Stochastic Models - Introduction to Derivatives

Stochastic ModelsIntroduction to Derivatives

Walt Pohl

Universitat ZurichDepartment of Business Administration

April 10, 2013

Decision Making, The Easy Case

There is one case where deciding between two projects iseasy: no matter what happens, one project is alwaysbetter than the other.

Taking advantage of this situation is an importantactivity in financial markets, known as arbitrage.

Walt Pohl (UZH QBA) Stochastic Models April 10, 2013 2 / 23

Example: Currency Forwards

A currency forward is a contract today that locks in anexchange rate for some date in the future. The futureexchange rate is known as the forward rate. (No moneychanges hand today.)

A typical example would be a contract to trade 1 CHFfor 1.05 USD in 3 months. Companies use forwards toavoid exchange rate risk – changes in the exchange ratethat hurt profitability.


Synthesizing Currency Forwards

Consider these two strategies.

Enter into a currency forward to trade 1 CHF for FUSD in 3 months. In 3 months you have -1 CHF,and +F USD.

Suppose that you can borrow in CHF for 3 monthsat an interest rate of r and lend in USD at aninterest rate of r f . Let today’s USD-CHF exchangerate (the spot rate) be S . Then do the following:

1 Borrow 1/(1 + r) CHF for 3 months.2 Convert it into S/(1 + r) USD.3 Lend the USD for 3 months.


Synthesizing Currency Forwards, cont’d

In 3 months, you have to pay back 1 CHF, but you make

S(1 + rf )/(1 + r)

USD.

If F > S(1 + rf )/(1 + r), everyone will wantforwards.

If F < S(1 + rf )/(1 + r), everyone will skip theforward, and borrow CHF and lend USD instead.


Arbitrage

But there’s more. In financial markets, you can sell theovervalued asset, and use the proceeds to buy theundervalued asset. This means you can make risk-freeprofits.

You can frequently do this even if you don’t own theunderlying asset. (For example, if you can borrow stockyou don’t own and sell it.)


Arbitrage, Overvalued Forward

If F > S(1 + rf )/(1 + r), then you can costlesslyarbitrage as follows:

Today TomorrowTransaction CHF USD CHF USDForward CHF to USD -1 F

Borrow USD S/(1 + rf ) −S (1+rf )(1+r)

Convert to CHF 1/(1 + rf ) −S/(1 + rf )Lend CHF -1/(1 + rf ) 1

Total 0 0 0 F − S (1+rf )(1+r)


Arbitrage, Undervalued Forward

If F < S(1 + rf )/(1 + r), then you can costlesslyarbitrage as follows:

Today TomorrowTransaction CHF USD CHF USDForward USD to CHF 1 −F

Borrow USD S/(1 + rf ) −S (1+rf )(1+r)

Convert to CHF 1/(1 + rf ) −S/(1 + rf )Lend CHF -1/(1 + rf ) 1

Total 0 0 0 F − S (1+rf )(1+r)


Calls

A call is a contract that gives you the option to buy astock at a future date at a fixed price X (known as thestrike price) by a fixed date T (the expiration date).

Let ST be the stock price on day T .

If ST > X , then you exercise the option, for a net payoffof ST − X .

If ST < X , then you let the option expire, for a netpayoff of zero.


Puts

A put is a contract that gives you the option to sell astock at a future date.

If ST < X , then you exercise the option, for a net payoffof X − ST .

If ST > X , then you let the option expire, for a netpayoff of zero.


American versus European

Puts and calls are both known as options, since they givethe option to do something later. There are somevariations on when you may exercise your option:

For American options, you can exercise any time upto the expiration date.

European options can only be exercised on theexpiration date.

Insert your joke about cultural differences here.


Put-Call Parity

Suppose that you buy a call and sell a put with the samestrike price and expiration date.

If ST > X , then you will exercise the call, for a payout ofST − X at expiration.

If ST < X , then your customer will exercise the put, fora payout of ST − X at expiration.


Put-Call Parity, cont’d

So buying a call and writing a put are a complicated wayof bringing about a payout of ST − X at date T . Youcan do the same by buying the stock today for S0 andborrowing X/(1 + r) until time T .

Suppose that the price of the put and call are p and c .Then put-call parity should hold,

p − c = X/(1 + r) − So

.


Extending the Idea

We can take this idea further. If we are willing topostulate a model for stocks, we can (under somecircumstances) use the same idea to figure out the priceof the option.We do this by replicating the payoffs from the option byusing:

The underlying stock.

A risk-free asset (like government bonds), whichhave a return of r .


One-Step Binary Trees

Suppose that tomorrow’s stock price can only be one oftwo values: Sd or Su. (d = down, u = up)

Suppose we have an option that pays either Pd or Pu

tomorrow. Then using the stock and the risk-free bond,we can replicate the payoff.

Pu = aSu + b(1 + r)

Pd = aSd + b(1 + r)


One-Step Binary Trees

We can solve this system of equations for a and b.

a =Pu − Pd

Su − Sd

b =1

1 + r(Pu − aSu)

=1

1 + r(Pd − aSd)

The price of the option should be aS + b.


Delta Hedging

The quantity a is known as the option’s ∆,

∆ =Pu − Pd

Su − Sd

This method of determining the price is sometimesknown as delta hedging. In general, hedging is theactivity of owning an asset to cancel out another risk youface. If you sell the option, you can hedge your riskcompletely by buying ∆ shares.


Delta Hedging, cont’d

The initial investment of the portfolio of 1 option, and−∆ stock is b, while the payoff is

b(1 + r).

The portfolio is risk-less, so has a return equal to therisk-free rate.


Risk Neutrality

Notice that we don’t need to know the probability of anup or down move in this calculation. In fact, we canback out a “market probability” that the price will be upor down.

Let p be the price of an option that pays 1 + r if uhappens, and 0 othewise.

Let q be the price of an option that pays 1 + r if dhappens, and 0 othewise.

If you own both options, you receive 1 + r no matterwhat happens, so owning both has the same payoff asthe risk-free rate, and should have the same price as therisk-free bond: 1.


Risk Neutrality, cont’d

Thus p + q = 1 satisfies the laws of probability. p and qare known as the risk-neutral probabilities of u and d ,respectively. In economics, these are known asArrow-Debreu securities.

Any other option can be priced using these two numbers.An option that pays off Pu and Pd can be replicated bybuying Pu/(1 + r) of the first security and Pd/(1 + r) ofthe second security.


Risk Neutrality, cont’d

The price is then

1

1 + r(Pup + Pdq) ,

which is the expected value of the payoff under thisprobability distribution, discounted by the risk-free rate.

This is known as risk-neutral pricing.


More than Two States

In this set-up we have two underlying securites (thestock and bond), and two states.

This generalizes to n securities and n states.This is notthe direction we will go.


More Than One Step

Instead, we consider multiple time steps, each allowing asingle up or down move. If the time steps are smallenough, the possible final states become large.

More importantly, you can approximate many generalstochastic processes this way.


Stochastic Models - Introduction to Derivatives

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