Risk Latte - FE Problem Set #08

FE Problem Set #08

Team Latte
Mar 05, 2006


Problem #1

A trader pays 11% volatility to go long on USD-JPY and 4% to go long on EUR-JPY. He shorts EUR-USD at 14% vol. He is:

  1. long correlation at 90%;

  2. short correlation 97%

  3. long correlation at 73%

  4. short correlation at 84%

Explain your answer.


Problem #2

An investor sells a straddle (an ATM call and an ATM put option) on the realized variance struck at 12 vol and collects 2.85 volatility points (i.e. with an initial vega of US$100,000 per vol point the investor would receive US$285,000). A variance unit (at least for the purposes of this problem) is defined as vega divided by two times the spot volatility. Determine the realized volatility levels at breakeven (VolBE).

( Hint: This is an option on variance swap and VolBE will be such that the value of the option at expiration should be equal to the upfront premium received. Also, the VolBE will be one number for the call and another for the put option. The upfront premium received is 2.85 times the vega .)


Problem #3

Show that in a Black-Scholes world any option (on a particular stock) is as expensive as any other option, regardless of strike and time to expiry. That is show, mathematically, any option on a stock is a perfect substitute for any other option.

( Hint: Every option (a call option) can be replicated as a stock plus a risk free bond with appropriate holdings in these assets. Use this equivalence relationship along with the standard Black-Scholes call or a put option formula to demonstrate the above .)


Problem #4

A market risk manager is estimating the VaR of a fixed income portfolio and for that he needs to map a cash flow (from a trade) on to two risk factors, a two year rate and a five year rate. The maturity of the cash flow is three and half years and its size is $1 million. The volatility of the two year rate is 10% and that of the five year rate is 15%. The correlation between these two rates is 0.67. What proportion of the cash flow should he map on to the two year risk factor?


Problem #5

The monthly moves of an asset price is given by:

Months
Price
1
$25
2
$29
3
$44
4
$63
5
$52
6
$48
7
$37
8
$49
9
$40
10
$64
11
$72
12
$58

From the above data (and in the absence of any other data) what can be said about the skew of the distribution? How are the volatility of the asset and its price correlated?

(Hint: Calculate the third moment of the distribution using non-centered volatility (mean is zero) and also estimate the correlation between and .)


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