APM466 Midterm 1 - March 3, 2010
Question 1. Consider the following two coupon-bearing government issued bonds:
Maturity date
June 4, 2010
December 4, 2010
Coupon
$2
$2
Price
$102
$102
Calculate the 9-month risk-free yield rate.
Question 2. A stock is val

A Worked-Out Example
Credit VaR
Goodrich-Rabobank Revisited
Portfolio Credit Risk II
Prof. Luis Seco
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 31, 2011
Prof. Luis Seco
Portfolio Credit Risk II

Review of Basic Concepts
Goodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit Loss
Portfolio Credit Risk
Prof. Luis Seco
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 27, 2011
Prof. Luis Seco
Po

Option Pricing
OptionPricing
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 23, 2011
Prof. Luis Seco (University of Toronto)
Option Pricing
July 23, 2011
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Table of Contents
1
Example
2
Discoun

Time is money
The theory of interest rates
The issuing
entity
Coupons are
paid every 6
months. The
amount
represents
the total for
one year
On this date,
the issuer
pays $100 to
the holder of
the bond
Price = - (annual coupon rate)
n
365
+ pi (1 + r )
cle

Risk Management
Bank Risks
Market Risk. The risk in reducing the value of the Banks positions due
to changes in markets.
Credit Risk. The risk in reducing the value of the Banks assets due to
changes in the credit quality of the counterparties.
Counterpar

The Business Case
I am holding a 15-year treasury zero on a $1,000 notional. What is my VaR?
(Note: this instrument does not exist)
1
The Model
No default risk
1 day VaR
95% condence
log-normally distributed yield
Present value: $370.
The yield chan

Risk Management
Exercises
Exercise
Value at Risk calculations
Problem
Consider a stock S valued at $1 today, which after
one period can be worth ST: $2 or $0.50.
Consider also a convertible bond B, which after one
period will be worth max(1, ST).
Determin

Pricing a European Cap (or Call)
1.25
2
U=1
B = -0.75
0.75
1.5
1
1
0.333
0.25
U = 0.8333
B = -0.5
0.125
0.75
U = 0.5
B = -0.25
0.5
0
Pricing an American Cap (or Call)
1.25
2
U=1
B = -0.75
1.5
0.75
1
1
0.3333
.75
0.5
0.25
U = 0.8333
B = -0. 5
0.125
It is n

Example
Example. (Ignore interest rates). Call Option. Pays f0 (S) = (S $1)+ .
$2
p
$1
1-p
$ 0.50
1Period Stock Tree
1
$1
p
$ ?
1-p
$0
Option Value = V
Assume p = 50%. Is V = 0.50?
2
Answer: No!.
V = 1/3$.
If we borrow $1/3, and we buy $2/3 of S, we will

Option Pricing
Luis A. Seco
Univ. of Toronto
1
Financial Instruments Equity
European Options Expire at a preset future time. Their pay-o f depends
on the price of the underlying ST at expiration.
Call options with strike K have pay-o given by
P (ST ) = (S

Interest Rate Theory.
1
Bonds, Yields
Consider a bond that, with a payment of P (t, T ) at time t, pays $1 at time T (and has no other
intermediate payments). If the interest rate r is assumed constant, then we would have
P (t, T ) = er(T t) .
Hence,
r=
l

MAT1856S/ APM466S Mathematical Theory of Finance
Final Exam - April 2003
Problem 1. Consider a call option on a stock. The stock value today is S , the option value
when the stock is worth S at time t is f (S, t) and the strike price for the option is K .

Credit Models
Exercise
KMV and Merton Model
Exercises and Examples
Portfolio Credit Risk III
Prof. Luis Seco
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 31, 2011
Prof. Luis Seco
Portfolio Credit