time to maturity
• For a long forward position, the value of the contract is: 𝐹0 − 𝐾 𝑒 −𝑟𝑇
3. Delta: Linear Products
• Example: A US bank has entered into a forward contract with a
corporate client where it agreed to sell the client 1 million euros for
$1.3 million in one year
Assume that the euro and dollar interest rates are 4% and 3%,
respectively, with annual compounding
The Present Value (PV) of 1.3 million dollars in one year is
1,300,000/1.03 = 1,262,136 dollars
The PV of a 1 million euro cash flow in one year is 1,000,000/1.04 =
961,538
Let S denote the value in dollars 1,262,136 - 961,538×S
3. Delta: Linear Products
• The delta of the contract is -961,538
• It can be hedged by buying 961,538 euros. Because of linearity, the
hedge provides protection against both small and large movements in
S
• When the bank enters into the opposite transaction and agrees to buy
one million euros in one year, the value of the contract is also linear in
S:
961,538×S – 1,262,136
• It must hedge by shorting 961,538 euros
3. Delta: Linear Products
• It can do so by borrowing the euros today at 4% and immediately
converting them to USD. The one million received in one year are
used to repay the loan.
• Linear products => hedges protect against large and small changes in S
=> hedge and forget
3. Delta: Nonlinear Products
• Options are a nonlinear product
• For nonlinear products, the relationship between the underlying asset
price and the value of the product is nonlinear
• Nonlinearity makes hedging more difficult for two reasons:
1. Making a portfolio delta neutral only protects against small
changes/movements in the price of the underlying asset
2. The hedge needs to be changed frequently => dynamic hedging
3. Delta: Nonlinear Products
Option
price
Slope = D
B
A
Stock price
3. Delta: Nonlinear Products
• Example: Trader sells 100,000 European call options on a nondividend paying stock
1. Stock price is $49
2. Strike price is $50
3. Risk-free rate is 5%
4. Stock price volatility is 20% per annum
5. Time to option maturity is 20 weeks
Amount received for the options is $300,000 and the trader has no position
dependent on the stock
Value/price/premium on the option?
3. Delta: Nonlinear Products
• Example (continued):
𝑐 = 𝑆0 𝑁 𝑑1 − 𝐾𝑒 −𝑟𝑇 𝑁 𝑑2
𝑑1 =
𝑑2 =
𝑙𝑛 𝑆0 Τ𝐾 + 𝑟 + 𝜎 2 Τ2 𝑇
𝜎 𝑇
𝑙𝑛 𝑆0 Τ𝐾 + 𝑟 − 𝜎 2 Τ2 𝑇
𝜎 𝑇
= 𝑑1 − 𝜎 𝑇
3. Aside: The (Standard) Normal Distribution
3. Aside: The (Standard) Normal Distribution
• Rules of the normal distribution
N(-d1)=1-N(d1)
or N(d1) = 1-N(-d1)
3. Delta: Nonlinear Products
• 𝑆0 : asset price today
• K: strike or exercise price
• r: continuously compounded risk-free rate
• 𝜎: stock price volatility
• T: Time to maturity
3. Delta: Nonlinear Products
• Example (continued):
Numerator of d1 = ln(49/50)+(0.05+0.22/2)×(20/52)
= -0.0202+0.0269
= 0.0067
Denominator of d1 = 0.2×
d1 =
0.0067
0.124
= 0.054
20
52
= 0.124
3. Delta: Nonlinear Products
• Example (continued):
N(d1) = N(0.054) = N(0.05)+0.4[N(0.06)-N(0.05)]
= 0.5199 + 0.4[0.5239-0.5199]
= 0.5199 + 0.0016
= 0.5215
𝑑2 = 𝑑1 − 𝜎 𝑇 = 0.054 − 0.2
20
52
= −0.07
N(d2) = N(-0.07) = 0.4721
c = 49×0.5215 - 50×e-0.05×20/52×0.4721
= 25.5535 - 50×0.981×0.4721
= 2.397
3. Delta: Nonlinear Products
• From your structured finance (or going back to the price of a call that I
have in the previous slide and realizing it is only a derivative with
respect to S) class ∆ 𝑐𝑎𝑙𝑙 = 𝑁 𝑑1 = 0.522
• Because the trader is short 100,000 options, the value of the trader’s
portfolio is -$240,000 and the delta of the portfolio is -$52,200
• To make the portfolio delta neutral, the trader needs to purchase
52,200 shares of the underlying asset
• To preserve delta neutrality, rebalancing is needed
3. Dynamic hedging: option closes in the money
(for a call option S0>K, put option S0<K)
3. Dynamic hedging: option closes out of
the money
4. Gamma
• Delta neutrality provides protection against small changes in the price
of the underlying asset
• Gamma (Γ) measures the extent to which large changes affect the
value of a portfolio
• Gamma is the rate of change of the portfolio’s delta (i.e., second
derivative of portfolio value) with respect to the change in the price of
the underlying asset
• 𝐺𝑎𝑚𝑚𝑎 =
𝜕2 𝑃
𝜕𝑆 2
4. Gamma
• If gamma is small, delta changes slowly and the adjustments to keep
the portfolio delta neutral need to be made only infrequently
• If gamma is large, delta is highly sensitive to the underlying asset price
• Gamma = curvature (for practitioners)
5. Making a Portfolio Gamma Neutral
• Linear product has zero gamma and cannot be used to change the gamma of
a portfolio
• Position in a nonlinear asset is required (e.g., option) is required to change
the gamma of a portfolio
• Suppose that a delta neutral portfolio has a gamma equal to Γ and traded
option has a gamma equal to Γ𝑇
• If the number of traded options added to the portfolio is 𝑤𝑇 , the gamma of
the portfolio is:
𝑤𝑇 Γ𝑇 + Γ
• The position in the traded option necessary to make the portfolio gamma
Γ
neutral is 𝑤𝑇 = −
Γ𝑇
5. Making a Portfolio Gamma Neutral
• Gamma neutrality can be maintained only if the position in
Γ
the traded option is adjusted so that it is always equal to −
Γ𝑇
• Delta neutrality provides protection against relatively small
changes in the asset price between rebalancing
• Gamma neutrality provides protection against larger
movements in the asset price between hedge rebalancing
5. Making a Portfolio Gamma Neutral
• Example: Suppose that a portfolio is delta neutral and has a gamma of
-3,000
∆ 𝑇 = 0.62
• Delta and gamma of a particular option (call) =
Γ𝑇 = 1.50
• The portfolio can be made gamma neutral by including in the portfolio
a long position of 3,000/1.5 = 2,000 in the call option
• Now, portfolio gamma -3,000+1.5×2,000 = 0
• However, the delta of the portfolio will then change from zero to:
2,000×0.62 = 1,240
• A quantity of 1,240 of the underlying asset must be sold to maintain
delta neutrality
6. Vega
• The vega of a portfolio, V, is the rate of change of the value of the portfolio
with respect to volatility 𝜎 of the underlying asset price
•𝑉=
𝜕𝑃
𝜕𝜎
• High vega => portfolio is sensitive to small changes in volatility
• The vega of a portfolio can only be changed by adding a position in a traded
option
• If V is the vega of the portfolio and 𝑉𝑇 is the vega of the traded option, a
position of − 𝑉Τ𝑉𝑇 makes the portfolio vega neutral
• A portfolio that is gamma neutral will not, in general, be vega neutral and
vice versa
• If a hedger requires a portfolio to be both gamma and vega neutral, at least
two options should be used
6. Vega, Gamma and Delta Neutrality
Delta
Gamma
Vega
Portfolio
0
-5,000
-8,000
Option 1
0.6
0.5
2.0
Option 2
0.5
0.8
1.2
6. Vega, Gamma and Delta Neutrality
• How to make the portfolio vega neutral?
• Long position of 4,000 in option 1
• New delta of the portfolio 2,400 meaning 2,400 units of the asset have
to be sold to maintain delta neutrality
• New gamma of portfolio -3,000
• To make the portfolio gamma and vega neutral, both option 1 and
option 2 can be used
6. Vega, Gamma and Delta Neutrality
• If 𝑤1 and 𝑤2 are quantities of option 1 and option 2 that are added to the
portfolio
−5,000 + 0.5𝑤1 + 0.8𝑤2 = 0
−8,000 + 2.0𝑤1 + 1.2𝑤2 = 0
• The solution to these equations 𝑤1 = 400 and 𝑤2 = 6,000
• The portfolio can therefore be made gamma and vega neutral by including
400 of option 1 and 6,000 of option 2
• Delta of portfolio after the addition of the traded options is
400×0.6+6,000×0.5=3,240
• Hence, 3,240 units of the underlying asset have to be sold to maintain delta
neutrality
• Delta neutrality protects against variations in volatility
7. Theta and Rho
• Theta of a portfolio, Θ, is the rate of change in the value of the
portfolio of value with respect to the passage of time
• Rho is rate of change of a portfolio with respect to interest rates
8. Taylor Series Expansions
• Consider a portfolio that is dependent on a single asset price, S
• If the volatility of the underlying asset and interest rates are assumed
to be constant, the value of the portfolio is a function of S and time, t:
∆𝑃 =
𝜕𝑃
∆𝑆
𝜕𝑆
+
𝜕𝑃
∆𝑡
𝜕𝑡
+
1 𝜕2 𝑃
2
∆𝑆
2 𝜕𝑆 2
∆𝑃 = ∆∆𝑆 + Θ∆𝑡
1 𝜕2 𝑃
𝜕2 𝑃
2
+
∆𝑡 +
∆𝑆∆𝑡 …
2 𝜕𝑡 2
𝜕𝑆𝜕𝑡
1
+ Γ∆𝑆 2 + ⋯
2
• Θ∆𝑡 is non-stochastic
• For a delta neutral portfolio, ∆𝑃 = Θ∆𝑡 +
1
Γ∆𝑆 2
2
8. Taylor Series Expansions
• When gamma is positive, the holder of the portfolio gains from large
movements in the asset price and loses when there are small
movements in asset prices
• When gamma is negative, large positive or negative movements in the
asset price leads to severe losses
8. Taylor Series Expansions
• Example 8.2: Delta neutral portfolio of options on an asset:
Γ = −10,000
∆𝑆 = +2
for a very small (assume zero) ∆𝑡
• Using the prior equation, the decrease in the value of the portfolio
would be approximately
0.5×10,000×$22 = $20,000
Note that the same decrease would occur if there was a change of -2