When you learn Quantitative Finance real late

To respond to Dimitri Bianco, FRM Christmas Quant Challenge, I tried to get my head around the real deal of Quantitative Finance - Paul Wilmott's book on QF. This is what I got after the 1st chapter. This is not a summary, as I have tried to note down key concepts and approaches while NOT reproducing verbatim the book.

Chapter 1. Products and Markets: This deals with the key products in Capital Markets around which we base our principles of Quant Finance

Sec 1.1 Equities, Commodities, FX and Indices

Equities/Stock/Share : Ownership of a small part of a company in exchange of cash to the company owner. Stock price is modelled by (Change in Stock Price ) = f(Current level of stock), assuming Stock Price >=0, and multiplicative power of stock prices instead of a fixed addition. So we use a lognormal/geometric random walk for stock prices.

Dividends: Value in holding the stock comes from dividends paid out by the company and any growth in the stock’s value.

Stock splits : A 3-for-1 stock split, on a stock priced at ￡120, results in 3 stocks each valued at ￡40.

Commodities : Raw products or precious materials, oil, food products whose prices are dictated by scarcity and show seasonal effects. These are traded by those who don’t need the product, on futures market, speculating to buy/sell in the future at an agreed price.

FX/Currencies : Trading in FX is done to make arbitrage profits by exploiting the mispricing between a currency pair. Exchange Rate and Interest rates in the currencies of 2 countries being traded are linked, as if interest rates in one are raised while the other stays the same, the 2nd currency will depreciate against the 1st for a while.

Indices : Indices measure how the stock market/economy has been doing as a whole, by making up a typical index from the weighted sum of selection/basket of representative stocks.

Sec 1.2 Time Value of Money

Compound interest is more relevant than Simple interest. There are 2 types of Compound interests.

Discretely Compounded : Investment of ￡ P in bank at a discrete rate of r/yr for t years amounts to P(1+r)^t

Continuously Compounded : Let there be m interest payments at a rate of r/m per year, for t years.? Investment of ￡ P amounts to P(1+r/m)^(mt). This can be approximated to pert. Thus the maturity value at time t is M(t) = M(0)e^(rt).

NPV at start i.e. M(0) = M(t) e^(-rt) .

NPV at time T (T<t) i.e. M(T) = M(0)e^{-r(t-T)}

Sec 1.3 Fixed and Floating Interest Rates

Bonds have either Fixed Interest Payment, with coupon-bearing bonds paying a fixed rate, or Floating Interest payments with variable interest payments. Inflation-proof bonds are related ?to the level of the Retail Price Index, by both Principal and Interest, with base movements being 8 months in the past.

Sec 1.4 Futures and Forwards:

These are used in speculation and hedging. A Forward Contract is an agreement for A to promise to buy an amount from B at a specified time in the future for an agreed price.

A Futures Contract is very similar to a Forward Contract. P&L is calculated every day and change in value of P&L is paid from 1 party to another on a daily basis.

Sec 1.5? No-arbitrage principle:

For a Forward Contract, let the delivery price = ￡ F at time T. Today’s date = t. Today’s asset price = ￡ S(t)

NPV(t) = S(t) e^{r(T-t)}

Net position at Maturity : S(t)e^{r(T-t)} ?- F

Since we start with NPV(0) = 0 and end with a predictable amount , NPV(t) = 0

This means S(t)e^{r(T-t)} ?- F = 0, ie. F = S(t)e^{r(T-t)}

Margin : ‘Marking to Market’ is the process of settling changes to the value of a futures contract everyday at exchanges. To prevent the risk of default, exchanges insist on traders depositing some money to cover these changes in the value of their positions, in a margin account.

Commodity Futures : Benefits from holding is Commodity is measured in terms of a Convenience yield c, with the No-arbitrage equation modified to

F = S(t)e^{(r+s-c)(T-t)} ?where s = storage cost of the commodity. Storage costs (s) and Convenience yield ( c ) work in opposite directions in the price of Commodity futures.

FX Futures : Let interest received on foreign currency rf

F = S(t) e^{(r-rf)(T-t)}

Index future: Future contracts on stock indices settled in cash with dividends (q = dividend yield) acting similar to interest rates on foreign currency in FX Futures.

F = S(t) e^{(r-q)(T-t)}

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I am happy to say this marks the start of a journey for me, as I move on to Chapter 2 : Derivatives.

Sec 2.1 Definition of basic derivative instruments :

Options give holder the right, but not the obligation, to trade in the future at a pre-agreed price.

Call Options give the right to buy a particular asset at a pre-agreed price at a specified time in the future. Value of call option at expiry , V = Max (S-E, 0) where S is the stock price, E is the strike price. This equation is also called the Payoff Function.

Put Options give the right to sell a particular asset at a pre-agreed price at a specified time in the future. V = Max (E-S, 0).

Sec 2.2 Option Jargon

This part deals with the definitions of components that quantify an Option : Premium, Underlying amount, Strike/Exercise Price, Expiration Date, Intrinsic Value, Time Value, In-the-money, Out-of-money, At-the-money, Long Position, Short Position. It also deals with the functions affecting Option Pricing e.g. Value of the underlying asset S, Time to expiry t, and parameters affecting the same e.g. interest rate, strike price, volatility. It also describes the different types of exercises for options i.e.

European Option : Exercise only at expiry

American Option : Determine when to exercise so that contract gets its highest value.

Bermuda Option : Exercise at a specified date at a specified period.

Sec 2.3 Payoff diagrams

Sec 2.4 No arbitrage and put-call parity

Buy 1 European call option with strike price E and expiry of T, with value of underlying asset at time T = S(T)

Write another European put option with strike price E and expiry of T. Today = t.? Payoff for the portfolio of 2 options max(s(T) – E,0) – max(E-S(T)), 0) = S(T) – E

Let C = today’s value of call option

Let P = today’s value of put option

C-P = S(t) - Ee^(-r(T-t))??

Binary /Digital Options

Binary call : Pays ￡1 at expiry, time T, if S(T) > E. Why buy ? If asset rise less faster than E.

Binary Put : Holder gets ￡1 at expiry, time T, if S(T) <E. Why buy ? if there is a modest fall in S(t). For Binary Put/call parity,

Binary call + Binary put = e^(-r(T-t))

Sec 2.5 Simple Option strategies

1.????? Bull and Bear Spreads:

I buy 1 call option with E1 = ￡100 and write another with E2 = ￡120. Both expire at T.

Payoff : ?????((max〖(S-E1,0)-max(S-E2,0))〗)/(E2-E1)

????????????????????????????????????????????? ?????????????????????????????????????????? where E2 > E1.

This is a Bull spread or a Call spread, because it benefits from a rising market.

I write a put option with E1 = ￡100, and buy a put with E2 = ￡120, with both expiring at T.

Payoff ? : ????((max〖(E2-S,0)-max(E1-S,0))〗)/(E2-E1)????????????????????????????????????????????????? ??????????????????????????????????????????

This is a Bear spread or a Put spread, because it benefits from a Bear , i.e. falling market

?

2.????? Straddles and Strangles:

Straddle : a call + a put with the same strike E. Bought at the money by those who expect the underlying price to rise or fall, but never stable; sold by someone who expects the underlying price to remain stable.

Strangle : similar to straddle, but strike prices on call and put options are different. The contract can be out-of-money or in-the-money. Bought by those who expect the underlying price to move larger than in straddles, when asset is around the middle of 2 strikes and is cheaper than a straddle.

3.????? Risk Reversal : Combination of long call (strike above current spot) and a short put (strike below current spot), with both having same expiry.

4.????? Butterflies and Condors : Butterfly spread involves the buying and sale of options with 3 different Strike prices e.g. buying a call option with a strike price of ￡90, writing 2 calls struck at ￡100, and buying another call option with a strike price of ￡110. When the asset is not going anywhere, either up or down, with no large upside potential, this kind of positions are likely to be entered. The Condor is like a Butterfly with 4 strikes, and 4 call options.

5.????? Calendar Spreads : Options with different expiry dates, to be entered into when trader can precisely time a market move and the direction of that move, with a view to increasing the payoff at asset values and times which the trader thinks are relevant, and reduce the payoff at those asset values and times which are irrelevant.

6.????? LEAPS and FLEX : Long-term Equity Anticipation (LEAP) securities are standardized exchange-traded calls and puts, with expiries up to 3 years, and coming with 3 strikes, corresponding to the money and c.20% in and out of money w.r.t underlying asset price when issued. Flexible Exchange traded options (FLEX) allow customisation in the expiry date, strike price and exercise style.

Warrants : ??????????????????????