PDF Math Finance summary Mich '10 7 - Dur

Math Finance summary Mich '10

7

Asset price behaviour The cornerstone of a great many models in mathematical finance H.6 is the assumption that asset prices S(t), t 0 (at least for non-dividend paying stocks) are distributed as a geometric Brownian motion with drift ? and volatility . Specifically,

log

S(u + t) S(u)

N

(? - 2/2)t, 2t

for all u 0, t > 0 and additionally S(u + t)/S(u) is independent of S(v), v u. There is considerable empirical evidence that this model is often reasonable (Taylor's

theorem for the log function links this model to our earlier observation about relative price changes when they are small). There is also much research into improving it e.g. by making the volatility time dependent.

Tuning the binary tree model It is possible to choose u, d and p to make the binary tree model emulate the geometric BM model of stock prices. There are various (essentially equivalent) ways to do this. Divide time period T into a large number n timesteps of length = T /n (with interest rate per time step equal to r) and set

u

=

exp(

+

(?

-

2/2)),

d

=

exp(-

+

(?

-

2/2)),

p

=

1 2

+

o()

where ? is the drift and 2 the volatility of the geometric BM stock price model. The parameters can be made risk neutral by setting (1+r-d)/(u-d) = p = 1/2+o()

which leads in the limit to a geometric BM model with volatility but with risk neutral drift r in place of ?. Assuming the limit can be passed through the expectation (correct but we do not prove it in this course) this leads to the call option price formula

H.12

C(K, T, , S, r) = e-rT Er [SeW - K]+

where S(0) = S and Er indicates that we use W N ((r - 2/2)T, 2T ).

3 The Black-Scholes formula

The expectation in the formula for the European call price can be evaluated in the form H.8.5

C = S(d1) - Ke-rT (d2)

(3.1)

where

d1

=

(r

+

2/2)T +

T

log(S/K )

,

d2

=

d1

-

T

=

(r

-

2/2)T +

T

log(S/K )

and denotes the standard Normal cdf. This is established by standard integration methods. If the option is purchased at t [0, T ) the formula still holds with T replaced by T - t throughout i.e. in d1 and d2 too.

We have already seen that C is increasing in T and decreasing and convex in K. We also find that C is increasing and convex in S and increasing in both r and .

Using put-call parity we immediately obtain a formula for the price of a European (K, T ) put option sold at t < T . It can be written

P = Ke-r(T -t)(-d2) - S(-d1)

by using the relation (z) + (-z) = 1 for the standard Normal cdf.

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4 Further results on options

Minor variations of calls and puts Pricing options as expectations of final values under risk neutral drift is a method that can also be applied to modified European calls and puts with various final payoffs. The standard options of this type are cash-or-nothing and assetor-nothing calls and puts.

Definition A European cash-or-nothing call option with strike value K and expiry date T pays the holder the amount A if S(T ) > K, 0 if S(T ) < K (a European (K, T ) cash-ornothing put pays 0 if S(T ) > K, A if S(T ) < K). The payout is A/2 if S(T ) = K for either option. Denote the no-arbitrage prices of these options by Cc, Pc respectively.

From the computations done to calculate the Black-Scholes formula we obtain

Cc(A, K, T

-

t; , S, r)

=

e-r(T -t)A(d1

-

T

-

t)

H.17

for the cash-or-nothing call cost at time t < T . The price of the put with the same parameters can be obtained from the put-call parity formula Cc + Pc = Ae-r(T -t). By linearity we have Cc(A, K, T - t; , S, r) = ACc(1, K, T - t; , S, r).

Asset-or-nothing options are defined in exactly the same way as cash-or-nothing except

that the amount A is replaced by S(T ), the stock price at the expiry date. The price Ca of the asset-or-nothing call at time t < T with time t stock price S is

Ca(K, T - t; , S, r) = S(d1)

(again using calculations done to find the Black-Scholes formula). By considering the portfolio with one asset-or-nothing call and short K unit cash-or-nothing calls we see that Ca -KCc = C, the cost of a standard (K, T ) call option.

A great many apparently complex European options can be replicated with portfolios of simpler options.

Option prices for dividend paying stocks Companies listed in stock exchanges normally pay dividends to their stock holders. Such payments affect the value of the underlying asset and so we will investigate how this affects our model for the value process (Higham omits dividends). Dividends are paid in various ways but we will only consider one case.

Discrete proportional payments Suppose a company announces that it will pay a dividend of size dS(Td) at Td. Suppress all the parameters of Cd that are not involved in this calculation and focus upon the time to expiry T - t and the stock price S at time t.

For the proportional dividend dS(Td) the stock price falls at Td (to avoid an arbitrage) but this can be incorporated into the standard model. We find that for t > Td or T < Td the dividend has no effect and Cd(T - t, S) = C(T - t, S) but for t < Td < T ,

Cd(T - t, S) = C(K, T - t; (1 - d)S, , r)

where C(. . .) is the Black-Scholes formula. We see that the option price is less for a dividend paying stock. This can be written in the form Cd(T -t, S) = (1-d)C(K/(1-d), T -t; S, , r) by scale invariance. The effect on an option price of multiple dividends within its life can be calculated by repeating the procedure for a single dividend.

The same arguments can be applied to European put options. Alternatively the put-call parity result (with minor adjustments) still holds as can be shown by varying the initial portfolios e.g. compare a call plus cash with a put plus 1 - d shares in the proportional dividend case.

Math Finance summary Mich '10

9

H.9 Hedging as used in the phrase "to hedge your bets", is a very important part of finance. Anybody who sells a call option may have to make very large payouts at the expiry date if S(T ) K. Any method of limiting or controlling such a risk by means of some other investments is known as hedging.

Delta hedging is particularly common and aims at reducing risk arising from stock price changes. Consider a portfolio of shares of a stock plus a fixed number of options on that stock which have been sold. We wish to adjust the number of shares over time to eliminate dependency upon the stock price movements.

We saw how to replicate European options on the binomial tree with a self-financing portfolio of cash and shares. The number of shares y is re-balanced after every stock price change. In the limit as the time step t 0 (which makes our tree model converge to the geometric BM model, though we haven't proved this) we find that for a call option

y=

#

shares

for

each

call

option

sold

C S

.

This derivative of C is known as and hence the name for this hedge. The argument extends

to other types of option and will be considered in more detail next term.

H.10 The greeks Various derivatives of C are used in hedging European options against a range

of risks and are referred to in the finance literature as "the greeks". They are

=

C S

,

=

2C S2

,

=

C r

,

=

C t

,

C

known as delta, gamma, rho, theta and vega (not a greek letter!). They can be calculated

directly from the Black-Scholes formula but it is quicker and clearer to differentiate under

the expectation in our risk neutral valuation formula. We find for example that

C S

= (d1),

2C S2

=

(d1) S T - t

(where

(x)

=

(x)

=

e-x2/2

/ 2

)

and

with

more

work

that

C t

= - S(d1) 2 T -t

- Kre-r(T -t)

d1 - T

-t

Summary We have studied the standard geometric BM model for stock prices as the limiting random walk on the binomial tree (with stock price jumps of order t over a time step of size t). For constant interest rate r and volatility and any finite t we found the risk neutral jump probabilities on the trees. Assuming everything works in the limit as t 0 we have

? calculated an explicit formula for the no-arbitrage price of European call and put options under the geometric BM model by finding the expected return of the option under the risk neutral drift ? = r;

? applied the idea to other European style options e.g. asset-or-nothing and cash-ornothing options;

? seen how to modify the model to be able to price European options on dividend paying stocks;

? met the idea of hedged portfolios which is important in practice but also provides a route to study the geometric BM model directly (more next term).

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