Applications of Stochastic Calculus in Finance

Argument for Continuous Time Model • Stock (log) price follows a diffusion equation, i.e. a continuous time, continuous

stochastic process such as: dxt = µ(xt , t)dt + σ(xt; t)dBt

• where: µ(xt , t) and σ(xt; t) are the drift and diffusion coefficient, respectively, and Bt is a standard Brownian motion (or Wiener process).

• In a complete market, use hedging to derive the price of an option (no arbitrage argument).

• In an incomplete market (e.g. existence of jumps), specify risk and a hedging strategy to minimize the risk.

Stochastic Calculus Edited: March 2011 Page 2 Professor: Nina Kajiji

Example 1: Portfolio Concept

Ito’s Process and a Risky Asset Let, the price of a risky asset (stock) follow a geometric Brownian motion of the form:

𝑋𝑡 = 𝑓(𝑡,𝐵𝑡) = 𝑋0 𝑒 (𝑐−0.5𝜎2)𝑡+ 𝜎𝐵𝑡

Then, we have a unique strong solution of the linear stochastic differential equation (see lecture 8) as:

𝑋𝑡 = 𝑋0 + 𝑐 �𝑋𝑠𝑑𝑠𝑡

0

+ 𝜎�𝑋𝑠𝑑𝐵𝑠

𝑡

0

Which we have shown already can be written as a generalized wiener process:

𝑑𝑋𝑡 = 𝑐𝑋𝑡 𝑑𝑡 + 𝜎𝑋𝑡𝑑𝐵𝑡

Or, for a small change in t say, [t, t + dt] we have:

𝑋𝑡+𝑑𝑡 − 𝑋𝑡 = 𝑐𝑋𝑡 𝑑𝑡 + 𝜎𝑋𝑡𝑑𝐵𝑡

Or, 𝑋𝑡+𝑑𝑡 − 𝑋𝑡

𝑋𝑡 = 𝑐 𝑑𝑡 + 𝜎𝑑𝐵𝑡

Therefore:

• Left hand side is the standard return computation

• Which is a function of some linear trend 𝑐 𝑑𝑡 (note: c is the mean rate of return)

• Which is disturbed by a stochastic noise term 𝜎𝑑𝐵𝑡 (note: σ is volatility)

• Thus, the larger the σ the larger the fluctuation in the price of the risky asset

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Deterministic Integrals and non-Risky Asset Let, the price of a non-risky asset (bond) follow an initial investment that is continuously compounded. That is,

𝛽𝑡 = 𝛽0𝑒𝑟𝑡

Then,

𝛽𝑡 = 𝛽0 + 𝑟 �𝛽𝑠

𝑡

0

𝑑𝑠

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The Portfolio Some combination of risky (at) and non-risky assets (bt) – a trading strategy will be implemented that will provide a value for your overall wealth. This can be stated as:

𝑉𝑡 = 𝑎𝑡𝑋𝑡 + 𝑏𝑡𝛽𝑡 ; 𝑡 ∈ [0,𝑇]

Some assumptions:

• at can be negative or positive and is unbounded – negative implies short sales

• bt can be negative or positive and is unbounded – negative implies borrowing

• ignore transaction costs

• no other consumption is allowed – that is your portfolio is not made smaller by any other consumption.

• The portfolio is self-financing – that is no additions and subtractions into the portfolio. Value changes only due to price movements.

Therefore using Ito’s arguments we have:

𝑉𝑡 − 𝑉0 = �𝑑(𝑎𝑠𝑋𝑠 + 𝑏𝑠𝛽𝑠)𝑡

0

Using the arguments developed in the “risky-asset” and “non-risky” asset section we have:

𝑉𝑡 − 𝑉0 = �𝑎𝑠𝑑𝑋𝑠

𝑡

0

+ �𝑏𝑠𝑑𝛽𝑠

𝑡

0

As shown earlier this can be written as an addition of two stochastic processes:

𝑉𝑡 − 𝑉0 = �𝑎𝑠(𝑐𝑋𝑠𝑑𝑠 + 𝜎𝑋𝑠𝑑𝐵𝑠)𝑡

0

+ �𝑏𝑠(𝑟𝛽𝑠𝑑𝑠)𝑡

0

Stochastic Calculus Edited: March 2011 Page 5 Professor: Nina Kajiji

Example 2: Option Pricing

Stock Options Financial options are tradable assets that are sold through regulated exchanges including major stock markets. Option contracts confer the right to a transaction but do not require the transaction to take place. Financial option contracts trade with standardized contract terms. This means that the counterparties know in advance the expiration date, exercise price, and quantity of the underlying should the holder of the option choose to exercise the contract. The two basic types of option contracts traded on the major exchanges are called: calls and puts

A call option contract confers the right to buy the underlying instrument at the specified exercise (strike) price any time prior to the close of trading on the option expiration date.

A put option contract confers the right to sell the underlying instrument at the specified exercise (strike) price any time prior to the close of trading on the option expiration date.

Option Style There are two well-known style descriptions for traded options. Note that the style description of American and European does not refer to where the options are traded. Style associations are made by the exchange on which the option trades.

American-style options can be exercised at any time between the purchase date (date the long position was created) and the last-trade time on the expiration date (the option end-of-life date).

European-style options can only be exercised on the expiration date.

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Basic Notation and Terminology

Intrinsic Value of a Call Option The intrinsic value for American-style call options is determined by the following relationship:

IValaC = C(X,T,K) = MAX[(X-K) , 0]

The definition hinges on the comparison of two entities: the difference between X and K as compared to the number zero (0).

The intrinsic value for European-style options requires applying a present-value factor [(1+r)-T] to the option’s strike price. Why? As previously defined, option exercise can only be done on the expiration date. Hence, to determine the today’s minimum value (or, IVal) it is necessary to obtain the present value of the exercise price. Using the estimated risk-free rate (r) and the time to expiration (T) the relationship is expresses as:

IValeC = C(X,T,K) = MAX[(X-K(1+r)-T, 0] .

Notation Explanation Xo –or- X Stock Price Today K Exercise price or Strike Price T Time to expiration in fraction

notation t A time period between today and T. XT Stock price at option’s expiration B Bond Portfolio (1), (2), (3) values (time, strike price) where (1)

< (2) < (3) r Risk-free rate (a) American style option (e) European style option

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Example – Hedging Recall: Some combination of risky (at) and non-risky assets (bt) – a trading strategy will be implemented that will provide a value for your overall wealth.

In this example the risky-asset is assumed to be a European call option. Then we have:

𝑉𝑡 = 𝑎𝑡𝑋𝑡 + 𝑏𝑡𝛽𝑡 = 𝑢(𝑇 − 𝑡,𝑋𝑡); 𝑡 ∈ [0,𝑇] (1.1)

Or from above the terminal condition can be stated as:

𝑉𝑇 = 𝑢(0,𝑋𝑇) = 𝑚𝑎𝑥(0,𝑋𝑇 − 𝐾); [𝑡 = 𝑇] (1.2)

Therefore to build a self-financing strategy (or Hedge) we apply the Ito lemma to the value process 𝑉𝑡.

Let: 𝑉𝑡 = 𝑓(𝑡, 𝑥) = 𝑢(𝑇 − 𝑡, 𝑥).

Then,

𝑓1(𝑡, 𝑥) = 𝜕𝜕𝑡𝑓(𝑡, 𝑥) = −𝑢1(𝑇 − 𝑡, 𝑥)

𝑓2(𝑡, 𝑥) = 𝜕𝜕𝑥

𝑓(𝑡, 𝑥) = 𝑢2(𝑇 − 𝑡, 𝑥)

𝑓22(𝑡, 𝑥) = 𝜕𝜕𝑥

𝜕𝜕𝑥

𝑓(𝑡, 𝑥) = 𝑢22(𝑇 − 𝑡, 𝑥)

Recall: X also satisfies the Ito integral equation:

𝑋𝑡 − 𝑋0 = 𝑐 �𝑋𝑠𝑑𝑠𝑡

0

+ 𝜎�𝑋𝑠𝑑𝐵𝑠

𝑡

0

Then by rewriting using Ito’s Lemma we have:

Stochastic Calculus Edited: March 2011 Page 8 Professor: Nina Kajiji

𝑉𝑡 − 𝑉0 = ��𝑓1(𝑠,𝑋𝑠) + 𝑐𝑋𝑠𝑓2(𝑠,𝑋𝑠) +12𝜎2𝑋𝑠2𝑓22(𝑠,𝑋𝑠)� 𝑑𝑠

𝑡

0

+ � 𝜎𝑋𝑠𝑓2(𝑠,𝑋𝑠)𝑑𝐵𝑠

𝑡

0

By substitution of the derivatives of 𝑢(𝑇 − 𝑡, 𝑥) we have:

𝑉𝑡 − 𝑉0 = ��−𝑢1(𝑇 − 𝑠,𝑋𝑠) + 𝑐𝑋𝑠𝑢2(𝑇 − 𝑠,𝑋𝑠)𝑡

0

+12𝜎2𝑋𝑠2𝑢22(𝑇 − 𝑠,𝑋𝑠)� 𝑑𝑠 + � 𝜎𝑋𝑠𝑢2(𝑇 − 𝑠,𝑋𝑠)𝑑𝐵𝑠

𝑡

0

(1.3)

But we assumed that the trading strategy (at, bt) was self-financing, so we have an

alternative formulation for 𝑉𝑡 − 𝑉0 = ∫ 𝑎𝑠𝑑𝑠𝑡0 + ∫ 𝑏𝑠𝑑𝐵𝑠

𝑡0 but 𝛽𝑡 = 𝛽0𝑒𝑟𝑡. Taking

derivatives of 𝛽𝑡 with respect to r, solving for bt using 1.1 and substituting into the alternative formulation we have:

𝑉𝑡 − 𝑉0 = �[(𝑐 − 𝑟)𝑎𝑠𝑋𝑠 + 𝑟𝑉𝑠]𝑑𝑠𝑡

0

+ �𝜎𝑎𝑠𝑋𝑠𝑑𝐵𝑠

𝑡

0

(1.4)

Comparing 1.3 & 1.4 and realizing that the coefficient functions of Ito coincide and after some derivation (see text) we have the continuous terminal value equation:

𝑢1(𝑡, 𝑥) =12𝜎2𝑥2𝑢22(𝑡, 𝑥) + 𝑟𝑥𝑢2(𝑡, 𝑥) − 𝑟𝑢(𝑡, 𝑥) 𝑥 > 0, 𝑡 ∈ [0,𝑇] (1.5)

Recall: 𝑉𝑇 = 𝑢(0,𝑋𝑇) = 𝑚𝑎𝑥(0,𝑋𝑇 − 𝐾) (1.6)

Which yields the deterministic terminal condition:

𝑢(0, 𝑥) = (𝑥 − 𝐾) 𝑥 > 0

Stochastic Calculus Edited: March 2011 Page 9 Professor: Nina Kajiji

So in conclusion: The Partial differential equation 1.5 has a terminal solution 1.6 and an explicit solution of:

𝑉0 = 𝑢(𝑇,𝑋0) = 𝑋0Φ(𝑔(𝑇,𝑋0)) − 𝐾𝑒−𝑟𝑇Φ(ℎ(𝑇,𝑋0))

• Which is the Black – Scholes option pricing formula and is the rational price at t=0 for a European call option with exercise price K

• NOTE: BS is independent of the mean rate of return; but it does depend on volatility

• The stochastic process 𝑉𝑡 = 𝑓(𝑡, 𝑥) = 𝑢(𝑇 − 𝑡, 𝑥) is the value of the self-financing portfolio at time t є [0,T]

• The self-financing strategy (at, bt) is given by: o 𝑎𝑡 = 𝑢2(𝑇 − 𝑡,𝑋𝑡)

o 𝑏𝑡 = 𝑢(𝑇−𝑡,𝑋𝑡)−𝑎𝑡𝑋𝑡 𝛽𝑡

Factors Affecting the Value of a Call • Stock price: the price of the underlying

• Time to Expiration: the premium (option price) is greater for a longer call than a shorter call

• Exercise Price: lower exercise price implies a greater premium

• Interest Rates: low interest rates stimulate purchase of calls.

• Volatility: higher volatility implies higher premium

Stochastic Calculus Edited: March 2011 Page 10 Professor: Nina Kajiji

Example 3: Jump Diffusion Proposed because the traditional stochastic diffusion models based on Brownian motion fails to explain some of the characteristics of asset returns and prices of their derivatives. One such example is the volatility smile.

Recall: Volatility smile is obtained because the implied volatility for both in- and out- of the money have higher volatilities; whereas at-the-money options have the lowest volatilities.

So what is a Jump Diffusion model? Assume that for a given stock, its price reacts to an event. Assume that the event follows a Poisson process:

𝑃(𝑋𝑡 = 𝑚) = 𝜆𝑚𝑡𝑚

𝑚!𝑒−𝜆𝑡 𝜆 > 0

This is a continuous-time discrete process. λ is the rate of intensity of the process.

Now that the price is not just a straightforward Brownian motion we need to modify the model to include the jump concept. These models are therefore called Jump Diffusion models. The original stochastic differential (generalized wiener process) can now be written as:

𝑑𝑋𝑡 = µ 𝑑𝑡 + 𝜎𝑑𝐵𝑡 + 𝑑 ��(𝐽𝑖 − 1)𝑛𝑡

𝑖=1

�

Where all parameters are as defined elsewhere; additionally,

• nt is a Poisson process with rate λ. It is the number of jumps in [0,t]

• {Ji} is a sequence of iid non-negative random variables such that X = ln(J) has a double exponential distribution with probability density function

o 𝑓(𝑥) = 12η𝑒−|𝑥−𝑘|/η

• At the ith jump, the proportion of price jump is (Ji -1)

• The equation for the price can be solved using the Ito’s Lemma as shown earlier.

• Similarly, the price of an option can be derived.

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Consequences of Jumps • Adding jumps while keeping other parameters fixed increases the price of the stock;

and / or the options (both calls and puts)

• Adding a jump process to the stock price often leads to different estimates for the stock volatility