Notesale: Turn your study into money

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FM03 Revision Notes
19 May 2016

Introduction
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Financial Markets
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Financial markets are where financial assets are bought and sold, allow market participants to reduce their
risks, facilitate efficient allocation of capital;
Well-functioning financial markets: (1) reduce transaction costs and (2) provide trading opportunities when
needed (liquidity);
Market participants: (1) individuals, (2) companies, (3) federal, state, and local governments, (4) central
banks, (5) supranational (International Monetary Fund, European central bank…);

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Depository instituoins
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Financial institutions are often classified: (1) depository, or (2) non-depository institutions;
Asset-liability management: invest deposits in financial assets that:
(1) Provide higher return than what is paid on the deposits,
(2) Are liquid enough to cover sudden withdraws;
Relevant risks include: (1) liquidity risk, (2) credit risk, (3) interest risk, (4) regulation risk;

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And the new NAV is
𝑉 + 𝑋
𝑛𝑒𝑤 𝑁𝐴𝑉 =
= 𝑁𝐴𝑉
𝑁 + 𝑋/𝑁𝐴𝑉
Closed-end funds: funded once by Initial public offering (IPO), do not accept additional investment, share
price is determined by supply and demand (may be higher/ lower than NAV), often traded on exchanges;
Exchange-traded funds (ETF): can be seen as a combination of open-end funds and closed-end funds, new
shares can be created/liquidated by ‘authorized participants’ (AP), most track an index;
Asset-liability management: increase its NAV;

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Primary markets
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Examples:
(1) A government or a corporate sells bonds;
(2) Initial public offer (IPO): equities of companies are sold to investors for the first time;
(3) Privatization: IPO of a government-owned company;
(4) Securitization: transformation of private assets into publicly traded assets;
Investment banks are often involved when new securities are issued;
Securities are usually issued through an auction where the price is determined by supply and demand;

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Double auctions
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Most exchanges are double auctions where both buyers and sellers bid for securities and the most generous
offers are selected to participate in trades;
Exchanges can be call auctions or continuous auctions:
(1) In a call auction, the market is cleared once at the end of a bidding period by matching the maximum
amount of buying and selling offers;
(2) In a continuous auction, a new offer is matched immediately with existing offers if possible, otherwise,
the new offer is added to the limit order book;
Call auctions:
- Selling offers: a piecewise constant non-decreasing function 𝑥 → 𝑠(𝑥), the supply cure;
- Buying offers: a piecewise constant non-increasing function 𝑥 → 𝑑(𝑥), the demand curve;
- The market is cleared by matching maximum number of trades: 𝑥 = sup {𝑥|𝑠(𝑥) ≤ 𝑑(𝑥)}, and the
interval [lim 𝑑 𝑥 , lim 𝑠 𝑥 ] consists of the market clearing prices;
J↑J

-

J↓J

Market clearing can be interpreted as finding the social optimum: 𝑆 𝑥 =
J
Q

•

J
Q

𝑠 𝑧 𝑑𝑧 and 𝐷 𝑥 =

𝑑 𝑧 𝑑𝑧 may be interpreted as the cost of producing x units, and the value of consuming x units;
- Market is cleared by minimizing the difference 𝑆 𝑥 − 𝐷(𝑥);
Convex analysis:
𝐹𝑜𝑟 𝑎 𝑟𝑒𝑎𝑙 − 𝑣𝑎𝑙𝑢𝑒𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓 𝑜𝑛 𝑎𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝐼, 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑎𝑟𝑒 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡:
𝑎 𝑓 𝑖𝑠 𝑐𝑜𝑛𝑣𝑒𝑥,
J

𝑏 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑛𝑜𝑛 − 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 ∅: 𝐼 → 𝑅 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑓 𝑥 = 𝑓 𝑥 +

∅ 𝑧 𝑑𝑧 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑥 ∈ 𝐼
J

𝑐 𝑓 𝑖𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑜𝑛 𝐼 𝑒𝑥𝑐𝑒𝑝𝑡 𝑝𝑒𝑟ℎ𝑎𝑝𝑠 𝑜𝑛 𝑎 𝑐𝑜𝑢𝑛𝑡𝑎𝑏𝑙𝑒 𝑠𝑒𝑡, 𝑖𝑡𝑠 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑓 ^ 𝑖𝑠 𝑛𝑜𝑛 − 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
𝑎𝑛𝑑 𝑏 ℎ𝑜𝑙𝑑𝑠 𝑤𝑖𝑡ℎ ∅ = 𝑓 ^ ;
𝑷𝒓𝒐𝒐𝒇 𝑜𝑓 𝑏 ⟹ 𝑎 : 𝐿𝑒𝑡 𝑥f ∈ 𝐼 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥g < 𝑥i 𝑎𝑛𝑑 𝛼f > 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝛼g + 𝛼i = 1
...

^
^
Note that ∅ in (b) is not unique: any ∅ with 𝑓u ≤ ∅ ≤ 𝑓v will do;
^
^
Subgradient: A 𝑣 ∈ 𝑅 is a subgradient at 𝑥 if 𝑣 ∈ [𝑓u 𝑥 , 𝑓v 𝑥 ]; and the set of subfradients of 𝑓 at 𝑥 is
known as the subdifferential of 𝑓 at 𝑥 and is denoted by 𝜕𝑓(𝑥); Note that an 𝑥 ∈ 𝐼 minimizes f over 𝐼 if
and only if 0 ∈ 𝜕𝑓(𝑥);
- Back to market clearing, an 𝑥 minimizes 𝑆 𝑥 − 𝐷(𝑥) iff 0 ∈ 𝜕 𝑆 − 𝐷 𝑥 , which means:
0 ∈ [𝑠u 𝑥 − 𝑑u 𝑥 , 𝑠v 𝑥 − 𝑑v 𝑥 ]
Limit order book (LOB): where the offers remaining after market clearing are recorded; LOB gives the
marginal prices for buying or selling a given quantity at the best available prices;
Flatter the curve s, more liquid the market;
Continuous auction: market is cleared very frequently;
Limit sell orders above the best ask-price and limit buy orders below the best bid-price increase liquidity; A
market order is an order to buy/sell a given amount at the best available prices, which reduce liquidity;
-

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Price per share 𝑆 𝑥 =

x J
J

, the convexity of 𝑆 gives us: if 𝑥g < 𝑥i
3

𝑆 𝑥g = 𝑆
•

Jp
Jq

𝑥i + 1 −

Jp
Jq

0 ≤

Jp
Jq

𝑆 𝑥i , so that 𝑆 𝑥g ≤ 𝑆 𝑥i ;

High-frequency trading and algorithmic trading: anticipate the market behaviour by studying the dynamics
of the limit order book and order flow – predictable phenomena;

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Stock markets
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Common stock: represents ownership in a company; capital structure refers to the mix of a company’s debt
and equity; market capitalization; heavily traded by institutional investors;
Private equity: common stock of companies that are not publicly traded in exchanges; less liquid; risky;
Most publicly traded stocks are traded in electronic limit order markets where limit orders are automatically
matched by market orders or recorded in limit order books;

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Bond markets

•

Bonds are long term debt instruments that give coupon payments at regular time intervals and principal
payments at maturity; often referred to as fixed income instruments;
If there is no credit risk, the replication cost is: Œ 𝑐Š 𝑃• ‹
Š•g
Š

•

The behaviours of bond prices can be described in terms of YTM, and we have 𝐵• = 𝐵 𝑡, 𝑌 :

•

Œ

𝑐Š 𝑒 u•(•• u•)

𝐵(𝑡, 𝑌) =
Š•g

•

Approximation of 𝑙𝑛𝐵 to Make a positive approximation of the price:
𝝏𝒍𝒏𝑩
𝝏𝒍𝒏𝑩
1
= 𝑌,
= −𝐷, 𝑤ℎ𝑒𝑟𝑒 𝑫𝒖𝒓𝒂𝒕𝒊𝒐𝒏 𝑫 ∶=
𝝏𝒕
𝝏𝒀
𝐵

Œ

𝑡Š − 𝑡 𝑒 u•

•• u•

𝑐Š

Š•g

𝝏 𝟐 𝒍𝒏𝑩
𝝏 𝟐 𝒍𝒏𝑩
𝝏 𝟐 𝒍𝒏𝑩
1
= 0,
= 1,
𝒕, 𝒀 = 𝐶 − 𝐷 i , 𝑤ℎ𝑒𝑟𝑒 𝑪𝒐𝒏𝒗𝒆𝒙𝒊𝒕𝒚 𝑪 ∶=
𝟐
𝟐
𝝏𝒕
𝝏𝒀𝝏𝒕
𝝏𝒀
𝐵
The second-order approximation becomes: ∆𝑙𝑛𝐵 ≈ 𝑌‚ ∆𝑡 − 𝐷• ∆𝑌 +

g
i

Œ

𝑡Š − 𝑡

i u• •• u•
𝑒

𝑐Š

Š•g

i
𝐶• − 𝐷• ∆𝑌 i

And the bond returns over a holding period [𝑡, 𝑠] can be approximated by:
𝐵‚
1
i
= 𝑒𝑥𝑝 ∆𝑙𝑛𝐵 ≈ 𝑒𝑥𝑝 𝑌‚ ∆𝑡 − 𝐷• ∆𝑌 +
𝐶 − 𝐷• ∆𝑌
𝐵•
2 •

i


4

The above analysis can be extended to index-linked bonds, then index 𝐼 can be used to model default losses
in corporate bonds, define the YTM by:
Œ

𝑐Š 𝐼• 𝑒 u•(•• u•)

𝐵(𝑡, 𝑌, 𝐼) =
Š•g

The second-order approximation becomes:
𝐵‚
1
i
= 𝑒𝑥𝑝 ∆𝑙𝑛𝐵 ≈ 𝑒𝑥𝑝 𝑌‚ ∆𝑡 − 𝐷• ∆𝑌 +
𝐶 − 𝐷• ∆𝑌
𝐵•
2 •

i

+ ∆𝑙𝑛𝐼

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Currency markets
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Also known as foreign exchange (FX) market;
Largest market measured by turnover; largely traded in OTC market;
Electronic limit order markets: Spot FX, Hotspot FX;
Consider a market with a finite set 𝐽 of currencies: denote the ask-price of buying currency 𝑗 with currency 𝑖
by 𝜋 f¨ , then from the point of view of currency 𝑖, the bid-price of currency 𝑗 is
𝜋

fp ,fq

𝜋

fq ,f¬

∙∙∙ 𝜋

f® ,fp

≥ 1

g
©ª«

, global market clearing:

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Derivatives
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A derivative is a financial contract whose payoff is a function of an underlying random variable;
A forward contract is an agreement to buy/sell an asset at a future time T (maturity) for price K (forward
price); nothing is paid at time 0;
In practice, there are no perfectly liquid assets: for non-financial assets, storage and lending costs may
increase the hedging costs even further;
A future contract: standardized forward contracts traded in exchanges; future prices;
The payoff 𝑆z − 𝐾Q of your contract can be replicated by entering a futures contract with payout 𝑆z − 𝐾g
and buying 𝐾g − 𝐾Q zero-coupon bonds with maturity T, and the replication cost is 𝐾g − 𝐾Q 𝑃z ;
Call and put options:
‹
°
- The put option can be replicated at cost: 𝐶 𝐾 ‹ + 𝐾𝑃z − 𝑆Q , if the bid-price of the put is higher than
°
‹
this, there is arbitrage; By a similar argument, arbitrage if the ask-price is lower than 𝐶 𝐾 ° + 𝐾𝑃z − 𝑆Q
- If all the assets are perfectly liquid, the put-call parity: 𝑃 𝐾 = 𝐶 𝐾 + 𝐾𝑃z − 𝑆Q
We can replicate any continuous payoff function 𝑓:
±

^

𝑓 𝑆 = 𝑓 0 + 𝑓 0 𝑆+
-

v

𝑆 − 𝐾f

^
^
∆𝑓 ^ 𝐾f , 𝑤ℎ𝑒𝑟𝑒 ∆𝑓 ^ 𝐾f = 𝑓v 𝐾 − 𝑓u 𝐾 = lim 𝑓 ^ 𝐿 − lim 𝑓 ^ 𝐿 ;
²↓±

f•g

²↑±

Under bid-ask spreads, the replication cost of 𝑓 is:
±

𝜋 𝑓 =

𝑓 0

v ‹
𝑃z

^

+ 𝑓 0

v ‹
𝑆Q

±

+

𝐶 𝐾f

‹

∆𝑓

^

𝐾f

v

−

𝑓 0

u °
𝑃z

^

+ 𝑓 0

u °
𝑆Q

𝐶 𝐾f ° ∆𝑓 ^ 𝐾f

+

f•g

𝜋 −𝑓 =

𝑓 0

^

+ 𝑓 0

u ‹
𝑆Q

+

±

𝐶 𝐾f
f•g



f•g

±
u ‹
𝑃z

u

‹

∆𝑓

^

𝐾f

u

−

𝑓 0

v °
𝑃z

^

+ 𝑓 0

v °
𝑆Q

𝐶 𝐾f ° ∆𝑓 ^ 𝐾f

+

v



f•g

5

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Bermudan option: an American with certain restrictions on the timing;
Swing option: gives the right to buy a total amount Q of the underlying at a fixed price K but only a maximum
of q units per day;
Interest rate derivatives: the net cash-flows of floating leg receiver can be replicated at cost:
Œ

𝑃

‚³‹´

𝑡Š − 𝑡Šug 𝑃••

𝑌 = 𝑃•† − 𝑃•µ − 𝑌
Š•g

The fixed leg can be replicated with zero-coupon bonds, the market price is 𝑌 Œ
Š•g 𝑡Š − 𝑡Šug 𝑃•• ;
The floating leg can be considered as follows: if we receive one pound at time 𝑡Q and pay it back at 𝑡Œ …
So the cost of implementing the above strategy is 𝑃•† − 𝑃•µ ;
Forward rate agreement (FRA): an interest rate swap with N=1, thus:
-

•

𝑃 ˆ¶· = 𝑃•† − 𝑃•p − 𝑌 𝑡g − 𝑡Q 𝑃•p = 𝑃•p 𝐹•† ,•p − 𝑌

𝑡g − 𝑡Q

Again, in perfectly liquid arbitrage price free markets, 𝐹•† ,•p = 𝑌;
•

Construct the yield curve 𝑇 → 𝑌z : (1) Libor rate 𝑃z , (2) market prices 𝐵 f of government bonds, (3) market
¨

•
•
•
•
•

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swap rates 𝑌 of interest rate swaps;
Credit derivatives: payout depends on defaults of one or more bond issuer;
Credit default swap (CDS): pays a fixed amount when a given issuer defaults within a given time period, the
owner of a CDS contract pays a fixed periodic premium until maturity or default;
Index CDS: payout depends on defaults of M issuers, if there are 𝐾• defaults in (𝑡 − 1, 𝑡] at time t; the
premium paid at time t is a multiple of the number 𝑀• = 𝑀 − •
‚•g 𝐾‚ of survivors;
Credit default obligation (CDO): is specified in terms of two numbers 𝑀 < 𝑀, the protection leg pays 𝐾• if
𝑀• ∈ 𝑀, 𝑀 and the premium leg pays a multiple of min {𝑀• − 𝑀, 𝑀 − 𝑀} while it stays positive;
Strong assumptions of pricing:
(1) Markets are perfectly liquid;
(2) There is no credit risk;
Credit valuation adjustment (CVA): tries to correct for credit risk that was ignored by the model;
Funding valuation adjustment (FVA): tries to correct for incorrect lending/borrowing rates

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